{"_id": "500001", "text": "proof (prove)\nusing this:\n length ps = length vs\n left_nesting f \\ left_nesting g\n is_const (fst (strip_comb f))\n\ngoal (1 subgoal):\n 1. match (list_comb f ps) (list_comb g vs) = None"} {"_id": "500002", "text": "proof (prove)\nusing this:\n \\?F \\ ?G \\ ?\\ \\\n AX10 \\\\<^sub>H\n ?H;\n ?\\ \\ AX10 \\\\<^sub>H ?F;\n ?\\ \\ AX10 \\\\<^sub>H ?G\\\n \\ ?\\ \\ AX10 \\\\<^sub>H ?H\n\ngoal (1 subgoal):\n 1. \\F_ \\<^bold>\\ G_ \\ S_;\n F_ \\ G_ \\ S_ \\\n AX10 \\\\<^sub>H\n \\\\\n \\ S_ \\ AX10 \\\\<^sub>H \\"} {"_id": "500003", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i. i < nr \\ f i \\ nc"} {"_id": "500004", "text": "proof (prove)\nusing this:\n monic v\n\ngoal (1 subgoal):\n 1. degree (#v * w') = degree (#v) + degree w'"} {"_id": "500005", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (\\ f j) k = f k"} {"_id": "500006", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wstate_of_dstate ((C', i) # N', P, Q, n) \\\\<^sub>w\\<^sup>*\n wstate_of_dstate ((C', i) # N', reduce_all C' P, Q, n)"} {"_id": "500007", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\P'. P \\\\ P' \\ M' = Fst P') \\\n (\\A B. P = SimplyTyped.Pair A B \\ M' = A)"} {"_id": "500008", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\regular (prim_EP w v e); e \\ w; injective w; w * v \\ v;\n e \\ v * - v\\<^sup>T; vector v\\\n \\ prim_W w v e = w"} {"_id": "500009", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OF_priority_match OF_match_fields_safe oft =\n OF_match_linear OF_match_fields_safe oft"} {"_id": "500010", "text": "proof (prove)\nusing this:\n wlp R1 w1 g1 f1 x1 l2 = wlp R2 w2 g2 f2 x2 l2\n wlp R1 w1 g1 f1 a l2 = wlp R2 w2 g2 f2 a l2\n\ngoal (1 subgoal):\n 1. wlp R1 w1 g1 f1 x1 l1 = wlp R2 w2 g2 f2 x2 (a # l2)"} {"_id": "500011", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. root n (f x)) expands_to zero_expansion) basis"} {"_id": "500012", "text": "proof (state)\nthis:\n \\A b nr.\n A \\ carrier_mat nr n \\\n b \\ carrier_vec nr \\ P = polyhedron A b\n\ngoal (2 subgoals):\n 1. polytope P \\\n \\A b nr bnd.\n A \\ carrier_mat nr n \\\n b \\ carrier_vec nr \\\n P = polyhedron A b \\ P \\ Bounded_vec bnd\n 2. \\A b nr bnd.\n A \\ carrier_mat nr n \\\n b \\ carrier_vec nr \\\n P = polyhedron A b \\\n P \\ Bounded_vec bnd \\\n polytope P"} {"_id": "500013", "text": "proof (prove)\nusing this:\n t1 \\ atrm\n t2 \\ atrm\n fst ` set txs \\ atrm\n snd ` set txs \\ var\n distinct (map snd txs)\n t1 \\ trm\n t2 \\ trm\n fst ` set txs \\ trm\n z \\ var\n z \\ xx\n z \\ yy\n z \\ FvarsT t1\n z \\ FvarsT t2\n z \\ \\ (FvarsT ` fst ` set txs)\n z \\ snd ` set txs\n\ngoal (1 subgoal):\n 1. psubst (LLs t1 t2) txs = LLs (psubstT t1 txs) (psubstT t2 txs)"} {"_id": "500014", "text": "proof (prove)\nusing this:\n is_quantum_predicate (adjoint (U2 * U1) * P * (U2 * U1))\n\ngoal (1 subgoal):\n 1. \\\\<^sub>p {adjoint (U2 * U1) * P * (U2 * U1)}\n Utrans U1;; Utrans U2 {P}"} {"_id": "500015", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f sums infsetsum f UNIV"} {"_id": "500016", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\3 < length l; n = length l; valid (tl l); l ! 0 = B\\\n \\ l \\ (#) B `\n {la.\n length la = length l - Suc 0 \\ valid la}"} {"_id": "500017", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length qs - 2 \\\n TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs)\n [x, y]"} {"_id": "500018", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a.\n \\S_inductive_set D; carrier D \\ {};\n a \\ carrier D\\\n \\ \\m. maximal m"} {"_id": "500019", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w * ((1::'a) \\ (v \\ - r) * top) =\n w * ((1::'a) \\ (v \\ - r) * (v \\ - r)\\<^sup>T)"} {"_id": "500020", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ y. \\\\<^sup>+ x. f x y \\M x\n \\M y =\n \\\\<^sup>+ z. f (fst z) (snd z)\n \\M x \\\\<^sub>M M y"} {"_id": "500021", "text": "proof (prove)\nusing this:\n C.ide w'\n \\\\' : f \\\\<^sub>C\n w' \\\\<^sub>C u\\\n \\\\' : F w \\\\<^sub>D F\n w'\\ \\\n D.inv (\\ (g, w')) \\\\<^sub>D\n F \\ \\\\<^sub>D \\ (g, w) =\n F g \\\\<^sub>D \\' \\\n F \\ \\\\<^sub>D \\ (f, w) =\n (F \\' \\\\<^sub>D \\ (f, w')) \\\\<^sub>D\n (F f \\\\<^sub>D \\')\n C.hseq ?\\ ?\\ \\\n F (?\\ \\\\<^sub>C ?\\) =\n \\ (C.cod ?\\, C.cod ?\\) \\\\<^sub>D\n (F ?\\ \\\\<^sub>D F ?\\) \\\\<^sub>D\n D.inv (\\ (C.dom ?\\, C.dom ?\\))\n src\\<^sub>C f = trg\\<^sub>C w \\ src\\<^sub>C f = trg\\<^sub>C w'\n (?h \\\\<^sub>D ?g) \\\\<^sub>D ?f =\n ?h \\\\<^sub>D ?g \\\\<^sub>D ?f\n \\D.arr ?h; ?f \\\\<^sub>D ?g = ?h; D.iso ?f\\\n \\ D.seq (D.inv ?f) ?h \\\n ?g = D.inv ?f \\\\<^sub>D ?h\n \\D.arr ?h; ?f \\\\<^sub>D ?g = ?h; D.iso ?g\\\n \\ D.seq ?h (D.inv ?g) \\\n ?f = ?h \\\\<^sub>D D.inv ?g\n\ngoal (1 subgoal):\n 1. F \\ =\n F \\' \\\\<^sub>D\n \\ (f, w') \\\\<^sub>D\n (F f \\\\<^sub>D \\') \\\\<^sub>D D.inv (\\ (f, w))"} {"_id": "500022", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ z \\ y \\ z"} {"_id": "500023", "text": "proof (prove)\nusing this:\n \\f : F y \\\\<^sub>C x\\\n\ngoal (1 subgoal):\n 1. C.ide x"} {"_id": "500024", "text": "proof (prove)\ngoal (1 subgoal):\n 1. VLambda I f \\\\<^sub>\\ vproduct I A"} {"_id": "500025", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\v'.\n (\\var\\V. v var = v' var) \\\n v' \\\\<^sub>a\\<^sub>s\n (snd `\n (set (Atoms (preprocess' t start)) \\ I \\ UNIV)) \\\n v' \\\\<^sub>t Tableau (preprocess' t start)"} {"_id": "500026", "text": "proof (prove)\nusing this:\n Ide a \\ Diag a\n Ide b \\ Diag b\n Ide c \\ Diag c\n C.ide f \\ C.arr f\n Ide b \\\n Diag b \\\n Arr b \\\n b \\ \\<^bold>\\ \\\n ide \\b\\ \\\n arr \\b\\ \\\n \\<^bold>\\b\\<^bold>\\ = b \\\n b\\<^bold>\\ = b \\ Dom b = b \\ Cod b = b\n Ide c \\\n Diag c \\\n Arr c \\\n c \\ \\<^bold>\\ \\\n ide \\c\\ \\\n arr \\c\\ \\\n \\<^bold>\\c\\<^bold>\\ = c \\\n c\\<^bold>\\ = c \\ Dom c = c \\ Cod c = c\n Diag (?a \\<^bold>\\ ?b) \\\n ?a \\<^bold>\\ ?b = ?a \\<^bold>\\ ?b\n\ngoal (1 subgoal):\n 1. \\\\<^bold>\\f\\<^bold>\\ \\<^bold>\\\n b\\ =\n V f \\ \\b\\"} {"_id": "500027", "text": "proof (prove)\ngoal (1 subgoal):\n 1. spec\\<^sub>2 p \\ spec\\<^sub>1 p"} {"_id": "500028", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x_ \\ y_; y_ \\ z_\\ \\ x_ \\ z_"} {"_id": "500029", "text": "proof (prove)\nusing this:\n B \\ \\\\<^sub>T\\<^sub>R K (\\ .\\. \\)\n \\ B \\ \\\n\ngoal (1 subgoal):\n 1. B \\ K .\\. \\"} {"_id": "500030", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\hom_boundary (int n) (subtopology (nsphere n) (upper n))\n (equator n) zn =\n z;\n subtopology (nsphere n) (equator n) = nsphere (n - Suc 0);\n subtopology (lsphere n) (equator n) = nsphere (n - Suc 0);\n subtopology (subtopology (nsphere n) (upper n)) (equator n) =\n nsphere (n - Suc 0)\\\n \\ hom_induced (int n - 1) (nsphere (n - Suc 0)) {}\n (nsphere (n - Suc 0)) {} id z =\n z"} {"_id": "500031", "text": "proof (prove)\ngoal (1 subgoal):\n 1. orthogonal (column i (fst (QR_decomposition A)))\n (column ia (fst (QR_decomposition A)))"} {"_id": "500032", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x =\n lincomb_list (lc(j := lc j - lc i * c))\n (us[i := us ! i + c \\\\<^sub>v us ! j])"} {"_id": "500033", "text": "proof (prove)\nusing this:\n a \\ carrier Zp\n b \\ carrier Zp\n a k = b k\n a \\ b\n ?b \\ carrier Zp \\\n (?a \\ ?b) ?k =\n ?a ?k \\\\<^bsub>residue_ring (p ^ ?k)\\<^esub> ?b ?k\n\ngoal (1 subgoal):\n 1. (a \\ b) k = 0"} {"_id": "500034", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\h e e' a z z'.\n h \\ valid_pub \\\n e \\ challenge_space \\\n e' \\ challenge_space \\\n e \\ e' \\\n check h a e z \\\n check h a e' z' \\\n lossless_spmf (ss_adversary h (a, e, z) (a, e', z')) \\\n (\\w'\\set_spmf (ss_adversary h (a, e, z) (a, e', z')).\n (h, w') \\ R_DL)"} {"_id": "500035", "text": "proof (prove)\ngoal (1 subgoal):\n 1. if v then R (tv \\ tp) q else R (at v \\ tp) q fi \\ R p q"} {"_id": "500036", "text": "proof (prove)\nusing this:\n X \\ E\n Int_stable E\n E \\ Pow \\\n ?X \\ E \\ emeasure M ?X = emeasure N ?X\n \\ \\ sets M\n\ngoal (1 subgoal):\n 1. emeasure (restrict_space M \\) X =\n emeasure (restrict_space N \\) X"} {"_id": "500037", "text": "proof (prove)\nusing this:\n Q + N = Q + I + J\n Q + M = Q + I + K\n\ngoal (1 subgoal):\n 1. (Q + M, Q + N) \\ mul r"} {"_id": "500038", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length xs = 2 ^ k \\ C_msort_bu xs \\ k * 2 ^ k"} {"_id": "500039", "text": "proof (prove)\nusing this:\n C.arr f\n\ngoal (1 subgoal):\n 1. constant_transformation (\\\\<^sub>J) (\\) f"} {"_id": "500040", "text": "proof (prove)\ngoal (1 subgoal):\n 1. qtable n C P\n (\\x.\n Q1 (restrict A x) \\\n (if b then Q2 (restrict B x) else \\ Q2 (restrict B x)))\n (join X b Y)"} {"_id": "500041", "text": "proof (prove)\nusing this:\n y \\ v1\n y = new_last\n ProofStepInduct G v0 v1 ?paths' ?P_new' sep_size \\\n H.distance (last P_new) v1 \\ H.distance (last ?P_new') v1\n\ngoal (1 subgoal):\n 1. ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y\n P_k_post"} {"_id": "500042", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fps_deriv (fps_ln c + fps_ln d) =\n fps_const ((1::'a) / c + (1::'a) / d) * inverse (1 + fps_X)"} {"_id": "500043", "text": "proof (prove)\ngoal (1 subgoal):\n 1. supp (ccTransform a e) \\ supp e"} {"_id": "500044", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (mk_rtrancl_list_main subsumes r init []) = mk_rtrancl_main init {}"} {"_id": "500045", "text": "proof (prove)\ngoal (1 subgoal):\n 1. weak_unit a'"} {"_id": "500046", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (map fi, f) \\ clw_rel A \\ clw_rel B"} {"_id": "500047", "text": "proof (prove)\ngoal (16 subgoals):\n 1. \\\\ e < s; \\ me < ms; e \\ me; ms = 0; ms \\ s;\n me = - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 2. \\\\ e < s; \\ me < ms; e \\ me; ms = 0; ms \\ s;\n me \\ - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 3. \\\\ e < s; \\ me < ms; e \\ me; ms = 0;\n \\ ms \\ s; me = - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 4. \\\\ e < s; \\ me < ms; e \\ me; ms = 0;\n \\ ms \\ s; me \\ - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 5. \\\\ e < s; \\ me < ms; e \\ me; ms \\ 0;\n ms \\ s; me = - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 6. \\\\ e < s; \\ me < ms; e \\ me; ms \\ 0;\n ms \\ s; me \\ - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 7. \\\\ e < s; \\ me < ms; e \\ me; ms \\ 0;\n \\ ms \\ s; me = - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 8. \\\\ e < s; \\ me < ms; e \\ me; ms \\ 0;\n \\ ms \\ s; me \\ - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 9. \\\\ e < s; \\ me < ms; \\ e \\ me; ms = 0;\n ms \\ s; me = - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\n 10. \\\\ e < s; \\ me < ms; \\ e \\ me; ms = 0;\n ms \\ s; me \\ - 1\\\n \\ wordinterval_to_set\n (if e < s \\ me < ms then WordInterval s e\n else if e \\ me\n then WordInterval (if ms = 0 then 1 else s)\n (min e (word_prev ms))\n else if ms \\ s\n then WordInterval (max s (word_next me))\n (if me = - 1 then 0 else e)\n else RangeUnion\n (WordInterval (if ms = 0 then 1 else s) (word_prev ms))\n (WordInterval (word_next me) (if me = - 1 then 0 else e))) =\n wordinterval_to_set (WordInterval s e) -\n wordinterval_to_set (WordInterval ms me)\nA total of 16 subgoals..."} {"_id": "500048", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_spmf R (p \\ f) (return_spmf x \\ (\\_. q))"} {"_id": "500049", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hn_refine (hn_val (A \\ B) f fi * hn_val A x xi)\n (return (pho_apply $ fi $ xi))\n (hn_val (A \\ B) f fi * hn_val A x xi) (pure B)\n (RETURN $ (pho_apply $ f $ x))"} {"_id": "500050", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ k \\ return\\\\ \\ k"} {"_id": "500051", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LstSeq s k z \\\n app (insf s (succ k) y) k' = (if k' = succ k then y else app s k')"} {"_id": "500052", "text": "proof (prove)\ngoal (1 subgoal):\n 1. invpst (mkNode c l a r) = (invpst l \\ invpst r)"} {"_id": "500053", "text": "proof (prove)\nusing this:\n \\\\<^sub>F i in sequentially. K \\ C i\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F i in sequentially. t i \\ K"} {"_id": "500054", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_integrable lborel {r..}\n (\\x. x powr s * exp (- a * x) / (1 - exp (- x)))"} {"_id": "500055", "text": "proof (prove)\nusing this:\n x \\ \\<^bold>V\n (y, x) \\ \\<^bold>E \\ h_plus ?i (y, x) = 0\n\ngoal (1 subgoal):\n 1. (y, x) \\ support_flow (\\e. \\n. h_plus n e)"} {"_id": "500056", "text": "proof (prove)\nusing this:\n reachable composition (s1, s2)\n invariant composition P7\n\ngoal (1 subgoal):\n 1. P7 (s1, s2)"} {"_id": "500057", "text": "proof (prove)\ngoal (1 subgoal):\n 1. certify cert =\n (subgraph (with_proj cert) (with_proj G) \\\n verts3 (with_proj cert) =\n verts3 (with_proj (pair_pre_digraph.slim cert)) \\\n (\\H.\n (K\\<^bsub>3,3\\<^esub> (with_proj H) \\\n K\\<^bsub>5\\<^esub> (with_proj H)) \\\n subdivision_pair H (pair_pre_digraph.slim cert)))"} {"_id": "500058", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\set (fos \\) \\ set bs1;\n set (sos \\) \\ set bs2;\n \\ = mk_I I1 I2 bs1 bs2\\\n \\ satisfies (dec_I1 \\ bs1) (dec_I2 \\ bs2)\n \\ =\n satisfies I1 I2 \\"} {"_id": "500059", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gmd pr + gmft m + gmnl sf =\n sum_list (map snd (digitToList pr @ toList m @ nlistToList sf))"} {"_id": "500060", "text": "proof (prove)\nusing this:\n r \\ r'\n\ngoal (1 subgoal):\n 1. \\s\\<^sub>3 s\\<^sub>3'.\n s\\<^sub>3 \\ s\\<^sub>3' \\\n (revision_step r' s\\<^sub>2 s\\<^sub>3 \\\n s\\<^sub>2 = s\\<^sub>3) \\\n (revision_step r s\\<^sub>2' s\\<^sub>3' \\ s\\<^sub>2' = s\\<^sub>3')"} {"_id": "500061", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pmf (sds R24) a = 0 &&&\n pmf (sds R24) b = 0 &&& pmf (sds R24) d = 1 - pmf (sds R24) c"} {"_id": "500062", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eq a'\n (\\ \\\\<^bsub>a'\\<^esub> e' \\\\<^bsub>\\1\\<^esub>)\n (\\ e' \\\\<^bsub>\\1\\<^esub>)"} {"_id": "500063", "text": "proof (prove)\ngoal (1 subgoal):\n 1. infdist y X = dist y x"} {"_id": "500064", "text": "proof (prove)\nusing this:\n \\?g \\ carrier G; ?h \\ carrier G\\\n \\ (?g \\ ?x = ?h \\ ?x) =\n (inv ?g \\ ?h \\ stabiliser ?x)\n inv g \\ h \\ stabiliser x\n g \\ carrier G\n h \\ carrier G\n\ngoal (1 subgoal):\n 1. g \\ x = h \\ x"} {"_id": "500065", "text": "proof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` local.span (set base) =\n basis_enum_of_vec ` mat_kernel Ag"} {"_id": "500066", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\A C B.\n \\M = zhmset_of A + C; N = zhmset_of B + C;\n A \\ B\\\n \\ thesis) \\\n thesis"} {"_id": "500067", "text": "proof (prove)\nusing this:\n Q u = enat d\n\ngoal (1 subgoal):\n 1. (\\v.\n \\\\ u = Some v; d = w {u, v}\\\n \\ thesis) \\\n thesis"} {"_id": "500068", "text": "proof (prove)\ngoal (1 subgoal):\n 1. igb.accept = accept &&& igba_rec.more G' = ecnv G"} {"_id": "500069", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (simple_fw rs1 p = Decision FinalAllow \\\n simple_fw rs2 p = Decision FinalAllow) =\n (simple_fw\n (rule_translate\n (generalized_fw_join (map simple_rule_dtor rs1)\n (map simple_rule_dtor rs2)))\n p =\n Decision FinalAllow)"} {"_id": "500070", "text": "proof (prove)\ngoal (1 subgoal):\n 1. val e"} {"_id": "500071", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^sub>\\i. P (i - Suc n)) =\n (\\\\<^sub>\\i. P (Suc i - Suc n))"} {"_id": "500072", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mcont luba orda lub_spmf (ord_spmf (=))\n (\\x. p \\ (\\y. g y x))"} {"_id": "500073", "text": "proof (prove)\nusing this:\n fst (if vars_term_ms (SCF (f (Suc i))) \\#\n vars_term_ms (SCF (f i)) \\\n weight (f (Suc i)) \\ weight (f i)\n then if weight (f (Suc i)) < weight (f i) then (True, True)\n else case f i of\n Var y \\\n (False,\n case f (Suc i) of Var x \\ x = y\n | Fun g ts \\ ts = [] \\ least g)\n | Fun fa ss \\\n case f (Suc i) of Var x \\ (True, True)\n | Fun g ts \\\n if pr_strict (fa, length ss) (g, length ts)\n then (True, True)\n else if pr_weak (fa, length ss) (g, length ts)\n then lex_ext_unbounded kbo ss ts\n else (False, False)\n else (False, False))\n\ngoal (1 subgoal):\n 1. weight (f (Suc i)) \\ weight (f i)"} {"_id": "500074", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sgn1 (Suc y) = 1"} {"_id": "500075", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C.dom (g.leg0 \\ fg.prj\\<^sub>0) = f.leg0 \\\\ g.leg1"} {"_id": "500076", "text": "proof (prove)\nusing this:\n finite A \\ A \\ {}\n\ngoal (1 subgoal):\n 1. Finite_Set.fold (\\x yo. yo \\ m x)\n (Some (SOME x. x \\ A)) (insert x A) =\n (if A = {} then Some x else Merge A \\ m x)"} {"_id": "500077", "text": "proof (prove)\nusing this:\n VVV.arr ?f =\n (arr (fst ?f) \\\n VV.arr (snd ?f) \\ src (fst ?f) = trg (fst (snd ?f)))\n VV.arr ?f =\n (arr (fst ?f) \\ arr (snd ?f) \\ src (fst ?f) = trg (snd ?f))\n src ?\\ = (if arr ?\\ then src\\<^sub>B ?\\ else null)\n trg ?\\ = (if arr ?\\ then trg\\<^sub>B ?\\ else null)\n B.arr ?f \\\n src\\<^sub>B ?f = a \\ trg\\<^sub>B ?f = a \\\n B.arr ?f\n arr ?f = (B.arr ?f \\ src\\<^sub>B ?f = a \\ trg\\<^sub>B ?f = a)\n\ngoal (1 subgoal):\n 1. VxVxV.arr f = VVV.arr f"} {"_id": "500078", "text": "proof (prove)\ngoal (1 subgoal):\n 1. create_gba \\ \\ SPEC (\\\\. nds_invars (g_V \\))"} {"_id": "500079", "text": "proof (prove)\nusing this:\n ?l < length xs \\\n pick (map (Pair 1) xs) (natural_of_nat ?l) = xs ! ?l\n l < length (x # xs)\n\ngoal (1 subgoal):\n 1. pick (map (Pair 1) (x # xs)) (natural_of_nat l) = (x # xs) ! l"} {"_id": "500080", "text": "proof (prove)\ngoal (1 subgoal):\n 1. i \\ mod_type_class.from_nat a"} {"_id": "500081", "text": "proof (prove)\ngoal (1 subgoal):\n 1. e \\ - v * - v\\<^sup>T \\\n - v * - v\\<^sup>T \\ top * v\\<^sup>T\n \\ v * - v\\<^sup>T \\ - v * - v\\<^sup>T"} {"_id": "500082", "text": "proof (state)\ngoal (2 subgoals):\n 1. bounded_linear (g' y)\n 2. \\e.\n (\\h.\n y + h \\ V \\\n g (y + h) = g y + g' y h + e h) \\\n (\\h. norm (e h) / norm h) \\0::'a\\ 0"} {"_id": "500083", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\t3.\n t2 \\\\<^sub>1\\<^sup>* t3 \\\n t1 \\\\<^sub>1\\<^sup>* t3"} {"_id": "500084", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AE x in MI. \\m n.\n uJ n (projI ((TI ^^ m) x)) = vI n ((TI ^^ m) x)"} {"_id": "500085", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ground_fm (WRP x y) = (ground x \\ ground y)"} {"_id": "500086", "text": "proof (prove)\nusing this:\n productF A B FAIL AB\n FAIL \\ succ AB w (p1, p2)\n well_formed A\n well_formed B\n p1 \\ nodes A \\\n p2 \\ nodes B \\\n fst w \\ inputs A \\ snd w \\ outputs A \\ outputs B\n\ngoal (1 subgoal):\n 1. succ B w p2 \\ nodes B"} {"_id": "500087", "text": "proof (prove)\nusing this:\n finite S'\n finite T'\n ?a1 \\ S' \\ g ?a1 = \\<^bold>1\n ?b1 \\ T' \\ h ?b1 = \\<^bold>1\n ?a1 \\ S \\ h (j ?a1) = g ?a1\n\ngoal (1 subgoal):\n 1. F (\\x. h (j x)) S = F h T"} {"_id": "500088", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\y. if t = 0 then g (ws' 0) y = 0 else h (ws' 0) y = 0"} {"_id": "500089", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subformulas\\<^sub>\\ (af \\ w) = subformulas\\<^sub>\\ \\"} {"_id": "500090", "text": "proof (chain)\npicking this:\n \\ \\ P'' \\ \\ \\ P'\n x \\ P''"} {"_id": "500091", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ennreal (1 - d2 / 2)\n < emeasure M\n {x \\ space M.\n \\B\\N1.\n real\n (card\n ({n. \\l\\{1..n}.\n u n x - u (n - l) x \\ - c0 * real l} \\\n {..\\"} {"_id": "500094", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b c.\n (b r\\ a) \\1 (a r\\ b) r\\\n c\n \\ c"} {"_id": "500095", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isnpolyh p n0 \\ isconstant p = polybound0 p"} {"_id": "500096", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n.\n prob\n {x \\ space M.\n \\\n < \\(\\i})\n \\ 0"} {"_id": "500097", "text": "proof (chain)\npicking this:\n as \\\\<^sub>i \\\\ s"} {"_id": "500098", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\<^bold>|w\\<^bold>| = 1 \\\n (\\l m.\n w = l \\ m \\ Lyndon l \\ Lyndon m \\ l i. iterate i\\f\\\\)"} {"_id": "500100", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\h ptr' sc thesis ptr.\n \\\\c.\n h \\ get_dom_component ptr'\n \\\\<^sub>r c \\\n set c \\ set sc \\ thesis;\n heap_is_wellformed h; DocumentClass.type_wf h;\n DocumentClass.known_ptrs h;\n h \\ get_scdom_component ptr \\\\<^sub>r sc;\n ptr' \\ set sc\\\n \\ thesis\n 2. \\h ptr' sc thesis ptr.\n \\\\owner_document.\n h \\ get_owner_document ptr'\n \\\\<^sub>r owner_document \\\n cast\\<^sub>d\\<^sub>o\\<^sub>c\\<^sub>u\\<^sub>m\\<^sub>e\\<^sub>n\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n owner_document\n \\ set sc \\\n thesis;\n heap_is_wellformed h; DocumentClass.type_wf h;\n DocumentClass.known_ptrs h;\n h \\ get_scdom_component ptr \\\\<^sub>r sc;\n ptr' \\ set sc\\\n \\ thesis\n 3. \\h ptr owner_document ptr' owner_document' x.\n \\heap_is_wellformed h; DocumentClass.type_wf h;\n DocumentClass.known_ptrs h;\n h \\ get_owner_document ptr\n \\\\<^sub>r owner_document;\n h \\ get_owner_document ptr'\n \\\\<^sub>r owner_document';\n owner_document \\ owner_document';\n x \\ set |h \\ get_scdom_component ptr|\\<^sub>r;\n x \\ set |h \\ get_scdom_component\n ptr'|\\<^sub>r\\\n \\ False"} {"_id": "500101", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f S.\n flow \\ f \\\n MFMC_Network.cut \\ S \\ orthogonal \\ f S"} {"_id": "500102", "text": "proof (prove)\nusing this:\n x \\ V\n\ngoal (1 subgoal):\n 1. \\- 1 \\ x\\ =\n \\- 1\\ * \\x\\"} {"_id": "500103", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n \\x \\ \\; y \\ \\; x \\ y;\n x \\ {x. x \\ i \\ c} =\n y \\ {x. x \\ i \\ c}\\\n \\ content (y \\ {x. x \\ i \\ c}) = 0"} {"_id": "500104", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P' r.\n \\t \\ g; t \\ r; s = t; ca = Guard f g c'; c = c';\n sa = t; a = AnnRec r P';\n \\, \\ \\\\<^bsub>/F\\<^esub> P' c' Q, A;\n r \\ g \\ pre P';\n r \\ - g \\ {} \\ f \\ F\\\n \\ t \\ pre P' \\\n (\\as.\n assertionsR \\ \\ Q A P' c'\n as \\\n assertionsR \\ \\ Q A (AnnRec r P')\n (Guard f g c') as) \\\n (\\pm cm.\n atomicsR \\ \\ P' c'\n (pm, cm) \\\n atomicsR \\ \\ (AnnRec r P')\n (Guard f g c') (pm, cm))"} {"_id": "500105", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rank 0 = 0"} {"_id": "500106", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\(a, b).\n hom_induced p (subtopology X S) {} X {} id\n a \\\\<^bsub>homology_group p X\\<^esub>\n hom_induced p (subtopology X T) {} X {} id b)\n \\ Group.iso\n (homology_group p (subtopology X S) \\\\\n homology_group p (subtopology X T))\n (homology_group p X)"} {"_id": "500107", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.left_complemented_monoid_algebra (*) (\\) (l\\)\n (\\) (<) (1::'a)"} {"_id": "500108", "text": "proof (prove)\nusing this:\n SecurityInvariant_preliminaries sinvar_spec\n \\SecurityInvariant_preliminaries ?sinvar; wf_graph ?G;\n \\ ?sinvar ?G ?nP\\\n \\ SecurityInvariant_withOffendingFlows.set_offending_flows\n ?sinvar ?G ?nP \\\n {}\n \\SecurityInvariant_withOffendingFlows.sinvar_mono ?sinvar;\n wf_graph \\nodes = ?N, edges = ?E\\; ?E' \\ ?E;\n ?sinvar \\nodes = ?N, edges = ?E\\ ?nP\\\n \\ ?sinvar \\nodes = ?N, edges = ?E'\\ ?nP\n \\SecurityInvariant_withOffendingFlows.sinvar_mono ?sinvar;\n wf_graph ?G; \\ ?sinvar ?G ?nP\\\n \\ ?sinvar \\nodes = nodes ?G, edges = {}\\\n ?nP =\n (SecurityInvariant_withOffendingFlows.set_offending_flows\n ?sinvar ?G ?nP \\\n {})\n SecurityInvariant_withOffendingFlows.sinvar_mono sinvar_spec\n {} \\ ?A\n nodes \\nodes = ?nodes, edges = ?edges, \\ = ?more\\ =\n ?nodes\n list_graph_to_graph ?G =\n \\nodes = set (nodesL ?G), edges = set (edgesL ?G)\\\n wf_list_graph G\n wf_list_graph ?G =\n (distinct (nodesL ?G) \\\n distinct (edgesL ?G) \\ wf_list_graph_axioms ?G)\n wf_graph (list_graph_to_graph ?G) = wf_list_graph_axioms ?G\n\ngoal (1 subgoal):\n 1. sinvar_spec \\nodes = set (nodesL G), edges = {}\\ nP"} {"_id": "500109", "text": "proof (prove)\nusing this:\n P \\* C\n bn \\ \\* C\n \\ \\ \\\\<^sub>P \\ Q \\ \\ \\ Q'\n \\ \\ Q' = \\ \\ ?P' \\\n Prop ?b (\\ \\ \\\\<^sub>P) Q ?P'\n extractFrame A\\<^sub>P = \\P, \\\\<^sub>P\\\n distinct P\n P \\* A\\<^sub>P\n P \\* Q\n P \\* \\\n P \\* \\\n P \\* Q'\n distinct (bn \\)\n bn \\ \\* \\\n bn \\ \\* \\\\<^sub>P\n bn \\ \\* A\\<^sub>P\n bn \\ \\* Q\n bn \\ \\* subject \\\n \\ \\ A\\<^sub>P \\ Q' = \\ \\ Q''\n\ngoal (1 subgoal):\n 1. Prop C \\ (A\\<^sub>P \\ Q) Q''"} {"_id": "500110", "text": "proof (state)\nthis:\n norm a ^ n * norm a ^ (k + i - j) = norm a ^ (n + (k + i - j))\n\ngoal (1 subgoal):\n 1. norm a ^ n * norm a ^ (k + i - j) \\ norm a ^ n * (c * norm a ^ k)"} {"_id": "500111", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. fml_sem fml x) = decide_universal fml &&&\n \\x. fml_sem fml x = decide_existential fml"} {"_id": "500112", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B \\ B' = {}; T \\ B \\ B'\\\n \\ (\\v\\T \\ B \\ T \\ B'.\n f v *\\<^sub>R v) =\n (\\v\\T. f v *\\<^sub>R v)"} {"_id": "500113", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.ring_1 = ring_1_ow UNIV"} {"_id": "500114", "text": "proof (prove)\nusing this:\n is_equalized_by e\n J.arr j\n \\a.\n if a = J.Zero then local.map a = C.dom f0\n else if a = J.One then local.map a = C.cod f0\n else if a = J.j0 then local.map a = f0\n else if a = J.j1 then local.map a = f1\n else local.map a = C.null\n local.map j = f1 \\ j = J.One \\ j = J.Zero \\ j = J.j0\n mkCone ?e \\\n \\j.\n if J.arr j then if j = J.Zero then ?e else f0 \\ ?e else C.null\n C.seq ?g ?f \\ C.cod (?g \\ ?f) = C.cod ?g\n\ngoal (1 subgoal):\n 1. j = J.Zero \\ local.map j \\ mkCone e (J.dom j) = mkCone e j"} {"_id": "500115", "text": "proof (prove)\nusing this:\n monotone (\\) (\\) (\\n. f^n \\)\n\ngoal (1 subgoal):\n 1. Ex (extreme_bound A (\\) {f^n \\ |. n})"} {"_id": "500116", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\us v vs. ys = us @ v # vs \\ P x v \\ list_emb P xs vs"} {"_id": "500117", "text": "proof (prove)\nusing this:\n k' \\ k \\ Abs_freeword\n proper_signed_list xs\n proper_signed_list ys\n\ngoal (1 subgoal):\n 1. k' (prappend_signed_list xs ys) = k' xs + k' ys"} {"_id": "500118", "text": "proof (prove)\ngoal (1 subgoal):\n 1. transformation_by_components.map (\\\\<^sub>C) (\\\\<^sub>C)\n C.map \\o a =\n G'\\F'o\\'.map a"} {"_id": "500119", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vertical_composite A B G F G \\'.map \\"} {"_id": "500120", "text": "proof (prove)\nusing this:\n length m = ?nc \\ Ball (set m) (vec nr) \\\n length (Matrix_Legacy.transpose nr m) = nr \\\n Ball (set (Matrix_Legacy.transpose nr m)) (vec ?nc)\n length (v # m) = nc \\ Ball (set (v # m)) (vec nr)\n nc = Suc ncc\n\ngoal (1 subgoal):\n 1. length (Matrix_Legacy.transpose nr m) = nr \\\n Ball (set (Matrix_Legacy.transpose nr m)) (vec ncc)"} {"_id": "500121", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\invar g; n \\ set (\\n g) - {Entry g}\\\n \\ isIdom g n (THE m. isIdom g n m)"} {"_id": "500122", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum content (extend ` \\) \\ content (cbox a b)"} {"_id": "500123", "text": "proof (prove)\nusing this:\n B \\ s \\ {}\n\ngoal (1 subgoal):\n 1. False"} {"_id": "500124", "text": "proof (prove)\nusing this:\n x \\ {x. order.greater_eq S (fst ` set (freeword x))}\n y \\ {x. order.greater_eq S (fst ` set (freeword x))}\n order.greater_eq (fst ` (set (freeword x) \\ set (freeword y)))\n (fst ` set (freeword (x - y)))\n\ngoal (1 subgoal):\n 1. x - y \\ {x. order.greater_eq S (fst ` set (freeword x))}"} {"_id": "500125", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [e * d = 1] (mod (P - 1) * (Q - 1))"} {"_id": "500126", "text": "proof (chain)\npicking this:\n infinite S"} {"_id": "500127", "text": "proof (state)\nthis:\n \\\\<^sup>+ x. ennreal (f x) \\M \\ \\\n\ngoal (1 subgoal):\n 1. \\epsilon>0.\n \\U\\sets M.\n emeasure M U < ennreal epsilon \\\n set_nn_integral M U h < ennreal delta"} {"_id": "500128", "text": "proof (prove)\nusing this:\n (\\x. ?P x = ?Q x) \\ {x. ?P x} = {x. ?Q x}\n\ngoal (1 subgoal):\n 1. {v. v \\ {uu_.\n \\\\ \\.\n uu_ = \\ \\\n (\\, \\) \\ prog_sem I \\ \\\n \\ \\ {v. v \\ fml_sem I \\}}} =\n {\\.\n \\\\.\n (\\, \\) \\ prog_sem I \\ \\\n \\ \\ fml_sem I \\}"} {"_id": "500129", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Suc 0 < k; m \\ NoMsg;\n \\t. event t = (t mod k = Suc 0); t0 = Suc (t * k);\n s =\n output_fun \\\n i_Exec_Comp_Stream_Init trans_fun (input \\ k) c\\\n \\ (i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun\n input c t =\n m) =\n (s t0 = m \\\n (\\ t' t0 [0\\].\n s t1 = NoMsg. t1 \\ t2 [0\\] \\ t'.\n event t2) \\\n (\\ t' t0 [0\\].\n \\ event t1.\n t1 \\ t2 [0\\] \\ t'.\n s t2 = m \\\n \\ event t2 \\\n (\\ t'' t2 [0\\].\n s t3 = NoMsg.\n t3 \\ t4 [0\\] \\ t''.\n event t4)))"} {"_id": "500130", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_offending_flows \\nodes = V, edges = E \\ X\\ nP\n \\ pow_combine X\n (set_offending_flows \\nodes = V, edges = E\\\n nP)"} {"_id": "500131", "text": "proof (prove)\ngoal (1 subgoal):\n 1. trace_core_eq core1 core2 E IA IU p q"} {"_id": "500132", "text": "proof (prove)\nusing this:\n Mapping.keys\n (fst (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n \\ set (\\n g) \\\n Mapping.keys\n (snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n \\ Mapping.keys (phis g) \\\n CFG_SSA_Transformed_notriv_linorder_code \\e \\n invar\n inEdges' Entry oldDefs oldUses defs\n (u g (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n s1_nodes_of_uses s1_phi_nodes_of))\n (p g (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n s1_nodes_of_uses s1_phi_nodes_of))\n var chooseNext_all\n uninst_code.triv_phis'\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n s1_triv_phis (snd (snd (snd s1))) =\n uninst_code.ssa.trivial_phis\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g\n uninst_code.nodes_of_uses' g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g)))\n (snd ` dom (Mapping.lookup (phis g))) (fst (snd (snd s1))) =\n uninst_code.nodes_of_uses' g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g)))\n (snd ` dom (Mapping.lookup s1_phis)) (fst (snd (snd s1)))\n\ngoal (1 subgoal):\n 1. Mapping.keys\n (fst (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n \\ set (\\n g) \\\n Mapping.keys\n (snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n \\ Mapping.keys (phis g) \\\n CFG_SSA_Transformed_notriv_linorder_code \\e \\n invar\n inEdges' Entry oldDefs oldUses defs\n (u g (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n (p g (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n var chooseNext_all \\\n uninst_code.triv_phis'\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g)\n (snd (snd (snd s1))) =\n uninst_code.ssa.trivial_phis\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g \\\n uninst_code.phi_equiv_mapping\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g (uninst_code.nodes_of_uses' g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (snd ` dom (Mapping.lookup (phis g))) (fst (snd (snd s1))))\n (uninst_code.ssa.useNodes_of\n (\\_.\n fst (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g) \\\n uninst_code.phi_equiv_mapping\n (\\_.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g (uninst_code.nodes_of_phis' g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis (\\_. snd (fst s1))\n g)))\n (snd (snd (snd s1))))\n (uninst_code.ssa.phiNodes_of\n (\\ga.\n snd (uninst_code.step_codem (u g (fst s1)) (p g (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g))\n (uninst_code.substitution_code (\\g. snd (fst s1)) g\n (Max (uninst_code.ssa.trivial_phis\n (\\_. snd (fst s1)) g)))\n (fst (snd (snd s1))) (snd (snd (snd s1)))))\n g)"} {"_id": "500133", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mat n n\n (\\(i, j).\n if i = j then D $$ (i, i) * adjoint D $$ (i, i) else 0) =\n mat n n (\\(i, j). if i = j then B $$ (i, i) else 0)"} {"_id": "500134", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n.\n norm (b - a) / 2 ^ n < e \\ thesis) \\\n thesis"} {"_id": "500135", "text": "proof (prove)\ngoal (1 subgoal):\n 1. outSetOfComponents level1 {data1, data10, data11} = {sA12, sA21}"} {"_id": "500136", "text": "proof (prove)\nusing this:\n ({#[entry fg p]#}, ww, {#u # ?r#} + ({#uh # rh#} + (ceh - {#u # r#})))\n \\ trcl (ntr fg) \\\n \\ww'.\n ({#[entry fg p]#}, ww',\n {#r' @ ?r#} + (csp' + ({#uh # rh#} + (ceh - {#u # r#}))))\n \\ trcl (ntr fg)\n ({#[entry fg p]#}, ww, ch) \\ trcl (ntr fg)\n ch = {#u # r#} + ({#uh # rh#} + (ceh - {#u # r#}))\n\ngoal (1 subgoal):\n 1. (\\ww'.\n ({#[entry fg p]#}, ww',\n {#r' @ r#} + (csp' + ({#uh # rh#} + (ceh - {#u # r#}))))\n \\ trcl (ntr fg) \\\n thesis) \\\n thesis"} {"_id": "500137", "text": "proof (prove)\ngoal (1 subgoal):\n 1. unique_on_cylinder t0 T x0 b f B B' &&& X \\ cball x0 b"} {"_id": "500138", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_rec (a # l) c h = h a l (list_rec l c h)"} {"_id": "500139", "text": "proof (prove)\nusing this:\n (poly (p div d) x = 0) = (poly p x = 0)\n d dvd p\n d dvd pderiv p\n sgn (poly (p * pderiv p) x) = (if x\\<^sub>0 < x then 1 else - 1) \\\n poly d x \\ 0\n\ngoal (1 subgoal):\n 1. sgn (poly (p div d * sturm_squarefree' p ! 1) x) =\n (if x\\<^sub>0 < x then 1 else - 1)"} {"_id": "500140", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Imon l (mkMinj i M) = Imon l (Minj i M)"} {"_id": "500141", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\ outs_\\\n (map_\\ rsuml lsumr\n (\\1 \\\\<^sub>\\\n (\\2 \\\\<^sub>\\ \\3)))) =\n (x \\ outs_\\\n ((\\1 \\\\<^sub>\\ \\2) \\\\<^sub>\\ \\3))"} {"_id": "500142", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\ e \\\\<^bsub>(\\\\\\\\)|\\<^sup>\\\\<^bsub>r\\<^esub>\\<^esub>)\\\n (C\\r) \\\n (\\\\ e \\\\<^bsub>\\\\delete x\n \\\\\\<^esub>)\\\n (C\\r)"} {"_id": "500143", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\valuation K v; x \\ carrier K; v x = 1;\n n_val K v x * Lv K v = v x; 0 < Lv K v\\\n \\ Lv K v = 1"} {"_id": "500144", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f \\ borel_measurable M) =\n (\\i\\Basis.\n \\a. {w \\ space M. a < f w \\ i} \\ sets M)"} {"_id": "500145", "text": "proof (prove)\nusing this:\n \\ \\ S\n\ngoal (1 subgoal):\n 1. False"} {"_id": "500146", "text": "proof (prove)\nusing this:\n vars_of (t \\ \\') = {}\n vars_of (v \\ \\') = {}\n replace_subterm t p v t'\n\ngoal (1 subgoal):\n 1. vars_of (t' \\ \\') = {}"} {"_id": "500147", "text": "proof (prove)\nusing this:\n Policy.sp_spec_subj_subj (Step.partition tid) (Step.partition partner)\n Policy.sp_spec_subj_obj (Step.partition partner) (PAGE mypage) WRITE\n\ngoal (1 subgoal):\n 1. \\ Policy.sp_spec_subj_obj u (PAGE mypage) READ"} {"_id": "500148", "text": "proof (prove)\ngoal (1 subgoal):\n 1. blinding_of_blindable h bo\n \\ vimage2p (hash_blindable h) (hash_blindable h) (=)"} {"_id": "500149", "text": "proof (state)\nthis:\n ?F1, \\<^bold>\\?G1, ?\\1 = \\<^bold>\\?G1, ?F1, ?\\1\n\ngoal (1 subgoal):\n 1. \\S' R'.\n \\\\.\n is_nnf_mset \\ \\\n is_disj R' \\\n is_nnf S' \\\n cnf R' = {R} \\\n cnf S' \\ S \\\n (R', S', \\ \\\\<^sub>n \\\n S', \\ \\\\<^sub>n)"} {"_id": "500150", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\<^sub>\\ r\n\ngoal (1 subgoal):\n 1. r\\a\\ = b"} {"_id": "500151", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_scc E U =\n (U \\ {} \\\n (\\u\\U.\n \\v\\U. (u, v) \\ (Restr E U)\\<^sup>*) \\\n (\\u\\U.\n \\v.\n v \\ U \\ (u, v) \\ E\\<^sup>* \\\n (\\u'\\U. (v, u') \\ E\\<^sup>*)))"} {"_id": "500152", "text": "proof (prove)\nusing this:\n a = ConstInt32 x1\n\ngoal (1 subgoal):\n 1. \\s;vs;vs_to_es ves @\n [$b_e]\\ \\_ i \\s';vs';es'\\"} {"_id": "500153", "text": "proof (prove)\nusing this:\n List.member \\ (\\ ! j)\n List.member \\ (\\ ! j)\n \\List.member ?\\ ?\\;\n ?\\ \\ fml_sem ?I ?\\\\\n \\ ?\\ \\ fml_sem ?I (foldr (||) ?\\ FF)\n\ngoal (1 subgoal):\n 1. \\ \\ fml_sem I (foldr (||) \\ FF)"} {"_id": "500154", "text": "proof (state)\nthis:\n simple_ownership_distinct ts\\<^sub>k\n\ngoal (1 subgoal):\n 1. safe_delayed (c' (n + k)) \\\n \\c' l.\n l \\ Suc k \\\n trace c' n l \\\n c' n = (u_ts, u_m, u_shared) \\\n (\\x\\l.\n length (fst (c' (n + x))) = length (fst (c (n + x)))) \\\n (\\x\n (l < Suc k \\ \\ safe_delayed (c' (n + l))) \\\n (\\x\\l.\n \\ts\\<^sub>x \\\\<^sub>x m\\<^sub>x ts\\<^sub>x'\n \\\\<^sub>x' m\\<^sub>x'.\n c (n + x) =\n (ts\\<^sub>x, m\\<^sub>x, \\\\<^sub>x) \\\n c' (n + x) =\n (ts\\<^sub>x', m\\<^sub>x', \\\\<^sub>x') \\\n ts\\<^sub>x' ! i = u_ts ! i \\\n (\\a\\u_owns. \\\\<^sub>x' a = u_shared a) \\\n (\\a\\u_owns. \\\\<^sub>x a = \\ a) \\\n (\\a\\u_owns. m\\<^sub>x' a = u_m a) \\\n (\\a\\u_owns. m\\<^sub>x a = m a)) \\\n (\\x\\l.\n \\ts\\<^sub>x \\\\<^sub>x m\\<^sub>x ts\\<^sub>x'\n \\\\<^sub>x' m\\<^sub>x'.\n c (n + x) =\n (ts\\<^sub>x, m\\<^sub>x, \\\\<^sub>x) \\\n c' (n + x) =\n (ts\\<^sub>x', m\\<^sub>x', \\\\<^sub>x') \\\n (\\jx.\n j \\ i \\\n ts\\<^sub>x' ! j = ts\\<^sub>x ! j) \\\n (\\a.\n a \\ u_owns \\\n a \\ owned (ts ! i) \\\n \\\\<^sub>x' a = \\\\<^sub>x a) \\\n (\\a.\n a \\ u_owns \\\n m\\<^sub>x' a = m\\<^sub>x a))"} {"_id": "500155", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (G ===> (=)) vars CFG_base2.vars"} {"_id": "500156", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 2 powr numeral i \\ float"} {"_id": "500157", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homogeneous_set (ideal {f} \\ P[X]) \\\n phull.subspace (ideal {f} \\ P[X])"} {"_id": "500158", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (- F \\ v) * (F \\ v)\\<^sup>T * top \\\n (- F \\ v) *\n (v\\<^sup>\\ \\ v\\<^sup>T\\<^sup>\\)\n \\ bot \\\n (- F \\ v) *\n (v\\<^sup>\\ \\ v\\<^sup>T\\<^sup>\\)"} {"_id": "500159", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ [a \\ b] =\n [\\ $ a \\ \\ $ b] \\ \\"} {"_id": "500160", "text": "proof (prove)\ngoal (1 subgoal):\n 1. u \\ PTA.inv_of A l'"} {"_id": "500161", "text": "proof (prove)\nusing this:\n J.ide j\n A_B.arr (\\' j) \\\n A_B.Map (A_B.cod (\\' j)) = A_B.Cod (\\' j)\n \\?f : ?a \\\\<^sub>J ?b\\ \\\n \\\\'\n ?f : \\'.A.map\n ?a \\\\<^sub>[\\<^sub>A\\<^sub>,\\<^sub>B\\<^sub>] D\n ?b\\\n\ngoal (1 subgoal):\n 1. A_B.Map (D (J.cod j)) = A_B.Cod (\\' j)"} {"_id": "500162", "text": "proof (prove)\nusing this:\n mbufn_take f z buf = (z', buf')\n list_all3 (\\P j xs. i \\ j \\ list_all2 P [i..i \\ ?k; Suc ?k \\ Mini i js;\n list_all2 (\\P. P ?k) Ps ?xs; U ?k ?z\\\n \\ U (Suc ?k) (f ?xs ?z)\n buf \\ []\n [] \\ set buf\n Mini i js = i\n\ngoal (1 subgoal):\n 1. U (Mini i js) z'"} {"_id": "500163", "text": "proof (prove)\nusing this:\n ab_spec ab\n D' =\n D \\\n (set hs \\ (set gs \\ set bs \\ set hs) \\\n set (ps -- sps) -\\<^sub>p\n set (ap gs bs (ps -- sps) hs data'))\n set ps \\ set bs \\ (set gs \\ set bs)\n D \\ (set gs \\ set bs) \\ (set gs \\ set bs)\n\ngoal (1 subgoal):\n 1. D' \\ (set gs \\ set (ab gs bs hs data')) \\\n (set gs \\ set (ab gs bs hs data'))"} {"_id": "500164", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (0 = \\\\<^sub>\\ A) = (\\x\\elts A. x = 0)"} {"_id": "500165", "text": "proof (prove)\nusing this:\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs e\\<^sub>2,\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1 (Val v),\n (h\\<^sub>2, ls\\<^sub>2,\n sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [Vs [\\] ls\\<^sub>2]\n P \\ \\e\\<^sub>2,(h, l, sh)\\ \\\n \\Val v,(h\\<^sub>2, l\\<^sub>2, sh\\<^sub>2)\\\n \\fv e\\<^sub>2 \\ set ?Vs;\n l \\\\<^sub>m [?Vs [\\] ?ls];\n length ?Vs + max_vars e\\<^sub>2 \\ length ?ls\\\n \\ \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 ?Vs\n e\\<^sub>2,\n (h, ?ls, sh)\\ \\\n \\fin\\<^sub>1 (Val v),\n (h\\<^sub>2, ls', sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [?Vs [\\] ls']\n \\b t. P \\ C has F,b:t in D\n fv (C\\\\<^sub>sF{D} := e\\<^sub>2) \\ set Vs\n l \\\\<^sub>m [Vs [\\] ls]\n length Vs + max_vars (C\\\\<^sub>sF{D} := e\\<^sub>2) \\ length ls\n\ngoal (1 subgoal):\n 1. \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs\n (C\\\\<^sub>sF{D} := e\\<^sub>2),\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1\n (THROW NoSuchFieldError),\n (h\\<^sub>2, ls', sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [Vs [\\] ls']"} {"_id": "500166", "text": "proof (state)\nthis:\n Cut .NotR u.M a v.NotL .N v{c:=y.P} =\n Cut .NotR u.M{c:=y.P} a v.NotL .N{c:=y.P} v\n\ngoal (1 subgoal):\n 1. \\b \\ y; b \\ c; b \\ P; v \\ y;\n v \\ c; v \\ P; u \\ y; u \\ c; u \\ P;\n a \\ y; a \\ c; a \\ P; b \\ a; v \\ u;\n v \\ M; v \\ N; u \\ N; u \\ v; a \\ M;\n a \\ N; a \\ b; b \\ M; v \\ u;\n b \\ a\\\n \\ Cut .NotR u.M a v.NotL .N v{c:=y.P} \\\\<^sub>a* Cut .N u.M{c:=y.P}"} {"_id": "500167", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (strip C = x ::= e) = (\\P. C = x ::= e {P})"} {"_id": "500168", "text": "proof (prove)\nusing this:\n (\\.P, \\.Q) \\ {(\\.P, \\.Q) |P Q. P \\ Q}\n\ngoal (1 subgoal):\n 1. \\.P \\ \\.Q"} {"_id": "500169", "text": "proof (prove)\ngoal (1 subgoal):\n 1. total_ordered_set A r = total_ordered_set A r'"} {"_id": "500170", "text": "proof (prove)\nusing this:\n antirange_semiring.bdia (\\) ar ?x ?y = ar (ar (?y \\ ?x))\n (antirange_semiring.bdia (\\) ar ?x ?y \\ ar (ar ?z)) =\n (ar (ar ?y) \\ ?x \\ ?x \\ ar (ar ?z))\n r ?x = ar (ar ?x)\n\ngoal (1 subgoal):\n 1. \\y.\n (\\i.\n \\q.\n i = r q \\\n r p \\ x \\ x \\ r q \\\n y \\ i) \\\n y \\ antirange_semiring.bdia (\\) ar x (r p)"} {"_id": "500171", "text": "proof (prove)\ngoal (1 subgoal):\n 1. desargues_config A B C A' B' C'\n (Pascal_Property.inter (line B C) (line B' C'))\n (Pascal_Property.inter (line A C) (line A' C'))\n (Pascal_Property.inter (line A B) (line A' B')) R"} {"_id": "500172", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\q1 q2.\n \\q1 + q2 \\ q; interaction_any_bounded_by \\1 q1;\n \\cipher \\.\n interaction_any_bounded_by (\\2 cipher \\) q2\\\n \\ thesis) \\\n thesis"} {"_id": "500173", "text": "proof (prove)\nusing this:\n \\generator g s = Skip ?x2.0;\n local.force ?x2.0 = Some (x, s')\\\n \\ local.unstream ?x2.0 = x # local.unstream s'\n local.force s = Some (x, s')\n\ngoal (1 subgoal):\n 1. local.unstream s = x # local.unstream s'"} {"_id": "500174", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\well_def a; well_def b\\\n \\ well_def (a ;; b)"} {"_id": "500175", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (?c, gbg_to_idx_ext)\n \\ (gbgv_impl_rel_ext Re Rv \\ Ri) \\\n gbgv_impl_rel_ext Re Rv \\\n \\igbgv_impl_rel_ext Ri Rv\\nres_rel"} {"_id": "500176", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_FGcontra L1\\\\ Co1\\\\\n Co2\\\\ Co3\\\\ Co4\\\\\n Co5\\\\ Contra1\\\\\n Contra2\\\\ Contra3\\\\\n Contra4\\\\ Contra5\\\\ =\n (rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4\n Contra5)\\\\"} {"_id": "500177", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\p\\<^sub>0\\set (take 8 PS).\n \\p\\<^sub>1\\set (drop 8 PS).\n snd p\\<^sub>0 \\ snd p\\<^sub>1"} {"_id": "500178", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ set xs \\ x \\ fold max xs z"} {"_id": "500179", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\GDERIV f x :> df; f x \\ 0\\\n \\ GDERIV (\\x. inverse (f x)) x\n :> - (inverse (f x))\\<^sup>2 *\\<^sub>R df"} {"_id": "500180", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_mset\n (r|map fst (snd (fst ((s, []), s, [])))| +\n r|map fst (snd (snd ((s, []), s, [])))|)\n \\ r \\m M"} {"_id": "500181", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fv\\<^sub>s\\<^sub>e\\<^sub>t (subterms (t \\ \\)) =\n fv\\<^sub>s\\<^sub>e\\<^sub>t\n (subterms t \\\\<^sub>s\\<^sub>e\\<^sub>t \\)"} {"_id": "500182", "text": "proof (prove)\nusing this:\n Arr t\n Arr u\n Src t = Trg u\n coherent t\n coherent u\n Arr t \\\n Ide (Dom t) \\\n Ide (Cod t) \\\n Ide \\<^bold>\\Dom t\\<^bold>\\ \\\n Ide \\<^bold>\\Cod t\\<^bold>\\ \\\n arr \\t\\ \\\n arr \\Dom t\\ \\\n ide \\Dom t\\ \\\n arr \\Cod t\\ \\ ide \\Cod t\\\n Arr u \\\n Ide (Dom u) \\\n Ide (Cod u) \\\n Ide \\<^bold>\\Dom u\\<^bold>\\ \\\n Ide \\<^bold>\\Cod u\\<^bold>\\ \\\n arr \\u\\ \\\n arr \\Dom u\\ \\\n ide \\Dom u\\ \\\n arr \\Cod u\\ \\ ide \\Cod u\\\n Arr ?t \\ Nml \\<^bold>\\?t\\<^bold>\\\n Arr ?t \\ Src \\<^bold>\\?t\\<^bold>\\ = Src ?t\n Arr ?t \\ Trg \\<^bold>\\?t\\<^bold>\\ = Trg ?t\n Arr ?t \\\n Dom \\<^bold>\\?t\\<^bold>\\ =\n \\<^bold>\\Dom ?t\\<^bold>\\\n Arr ?t \\\n Cod \\<^bold>\\?t\\<^bold>\\ =\n \\<^bold>\\Cod ?t\\<^bold>\\\n Arr ?t \\\n \\<^bold>\\?t\\<^bold>\\ \\ HHom (Src ?t) (Trg ?t)\n Arr ?t \\\n \\<^bold>\\?t\\<^bold>\\\n \\ VHom \\<^bold>\\Dom ?t\\<^bold>\\\n \\<^bold>\\Cod ?t\\<^bold>\\\n \\Nml \\<^bold>\\t\\<^bold>\\;\n Nml \\<^bold>\\u\\<^bold>\\;\n Src \\<^bold>\\t\\<^bold>\\ =\n Trg \\<^bold>\\u\\<^bold>\\\\\n \\ \\Cod \\<^bold>\\t\\<^bold>\\ \\<^bold>\\\n Cod \\<^bold>\\u\\<^bold>\\\\ \\\n (\\\\<^bold>\\t\\<^bold>\\\\ \\\n \\\\<^bold>\\u\\<^bold>\\\\) =\n \\\\<^bold>\\t\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n \\<^bold>\\u\\<^bold>\\\\ \\\n \\Dom \\<^bold>\\t\\<^bold>\\ \\<^bold>\\\n Dom \\<^bold>\\u\\<^bold>\\\\\n\ngoal (1 subgoal):\n 1. (\\\\<^bold>\\Cod t\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\Cod u\\<^bold>\\\\ \\\n (\\\\<^bold>\\t\\<^bold>\\\\ \\\n \\\\<^bold>\\u\\<^bold>\\\\)) \\\n (\\Dom t\\<^bold>\\\\ \\\n \\Dom u\\<^bold>\\\\) =\n (\\\\<^bold>\\t\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n \\<^bold>\\u\\<^bold>\\\\ \\\n \\\\<^bold>\\Dom t\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\Dom u\\<^bold>\\\\) \\\n (\\Dom t\\<^bold>\\\\ \\\n \\Dom u\\<^bold>\\\\)"} {"_id": "500183", "text": "proof (prove)\nusing this:\n g1 \\ g2\n\ngoal (1 subgoal):\n 1. f bl index {(x, ix), (y, iy)} = 1"} {"_id": "500184", "text": "proof (prove)\nusing this:\n (?\\1, ?Q1) mem CsQ \\\n \\P.\n (?\\1, P) mem CsP \\\n guarded P \\ (\\\\. \\ \\ P \\ ?Q1)\n ?\\1 \\ ?P1 \\ ?Q1 \\\n ?\\1 \\ ?P1 \\ ?Q1\n ?\\ \\ ?P \\ ?Q \\\n ?\\ \\ ?P \\ ?Q\n ?\\ \\ ?P \\ ?Q \\\n ?\\ \\ ?P \\\\weakBisim\\ ?Q\n\ngoal (1 subgoal):\n 1. \\ \\ Cases CsP \\ Cases CsQ"} {"_id": "500185", "text": "proof (prove)\ngoal (1 subgoal):\n 1. col (- A) i = - col A i"} {"_id": "500186", "text": "proof (prove)\nusing this:\n \\\\v. v \\ x \\ wf_tuple n A_x v;\n \\v. v \\ y \\ wf_tuple n A_y v;\n \\ True \\ A_y \\ A_x\\\n \\ (?v \\ join x True y) =\n (wf_tuple n (A_x \\ A_y) ?v \\\n restrict A_x ?v \\ x \\\n join_cond True y (restrict A_y ?v))\n \\wf_tuple n ?B ?v; A_x \\ A_y \\ ?B\\\n \\ wf_tuple n (A_x \\ A_y)\n (restrict (A_x \\ A_y) ?v)\n table n A_x x\n table n A_y y\n\ngoal (1 subgoal):\n 1. \\z Z.\n \\wf_tuple n Z z; A_x \\ A_y \\ Z\\\n \\ (restrict (A_x \\ A_y) z\n \\ join x True y) =\n (restrict A_x z \\ x \\\n restrict A_y z \\ y)"} {"_id": "500187", "text": "proof (prove)\nusing this:\n \\m\\set M.\n set ` set (implc_offending_flows (fst m) G) =\n c_offending_flows (snd m) (list_graph_to_graph G)\n\ngoal (1 subgoal):\n 1. set ` set (implc_get_offending_flows (get_impl M) G) =\n get_offending_flows (get_spec M) (list_graph_to_graph G)"} {"_id": "500188", "text": "proof (prove)\nusing this:\n subgroup H (homology_group p (subtopology X S))\n\ngoal (1 subgoal):\n 1. (\\(x, y).\n x \\\\<^bsub>homology_group p (subtopology X S)\\<^esub> y)\n \\ Group.iso\n (subgroup_generated (homology_group p (subtopology X S))\n H \\\\\n subgroup_generated (homology_group p (subtopology X S)) K)\n (homology_group p (subtopology X S))"} {"_id": "500189", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nfoldli [a..x s.\n ASSERT (ofs \\ x) \\ (\\_. f (x - ofs) s))"} {"_id": "500190", "text": "proof (state)\nthis:\n partition_on A ((\\b. {x \\ A. f' x = b}) ` B - {{}})\n\ngoal (1 subgoal):\n 1. (\\p\\<^sub>A p\\<^sub>B.\n \\p\\<^sub>A permutes A; p\\<^sub>B permutes B;\n \\x\\A. f x = p\\<^sub>B (f' (p\\<^sub>A x))\\\n \\ thesis) \\\n thesis"} {"_id": "500191", "text": "proof (state)\nthis:\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\ (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\2)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\ (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\2))\n\ngoal (4 subgoals):\n 1. \\t S.\n \\send\\t\\\\<^sub>s\\<^sub>t # S \\ \\1;\n \\2 =\n update\\<^sub>s\\<^sub>t \\1\n (send\\t\\\\<^sub>s\\<^sub>t # S);\n \\2 = \\1 @ [Step (receive\\t\\\\<^sub>s\\<^sub>t)];\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1))\\\n \\ tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2))\n 2. \\t S.\n \\receive\\t\\\\<^sub>s\\<^sub>t # S \\ \\1;\n \\2 =\n update\\<^sub>s\\<^sub>t \\1\n (receive\\t\\\\<^sub>s\\<^sub>t # S);\n \\2 = \\1 @ [Step (send\\t\\\\<^sub>s\\<^sub>t)];\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1))\\\n \\ tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2))\n 3. \\a t t' S.\n \\\\a: t \\ t'\\\\<^sub>s\\<^sub>t # S\n \\ \\1;\n \\2 =\n update\\<^sub>s\\<^sub>t \\1\n (\\a: t \\ t'\\\\<^sub>s\\<^sub>t # S);\n \\2 =\n \\1 @ [Step \\a: t \\ t'\\\\<^sub>s\\<^sub>t];\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1))\\\n \\ tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2))\n 4. \\X F S.\n \\\\X\\\\\\: F\\\\<^sub>s\\<^sub>t #\n S\n \\ \\1;\n \\2 =\n update\\<^sub>s\\<^sub>t \\1\n (\\X\\\\\\: F\\\\<^sub>s\\<^sub>t # S);\n \\2 =\n \\1 @\n [Step \\X\\\\\\: F\\\\<^sub>s\\<^sub>t];\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\1) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p ` set \\1))\\\n \\ tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>s\\<^sub>t ` \\2) \\\n \\\n (trms\\<^sub>e\\<^sub>s\\<^sub>t\\<^sub>p `\n set \\2))"} {"_id": "500192", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mon_c fg csp = {}"} {"_id": "500193", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\ fmla; set ts \\ trm;\n set xs \\ var; distinct xs; set us \\ var;\n distinct us; set us \\ Fvars \\ = {};\n set us \\ \\ (FvarsT ` set ts) = {};\n set us \\ set xs = {}; set vs \\ var; distinct vs;\n set vs \\ Fvars \\ = {};\n set vs \\ \\ (FvarsT ` set ts) = {};\n set vs \\ set xs = {}; length us = length xs;\n length vs = length xs; length ts = length xs\\\n \\ set (getFrN (xs @ us @ vs) ts [\\]\n (length xs)) \\\n Fvars \\ =\n {}"} {"_id": "500194", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(1::'a) + x \\ y; y \\ y = y\\\n \\ x\\<^sup>\\ \\ y"} {"_id": "500195", "text": "proof (prove)\nusing this:\n g permutes A\n a < b\n\ngoal (1 subgoal):\n 1. distr (Pi\\<^sub>M A (\\_. uniform_measure lborel {a..b}))\n (Pi\\<^sub>M A (\\_. lborel)) (\\f. f \\ g) =\n distr (Pi\\<^sub>M A (\\_. uniform_measure lborel {a..b}))\n (Pi\\<^sub>M A (\\_. uniform_measure lborel {a..b}))\n (\\f. \\x\\A. f (g x))"} {"_id": "500196", "text": "proof (prove)\ngoal (2 subgoals):\n 1. conucleus f \\ Sup_closed_set (Fix f)\n 2. conucleus f \\ comp_closed_set (Fix f)"} {"_id": "500197", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (top * e * (w\\<^sup>T \\ - v\\<^sup>T)\\<^sup>\\ \\\n - - v\\<^sup>T) *\n w\\<^sup>T =\n top * e * (w\\<^sup>T \\ - v\\<^sup>T)\\<^sup>\\ *\n w\\<^sup>T \\\n - - v\\<^sup>T * w\\<^sup>T"} {"_id": "500198", "text": "proof (prove)\ngoal (1 subgoal):\n 1. closure (cspan (range ket)) = UNIV"} {"_id": "500199", "text": "proof (state)\nthis:\n (ts, m, \\) \\\\<^sub>d\\<^sup>* (ts, m, \\)\n\ngoal (3 subgoals):\n 1. \\x21 x22 x23 x24.\n r = Read\\<^sub>s\\<^sub>b x21 x22 x23 x24 \\\n valid_ownership \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_reads m\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_history program_step ts\\<^sub>s\\<^sub>b' \\\n valid_sharing \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n tmps_distinct ts\\<^sub>s\\<^sub>b' \\\n valid_data_dependency ts\\<^sub>s\\<^sub>b' \\\n valid_sops ts\\<^sub>s\\<^sub>b' \\\n load_tmps_fresh ts\\<^sub>s\\<^sub>b' \\\n enough_flushs ts\\<^sub>s\\<^sub>b' \\\n valid_program_history ts\\<^sub>s\\<^sub>b' \\\n valid ts\\<^sub>s\\<^sub>b' \\\n (\\ts' \\' m'.\n (ts, m,\n \\) \\\\<^sub>d\\<^sup>* (ts', m', \\') \\\n (ts\\<^sub>s\\<^sub>b', m\\<^sub>s\\<^sub>b',\n \\\\<^sub>s\\<^sub>b') \\ (ts', m', \\'))\n 2. \\x31 x32 x33.\n r = Prog\\<^sub>s\\<^sub>b x31 x32 x33 \\\n valid_ownership \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_reads m\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_history program_step ts\\<^sub>s\\<^sub>b' \\\n valid_sharing \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n tmps_distinct ts\\<^sub>s\\<^sub>b' \\\n valid_data_dependency ts\\<^sub>s\\<^sub>b' \\\n valid_sops ts\\<^sub>s\\<^sub>b' \\\n load_tmps_fresh ts\\<^sub>s\\<^sub>b' \\\n enough_flushs ts\\<^sub>s\\<^sub>b' \\\n valid_program_history ts\\<^sub>s\\<^sub>b' \\\n valid ts\\<^sub>s\\<^sub>b' \\\n (\\ts' \\' m'.\n (ts, m,\n \\) \\\\<^sub>d\\<^sup>* (ts', m', \\') \\\n (ts\\<^sub>s\\<^sub>b', m\\<^sub>s\\<^sub>b',\n \\\\<^sub>s\\<^sub>b') \\ (ts', m', \\'))\n 3. \\x41 x42 x43 x44.\n r = Ghost\\<^sub>s\\<^sub>b x41 x42 x43 x44 \\\n valid_ownership \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_reads m\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n valid_history program_step ts\\<^sub>s\\<^sub>b' \\\n valid_sharing \\\\<^sub>s\\<^sub>b' ts\\<^sub>s\\<^sub>b' \\\n tmps_distinct ts\\<^sub>s\\<^sub>b' \\\n valid_data_dependency ts\\<^sub>s\\<^sub>b' \\\n valid_sops ts\\<^sub>s\\<^sub>b' \\\n load_tmps_fresh ts\\<^sub>s\\<^sub>b' \\\n enough_flushs ts\\<^sub>s\\<^sub>b' \\\n valid_program_history ts\\<^sub>s\\<^sub>b' \\\n valid ts\\<^sub>s\\<^sub>b' \\\n (\\ts' \\' m'.\n (ts, m,\n \\) \\\\<^sub>d\\<^sup>* (ts', m', \\') \\\n (ts\\<^sub>s\\<^sub>b', m\\<^sub>s\\<^sub>b',\n \\\\<^sub>s\\<^sub>b') \\ (ts', m', \\'))"} {"_id": "500200", "text": "proof (prove)\nusing this:\n fic M' a\n y':N'' \\ \\(A)\\\n M' = NotR y'.N'' a\n\ngoal (1 subgoal):\n 1. :M' \\ NOTRIGHT (NOT A) (\\(A)\\)"} {"_id": "500201", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Mapping.lookup (fold (addN g) xs m) v =\n (case Mapping.lookup m v of\n None \\\n if \\n\\set xs. v \\ uses g n\n then Some {n \\ set xs. v \\ uses g n} else None\n | Some N \\\n Some ({n \\ set xs. v \\ uses g n} \\ N))"} {"_id": "500202", "text": "proof (chain)\npicking this:\n D' \\ inferred_clause_sets R S M"} {"_id": "500203", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (w \\ w\\<^sup>T\\<^sup>+) * r = bot"} {"_id": "500204", "text": "proof (prove)\nusing this:\n Q_invar Qi \\\n M_invar \\i \\ prim_invar2 (Q_\\ Qi) (M_lookup \\i)\n\ngoal (1 subgoal):\n 1. Q_invar Qi &&& M_invar \\i"} {"_id": "500205", "text": "proof (prove)\ngoal (10 subgoals):\n 1. \\SK KK.\n \\SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK []) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy []) \\\n SK \\ KK \\ Key SK \\ analz (knows Spy [])\n 2. \\evsf X B SK KK.\n \\evsf \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evsf) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evsf)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evsf));\n X \\ synth (analz (knows Spy evsf)); SK \\ symKeys;\n KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says Spy B X #\nevsf)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says Spy B X # evsf)) \\\n SK \\ KK \\\n Key SK\n \\ analz (knows Spy (Says Spy B X # evsf))\n 3. \\evs1 A SK KK.\n \\evs1 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs1) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs1)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs1));\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says A Kas\n \\Agent A, Agent Tgs, Number (CT evs1)\\ #\nevs1)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says A Kas\n \\Agent A, Agent Tgs,\nNumber (CT evs1)\\ #\n evs1)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says A Kas\n \\Agent A, Agent Tgs,\nNumber (CT evs1)\\ #\n evs1))\n 4. \\evs2 authK A' A T1 SK KK.\n \\evs2 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs2) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs2)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs2));\n Key authK \\ used evs2; authK \\ symKeys;\n Says A' Kas \\Agent A, Agent Tgs, Number T1\\\n \\ set evs2;\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says Kas A\n \\Crypt (shrK A)\n \\Key authK, Agent Tgs, Number (CT evs2)\\,\n Crypt (shrK Tgs)\n \\Agent A, Agent Tgs, Key authK,\n Number (CT evs2)\\\\ #\nevs2)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says Kas A\n \\Crypt (shrK A)\n \\Key authK, Agent Tgs, Number (CT evs2)\\,\nCrypt (shrK Tgs)\n \\Agent A, Agent Tgs, Key authK,\n Number (CT evs2)\\\\ #\n evs2)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says Kas A\n \\Crypt (shrK A)\n \\Key authK, Agent Tgs, Number (CT evs2)\\,\nCrypt (shrK Tgs)\n \\Agent A, Agent Tgs, Key authK,\n Number (CT evs2)\\\\ #\n evs2))\n 5. \\evs3 A T1 Kas' authK Ta authTicket B SK KK.\n \\evs3 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs3) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs3)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs3));\n A \\ Kas; A \\ Tgs;\n Says A Kas \\Agent A, Agent Tgs, Number T1\\\n \\ set evs3;\n valid Ta wrt T1; authTicket \\ analz (knows Spy evs3);\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says A Tgs\n \\authTicket,\n Crypt authK \\Agent A, Number (CT evs3)\\,\n Agent B\\ #\nevs3)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says A Tgs\n \\authTicket,\nCrypt authK \\Agent A, Number (CT evs3)\\, Agent B\\ #\n evs3)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says A Tgs\n \\authTicket,\nCrypt authK \\Agent A, Number (CT evs3)\\, Agent B\\ #\n evs3))\n 6. \\evs4 servK B authK A' A Ta T2 SK KK.\n \\evs4 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs4) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs4)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs4));\n Key servK \\ used evs4; servK \\ symKeys; B \\ Tgs;\n authK \\ symKeys;\n Says A' Tgs\n \\Crypt (shrK Tgs)\n \\Agent A, Agent Tgs, Key authK,\n Number Ta\\,\n Crypt authK \\Agent A, Number T2\\,\n Agent B\\\n \\ set evs4;\n \\ expiredAK Ta evs4; \\ expiredA T2 evs4;\n servKlife + CT evs4 \\ authKlife + Ta; SK \\ symKeys;\n KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says Tgs A\n \\Crypt authK\n \\Key servK, Agent B, Number (CT evs4)\\,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number (CT evs4)\\\\ #\nevs4)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says Tgs A\n \\Crypt authK\n \\Key servK, Agent B, Number (CT evs4)\\,\nCrypt (shrK B)\n \\Agent A, Agent B, Key servK, Number (CT evs4)\\\\ #\n evs4)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says Tgs A\n \\Crypt authK\n \\Key servK, Agent B, Number (CT evs4)\\,\nCrypt (shrK B)\n \\Agent A, Agent B, Key servK, Number (CT evs4)\\\\ #\n evs4))\n 7. \\evs5 authK servK A authTicket T2 B Tgs' Ts servTicket SK KK.\n \\evs5 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs5) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs5)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs5));\n authK \\ symKeys; servK \\ symKeys; A \\ Kas;\n A \\ Tgs;\n Says A Tgs\n \\authTicket,\n Crypt authK \\Agent A, Number T2\\,\n Agent B\\\n \\ set evs5;\n valid Ts wrt T2; servTicket \\ analz (knows Spy evs5);\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says A B\n \\servTicket,\n Crypt servK \\Agent A, Number (CT evs5)\\\\ #\nevs5)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says A B\n \\servTicket,\nCrypt servK \\Agent A, Number (CT evs5)\\\\ #\n evs5)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says A B\n \\servTicket,\nCrypt servK \\Agent A, Number (CT evs5)\\\\ #\n evs5))\n 8. \\evs6 B A' A servK Ts T3 Ta2 SK KK.\n \\evs6 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK. \\ AKcryptSK K SK evs6) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evs6)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evs6));\n B \\ Kas; B \\ Tgs;\n Says A' B\n \\Crypt (shrK B)\n \\Agent A, Agent B, Key servK, Number Ts\\,\n Crypt servK \\Agent A, Number T3\\\\\n \\ set evs6;\n \\ expiredSK Ts evs6; \\ expiredA T3 evs6; SK \\ symKeys;\n KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Says B A (Crypt servK (Number Ta2)) #\nevs6)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Says B A (Crypt servK (Number Ta2)) #\n evs6)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Says B A (Crypt servK (Number Ta2)) #\n evs6))\n 9. \\evsO1 A authK Ta authTicket SK KK.\n \\evsO1 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK.\n \\ AKcryptSK K SK evsO1) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evsO1)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evsO1));\n A \\ Spy;\n Says Kas A\n \\Crypt (shrK A)\n \\Key authK, Agent Tgs, Number Ta\\,\n authTicket\\\n \\ set evsO1;\n expiredAK Ta evsO1; authK \\ range shrK; authK \\ symKeys;\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n (Notes Spy\n \\Agent A, Agent Tgs, Number Ta, Key authK\\ #\nevsO1)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Notes Spy\n \\Agent A, Agent Tgs, Number Ta,\nKey authK\\ #\n evsO1)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Notes Spy\n \\Agent A, Agent Tgs, Number Ta,\nKey authK\\ #\n evsO1))\n 10. \\evsO2 A authK servK B Ts servTicket SK KK.\n \\evsO2 \\ kerbV;\n \\SK KK.\n SK \\ symKeys \\\n KK \\ - range shrK \\\n (\\K\\KK.\n \\ AKcryptSK K SK evsO2) \\\n (Key SK \\ analz (Key ` KK \\ knows Spy evsO2)) =\n (SK \\ KK \\ Key SK \\ analz (knows Spy evsO2));\n A \\ Spy;\n Says Tgs A\n \\Crypt authK\n \\Key servK, Agent B, Number Ts\\,\n servTicket\\\n \\ set evsO2;\n expiredSK Ts evsO2; servK \\ range shrK; servK \\ symKeys;\n SK \\ symKeys; KK \\ - range shrK\\\n \\ (\\K\\KK.\n \\ AKcryptSK K SK\n(Notes Spy \\Agent A, Agent B, Number Ts, Key servK\\ #\n evsO2)) \\\n Key SK\n \\ analz\n (Key ` KK \\\n knows Spy\n (Notes Spy\n \\Agent A, Agent B, Number Ts,\n Key servK\\ #\n evsO2)) \\\n SK \\ KK \\\n Key SK\n \\ analz\n (knows Spy\n (Notes Spy\n \\Agent A, Agent B, Number Ts,\n Key servK\\ #\n evsO2))"} {"_id": "500206", "text": "proof (prove)\nusing this:\n \\ g \\ Bij M\n \\ h \\ Bij M\n\ngoal (1 subgoal):\n 1. (\\\n g \\\\<^bsub>\\carrier = Bij M, monoid.mult = \\g\\Bij M. restrict (compose M g) (Bij M), one = \\x\\M. x\\\\<^esub>\n \\ h)\n m =\n compose M (\\ g) (\\ h) m"} {"_id": "500207", "text": "proof (prove)\nusing this:\n \\fmlookup \\ ?n3 = Some ?v3;\n value_pred.pred\n (\\_.\n list_all (\\(uu_, t). consts t |\\| all_consts))\n (\\name. name |\\| C) (\\dom. dom |\\| heads)\n ?v3\\\n \\ consts (value_to_sterm ?v3) |\\| all_consts\n fmpred\n (\\_.\n value_pred.pred\n (\\_.\n list_all (\\(uu_, t). consts t |\\| all_consts))\n (\\name. name |\\| C)\n (\\dom. dom |\\| heads))\n \\\n list_all (\\(uu_, t). consts t |\\| all_consts) cs\n\ngoal (1 subgoal):\n 1. \\xb y.\n \\x_ =\n consts\n (subst b_\n (fmmap value_to_sterm\n (fmdrop_fset (frees a_) \\)));\n (a_, b_) \\ set cs; x = (a_, b_);\n fmlookup \\ xb = Some y\\\n \\ consts (value_to_sterm y) |\\| all_consts"} {"_id": "500208", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\g x n fa sa t ga.\n \\A \\ {}; fgmodule_condition R f i s A z;\n free_generator R (fgmodule R A z i f s) A; R module M;\n free_generator R M A; R module fgmodule R A z i f s;\n g \\ mHom R M (fgmodule R A z i f s); \\x\\A. g x = x;\n l_comb R M (card (fa ` {j. j \\ n}) - Suc 0) t ga\n \\ carrier M;\n g (l_comb R M (card (fa ` {j. j \\ n}) - Suc 0) t ga) =\n \\\\<^bsub>fgmodule R A z i f s\\<^esub>;\n A \\ carrier M; fa \\ {j. j \\ n} \\ A;\n sa \\ {j. j \\ n} \\ carrier R;\n x = l_comb R M (card (fa ` {j. j \\ n}) - Suc 0) t ga;\n ideal R (carrier R);\n t \\ {j. j \\ card (fa ` {j. j \\ n}) -\n Suc 0} \\\n carrier R;\n ga \\ {j. j \\ card (fa ` {j. j \\ n}) -\n Suc 0} \\\n fa ` {j. j \\ n};\n surj_to ga {j. j \\ card (fa ` {j. j \\ n}) - Suc 0}\n (fa ` {j. j \\ n});\n l_comb R M n sa fa =\n l_comb R M (card (fa ` {j. j \\ n}) - Suc 0) t ga;\n l_comb R (fgmodule R A z i f s) (card (fa ` {j. j \\ n}) - Suc 0)\n t ga =\n \\\\<^bsub>fgmodule R A z i f s\\<^esub>;\n l_comb R (fgmodule R A z i f s) (card (fa ` {j. j \\ n}) - Suc 0)\n t (cmp g ga) =\n \\\\<^bsub>fgmodule R A z i f s\\<^esub>;\n finite {na. na \\ n};\n finite {na. na \\ card (fa ` {j. j \\ n}) - Suc 0};\n finite (fa ` {j. j \\ n}); fa 0 \\ fa ` {j. j \\ n};\n fa ` {j. j \\ n} \\ {}; 0 < card (fa ` {j. j \\ n});\n inj_on ga {i. i \\ card (fa ` {j. j \\ n}) - Suc 0};\n A \\ carrier (fgmodule R A z i f s);\n \\j\\{j. j \\ card (fa ` {j. j \\ n}) - Suc 0}.\n t j = \\\\\n \\ l_comb R M (card (fa ` {j. j \\ n}) - Suc 0) t\n ga =\n \\\\<^bsub>M\\<^esub>\n 2. \\g.\n \\A \\ {}; fgmodule_condition R f i s A z;\n free_generator R (fgmodule R A z i f s) A; R module M;\n free_generator R M A; R module fgmodule R A z i f s;\n g \\ mHom R M (fgmodule R A z i f s);\n \\x\\A. g x = x\\\n \\ {\\\\<^bsub>M\\<^esub>}\n \\ ker\\<^bsub>M,fgmodule R A z i f\n s\\<^esub> g\n 3. \\g.\n \\A \\ {}; fgmodule_condition R f i s A z;\n free_generator R (fgmodule R A z i f s) A; R module M;\n free_generator R M A; R module fgmodule R A z i f s;\n g \\ mHom R M (fgmodule R A z i f s);\n \\x\\A. g x = x\\\n \\ surjec\\<^bsub>M,fgmodule R A z i f s\\<^esub> g"} {"_id": "500209", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Col U Q P"} {"_id": "500210", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map (\\i. x) [0..v\n (M_mat_R signs subsets *\\<^sub>v w_vec_R p qs signs) =\n w_vec_R p qs signs\n\ngoal (1 subgoal):\n 1. w_vec_R p qs signs =\n matr_option (dim_row (M_mat_R signs subsets))\n (mat_inverse (M_mat_R signs subsets)) *\\<^sub>v\n v_vec_R p qs subsets"} {"_id": "500212", "text": "proof (prove)\nusing this:\n rel_fun\n (rel_fun (BNF_Def.Grp UNIV id)\n (rel_fun (BNF_Def.Grp UNIV f)\\\\\n (rel_spmf (rel_prod (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV id)))))\n (rel_fun (BNF_Def.Grp UNIV id)\n (rel_resource (BNF_Def.Grp UNIV f)\\\\\n (BNF_Def.Grp UNIV g)))\n resource_of_oracle resource_of_oracle\n\ngoal (1 subgoal):\n 1. map_resource f g (resource_of_oracle oracle s) =\n resource_of_oracle\n (map_fun id (map_fun f (map_spmf (map_prod g id))) oracle) s"} {"_id": "500213", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\is1 | is1 \\ input_sizes m1.\n (\\kis2 | is2 \\ input_sizes m2.\n (\\knl. nl = xs @ 0 \\ (fp - arity) @ anything} ap\n {\\nl.\n nl =\n xs @ rec_exec (Mn n f) xs # 0 \\ (fp - Suc arity) @ anything}"} {"_id": "500216", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nontriv_solution fund_sol &&&\n (nontriv_solution (a, b) \\\n pell_valuation (case fund_sol of (x, y) \\ (int x, int y))\n \\ pell_valuation (int a, int b)) &&&\n (nontriv_solution z \\\n pell_valuation (case fund_sol of (x, y) \\ (int x, int y))\n \\ pell_valuation (case z of (x, y) \\ (int x, int y)))"} {"_id": "500217", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj undual"} {"_id": "500218", "text": "proof (prove)\ngoal (1 subgoal):\n 1. int (N ^ m * 2 ^ (m - 1) * m) \\ int (N ^ (2 * m) * 2 ^ m * m)"} {"_id": "500219", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (f ` set_of_vector X) = card (set_of_vector X)"} {"_id": "500220", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pred.eval (Anno_code_i_i_i_o P E e) = Anno_code P E e"} {"_id": "500221", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\.rank_of (insert x X) = rank_of (insert x X)"} {"_id": "500222", "text": "proof (prove)\nusing this:\n count (image_mset f A) x = sum (count A) (f -` {x} \\ set_mset A)\n (if x = ?y then Suc ?n else ?n) = ?n + (if x = ?y then 1 else 0)\n\ngoal (1 subgoal):\n 1. count (image_mset f (add_mset x A)) x =\n sum (count (add_mset x A)) (f -` {x} \\ set_mset (add_mset x A))"} {"_id": "500223", "text": "proof (prove)\nusing this:\n P,a\\compareAndSwap(D\\F, i, e),h' \\ (e''\\compareAndSwap(D\\F, i, e),\n xs'') \\ (stk', loc', pc', xcp')\n no_call2 (a\\compareAndSwap(D\\F, i, e)) pc = no_call2 a pc\n if \\move2 (compP2 P) h stk a pc xcp\n then h' = h \\\n (if xcp' = None \\ pc < pc' then \\red1r\n else \\red1t)\n P t h (a', xs) (e'', xs'')\n else \\ta' e' xs'.\n \\red1r P t h (a', xs) (e', xs') \\\n True,P,t \\1 \\e',\n (h, xs')\\ -ta'\\\n \\e'',(h', xs'')\\ \\\n ta_bisim wbisim1 (extTA2J1 P ta') ta \\\n \\ \\move1 P h e' \\\n (call1 a' = None \\\n no_call2 a pc \\ e' = a' \\ xs' = xs)\n \\move2 (compP2 P) h stk (a\\compareAndSwap(D\\F, i, e))\n pc xcp =\n \\move2 (compP2 P) h stk a pc xcp\n\ngoal (1 subgoal):\n 1. \\e'' xs''.\n P,a\\compareAndSwap(D\\F, i, e),h' \\ (e'',\n xs'') \\ (stk', loc', pc',\n xcp') \\\n (if \\move2 (compP2 P) h stk\n (a\\compareAndSwap(D\\F, i, e)) pc xcp\n then h' = h \\\n (if xcp' = None \\ pc < pc' then \\red1r\n else \\red1t)\n P t h (a'\\compareAndSwap(D\\F, i, e), xs)\n (e'', xs'')\n else \\ta' e' xs'.\n \\red1r P t h\n (a'\\compareAndSwap(D\\F, i, e), xs)\n (e', xs') \\\n True,P,t \\1 \\e',\n (h, xs')\\ -ta'\\\n \\e'',(h', xs'')\\ \\\n ta_bisim wbisim1 (extTA2J1 P ta') ta \\\n \\ \\move1 P h e' \\\n (call1 (a'\\compareAndSwap(D\\F, i, e)) =\n None \\\n no_call2 (a\\compareAndSwap(D\\F, i, e))\n pc \\\n e' = a'\\compareAndSwap(D\\F, i, e) \\\n xs' = xs))"} {"_id": "500224", "text": "proof (prove)\ngoal (1 subgoal):\n 1. design \\ \\"} {"_id": "500225", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f \\ g) e \\ capacity (residual_network g) e"} {"_id": "500226", "text": "proof (prove)\ngoal (1 subgoal):\n 1. natural_transformation (\\\\<^sub>A\\<^sub>1\\<^sub>x\\<^sub>A\\<^sub>2)\n (\\\\<^sub>B) (uncurry F) (uncurry G) (uncurry \\)"} {"_id": "500227", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mpoly_to_poly v (p1 + p2) = mpoly_to_poly v p1 + mpoly_to_poly v p2"} {"_id": "500228", "text": "proof (prove)\nusing this:\n a = Decomp t\n\ngoal (1 subgoal):\n 1. (\\f T. t = Fun f T \\ thesis) \\\n thesis"} {"_id": "500229", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\M; map (\\d.\n \\X\\\\\\: [(pair (t, s),\n pair (snd d))]\\\\<^sub>s\\<^sub>t)\n (filter (\\d. d \\ set Di)\n (dl # D))\\\\<^sub>d\n \\"} {"_id": "500230", "text": "proof (state)\nthis:\n x' = h \\ x\n\ngoal (1 subgoal):\n 1. x' \\ H #> x"} {"_id": "500231", "text": "proof (prove)\nusing this:\n (\\f. f \\ M \\ f \\ M) \\\n (\\x\\M. \\ (x ` Y)) =\n (\\x\\Y. \\f\\M. f x)\n\ngoal (1 subgoal):\n 1. fun_lub Sup M (\\ Y) = \\ (fun_lub Sup M ` Y)"} {"_id": "500232", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ t\\<^sub>1\\<^sub>2[0 \\\\<^sub>\\ T\\<^sub>2] : T"} {"_id": "500233", "text": "proof (state)\nthis:\n \\ treach \\ ref k \\ ref l\n\ngoal (1 subgoal):\n 1. \\l k.\n \\s\\ l reachable_from (s@@k);\n \\ treach \\ s@@k \\ ref l \\ False;\n ref k \\ nullV;\n \\ treach \\ ref k \\ ref l\\\n \\ False"} {"_id": "500234", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distinct (map basis_enum_of_vec gs)"} {"_id": "500235", "text": "proof (prove)\nusing this:\n fst (exec_step P h stk loc C M pc ics frs sh) = \\xcp\\\n P \\ cname_of h xcp \\\\<^sup>* D\n pc \\ {f..i. iterates body G i P s + iterates body G i Q s)\n \\ wp (\\x.\n body ;;\n x \\<^bsub>\\ G \\\\<^esub>\\ Skip)\n P s +\n wp (\\x.\n body ;;\n x \\<^bsub>\\ G \\\\<^esub>\\ Skip)\n Q s\n\ngoal (1 subgoal):\n 1. \\P Q s.\n \\sound P; sound Q\\\n \\ wp (\\x.\n body ;;\n x \\<^bsub>\\ G \\\\<^esub>\\ Skip)\n P s +\n wp (\\x.\n body ;;\n x \\<^bsub>\\ G \\\\<^esub>\\ Skip)\n Q s\n \\ wp (\\x.\nbody ;; x \\<^bsub>\\ G \\\\<^esub>\\ Skip)\n (\\s'. P s' + Q s') s"} {"_id": "500237", "text": "proof (prove)\nusing this:\n a \\ carrier G \\ a \\ Units G\n\ngoal (1 subgoal):\n 1. a \\ Units G"} {"_id": "500238", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s.\n k pfixGe g s \\\n {sa. g s \\ f sa} \\ {s. k pfixGe f s}"} {"_id": "500239", "text": "proof (prove)\nusing this:\n ?k < j \\ f' ?k = f ?k\n\ngoal (1 subgoal):\n 1. matrix_sum d f j = matrix_sum d f' j"} {"_id": "500240", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\X Y. (X, Y) \\ rel \\ qSkel Y = qSkel X\n 2. \\xs x. chi (termIn (qVar xs x))\n 3. \\delta inp binp.\n \\liftAll (\\a. chi (termIn a)) inp;\n liftAll (\\a. chi (absIn a)) binp\\\n \\ chi (termIn (qOp delta inp binp))\n 4. \\xs x X.\n (\\Y.\n (X, Y) \\ rel \\\n chi (termIn Y)) \\\n chi (absIn (qAbs xs x X))"} {"_id": "500241", "text": "proof (prove)\ngoal (1 subgoal):\n 1. category (set_cat U)"} {"_id": "500242", "text": "proof (prove)\nusing this:\n x = tau_steps y nx\n y = tau_steps x ny\n\ngoal (1 subgoal):\n 1. ny + nx = 0"} {"_id": "500243", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\clist xa.\n \\xs \\ [];\n \\i\n (\\j\\{j. j < length xs \\ j \\ i}.\n Guar (xs ! j))\n \\ Rely (xs ! i);\n pre \\ (\\i\\{i. i < length xs}. Pre (xs ! i));\n x \\ par_cp (map (Some \\ fst) xs) s;\n x \\ par_assum (pre, rely); All_None (fst (last x));\n length clist = length xs; xa < length xs;\n \\i assum (Pre (xs ! i), Rely (xs ! i));\n clist ! xa \\ cp (Some (fst (xs ! xa))) s; same_length x clist;\n \\i.\n Suc i < length (clist ! xa) \\\n clist ! xa !\n i -c\\ clist ! xa ! Suc i \\\n (snd (clist ! xa ! i), snd (clist ! xa ! Suc i))\n \\ Guar (xs ! xa);\n fst (last (clist ! xa)) = None \\\n snd (last (clist ! xa)) \\ Post (xs ! xa);\n same_state x clist; compat_label x clist;\n length x - 1 < length x \\\n fst (x ! (length x - 1)) =\n map (\\xa. fst (xa ! (length x - 1))) clist\\\n \\ snd (last x) \\ Post (xs ! xa)\n 2. \\clist xa.\n \\xs \\ [];\n \\i\n (\\j\\{j. j < length xs \\ j \\ i}.\n Guar (xs ! j))\n \\ Rely (xs ! i);\n pre \\ (\\i\\{i. i < length xs}. Pre (xs ! i));\n \\i Com (xs !\n i) sat [Pre\n(xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)];\n x \\ par_cp (map (Some \\ fst) xs) s;\n x \\ par_assum (pre, rely); All_None (fst (last x));\n length clist = length xs;\n \\i cp (map (Some \\ fst) xs ! i) s;\n x \\ clist; xa < length xs\\\n \\ \\i assum (Pre (xs ! i), Rely (xs ! i))"} {"_id": "500244", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\poly_cmul c p\\\\<^sub>p\\<^bsup>bs\\<^esup> =\n \\Mul (C c) p\\\\<^sub>p\\<^bsup>bs\\<^esup>"} {"_id": "500245", "text": "proof (prove)\ngoal (1 subgoal):\n 1. y ! index_of x y = x"} {"_id": "500246", "text": "proof (prove)\ngoal (1 subgoal):\n 1. SAMS D' E' F' A B C"} {"_id": "500247", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (F1 \\<^bold>\\\n F2, F1 \\<^bold>\\ F2, \\ \\\n \\ \\\n F1 \\<^bold>\\ F2, \\ \\ \\) \\\n (\\ \\\n F1 \\<^bold>\\\n F2, F1 \\<^bold>\\ F2, \\ \\\n \\ \\ F1 \\<^bold>\\ F2, \\)"} {"_id": "500248", "text": "proof (prove)\nusing this:\n \\ (cmod la = 1 \\ k = m \\ i = 0 \\ j = k - 1)\n\ngoal (1 subgoal):\n 1. \\la = 0 \\ thesis;\n \\la \\ 0; cmod la < 1\\ \\ thesis;\n \\cmod la = 1;\n k < m \\ i \\ 0 \\ j \\ k - 1\\\n \\ thesis\\\n \\ thesis"} {"_id": "500249", "text": "proof (prove)\nusing this:\n [\\ (y\\<^sup>P) in dw]\n\ngoal (1 subgoal):\n 1. [\\ (\\<^bold>\\x. \\ (x\\<^sup>P)) in dw]"} {"_id": "500250", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f abs_summable_on A"} {"_id": "500251", "text": "proof (prove)\nusing this:\n P \\ C sees clinit, Static : []\\Void = m' in C\n P \\ C sees clinit, Static : []\\Void = m' in C\n wf_prog wf_md P\n\ngoal (1 subgoal):\n 1. b = Static \\ Ts = [] \\ T = Void \\ D = C"} {"_id": "500252", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly_roots (\\x\\#A. [:- x, 1::'a:]) = A"} {"_id": "500253", "text": "proof (prove)\nusing this:\n m * m \\ 0\n\ngoal (1 subgoal):\n 1. N \\ 0"} {"_id": "500254", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Col C X Y"} {"_id": "500255", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_in_bicategory (\\) (\\) \\ \\ src trg u"} {"_id": "500256", "text": "proof (prove)\nusing this:\n t \\ t'\n\ngoal (1 subgoal):\n 1. t[x::=v] \\ t'[x::=v]"} {"_id": "500257", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Rep_process\n (Abs_process\n ({(s, X). s = [] \\ (s, X) \\ F P \\ F Q} \\\n {(s, X).\n s \\ [] \\ (s, X) \\ F P \\ F Q} \\\n {(s, X). s = [] \\ s \\ D P \\ D Q} \\\n {(s, X).\n s = [] \\\n tick \\ X \\ [tick] \\ T P \\ T Q},\n D P \\ D Q)) =\n ({(s, X). s = [] \\ (s, X) \\ F P \\ F Q} \\\n {(s, X). s \\ [] \\ (s, X) \\ F P \\ F Q} \\\n {(s, X). s = [] \\ s \\ D P \\ D Q} \\\n {(s, X).\n s = [] \\ tick \\ X \\ [tick] \\ T P \\ T Q},\n D P \\ D Q)"} {"_id": "500258", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hausdorff_distance (d ` {A..B}) {d A--d B}\n \\ 92 * lambda\\<^sup>2 * (C + deltaG TYPE('a))"} {"_id": "500259", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n (\\(a, b, c)\\set eq. a * x\\<^sup>2 + b * x + c = 0) \\\n (\\(a, b, c)\\set les. a * x\\<^sup>2 + b * x + c < 0)"} {"_id": "500260", "text": "proof (prove)\nusing this:\n \\ = r \\ a \\ i\n r \\ carrier R\n a \\ carrier R\n i \\ carrier R\n\ngoal (1 subgoal):\n 1. a \\ r = \\ i \\ \\"} {"_id": "500261", "text": "proof (prove)\nusing this:\n isrlfm\n (foldr MIR.disj\n (map (\\(\\, n, s). MIR.conj \\ (f n s)) (rsplit0 a)) F)\n rsplit ?f ?a \\\n foldr MIR.disj\n (map (\\(\\, n, s). MIR.conj \\ (?f n s)) (rsplit0 ?a)) F\n\ngoal (1 subgoal):\n 1. isrlfm (rsplit f a)"} {"_id": "500262", "text": "proof (prove)\nusing this:\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs e,\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1 (addr a),\n (h', ls\\<^sub>2, sh')\\ \\\n l' \\\\<^sub>m [Vs [\\] ls\\<^sub>2]\n P \\ \\e,(h, l, sh)\\ \\\n \\addr a,(h', l', sh')\\\n \\fv e \\ set ?Vs;\n l \\\\<^sub>m [?Vs [\\] ?ls];\n length ?Vs + max_vars e \\ length ?ls\\\n \\ \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 ?Vs e,\n (h, ?ls, sh)\\ \\\n \\fin\\<^sub>1 (addr a),(h', ls', sh')\\ \\\n l' \\\\<^sub>m [?Vs [\\] ls']\n h' a = \\(C, fs)\\\n P \\ C has F,Static:t in D\n fv (e\\F{D}) \\ set Vs\n l \\\\<^sub>m [Vs [\\] ls]\n length Vs + max_vars (e\\F{D}) \\ length ls\n\ngoal (1 subgoal):\n 1. \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs (e\\F{D}),\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1\n (THROW IncompatibleClassChangeError),\n (h', ls', sh')\\ \\\n l' \\\\<^sub>m [Vs [\\] ls']"} {"_id": "500263", "text": "proof (prove)\nusing this:\n gs \\ carrier G\n (\\i\\gs. generate G {i} \\ G) \\\n A = generate G (\\g\\gs. generate G {g}) \\\n compl_fam (\\g. generate G {g}) gs\n ?A \\ carrier G \\\n generate G (\\x\\?A. generate G {x}) = generate G ?A\n\ngoal (1 subgoal):\n 1. A = generate G gs"} {"_id": "500264", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\d.\n induced_modulus d \\ a \\ d) \\\n a \\ conductor"} {"_id": "500265", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite {v. atom v \\ supp e} \\\n (atom x \\ supp {v. atom v \\ supp e}) =\n (atom x \\ supp e)"} {"_id": "500266", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\n A form m na y.\n \\infinite (deriv s); init s; Suc (size form) = n;\n A = FAll form; contains f na (0, FAll form); m = 0;\n contains f (Suc (na + y))\n (0, finst form (newvar (sfv (s_of_ns (snd (f (na + y)))))));\n \\ FEval mo\n (case_nat (ntou (newvar (sfv (s_of_ns (snd (f (na + y)))))))\n ntou)\n form\\\n \\ ntou (newvar (sfv (s_of_ns (snd (f (na + y))))))\n \\ fst mo\n 2. \\n A x6.\n \\infinite (deriv s); init s;\n \\mA.\n size A = m \\\n (\\m n.\n contains f n (m, A) \\\n \\ FEval mo ntou A);\n size A = n; A = FEx x6\\\n \\ \\m n.\n contains f n (m, A) \\\n \\ FEval mo ntou A"} {"_id": "500267", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\State_Idle localState output_fun trans_fun\n (localState\n (f_Exec_Comp trans_fun (xs @ NoMsg\\<^bsup>m\\<^esup>) c));\n m < n\\\n \\ output_fun\n (f_Exec_Comp trans_fun\n ((xs @ NoMsg\\<^bsup>m\\<^esup>) @\n NoMsg\\<^bsup>n - m\\<^esup>)\n c) =\n NoMsg"} {"_id": "500268", "text": "proof (prove)\ngoal (1 subgoal):\n 1. final e \\ compE1 Vs e = fin1 e"} {"_id": "500269", "text": "proof (prove)\ngoal (1 subgoal):\n 1. n < length qs \\\n t_A' n = int (t (s_A' n) (qs ! n) (acts ! n))"} {"_id": "500270", "text": "proof (prove)\ngoal (1 subgoal):\n 1. locally path_connected S"} {"_id": "500271", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (local.span (set ws) = local.span (set (rev us)) &&&\n corthogonal (rev us)) &&&\n set us \\ carrier_vec n &&&\n length us = length ws &&& distinct us"} {"_id": "500272", "text": "proof (state)\ngoal (1 subgoal):\n 1. open (Proj ` X)"} {"_id": "500273", "text": "proof (state)\nthis:\n C.reachable (cs i)\n\ngoal (1 subgoal):\n 1. \\n.\n \\m\n A.reachable (arun cl cs m) \\\n R (arun cl cs (Suc m)) (cs (Suc m)) \\\n alstep (cl n) (arun cl cs n) (arun cl cs (Suc n)) \\\n A.reachable (arun cl cs n) \\ R (arun cl cs (Suc n)) (cs (Suc n))"} {"_id": "500274", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (lcount 0 = 1 &&& lcount 1 = 1) &&& lcount 2 = 1 &&& lcount 3 = 2"} {"_id": "500275", "text": "proof (prove)\nusing this:\n 0 < degree g \\\n \\q r.\n irreducible\\<^sub>d q \\ g = q * r \\ degree r < degree g\n i \\ 0\n degree g = i\n \\ irreducible g\n\ngoal (1 subgoal):\n 1. (\\h1 h2.\n \\irreducible h1; g = h1 * h2; degree h2 < degree g\\\n \\ thesis) \\\n thesis"} {"_id": "500276", "text": "proof (state)\ngoal (4 subgoals):\n 1. \\C g f.\n local.bind_state (local.altc_state C g) f =\n local.altc_state C (\\c. local.bind_state (g c) f)\n 2. \\x f. local.altc_state (csingle x) f = f x\n 3. \\C f g.\n local.altc_state (cUnion (cimage f C)) g =\n local.altc_state C (\\x. local.altc_state (f x) g)\n 4. \\R.\n bi_unique R \\\n (rel_cset R ===> (R ===> (=)) ===> (=)) local.altc_state\n local.altc_state"} {"_id": "500277", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\c. ccompare = Some c \\ comparator c"} {"_id": "500278", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cid \\ cid'"} {"_id": "500279", "text": "proof (prove)\nusing this:\n l_new_document ShadowRootClass.type_wf\n l_get_shadow_root ShadowRootClass.type_wf get_shadow_root\n get_shadow_root_locs\n\ngoal (1 subgoal):\n 1. l_new_document_get_shadow_root ShadowRootClass.type_wf get_shadow_root\n get_shadow_root_locs"} {"_id": "500280", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f \\ borel_measurable M) =\n (\\a. {w \\ space M. f w < a} \\ sets M)"} {"_id": "500281", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s\\<^sub>0 \\ s \\\n pred_owner_iii n (run_asset_ii n s) (run_owner_iii n s)"} {"_id": "500282", "text": "proof (prove)\nusing this:\n bij_betw h (carrier H) (carrier G)\n\ngoal (1 subgoal):\n 1. card (carrier H) = card (carrier G)"} {"_id": "500283", "text": "proof (prove)\ngoal (1 subgoal):\n 1. irreflexive\n (prim_W w v e * (w \\ - prim_EP w v e)\\<^sup>\\ * e *\n prim_P w v e\\<^sup>T\\<^sup>\\ *\n (w \\ - prim_EP w v e)\\<^sup>\\)"} {"_id": "500284", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\x. PP' x \\ P x \\ Q x\n 2. \\x y. fst R x y \\\\<^sub>t fst R x y\n 3. \\x y. snd R x y \\\\<^sub>t snd R x y\n 4. \\x y. S x y \\\\<^sub>t S x y"} {"_id": "500285", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distinct (map fst y) \\\n count_of y x = count (mset (ass_list_to_single_list y)) x"} {"_id": "500286", "text": "proof (prove)\nusing this:\n \\K rev_K rev_H.\n subdivision (K, rev_K)\n (case (H, rev_H) of (x, y) \\ (with_proj x, y)) \\\n (K\\<^bsub>3,3\\<^esub> K \\ K\\<^bsub>5\\<^esub> K)\n subgraph (with_proj H) (with_proj K)\n subgraph (with_proj K) (with_proj G)\n\ngoal (1 subgoal):\n 1. \\ (\\H.\n subgraph H (with_proj G) \\\n (\\K rev_K rev_H.\n subdivision (K, rev_K) (H, rev_H) \\\n (K\\<^bsub>3,3\\<^esub> K \\ K\\<^bsub>5\\<^esub> K)))"} {"_id": "500287", "text": "proof (prove)\ngoal (1 subgoal):\n 1. phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) =\n phull (set ps)"} {"_id": "500288", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fetch (adjust0 ap) sa (read ra) = (ac, Suc (length ap div 2))"} {"_id": "500289", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\x i vr vl.\n x \\ ifex_var_set\n (restrict_top i vr vl) \\\n x \\ ifex_var_set i;\n x = a\\\n \\ x \\ ifex_var_set i \\\n x \\ ifex_var_set t \\ x \\ ifex_var_set e"} {"_id": "500290", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_eq (RBT_set rbt1) (RBT_set rbt2) =\n (case ID ccompare of\n None \\\n Code.abort STR ''set_eq RBT_set RBT_set: ccompare = None''\n (\\_. set_eq (RBT_set rbt1) (RBT_set rbt2))\n | Some c \\\n case ID CEQ('b) of\n None \\\n list_all2_fusion (\\x y. c x y = Eq) rbt_keys_generator\n rbt_keys_generator (RBT_Set2.init rbt1) (RBT_Set2.init rbt2)\n | Some eq \\\n list_all2_fusion eq rbt_keys_generator rbt_keys_generator\n (RBT_Set2.init rbt1) (RBT_Set2.init rbt2))"} {"_id": "500291", "text": "proof (prove)\ngoal (1 subgoal):\n 1. m \\ LENGTH('a)"} {"_id": "500292", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ys\\#YS1. {#y \\# mset ys. y < x#}) +\n (\\ys\\#YS2. {#y \\# mset ys. y < x#}) \\#\n \\\\<^sub># (image_mset mset YS1) +\n (\\ys\\#YS2. {#y \\# mset ys. y < med ys#})"} {"_id": "500293", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xs \\ Joints XS \\\n bounded_by xs (ivls_of_aforms prec XS)"} {"_id": "500294", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ([a]lst. x = [a']lst. x') = ([a]lst. y = [a']lst. y')"} {"_id": "500295", "text": "proof (state)\nthis:\n terminal_arrow_from_functor (\\\\<^sub>D) (\\\\<^sub>C) F (G x) x\n (\\ x)\n\ngoal (1 subgoal):\n 1. \\y' f.\n arrow_from_functor (\\\\<^sub>D) (\\\\<^sub>C) F y' x\n f \\\n \\x.the_coext y' f = \\ y' f"} {"_id": "500296", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sorted xs =\n (\\i j.\n i \\ j \\\n j < length xs \\ xs ! i \\ xs ! j)"} {"_id": "500297", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n. S (Suc n) = x * S n /\\<^sub>R real (Suc n)"} {"_id": "500298", "text": "proof (prove)\nusing this:\n e1 \\ isubexprs(e0)\n\ngoal (1 subgoal):\n 1. \\C.\n CT;Map.empty \\ e0 : C \\\n \\D. CT;Map.empty \\ e1 : D"} {"_id": "500299", "text": "proof (prove)\nusing this:\n trace (c \\\\<^sub>m (f ?n - f 0))\n \\ trace (c \\\\<^sub>m (A - f 0))\n\ngoal (1 subgoal):\n 1. trace (c \\\\<^sub>m (f n - f 0)) < 1"} {"_id": "500300", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\x1.\n \\\\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n a = dir.Left;\n \\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n t = Hd x1\\\n \\ emb_step_at ((a # p) @ q) d2\n (emb_step_at (a # p) d1 t) =\n emb_step_at (a # p) d1\n (emb_step_at ((a # p) @ [d1] @ q) d2 t)\n 2. \\x21 x22.\n \\\\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n a = dir.Left;\n \\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n t = App x21 x22\\\n \\ emb_step_at ((a # p) @ q) d2\n (emb_step_at (a # p) d1 t) =\n emb_step_at (a # p) d1\n (emb_step_at ((a # p) @ [d1] @ q) d2 t)\n 3. \\x1.\n \\\\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n a = dir.Right;\n \\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n t = Hd x1\\\n \\ emb_step_at ((a # p) @ q) d2\n (emb_step_at (a # p) d1 t) =\n emb_step_at (a # p) d1\n (emb_step_at ((a # p) @ [d1] @ q) d2 t)\n 4. \\x21 x22.\n \\\\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n a = dir.Right;\n \\t.\n emb_step_at (p @ q) d2 (emb_step_at p d1 t) =\n emb_step_at p d1 (emb_step_at (p @ [d1] @ q) d2 t);\n t = App x21 x22\\\n \\ emb_step_at ((a # p) @ q) d2\n (emb_step_at (a # p) d1 t) =\n emb_step_at (a # p) d1\n (emb_step_at ((a # p) @ [d1] @ q) d2 t)"} {"_id": "500301", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\F. (\\n. compact (F n)) \\ \\ (range F) = C m"} {"_id": "500302", "text": "proof (state)\nthis:\n \\rk {A, B, M} = 2; rk {C, D, M} = 2\\\n \\ rk {A, B, C, D} \\ 3\n\ngoal (1 subgoal):\n 1. \\M. rk {A, B, M} \\ 2 \\ rk {C, D, M} \\ 2"} {"_id": "500303", "text": "proof (prove)\nusing this:\n es_is_trap es\n\ngoal (1 subgoal):\n 1. case (s, vs, ves, e) of\n (s, vs, ves, e) \\\n \\s;vs;vs_to_es ves @\n [e]\\ \\_ i \\s';vs';es'\\"} {"_id": "500304", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f(i := x) \\ u j) (at_right j)"} {"_id": "500305", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fmapU\\f\\\\ = \\ &&&\n fmapU\\f\\(Writer\\w\\x) =\n Writer\\w\\(f\\x)"} {"_id": "500306", "text": "proof (prove)\nusing this:\n j (h ?x) = ?x\n h (j ?y) = ?y\n\ngoal (1 subgoal):\n 1. {x. \\ ((j \\ f \\ h) x = x \\\n (j \\ g \\ h) x = x)} =\n j ` {x. \\ (f x = x \\ g x = x)}"} {"_id": "500307", "text": "proof (prove)\nusing this:\n merge_guards c = Guard f' g' c'\n f = f'\n\ngoal (1 subgoal):\n 1. merge_guards (Guard f g c) = Guard f (g \\ g') c'"} {"_id": "500308", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x.\n (\\g' x\\ *\\<^sub>R f (g x)) $ i) absolutely_integrable_on\n S \\\n ((\\x. (\\g' x\\ *\\<^sub>R f (g x)) $ i) has_integral\n b $ i)\n S) =\n ((\\x. f x $ i) absolutely_integrable_on g ` S \\\n ((\\x. f x $ i) has_integral b $ i) (g ` S))"} {"_id": "500309", "text": "proof (prove)\ngoal (1 subgoal):\n 1. StdOMap dflt_oops"} {"_id": "500310", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\e. res = RSCrash e"} {"_id": "500311", "text": "proof (prove)\nusing this:\n \\x\\{0..1}. finite (\\ -` {x})\n\ngoal (1 subgoal):\n 1. finite (\\ -` (s' \\ {0..1}))"} {"_id": "500312", "text": "proof (prove)\ngoal (1 subgoal):\n 1. int (nat x choose 2) = x * (x - 1) div 2"} {"_id": "500313", "text": "proof (prove)\nusing this:\n path ?x = path (?x\\<^sup>T)\n\ngoal (1 subgoal):\n 1. no_start_points_path x = no_end_points_path (x\\<^sup>T)"} {"_id": "500314", "text": "proof (prove)\nusing this:\n A.arr a\n J.arr j\n A_B.arr (\\ j)\n Curry.uncurry ?\\ ?f \\\n if JxA.arr ?f then E.map (?\\ (fst ?f), snd ?f) else B.null\n A_BxA.arr ?Fg \\ E.map ?Fg = A_B.Map (fst ?Fg) (snd ?Fg)\n\ngoal (1 subgoal):\n 1. at a \\ j = A_B.Map (\\ j) a"} {"_id": "500315", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly_eval a (extend_indets p) = poly_eval (a \\ Some) p"} {"_id": "500316", "text": "proof (chain)\npicking this:\n h \\ get_dom_component parent_ptr \\\\<^sub>r c"} {"_id": "500317", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ts_ok wfx (thr (h.start_state f P C M vs)) h.start_heap;\n wf_syscls P;\n legal_execution P (h.\\_start f P C M vs status) (E, ws);\n enat a < llength E;\n action_obs E a = NormalAction (ReadMem ad al v)\\\n \\ \\T.\n P \\ ad@al : T \\\n P \\ v :\\ T"} {"_id": "500318", "text": "proof (prove)\nusing this:\n healthy (iterates body ?G ?i)\n\ngoal (1 subgoal):\n 1. le_trans\n (wp (body ;;\n Embed\n (iterates body G\n i) \\<^bsub>\\ G \\\\<^esub>\\ Skip))\n (wp (body ;;\n Embed\n (iterates body G\n (Suc i)) \\<^bsub>\\ G \\\\<^esub>\\ Skip))"} {"_id": "500319", "text": "proof (prove)\nusing this:\n mod_subset ?p ?X \\ {(a, b). singular_chain ?p ?X (a - b)}\n singular_chain ?p ?X (- ?c) = singular_chain ?p ?X ?c\n\ngoal (1 subgoal):\n 1. singular_chain p (subtopology X S) c \\\n \\d.\n singular_chain (Suc p) X d \\\n (chain_boundary (Suc p) d, c) \\ mod_subset p (subtopology X S)"} {"_id": "500320", "text": "proof (prove)\nusing this:\n ?v \\?xs\\\\<^bsub>T\\<^esub> ?w \\\n ?xs = ?v \\\\<^bsub>T\\<^esub> ?w\n s \\(x # xs)\\\\<^bsub>T\\<^esub> t\n xs \\ []\n\ngoal (1 subgoal):\n 1. hd xs \\ set (s \\\\<^bsub>T\\<^esub> t)"} {"_id": "500321", "text": "proof (prove)\ngoal (1 subgoal):\n 1. DF UNIV \\\\<^sub>F\\<^sub>D SYSTEM"} {"_id": "500322", "text": "proof (prove)\nusing this:\n VoV.seq (\\, \\) (\\, \\)\n\ngoal (1 subgoal):\n 1. \\ \\ \\ \\ \\ \\ \\ =\n (\\ \\ \\) \\ (\\ \\ \\)"} {"_id": "500323", "text": "proof (prove)\nusing this:\n \\ isACT sys (PROC p)\n p' = solution sys (PROC p)\n isAction ?p \\ isChoice ?p\n\ngoal (1 subgoal):\n 1. isAction p = isAction p' \\ isChoice p = isChoice p'"} {"_id": "500324", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\hbt l; hbt r; height l \\ height r + m + 1\\\n \\ height (balL l a r) = 1 + max (height l) (height r)"} {"_id": "500325", "text": "proof (prove)\ngoal (1 subgoal):\n 1. add_local_ring_at zero_local_ring_at a = a"} {"_id": "500326", "text": "proof (prove)\nusing this:\n X' \\ T11\n X' \\ X\n X' \\ P\n X' \\ (D' @ TVarB X Q # G)\n X' \\ t1\n X' \\ T2\n D' @ TVarB X Q # G \\ t1 : (\\X'<:T11. T12)\n \\D' @ TVarB X Q # G = ?D @ TVarB ?b Q # G;\n G\\?ba<:Q\\\n \\ ?D[?b \\ ?ba]\\<^sub>e @\n G \\ t1[?b \\\\<^sub>\\ ?ba] : (\\X'<:T11.\n T12)[?b \\ ?ba]\\<^sub>\\\n (D' @ TVarB X Q # G)\\T2<:T11\n G\\P<:Q\n D'[X \\ P]\\<^sub>e @\n G \\ t1[X \\\\<^sub>\\ P] : (\\X'<:T11.\n T12)[X \\ P]\\<^sub>\\\n\ngoal (1 subgoal):\n 1. D'[X \\ P]\\<^sub>e @\n G \\ (t1 \\\\<^sub>\\\n T2)[X \\\\<^sub>\\ P] : T12[X' \\ T2]\\<^sub>\\[X \\ P]\\<^sub>\\"} {"_id": "500327", "text": "proof (prove)\nusing this:\n V \\ {}\n \\ {bag t |t. t \\ V\\<^bsub>T\\<^esub>} = V\n\ngoal (1 subgoal):\n 1. \\v\\V. \\t\\V\\<^bsub>T\\<^esub>. v \\ bag t"} {"_id": "500328", "text": "proof (prove)\ngoal (1 subgoal):\n 1. linear (*s) (*s) g &&& \\x\\set_of_vector X. g x = f x"} {"_id": "500329", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monoid_cancel G"} {"_id": "500330", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(Enum.finite_2, cenum_class)"} {"_id": "500331", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Quotient eq_fp64 fp64_of_float float_of_fp64 rel_fp64"} {"_id": "500332", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (IEEE.exponent 1 = emax TYPE(('a, 'b) IEEE.float) \\\n fraction 1 \\ 0)"} {"_id": "500333", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\v. lookup \\ x = Some v \\ v \\ good A"} {"_id": "500334", "text": "proof (prove)\nusing this:\n ?x \\ A \\ integrable (count_space B) (f ?x)\n\ngoal (1 subgoal):\n 1. integrable (count_space B) (\\y. \\x\\A. f x y)"} {"_id": "500335", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_spmf\n (\\(b'1, bad1, L1) (b'2, bad2, L2).\n bad1 = bad2 \\ (\\ bad2 \\ b'1 = b'2))\n (exec_gpv\n (\\(ind_cca'.oracle_encrypt k b) \\\\<^sub>O\n oracle_decrypt1' k)\n \\ (False, {}))\n (exec_gpv\n (\\(ind_cca'.oracle_encrypt k b) \\\\<^sub>O\n oracle_decrypt0' k)\n \\ (False, {}))"} {"_id": "500336", "text": "proof (prove)\nusing this:\n \\0 \\ ?s''2; ?s''2 < s'\\\n \\ flow0 x ?s''2 \\ {x. c \\ x \\ n}\n flow0 x s' \\ {x. c \\ x \\ n}\n 0 \\ s'\n s' \\ s\n flow0 x s' \\ frontier {x. c \\ x \\ n}\n flowpipe X0 hl t CX Y\n fst ` X0 \\ {x. c \\ x \\ n} = {}\n n \\ (0::'a)\n closed R\n R \\ plane n c\n \\(?x2, ?d2) \\ CX; ?x2 \\ n = c\\\n \\ (?x2,\n ?d2 -\n (blinfun_scaleR_left (f ?x2) o\\<^sub>L\n (blinfun_scaleR_left\n (inverse (f ?x2 \\ n)) o\\<^sub>L\n (blinfun_inner_left n o\\<^sub>L ?d2))))\n \\ PDP\n (?x2, ?d2) \\ PDP \\ f ?x2 \\ n \\ 0\n (?x2, ?d2) \\ PDP \\ ?x2 \\ R\n (?x2, ?d2) \\ PDP \\\n \\\\<^sub>F x in at ?x2 within plane n c. x \\ R\n (x, d) \\ X0\n 0 \\ s\n s \\ t\n flow0 x s \\ {x. c \\ x \\ n}\n t \\ existence_ivl0 x\n \\0 \\ ?s2; ?s2 \\ t\\\n \\ (flow0 x ?s2, Dflow x ?s2 o\\<^sub>L d) \\ CX\n (flow0 x t, Dflow x t o\\<^sub>L d) \\ Y\n 0 < s'\n {0..t} \\ existence_ivl0 x\n frontier {x. c \\ x \\ n} \\ {x. c \\ x \\ n}\n\ngoal (1 subgoal):\n 1. (x, d)\n \\ {x \\ X0.\n flowsto {x} {0<..t}\n (CX \\ {x. x \\ n < c} \\ UNIV)\n (CX \\ {x \\ R. x \\ n = c} \\ UNIV)}"} {"_id": "500337", "text": "proof (prove)\ngoal (1 subgoal):\n 1. D.iso\n (\\ (g, f) \\\\<^sub>D\n (F g \\\\<^sub>D F f) \\\\<^sub>D\n (\\ \\\\<^sub>D F f) \\\\<^sub>D\n \\ \\\\<^sub>D D.inv (unit (src\\<^sub>C f)))"} {"_id": "500338", "text": "proof (prove)\nusing this:\n K \\ carrier R\n polynomial ?K ?p \\ set ?p \\ ?K\n polynomial K p\n\ngoal (1 subgoal):\n 1. set p \\ carrier R"} {"_id": "500339", "text": "proof (prove)\ngoal (1 subgoal):\n 1. shd (sdrop (LEAST n. enabled (shd (sdrop n rs)) s) rs) \\ R &&&\n enabled (shd (sdrop (LEAST n. enabled (shd (sdrop n rs)) s) rs)) s &&&\n fair (sdrop (LEAST n. enabled (shd (sdrop n rs)) s) rs)"} {"_id": "500340", "text": "proof (prove)\nusing this:\n wtL l\n I.wtE (kE \\)\n\ngoal (1 subgoal):\n 1. I.satL (kE \\) l = F.satL \\ l"} {"_id": "500341", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P,E,s \\ e : T \\;\n \\ e \\dom (lcl s)\\;\n P,shp s \\\\<^sub>b (e,b) \\; \\ final e\\\n \\ \\e' s' b'.\n P \\ \\e,s,\n b\\ \\\n \\e',s',b'\\"} {"_id": "500342", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\v \\ b.\n \\finite (dom (Mapping.lookup (phis'_code g next)));\n v \\ snd ` dom (Mapping.lookup (phis g));\n snd next \\ set (the (Mapping.lookup (phis g) \\));\n Mapping.lookup (phis g) \\ = Some b\\\n \\ (case Mapping.lookup\n (uninst_code.nodes_of_phis' g next\n (uninst_code.substitution_code phis g\n next)\n nodes_of_phis)\n v of\n None \\ {}\n | Some x \\ id x) =\n (case if \\n\n \\dom (Mapping.lookup (phis g)) - {v. snd v = snd next}.\n v \\ set\n (the (if snd n = snd next then None\n else map_option\n (map (\\v.\n if v = snd next\n then uninst_code.substitution_code phis g next\n else v))\n (Mapping.lookup (phis g) n)))\n then Some\n {n\n\\ dom (Mapping.lookup (phis g)) - {v. snd v = snd next}.\nv \\ set (the (if snd n = snd next then None\n else map_option\n (map (\\v.\n if v = snd next\n then uninst_code.substitution_code phis g\n next\n else v))\n (Mapping.lookup (phis g) n)))}\n else None of\n None \\ {} | Some x \\ id x)\n 2. \\v.\n \\finite (dom (Mapping.lookup (phis'_code g next)));\n v \\ snd ` dom (Mapping.lookup (phis g));\n \\ (\\\\\\Mapping.keys (phis g).\n snd next\n \\ set (the (Mapping.lookup (phis g)\n \\)))\\\n \\ (case Mapping.lookup\n (uninst_code.nodes_of_phis' g next\n (uninst_code.substitution_code phis g\n next)\n nodes_of_phis)\n v of\n None \\ {}\n | Some x \\ id x) =\n (case if \\n\n \\dom (Mapping.lookup (phis g)) - {v. snd v = snd next}.\n v \\ set\n (the (if snd n = snd next then None\n else map_option\n (map (\\v.\n if v = snd next\n then uninst_code.substitution_code phis g next\n else v))\n (Mapping.lookup (phis g) n)))\n then Some\n {n\n\\ dom (Mapping.lookup (phis g)) - {v. snd v = snd next}.\nv \\ set (the (if snd n = snd next then None\n else map_option\n (map (\\v.\n if v = snd next\n then uninst_code.substitution_code phis g\n next\n else v))\n (Mapping.lookup (phis g) n)))}\n else None of\n None \\ {} | Some x \\ id x)"} {"_id": "500343", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\A \\ (\\xs.\n wordinterval_sort\n (wordinterval_compress\n (foldr wordinterval_union xs\n Empty_WordInterval))) `\n set (groupWIs c rs);\n B \\ (\\xs.\n wordinterval_sort\n (wordinterval_compress\n (foldr wordinterval_union xs Empty_WordInterval))) `\n set (groupWIs c rs);\n A \\ B\\\n \\ \\ip1\\wordinterval_to_set A.\n \\ip2\\wordinterval_to_set B.\n \\ same_fw_behaviour_one ip1 ip2 c rs"} {"_id": "500344", "text": "proof (prove)\nusing this:\n \\y.\n |x\\ y\n \\ |x\\ |x\\<^sup>\\\\ \\ x y\n\ngoal (1 subgoal):\n 1. \\y. |x\\ y \\ |x\\ \\ x y"} {"_id": "500345", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LIM x F. f x * g x :> at_bot"} {"_id": "500346", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i\\UNIV. (f i)\\<^sup>2 * (x $ i)\\<^sup>2)\n \\ Max {(f i)\\<^sup>2 |i. i \\ UNIV} *\n (\\i\\UNIV. (x $ i)\\<^sup>2)"} {"_id": "500347", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n. \\.expectation n (indicat_real A))\n \\ M.expectation (indicat_real A)"} {"_id": "500348", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w1 = basic [brick.under] \\ basic [over] \\\n w2 = basic [vert, vert] \\ basic [vert, vert] \\\n kauff_mat w1 = kauff_mat w2"} {"_id": "500349", "text": "proof (prove)\nusing this:\n i \\ agents\n total_preorder_on alts Ri'\n finite_total_preorder_on alts Ri'\n\ngoal (1 subgoal):\n 1. pref_profile_wf agents alts (R(i := Ri'))"} {"_id": "500350", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sorted1 (inorder (Splay_Map.splay x t));\n Splay_Map.splay x t = \\l, a, r\\\\\n \\ Sorted_Less.sorted\n (map fst (inorder l) @ x # map fst (inorder r))"} {"_id": "500351", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (mod_ring_rel32 ===> mod_ring_rel32) (inverse_p32 pp) inverse"} {"_id": "500352", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f has_derivative\n (\\x.\n x *\\<^sub>R\n (SOME f'. (f has_derivative (\\x. x *\\<^sub>R f')) net)))\n net"} {"_id": "500353", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b.\n \\(gx a b, gy a b) = 0; (a, b) \\ trapG1;\n p1 a b \\ 0\\\n \\ False"} {"_id": "500354", "text": "proof (prove)\nusing this:\n ?i1 \\ I \\ H ?i1 \\ carrier (G ?i1)\n\ngoal (1 subgoal):\n 1. subgroup (carrier (sum_group I G) \\ Pi\\<^sub>E I H)\n (sum_group I G) \\\n {f \\ \\\\<^sub>E i\\I.\n carrier (subgroup_generated (G i) (H i)).\n finite {i \\ I. f i \\ \\\\<^bsub>G i\\<^esub>}} =\n carrier (sum_group I G) \\ Pi\\<^sub>E I H"} {"_id": "500355", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homology_group 0 X \\\n sum_group (path_components_of X)\n (\\S. homology_group 0 (subtopology X S))"} {"_id": "500356", "text": "proof (prove)\nusing this:\n v = - (w 0 - v)\n\ngoal (1 subgoal):\n 1. v \\ space\\<^sub>N (\\ \\ M)"} {"_id": "500357", "text": "proof (prove)\nusing this:\n (*f* (\\h. f (a + h))) (- star_of a + x) \\ star_of L\n\ngoal (1 subgoal):\n 1. (*f* f) x \\ star_of L"} {"_id": "500358", "text": "proof (prove)\nusing this:\n S.inv (S.cmp\\<^sub>U\\<^sub>P (f, g)) \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (f, g) =\n S.UP f \\\\<^sub>S S.UP g\n \\S.arr ?f; S.dom ?f = ?a\\\n \\ ?f \\\\<^sub>S ?a = ?f\n \\\\' : S.UP f \\\\<^sub>S\n S.UP\n g \\\\<^sub>S S.trg\n (S.UP f)\\\n\ngoal (1 subgoal):\n 1. \\' \\\\<^sub>S\n S.inv (S.cmp\\<^sub>U\\<^sub>P (f, g)) \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (f, g) =\n \\'"} {"_id": "500359", "text": "proof (prove)\ngoal (1 subgoal):\n 1. np f = (\\t. - f t) &&& pp f = (\\t. 0)"} {"_id": "500360", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Octonions.cnj q =\n Ree q *\\<^sub>R e0 - Im1 q *\\<^sub>R e1 - Im2 q *\\<^sub>R e2 -\n Im3 q *\\<^sub>R e3 -\n Im4 q *\\<^sub>R e4 -\n Im5 q *\\<^sub>R e5 -\n Im6 q *\\<^sub>R e6 -\n Im7 q *\\<^sub>R e7"} {"_id": "500361", "text": "proof (state)\ngoal (1 subgoal):\n 1. the_elem (Set t) =\n (case rbt.impl_of t of\n rbt.Empty \\\n Code.abort STR ''not_a_singleton_tree''\n (\\_. the_elem (Set t))\n | Branch color.R rbt1 x b rbt2 \\\n Code.abort STR ''not_a_singleton_tree''\n (\\_. the_elem (Set t))\n | Branch color.B rbt.Empty x () rbt.Empty \\ x\n | Branch color.B rbt.Empty x ()\n (Branch color rbt1 aa b rbt2a) \\\n Code.abort STR ''not_a_singleton_tree''\n (\\_. the_elem (Set t))\n | Branch color.B (Branch color rbt1a ab bb rbt2b) x b\n rbt2 \\\n Code.abort STR ''not_a_singleton_tree''\n (\\_. the_elem (Set t)))"} {"_id": "500362", "text": "proof (prove)\nusing this:\n \\a' \\ b' \\ carrier G; a' \\ b \\ carrier G;\n set (as @ bs) \\ carrier G; a' \\ b \\ carrier G;\n a \\ b \\ carrier G;\n set (as @ bs) \\ carrier G\\\n \\ wfactors G (as @ bs) (a \\ b)\n\ngoal (1 subgoal):\n 1. wfactors G (as @ bs) (a \\ b)"} {"_id": "500363", "text": "proof (prove)\nusing this:\n (inv_msog (ms_of_greek s), inv_msog (ms_of_greek t))\n \\ mult (letter_less r <*lex*> greek_less r)\n\ngoal (1 subgoal):\n 1. (inv_greek s, inv_greek t) \\ greek_less r"} {"_id": "500364", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fmap_of_list m \\\\<^sub>f fmap_of_list n =\n fmap_of_list (AList.compose n m)"} {"_id": "500365", "text": "proof (prove)\nusing this:\n \\i\\n. \\ FW M n i i < (0::'a)\n i_ \\ n\n j_ \\ n\n \\ cycle_free ?m ?n \\\n \\i\\?n. FW ?m ?n i i < (0::?'a)\n\ngoal (1 subgoal):\n 1. FW M n i_ j_ = D M i_ j_ n"} {"_id": "500366", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AR S"} {"_id": "500367", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\A C B.\n \\M = zmset_of A + C; N = zmset_of B + C;\n A \\# B\\\n \\ thesis) \\\n thesis"} {"_id": "500368", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Thm (Abs_idx (0, 0)) \\"} {"_id": "500369", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\u'.\n (u', u) \\ cf.E \\\n l u' \\ Suc (Min {l v |v. (u, v) \\ cf.E});\n \\v.\n (u, v) \\ cf.E \\\n Min {l v |v. (u, v) \\ cf.E} \\ l v\\\n \\ (l(u := Suc (Min {l v |v. (u, v) \\ cf.E}))) t = 0"} {"_id": "500370", "text": "proof (state)\nthis:\n negligible (S - \\ (range C))\n\ngoal (2 subgoals):\n 1. \\n. compact (?C n)\n 2. S = \\ (range ?C) \\ (S - \\ (range C))"} {"_id": "500371", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 \\ numgcdh t g"} {"_id": "500372", "text": "proof (prove)\nusing this:\n finite pid.reachable\n\ngoal (1 subgoal):\n 1. finite (lih'.pid_step\\<^sup>* `` {pid_init_gc})"} {"_id": "500373", "text": "proof (prove)\nusing this:\n p = [- a, 1::'a] *** q\n\ngoal (1 subgoal):\n 1. poly p a = (0::'a)"} {"_id": "500374", "text": "proof (prove)\nusing this:\n \\\\<^sub>\\i.\n {\\ \\ subformulas\\<^sub>\\ \\.\n suffix i w \\\\<^sub>n F\\<^sub>n (G\\<^sub>n \\)} =\n {\\ \\ subformulas\\<^sub>\\ \\.\n suffix i w \\\\<^sub>n G\\<^sub>n \\}\n\ngoal (1 subgoal):\n 1. \\\\<^sub>\\j. \\_stable \\ (suffix j w)"} {"_id": "500375", "text": "proof (prove)\nusing this:\n lt ox k (fst z)\n\ngoal (1 subgoal):\n 1. le ox (fst (hd xs')) (fst z)"} {"_id": "500376", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\l a aa b ab ac ba h v n2 k w.\n \\((a, aa, b), l, (ab, ac, ba), 0, h, v) \\ Exec;\n ((a, aa, b), l, (ab, ac, ba), n2, k, w) \\ Exec\\\n \\ n2 = 0 \\ h = k \\ v = w\n 2. \\n.\n (\\n1 l M s l1 t.\n n1 \\ n \\\n (M, l, s, n1, l1, t) \\ Step \\\n (\\n2 l2 r.\n (M, l, s, n2, l2, r) \\ Step \\\n n1 = n2 \\ t = r \\ l1 = l2)) \\\n (\\n1 l M s h v.\n n1 \\ n \\\n (M, l, s, n1, h, v) \\ Exec \\\n (\\n2 k w.\n (M, l, s, n2, k, w) \\ Exec \\\n n1 = n2 \\ h = k \\ v = w)) \\\n (\\n1 l M s l1 t.\n n1 \\ Suc n \\\n (M, l, s, n1, l1, t) \\ Step \\\n (\\n2 l2 r.\n (M, l, s, n2, l2, r) \\ Step \\\n n1 = n2 \\ t = r \\ l1 = l2)) \\\n (\\n1 l M s h v.\n n1 \\ Suc n \\\n (M, l, s, n1, h, v) \\ Exec \\\n (\\n2 k w.\n (M, l, s, n2, k, w) \\ Exec \\\n n1 = n2 \\ h = k \\ v = w))"} {"_id": "500377", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mod_exp_aux m y x 0 = y &&& mod_exp_aux m y x (Suc 0) = x * y mod m"} {"_id": "500378", "text": "proof (prove)\nusing this:\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t A \\\n pair ` setops\\<^sub>s\\<^sub>s\\<^sub>t (unlabel A))\n\ngoal (1 subgoal):\n 1. wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t (proj l A) \\\n pair ` setops\\<^sub>s\\<^sub>s\\<^sub>t (proj_unl l A))"} {"_id": "500379", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pi \\ t[X \\\\<^sub>\\ T] =\n (pi \\ t)[pi \\ X \\\\<^sub>\\ pi \\ T]"} {"_id": "500380", "text": "proof (prove)\nusing this:\n extended_to_set p \\ extended_to_set q\n \\ I ` insert_messages M\n create_insert s n \\ new_id =\n Inr (Insert (InsertMessage p new_id q \\))\n Inr m = create_insert s n \\ new_id\n\ngoal (1 subgoal):\n 1. deps m \\ I ` insert_messages M"} {"_id": "500381", "text": "proof (prove)\nusing this:\n \\arr t; arr \\.the_\\;\n arr r\\<^sub>0s\\<^sub>1.p\\<^sub>1; src t = trg \\.the_\\;\n src \\.the_\\ = trg r\\<^sub>0s\\<^sub>1.p\\<^sub>1\\\n \\ \\[cod t, cod \\.the_\\, cod\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1] \\\n ((t \\ \\.the_\\) \\\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1) =\n (t \\\n \\.the_\\ \\\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1) \\\n \\[local.dom\n t, local.dom\n \\.the_\\, local.dom\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1]\n trg \\.the_\\ = src t\n hseq r\\<^sub>1 r\\<^sub>0s\\<^sub>1.p\\<^sub>1\n hseq ?\\ ?\\ =\n (arr ?\\ \\ arr ?\\ \\ src ?\\ = trg ?\\)\n\ngoal (1 subgoal):\n 1. (t \\\n \\.the_\\ \\ r\\<^sub>0s\\<^sub>1.p\\<^sub>1) \\\n \\[t, t\\<^sub>0 \\\n \\.chine, r\\<^sub>0s\\<^sub>1.p\\<^sub>1] =\n \\[t, r\\<^sub>0, r\\<^sub>0s\\<^sub>1.p\\<^sub>1] \\\n ((t \\ \\.the_\\) \\ r\\<^sub>0s\\<^sub>1.p\\<^sub>1)"} {"_id": "500382", "text": "proof (prove)\nusing this:\n finite A\n x \\ A\n ?k \\ card A \\ \\B\\A. card B = ?k\n k \\ card (insert x A)\n\ngoal (1 subgoal):\n 1. \\B\\A. card B = k - 1"} {"_id": "500383", "text": "proof (state)\nthis:\n \\rel_sum boa bob ?x ?y; rel_sum boa bob ?y ?x;\n ?x \\ {x. setl x \\ A \\ setr x \\ B}\\\n \\ ?x = ?y\n\ngoal (1 subgoal):\n 1. rel_sum boa bob \\ vimage2p (map_sum rha rhb) (map_sum rha rhb) (=)"} {"_id": "500384", "text": "proof (prove)\nusing this:\n S,kind \\ (m # ms,s) =as\\\\<^sub>\\ (m' # ms',s')\n as = as' @ [a']\n as' = asx @ ax # asx'\n\ngoal (1 subgoal):\n 1. (\\msx sx.\n \\S,kind \\ (m #\nms,s) =asx\\\\<^sub>\\ (msx,sx);\n S,kind \\ (msx,sx) =ax #\n asx' @ [a']\\\\<^sub>\\ (m' # ms',s')\\\n \\ thesis) \\\n thesis"} {"_id": "500385", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Bseq X; monoseq X\\ \\ convergent X"} {"_id": "500386", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ffold f z (finsert x A) = f x (ffold f z A)"} {"_id": "500387", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P"} {"_id": "500388", "text": "proof (prove)\nusing this:\n \\\\<^sup>+ x. ennreal (pmf pq (None, x))\n \\count_space UNIV =\n ennreal (pmf p None) - ennreal (pmf q None) + ennreal (pmf q None)\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (pmf pq (i, x))\n \\count_space UNIV =\n ennreal (pmf p i)"} {"_id": "500389", "text": "proof (prove)\nusing this:\n dip \\ kD rt\n dip \\ kD (invalidate rt dests)\n\ngoal (1 subgoal):\n 1. rt \\\\<^bsub>dip\\<^esub> invalidate rt dests"} {"_id": "500390", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\F.\n \\\\ y = \\\\ x \\\n [\\ (y\\<^sup>P) in v] \\\n (\\r o\\<^sub>1.\n Some r = Semantics.d\\<^sub>1 F \\\n Some o\\<^sub>1 = Semantics.d\\<^sub>\\ (x\\<^sup>P) \\\n o\\<^sub>1 \\ Semantics.ex1 r v) =\n (\\r o\\<^sub>1.\n Some r = Semantics.d\\<^sub>1 F \\\n Some o\\<^sub>1 = Semantics.d\\<^sub>\\ (y\\<^sup>P) \\\n o\\<^sub>1 \\ Semantics.ex1 r v)"} {"_id": "500391", "text": "proof (prove)\nusing this:\n wf_matchF_invar \\ V n R I r st (Suc i)\n to_mregex r = (mr, \\s)\n st = (aux, X)\n ms \\ LPDs mr\n\ngoal (1 subgoal):\n 1. qtable n (fv_regex r) (mem_restr R)\n (\\v.\n Regex.match (Formula.sat \\ V (map the v))\n (from_mregex ms \\s) i (i + length aux))\n (l\\ (\\x. x) X rels ms)"} {"_id": "500392", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bisimulation trsys1 trsys2 bisim tlsim"} {"_id": "500393", "text": "proof (prove)\nusing this:\n ord_spmf (rel_generat (=) (=) (rel_fun (=) ord_gpv)) (the_gpv f)\n (the_gpv g)\n ord_spmf (rel_generat (=) (=) (rel_fun (=) ord_gpv)) (the_gpv g)\n (the_gpv f)\n\ngoal (1 subgoal):\n 1. rel_spmf\n (rel_generat (=) (=) (rel_fun (=) ord_gpv) \\\n (rel_generat (=) (=) (rel_fun (=) ord_gpv))\\\\)\n (the_gpv f) (the_gpv g)"} {"_id": "500394", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ (c, s) \\\\<^sup>* (Skip, Fault f)"} {"_id": "500395", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bij_on_trancl (\\) Bag"} {"_id": "500396", "text": "proof (prove)\nusing this:\n \\ t\n \\ t \\\n let s' = assert_bound_loop ats (init t)\n in \\ (\\ s') \\\n \\ s' \\\n (\\ \\ s' \\\n \\\\<^sub>n\\<^sub>o\\<^sub>l\\<^sub>h\\<^sub>s s' \\\n \\ s')\n \\\\ \\ ?s;\n \\\\<^sub>n\\<^sub>o\\<^sub>l\\<^sub>h\\<^sub>s ?s; \\ ?s;\n \\ (\\ ?s); \\ ?s; \\ \\ (check ?s)\\\n \\ \\ (check ?s)\n \\ ?s \\ check ?s = ?s\n\ngoal (1 subgoal):\n 1. \\ \\ (check (assert_bound_loop ats (init t))) \\\n \\ (check (assert_bound_loop ats (init t)))"} {"_id": "500397", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vertical_composite B D (K \\ F) (L \\ G) (M \\ H)\n (\\ \\ \\) (\\ \\ \\)"} {"_id": "500398", "text": "proof (prove)\nusing this:\n qSkel (X #[[y \\ x]]_xs) = qSkel (X' #[[y \\ x']]_xs)\n\ngoal (1 subgoal):\n 1. qSkel X = qSkel X'"} {"_id": "500399", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s\\setops\\<^sub>s\\<^sub>s\\<^sub>t S.\n \\X.\n set X \\ bvars\\<^sub>s\\<^sub>s\\<^sub>t S \\\n p = s \\\\<^sub>p rm_vars (set X) \\"} {"_id": "500400", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fs.gs.\\ i (i - 1)\\ \\ inverse 2"} {"_id": "500401", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\s args.\n \\strip_comb pat = (Const s, args);\n pat = s $$ args\\\n \\ thesis;\n \\s.\n \\strip_comb pat = (Free s, []); pat = Free s\\\n \\ thesis\\\n \\ thesis"} {"_id": "500402", "text": "proof (prove)\nusing this:\n \\ \\ fs [:] fTs\n\ngoal (1 subgoal):\n 1. (l, T) \\ set fTs \\\n \\t.\n fs\\l\\\\<^sub>? = \\t\\ \\\n \\ \\ t : T"} {"_id": "500403", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\e\\edges.\n valid_out_port (fst e) \\ valid_in_port (snd e)"} {"_id": "500404", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\C0'.\n B' Out A' C0' \\ Cong B' C0' B C0 \\\n thesis) \\\n thesis"} {"_id": "500405", "text": "proof (prove)\ngoal (1 subgoal):\n 1. smult s p = monom 0 s * p"} {"_id": "500406", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (typeof\\<^bsub>h\\<^esub> v = \\Boolean\\) =\n (\\b. v = Bool b)"} {"_id": "500407", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (rel_blindable\\<^sub>m (rel_prod (=) (list_all2 pcr_view\\<^sub>m))\n (rel_prod (=) (list_all2 pcr_view\\<^sub>h)) ===>\n pcr_view\\<^sub>m)\n Tree\\<^sub>m View\\<^sub>m"} {"_id": "500408", "text": "proof (state)\nthis:\n J \\ gen P I\n\ngoal (2 subgoals):\n 1. \\x.\n x \\ gen P I \\\n x \\ \\ {J \\ P. J \\ gen P I}\n 2. \\x X.\n \\x \\ X; X \\ P; X \\ gen P I\\\n \\ x \\ gen P I"} {"_id": "500409", "text": "proof (prove)\nusing this:\n Ball F is_monomial\n f0 \\ F\n\ngoal (1 subgoal):\n 1. is_monomial f0"} {"_id": "500410", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a_transform (ae, a, as) (\\, App e x, S) \\\n a_transform (ae, inc\\a, as) (\\, e, Arg x # S)"} {"_id": "500411", "text": "proof (prove)\nusing this:\n C0 \\ D0\n f ` D0 = C0\n D0 \\ folding_g.\\ - f \\ folding_g.\\\n C0 \\ folding_g.\\ - g \\ folding_g.\\\n C0 \\ folding_g.\\ - g \\ folding_g.\\ \\\n g \\ folding_g.\\ =\n {B \\ folding_g.\\.\n order.greater (chamber_distance B C0) (chamber_distance B D0)}\n C0 \\ folding_g.\\ - g \\ folding_g.\\ \\\n folding_g.\\ - g \\ folding_g.\\ =\n {B \\ folding_g.\\.\n order.greater (chamber_distance B D0) (chamber_distance B C0)}\n D0 \\ folding_g.\\ - f \\ folding_g.\\ \\\n f \\ folding_g.\\ =\n {B \\ folding_g.\\.\n order.greater (chamber_distance B D0) (chamber_distance B (f ` D0))}\n D0 \\ folding_g.\\ - f \\ folding_g.\\ \\\n folding_g.\\ - f \\ folding_g.\\ =\n {B \\ folding_g.\\.\n order.greater (chamber_distance B (f ` D0)) (chamber_distance B D0)}\n C0 \\ D0 \\ D0 \\ C0\n\ngoal (1 subgoal):\n 1. folding_g.\\ - f \\ folding_g.\\ =\n g \\ folding_g.\\"} {"_id": "500412", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gcd (e - e') (order \\) = 1"} {"_id": "500413", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((Out ===> (In ===> M) ===> M) ===>\n Out ===> (In ===> rel_stateT S M) ===> rel_stateT S M)\n pause_state pause_state"} {"_id": "500414", "text": "proof (prove)\nusing this:\n \\e>0.\n \\x\\S.\n \\y\\S. dist x y < e \\ x = y\n\ngoal (1 subgoal):\n 1. (\\e1.\n \\0 < e1;\n \\x\\S.\n \\y\\S. dist y x < e1 \\ y = x\\\n \\ thesis) \\\n thesis"} {"_id": "500415", "text": "proof (prove)\nusing this:\n \\ \\ ub\n\ngoal (1 subgoal):\n 1. \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>\\\\<^esub> \\\n \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>ub\\<^esub>"} {"_id": "500416", "text": "proof (state)\nthis:\n ?y < xa \\ hoare (p ?y) x q\n\ngoal (1 subgoal):\n 1. \\xa.\n \\F x = x;\n \\w f.\n (\\v.\n v < w \\ hoare (p v) f q) \\\n hoare (p w) (F f) q;\n \\y. y < xa \\ hoare (p y) x q\\\n \\ hoare (p xa) x q"} {"_id": "500417", "text": "proof (state)\ngoal (1 subgoal):\n 1. M = uniform_measure (count_space (Pow (B \\ B))) (linorders_on A)"} {"_id": "500418", "text": "proof (prove)\nusing this:\n (u expands_to const_expansion (- 1)) bs\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. u x = - 1"} {"_id": "500419", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coprime (inv hom x') y = coprime x' (hom y)"} {"_id": "500420", "text": "proof (prove)\nusing this:\n k \\ 0\n similarity I M1 = k *\\<^sub>s\\<^sub>m M2\n\ngoal (1 subgoal):\n 1. moebius_cmat_eq M1 (moebius_mb_cmat (moebius_inv_cmat I) M2)"} {"_id": "500421", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Subobjs\\<^sub>R x xa xb;\n \\P C.\n \\x = P; xa = C; xb = [C]; is_class P C\\\n \\ thesis;\n \\Pa_ Ca_ Cs_ D_.\n \\x = Pa_; xa = Ca_; xb = Ca_ # Cs_; subclsRp Pa_ Ca_ D_;\n Subobjs\\<^sub>R Pa_ D_ Cs_\\\n \\ thesis\\\n \\ thesis"} {"_id": "500422", "text": "proof (prove)\nusing this:\n u \\ r = r \\ v\n u \\ \\\n\ngoal (1 subgoal):\n 1. u \\ v"} {"_id": "500423", "text": "proof (prove)\nusing this:\n n \\ 0\n\ngoal (1 subgoal):\n 1. ennreal (prod (prob_component p x) {0..n}) =\n ennreal (prob_component p x 0)"} {"_id": "500424", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.put_state s (local.put_state s' m) = local.put_state s' m"} {"_id": "500425", "text": "proof (prove)\nusing this:\n castAbs xs s (Abs xs x X) = (x', X')\n\ngoal (1 subgoal):\n 1. absCase xs s f (Abs xs x X) = f x' X'"} {"_id": "500426", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\U\\{{u..next_in A u} |u. u \\ A - {b}}. U)\n \\ {a..b}"} {"_id": "500427", "text": "proof (prove)\nusing this:\n b = a + int n * k\n gauss_sum_int (a + of_int k * int n) = gauss_sum_int a\n\ngoal (1 subgoal):\n 1. gauss_sum_int a = gauss_sum_int b"} {"_id": "500428", "text": "proof (state)\nthis:\n {x0} \\ C \\ W\n\ngoal (1 subgoal):\n 1. (\\X0.\n \\x0 \\ X0; open X0;\n \\x\\X0 \\ U.\n \\t\\C. dist (fx (x, t)) (fx (x0, t)) \\ e\\\n \\ thesis) \\\n thesis"} {"_id": "500429", "text": "proof (prove)\nusing this:\n \\ \\ \\ \\ \\\n eligible_literal L1 P1 \\\n \\eligible_literal ?L ?C ?\\;\n ?\\ \\ ?\\ \\ ?\\\\\n \\ eligible_literal ?L ?C ?\\\n\ngoal (1 subgoal):\n 1. eligible_literal L1 P1 \\"} {"_id": "500430", "text": "proof (prove)\nusing this:\n C.obj a\n C.obj a'\n \\g : map\\<^sub>0\n a \\\\<^sub>D map\\<^sub>0\n a'\\\n C.obj ?a \\ D.ide (\\\\<^sub>0 ?a)\n C.obj ?a \\\n trg\\<^sub>D (\\'.map\\<^sub>0 ?a) = F.map\\<^sub>0 ?a\n C.obj ?a \\ C.arr ?a\n C.obj ?a \\ src\\<^sub>C ?a = ?a\n C.obj ?a \\ trg\\<^sub>C ?a = ?a\n C.obj ?a \\ C.dom ?a = ?a\n C.obj ?a \\ C.cod ?a = ?a\n\ngoal (1 subgoal):\n 1. \\\\'.map\\<^sub>0 a' \\\\<^sub>D\n g \\\\<^sub>D\n \\\\<^sub>0\n a : F.map\\<^sub>0\n a \\\\<^sub>D F.map\\<^sub>0\n a'\\"} {"_id": "500431", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (pcr_stream A ===> rel_set A) lset sset"} {"_id": "500432", "text": "proof (prove)\nusing this:\n f \\\\<^sub>C\n F (\\'.map\\<^sub>0 b' \\\\<^sub>D\n G f \\\\<^sub>D \\\\<^sub>0 b)\n\ngoal (1 subgoal):\n 1. \\g.\n \\g : b \\\\<^sub>D b'\\ \\\n D.ide g \\ F g \\\\<^sub>C f"} {"_id": "500433", "text": "proof (prove)\ngoal (1 subgoal):\n 1. natural_isomorphism (\\\\<^sub>A) (\\\\<^sub>B) G F \\'"} {"_id": "500434", "text": "proof (prove)\nusing this:\n (?q1, ?ri1)\n \\ set todo \\ set ((q1, ri1) # old') \\\n ?q1 \\ 0\n (?q1, ?ri1)\n \\ set todo \\ set ((q1, ri1) # old') \\\n root_info_cond ?ri1 ?q1\n (q2, ri2) \\ set old'\n root_info.l_r ri2 l r \\ 0\n\ngoal (1 subgoal):\n 1. root_info_cond ri2 q2"} {"_id": "500435", "text": "proof (prove)\nusing this:\n finite (Units S)\n \\ (2, 1) \\ Units S\n\ngoal (1 subgoal):\n 1. card (generate S' {\\ (2, 1)}) \\ card (carrier S')"} {"_id": "500436", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z.\n x \\ carrier rec_rng_of_frac \\\n y \\ carrier rec_rng_of_frac \\\n z \\ carrier rec_rng_of_frac \\\n (x \\\\<^bsub>rec_rng_of_frac\\<^esub>\n y) \\\\<^bsub>rec_rng_of_frac\\<^esub>\n z =\n x \\\\<^bsub>rec_rng_of_frac\\<^esub>\n z \\\\<^bsub>rec_rng_of_frac\\<^esub>\n y \\\\<^bsub>rec_rng_of_frac\\<^esub> z"} {"_id": "500437", "text": "proof (prove)\nusing this:\n ((=) ===> pcr_word) of_int of_int\n\ngoal (1 subgoal):\n 1. ((=) ===> pcr_word) (\\k. k) of_int"} {"_id": "500438", "text": "proof (prove)\ngoal (1 subgoal):\n 1. R\\<^sup>*\\<^sup>* (\\_Fbcb x) y \\\n \\z. Ebc x z \\ y = \\_Fbcb z"} {"_id": "500439", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F y in at x.\n \\f y\\ = \\f x\\"} {"_id": "500440", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fst instr = call_type CALL;\n \\ snd (call_instr instr s1) \\\n \\ snd (call_instr instr s2) \\\n snd (fst (dispatch_instruction instr s1)) =\n snd (fst (call_instr instr s1)) \\\n snd (fst (dispatch_instruction instr s2)) =\n snd (fst (call_instr instr s2));\n low_equal s1 s2 \\\n get_S (cpu_reg_val PSR s1) = 0 \\\n get_S (cpu_reg_val PSR s2) = 0 \\\n \\ snd (dispatch_instruction instr s1) \\\n \\ snd (dispatch_instruction instr s2) \\\n t1 = snd (fst (call_instr instr s1)) \\\n t2 = snd (fst (call_instr instr s2));\n get_trap_set s1 = {} \\ get_trap_set s2 = {}\\\n \\ low_equal (snd (fst (call_instr instr s1)))\n (snd (fst (call_instr instr s2)))"} {"_id": "500441", "text": "proof (prove)\ngoal (1 subgoal):\n 1. normal_form F 0 = 0"} {"_id": "500442", "text": "proof (prove)\nusing this:\n reduced_run ?q ?v\\<^sub>1 ?v\\<^sub>2 ?l ?w ?w\\<^sub>1 ?w\\<^sub>2\n ?u \\\n ?v\\<^sub>1 @ ?l =\\<^sub>F ?w\\<^sub>1\n reduced_run q v\\<^sub>1 v\\<^sub>2 l w w\\<^sub>1 w\\<^sub>2 u\n\ngoal (1 subgoal):\n 1. length v\\<^sub>1 \\ length w\\<^sub>1"} {"_id": "500443", "text": "proof (prove)\nusing this:\n card (carrier G) = card (carrier H)\n\ngoal (1 subgoal):\n 1. finite (carrier H)"} {"_id": "500444", "text": "proof (prove)\ngoal (1 subgoal):\n 1. y.C 0 m (\\_. undefined) \\\n (\\x.\n distr (y.eP m x) (Pi\\<^sub>M {0..i. M))\n (case_nat y)) =\n distr (y.C 0 m (\\_. undefined) \\ y.eP m)\n (Pi\\<^sub>M {0..i. M)) (case_nat y)"} {"_id": "500445", "text": "proof (prove)\ngoal (1 subgoal):\n 1. false\\<^sub>n \\ \\ \\\n \\ \\\\<^sub>P \\[X]\\<^sub>\\ =\n {\\. \\[X]\\<^sub>\\ \\ \\} \\\\<^sub>P \\"} {"_id": "500446", "text": "proof (prove)\ngoal (1 subgoal):\n 1. norm (x + y) \\ norm x + norm y"} {"_id": "500447", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((A ===> A ===> A) ===> (A ===> A ===> A) ===> (=))\n (semiring_ow (Collect (Domainp A))) class.semiring"} {"_id": "500448", "text": "proof (state)\nthis:\n \\?x \\ carrier G; ?y \\ carrier G;\n N #> ?x = N #> ?y\\\n \\ h ?x = h ?y\n\ngoal (1 subgoal):\n 1. (\\g.\n \\g \\ hom (G Mod N) H;\n \\x.\n x \\ carrier G \\ g (N #> x) = h x\\\n \\ thesis) \\\n thesis"} {"_id": "500449", "text": "proof (state)\nthis:\n k \\ \\x. C x g\n\ngoal (1 subgoal):\n 1. \\x y.\n resumption.le_fun x y \\\n resumption_ord (B x \\ (\\y. C y x))\n (B y \\ (\\ya. C ya y))"} {"_id": "500450", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rsquarefree p"} {"_id": "500451", "text": "proof (prove)\ngoal (1 subgoal):\n 1. integrable (Pi\\<^sub>M I (\\i. distr M borel (Y i)))\n (\\\\. prod \\ I)"} {"_id": "500452", "text": "proof (prove)\nusing this:\n check_types P mxs mxl (map OK \\s)\n length \\s = length is\n\ngoal (1 subgoal):\n 1. map OK \\s \\ list (length is) A"} {"_id": "500453", "text": "proof (prove)\nusing this:\n b \\ \\ s\n (x, a) \\ set_pmf b\n b = return_pmf (l, R'_)\n s \\ {(l, R).\n l \\ L \\\n R \\ \\ \\\n R \\ {u. u \\ PTA.inv_of A l}}\n s = (l, R)\n R'_ \\ Succ \\ R\n R'_ \\ {v. v \\ PTA.inv_of A l}\n\ngoal (1 subgoal):\n 1. (x, a)\n \\ {(l, R).\n l \\ L \\\n R \\ \\ \\\n R \\ {u. u \\ PTA.inv_of A l}}"} {"_id": "500454", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\c c' x.\n \\finite X; strip c = strip c';\n \\S\\set (annos c). domo S \\ X;\n \\S\\set (annos c'). domo S \\ X;\n c' \\ c; \\ c \\ c \\\\<^sub>c c';\n length (annos c') = length (annos c); x < length (annos c)\\\n \\ n_o (n_st n_ivl X)\n (annos c ! x \\ annos c' ! x)\n \\ n_o (n_st n_ivl X) (annos c ! x)\n 2. \\c c'.\n \\finite X; strip c = strip c';\n \\S\\set (annos c). domo S \\ X;\n \\S\\set (annos c'). domo S \\ X;\n c' \\ c; \\ c \\ c \\\\<^sub>c c';\n length (annos c') = length (annos c)\\\n \\ \\a\\{0.. annos c' ! a)\n < n_o (n_st n_ivl X) (annos c ! a)"} {"_id": "500455", "text": "proof (prove)\nusing this:\n compare\\x\\y = EQ\n\ngoal (1 subgoal):\n 1. x = y"} {"_id": "500456", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (ffact (Suc 0)) k = of_nat (Suc 0) * ffact 0 (of_int k)"} {"_id": "500457", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>\\carrier = a_rcosets\\<^bsub>poly_ring L\\<^esub> (PIdl\\<^bsub>poly_ring L\\<^esub> q), monoid.mult = rcoset_mult (poly_ring L) (PIdl\\<^bsub>poly_ring L\\<^esub> q), one = L.rupture_surj (carrier L) q \\\\<^bsub>poly_ring L\\<^esub>, zero = PIdl\\<^bsub>poly_ring L\\<^esub> q, add = (<+>\\<^bsub>poly_ring L\\<^esub>)\\\\<^esub> =\n L.rupture_surj (carrier L) q \\\\<^bsub>poly_ring L\\<^esub>"} {"_id": "500458", "text": "proof (prove)\ngoal (1 subgoal):\n 1. All ge_game_dom"} {"_id": "500459", "text": "proof (prove)\nusing this:\n (upd ^^ 0) init = ((L1, R1, SR1), L2, R2, SR2)\n g 0 = (L, R)\n\ngoal (1 subgoal):\n 1. I L1 \\\n I L2 \\\n root_cond (p1, L1, R1) x \\\n root_cond (p2, L2, R2) y \\\n (\\!a. root_cond (p1, L1, R1) a) \\\n (\\!a. root_cond (p2, L2, R2) a) \\\n in_interval (L, R) (f x y) \\\n (0 = Suc j \\\n sub_interval (g 0) (g j) \\\n R - L \\ 3 / 4 * (snd (g j) - fst (g j))) \\\n SR1 = sgn (ipoly p1 R1) \\ SR2 = sgn (ipoly p2 R2)"} {"_id": "500460", "text": "proof (prove)\nusing this:\n distinct ys\n inj_on fst (set ys)\n inj_on snd (set ys)\n\ngoal (1 subgoal):\n 1. foldl (\\t' (x, p). ins x p t') \\\\ ys =\n treap_of (set ys)"} {"_id": "500461", "text": "proof (prove)\nusing this:\n x \\\\<^sub>\\ (\\r\\\\<^sub>\\A.\n set {\\b, a\\ |a b.\n \\a, b\\\n \\\\<^sub>\\ r})\n\ngoal (1 subgoal):\n 1. (\\r.\n \\x =\n \\r, (\\r\\\\<^sub>\\set {r}.\n set {\\b, a\\ |a b.\n\\a, b\\ \\\\<^sub>\\ r})\\r\\\\;\n r \\\\<^sub>\\ A\\\n \\ thesis) \\\n thesis"} {"_id": "500462", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Col X Y Z"} {"_id": "500463", "text": "proof (prove)\nusing this:\n Formula.future_bounded (formula.Prev I \\)\n dom P = S\n range_mapping i j P\n i \\ Monitor.progress \\ P \\ j\n\ngoal (1 subgoal):\n 1. dom P = S \\\n range_mapping i (max i j) P \\\n i \\ Monitor.progress \\ P (formula.Prev I \\) (max i j)"} {"_id": "500464", "text": "proof (prove)\nusing this:\n P \\ {P. partition_on C P \\ card P = j}\n B' \\ {B'. B' \\ B \\ card B' = k}\n\ngoal (1 subgoal):\n 1. disjoint_family_on\n (\\Q.\n B - B' \\\\<^sub>E P \\\n (\\f.\n {(\\X. X \\ {x \\ B - B'. f x = X}) `\n P} \\\n (\\P'. {P' \\ Q})))\n {Q. partition_on B' Q}"} {"_id": "500465", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sat\\<^sub>b \\ \\ = (\\ \\\\<^sub>b RESTRICT \\)"} {"_id": "500466", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\a \\ b;\n qs \\ lang (seq [question (Atom a), Atom b, Atom b])\\\n \\ real (T\\<^sub>p [a, b] qs (OPT2 qs [a, b])) = ?opt\n 2. qs \\ lang\n (seq [question (Atom a), Atom b, Atom b]) \\\n 2 / 10 * 2 + 8 / 10 * (15 / 10) \\ 16 / 10 * ?opt"} {"_id": "500467", "text": "proof (prove)\nusing this:\n (f has_real_derivative D) (at x)\n\ngoal (1 subgoal):\n 1. (\\h. (f (x + h) - f x) / h) \\0\\ D"} {"_id": "500468", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\I s; (ni, n) \\ R s\\\n \\ ospec (DESTRimpl ni s)\n (\\r.\n case n of Trueif \\ r = TD\n | Falseif \\ r = FD\n | IF v t e \\\n \\tn en.\n r = IFD v tn en \\\n (tn, t) \\ R s \\\n (en, e) \\ R s)"} {"_id": "500469", "text": "proof (prove)\nusing this:\n Outside' {n}\n (the_elem\n (Outside' {seller} `\n (argmax \\ sum)\n (summedBidVector\n (pseudoAllocation\n (randomEl\n (takeAll\n (\\x.\n winningAllocationRel (N \\ {seller}) (set \\)\n ((\\) x) b)\n (allAllocationsAlg (N \\ {seller}) \\))\n r) <|\n ((N \\ {seller}) \\ finestpart (set \\)))\n (N \\ {seller}) (set \\))\n ((argmax \\ sum) b\n (allAllocations (N \\ {seller}) (set \\)))))\n \\ soldAllocations (N - {n}) (set \\)\n finite (soldAllocations (N - {n}) (set \\))\n\ngoal (1 subgoal):\n 1. sum b\n (Outside' {n}\n (the_elem\n (Outside' {seller} `\n (argmax \\ sum)\n (summedBidVector\n (pseudoAllocation\n (randomEl\n (takeAll\n (\\x.\n winningAllocationRel (N \\ {seller})\n (set \\) ((\\) x) b)\n (allAllocationsAlg (N \\ {seller}) \\))\n r) <|\n ((N \\ {seller}) \\ finestpart (set \\)))\n (N \\ {seller}) (set \\))\n ((argmax \\ sum) b\n (allAllocations (N \\ {seller}) (set \\))))))\n \\ Max (sum b ` soldAllocations (N - {n}) (set \\))"} {"_id": "500470", "text": "proof (prove)\nusing this:\n xcpt_app i P pc m xt \\\\<^sub>2\n length (fst \\\\<^sub>1) = length (fst \\\\<^sub>2)\n\ngoal (1 subgoal):\n 1. xcpt_app i P pc m xt \\\\<^sub>1"} {"_id": "500471", "text": "proof (prove)\nusing this:\n xs \\ []\n last xs = t\n T.path xs\n hd xs \\ vertex_subtree v\n \\v \\ V; ?s \\ vertex_subtree v; t \\ vertex_subtree v;\n ?s \\xs\\\\<^bsub>T\\<^esub> t\\\n \\ set xs \\ vertex_subtree v\n v \\ V\n t \\ vertex_subtree v\n\ngoal (1 subgoal):\n 1. set xs \\ vertex_subtree v"} {"_id": "500472", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (h * M) Amic (h * N)"} {"_id": "500473", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s\\{x. distinct x \\ x \\ []}.\n \\b\\0.\n \\rs.\n set rs \\ set s \\\n T\\<^sub>p_on_rand' (COMB [])\n (fst (COMB []) s \\ (\\is. return_pmf (s, is))) rs\n \\ real 8 / real 5 * real (T\\<^sub>p_opt s rs) + b"} {"_id": "500474", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\binop_type op k \\ \\ \\\\<^sub>1;\n binop_type op k \\ \\ \\\\<^sub>2\\\n \\ \\\\<^sub>1 = \\\\<^sub>2"} {"_id": "500475", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\prj\\<^sub>0 h\n k : src (prj\\<^sub>0 h\n k) \\ src h\\ &&&\n \\prj\\<^sub>1 h\n k : src (prj\\<^sub>0 h\n k) \\ src k\\) &&&\n \\prj\\<^sub>0 h\n k : prj\\<^sub>0 h\n k \\ prj\\<^sub>0 h\n k\\ &&&\n \\prj\\<^sub>1 h\n k : prj\\<^sub>1 h\n k \\ prj\\<^sub>1 h k\\"} {"_id": "500476", "text": "proof (state)\nthis:\n finprod G f {x} = f x \\ finprod G f {}\n\ngoal (1 subgoal):\n 1. f x = a"} {"_id": "500477", "text": "proof (state)\nthis:\n \\t'. t' \\\\<^sub>A frontier (c_imp c loc) \\ t' \\ t\n n \\ 0\n\ngoal (1 subgoal):\n 1. the_cm c loc t n = apply_cm c loc t n"} {"_id": "500478", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\g.\n \\continuous_on UNIV g; range g \\ sphere (0::'a) 1;\n \\x. x \\ S \\ g x = f x\\\n \\ thesis) \\\n thesis"} {"_id": "500479", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ly = layout_of aprog; abc_fetch as aprog = Some ins\\\n \\ \\tp1 tp2.\n concat (tms_of aprog) =\n tp1 @ ci ly (start_of ly as) ins @ tp2 \\\n tp1 = concat (take as (tms_of aprog)) \\\n tp2 = concat (drop (Suc as) (tms_of aprog))"} {"_id": "500480", "text": "proof (prove)\nusing this:\n p' = snd (mkVarChannel d (apsnd \\ pState.vars_update) g p)\n pState_inv prog p\n sidx = pState.idx p\n cl_inv (g, p)\n gState_inv prog g\n varDeclName ` set (d # ds)\n \\ process_names (states prog !! sidx) (processes prog !! sidx)\n\ngoal (1 subgoal):\n 1. pState.idx p' = sidx"} {"_id": "500481", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Hom_FopxC.map gf =\n S.mkArr (HomC.set (F (D.cod (fst gf)), C.dom (snd gf)))\n (HomC.set (F (D.dom (fst gf)), C.cod (snd gf)))\n (\\C (F (D.dom (fst gf)), C.cod (snd gf)) \\\n (\\h.\n snd gf \\\\<^sub>C h \\\\<^sub>C F (fst gf)) \\\n \\C (F (D.cod (fst gf)), C.dom (snd gf)))"} {"_id": "500482", "text": "proof (prove)\ngoal (1 subgoal):\n 1. GrdSymSystem init r = SymSystem init (inpt_st r) r"} {"_id": "500483", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\\\. sat_models \\ \\ Pow P)\n {abs_ltln\\<^sub>P \\ |\\. prop_atoms \\ \\ P}"} {"_id": "500484", "text": "proof (prove)\ngoal (1 subgoal):\n 1. null \\ UML_Types.bot_class.bot"} {"_id": "500485", "text": "proof (prove)\nusing this:\n A B Lt C D \\ B A Lt C D \\ A B Lt D C \\ B A Lt D C\n ?A ?B Lt ?C ?D \\ ?B ?A Lt ?D ?C\n ?A ?B Lt ?C ?D \\ ?B ?A Lt ?C ?D\n\ngoal (1 subgoal):\n 1. A B Lt C D"} {"_id": "500486", "text": "proof (prove)\nusing this:\n Auth_PriK n = Auth_PriK m\n (Asset m, \\Num 2, PubKey B\\) \\ u\n\ngoal (1 subgoal):\n 1. Auth_PriKey m \\ spied u \\\n (\\C SK. (Asset m, Token m (Auth_PriK m) B C SK) \\ u)"} {"_id": "500487", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n PRJ('b\\<^sub>\\)\\(EMB('a\\<^sub>\\)\\x) =\n u_map\\(PRJ('b) oo EMB('a))\\x"} {"_id": "500488", "text": "proof (prove)\nusing this:\n u \\ RSpan us\n v \\ RSpan vs\n\ngoal (1 subgoal):\n 1. (\\as bs.\n \\length as = length us; u = as \\\\ us;\n length bs = length vs; v = bs \\\\ vs\\\n \\ thesis) \\\n thesis"} {"_id": "500489", "text": "proof (prove)\nusing this:\n 0 < p x y n\n 0 < p y x m\n \\0 < p ?xa6 ?ya6 ?na6; 0 < p ?ya6 ?xa6 ?ma6;\n G ?ya6 ?ya6 = \\\\\n \\ G ?xa6 ?xa6 = \\\n\ngoal (1 subgoal):\n 1. recurrent x = recurrent y"} {"_id": "500490", "text": "proof (prove)\nusing this:\n oalist_inv_raw ox xs'\n oalist_inv_raw ?ko (sort_oalist_aux ?ko ys)\n prod_ord_pair ox P xs' (sort_oalist_aux ox ys)\n k \\ fst ` set xs' \\ fst ` set (sort_oalist_aux ox ys)\n\ngoal (1 subgoal):\n 1. P k (lookup_pair ox xs' k) (lookup_pair ox (sort_oalist_aux ox ys) k)"} {"_id": "500491", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..pdevs_apply (pdevs_of_ivl l u) i\\) =\n (\\i = 0..((u - l) \\ Basis_list ! i / 2) *\\<^sub>R\n Basis_list ! i\\)"} {"_id": "500492", "text": "proof (prove)\nusing this:\n cfg' \\ cont (repcs (l, u) cfg) ` set_pmf (action (repcs (l, u) cfg))\n state cfg' = (l', u')\n (l, u) \\ S\n cfg \\ R_G.valid_cfg\n abss (l, u) = state cfg\n action (repcs (l, u) cfg) \\ K (l, u)\n action (repcs (l, u) cfg) = return_pmf (l, u \\ t)\n (l, u) \\ S\n (l, u) = (l, u)\n 0 \\ t\n u \\ t \\ PTA.inv_of A l\n\ngoal (1 subgoal):\n 1. u' \\ [u']\\<^sub>\\"} {"_id": "500493", "text": "proof (state)\nthis:\n dim ((+) (- a) ` S) \\ dim U\n\ngoal (1 subgoal):\n 1. \\\\T.\n \\subspace T; T \\ U;\n S \\ {} \\ aff_dim T = aff_dim S;\n rel_frontier S homeomorphic\n sphere (0::'b) 1 \\ T\\\n \\ thesis;\n S \\ {}\\\n \\ thesis"} {"_id": "500494", "text": "proof (prove)\nusing this:\n Extreme (\\i a. if a = b then 0 else 1) b\n b \\< F (\\_ a. if a = b then 0 else 1)\n\ngoal (1 subgoal):\n 1. \\i. pivotal i b"} {"_id": "500495", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\t. pbernpoly 1 t *\\<^sub>R f' t) has_integral\n S - I - (f (real_of_int b) - f (real_of_int a)) /\\<^sub>R 2)\n {real_of_int a..real_of_int b}"} {"_id": "500496", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\s.\n \\0 < l s \\\n l s \\ carrots s \\\n (mu \\ (carrots_total \\ carrots)\n s \\\n l s \\ lambda) \\\n (\\x. carrots s = 2 ^ x) \\\n tortoise s =\n (seq \\ carrots_total \\ carrots) s \\\n hare s =\n (seq \\\n (\\s. l s + (carrots_total \\ carrots) s))\n s;\n hare s \\ tortoise s\\\n \\ (((\\s.\n if carrots s = l s\n then s\\tortoise := hare s,\n carrots := 2 * carrots s, l := 0\\\n else id s) ;;\n (\\s. s\n \\hare := f (hare s),\n l := Suc (l s)\\))\n s,\n s)\n \\ inv_image find_lambda_measures\n (\\x. (l x, carrots x))\n 2. \\?R1\\\n \\s. s\n \\carrots := Suc 0, l := Suc 0, tortoise := x0,\n hare := f x0\\\n \\\\s.\n 0 < l s \\\n l s \\ carrots s \\\n (mu \\ (carrots_total \\ carrots)\n s \\\n l s \\ lambda) \\\n (\\x. carrots s = 2 ^ x) \\\n tortoise s =\n (seq \\ carrots_total \\ carrots) s \\\n hare s =\n (seq \\\n (\\s. l s + (carrots_total \\ carrots) s))\n s\\\n 3. \\s. True \\ ?R1 s"} {"_id": "500497", "text": "proof (prove)\ngoal (1 subgoal):\n 1. restrict A X ! i = None"} {"_id": "500498", "text": "proof (prove)\nusing this:\n C' =\n (\\i j.\n if i \\ S \\ j \\ S \\ seq (\\ i) (\\ j)\n then \\ (\\ i \\ \\ j) else n)\n i \\ S\n bij_betw \\ S (Collect arr)\n n \\ S\n bij_betw \\ (Collect arr) S\n ?i \\ S \\ \\ (\\ ?i) = ?i\n \\ = (\\f. if arr f then inv_into S \\ f else n)\n bij_betw ?f ?A ?B = (inj_on ?f ?A \\ ?f ` ?A = ?B)\n\ngoal (1 subgoal):\n 1. C' (\\ (local.dom (\\ i))) (\\ (local.dom (\\ i))) =\n \\ (local.dom (\\ i))"} {"_id": "500499", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bit (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"} {"_id": "500500", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {l2_inv8 \\ l2_inv1 \\ l2_inv3} TS.trans l2 {> l2_inv8}"} {"_id": "500501", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (A ===> pcr_stream A ===> pcr_stream A) LCons (##)"} {"_id": "500502", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rotater1 (rev xs) = rev (rotate1 xs)"} {"_id": "500503", "text": "proof (state)\nthis:\n subspace.dependent = dependent\n\ngoal (1 subgoal):\n 1. subspace.subspace = subspace"} {"_id": "500504", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wp\\<^sub>t (Do f) Q =\n (\\s. (\\t\\f s. Q t) \\ f s \\ {})"} {"_id": "500505", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Product_Type.product (Set t1) (Set t2) =\n fold_keys (\\x. fold_keys (\\y. insert (x, y)) t2) t1 {}"} {"_id": "500506", "text": "proof (chain)\npicking this:\n l = treap_of {p \\ A. fst p < fst (arg_min_on snd A)}\n r = treap_of {p \\ A. fst (arg_min_on snd A) < fst p}"} {"_id": "500507", "text": "proof (prove)\nusing this:\n invar (build ks l)\n invar (build ks r)\n \\p\\set l. p $ k \\ m\n \\p\\set r. m < p $ k\n widest_spread_axis k UNIV (set l \\ set r)\n build ks ps = Node k m (build ks l) (build ks r)\n set l = set_kdt (build ?ks l)\n set r = set_kdt (build ?ks r)\n\ngoal (1 subgoal):\n 1. invar (build ks ps)"} {"_id": "500508", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. exp (x *\\<^sub>R - a)) has_integral\n exp (c *\\<^sub>R - a) / a)\n {c..}"} {"_id": "500509", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (pcr_nonzero A ===> pcr_nonzero A ===> (=))\n (\\x a. \\c. a = c *\\<^sub>R x \\ c \\ 0)\n (\\x a. \\c. a = c *\\<^sub>R x \\ c \\ 0)"} {"_id": "500510", "text": "proof (prove)\nusing this:\n h \\ get_root_node ptr \\\\<^sub>r root_ptr\n h \\ to_tree_order root_ptr \\\\<^sub>r c\n heap_is_wellformed h\n type_wf h\n known_ptrs h\n h \\ get_dom_component ptr \\\\<^sub>r c\n h \\ to_tree_order ptr'' \\\\<^sub>r to''\n ptr'' \\ set c\n h \\ get_dom_component ptr' \\\\<^sub>r c'\n ptr' \\ set c\n \\heap_is_wellformed ?h; type_wf ?h; known_ptrs ?h;\n ?h \\ get_dom_component ?ptr \\\\<^sub>r ?c;\n ?h \\ to_tree_order ?ptr \\\\<^sub>r ?to;\n ?h \\ get_dom_component ?ptr' \\\\<^sub>r ?c';\n ?ptr' \\ set ?c\\\n \\ set ?to \\ set ?c' = {}\n \\heap_is_wellformed ?h; type_wf ?h; known_ptrs ?h;\n ?h \\ get_dom_component ?ptr \\\\<^sub>r ?c;\n ?ptr' \\ set ?c\\\n \\ ?h \\ get_dom_component ?ptr'\n \\\\<^sub>r ?c\n\ngoal (1 subgoal):\n 1. set to'' \\ set c' = {}"} {"_id": "500511", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ range c ; \\ range d =\n (\\(x::'b) y::'c. c x ; d y)\nvariables:\n d :: 'c \\ 'a\n c :: 'b \\ 'a\n (;) :: 'a \\ 'a \\ 'a\ntype variables:\n 'a :: refinement_lattice\n 'b, 'c :: type"} {"_id": "500512", "text": "proof (prove)\ngoal (1 subgoal):\n 1. conv_radius f = c"} {"_id": "500513", "text": "proof (prove)\nusing this:\n p dvd smult (of_nat (Suc k)) (pderiv a)\n \\?p dvd smult ?a ?q; ?a \\ (0::?'a)\\\n \\ ?p dvd ?q\n (of_nat ?m = (0::?'a)) = (?m = 0)\n\ngoal (1 subgoal):\n 1. p dvd pderiv a"} {"_id": "500514", "text": "proof (prove)\nusing this:\n 0 < dim (sub.tangent_space p)\n\ngoal (1 subgoal):\n 1. (\\basis.\n \\independent basis;\n span basis = span (sub.tangent_space p)\\\n \\ thesis) \\\n thesis"} {"_id": "500515", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pivot_fun A (\\_. 0) 0"} {"_id": "500516", "text": "proof (prove)\nusing this:\n \\ \\ [e] : [t] _> [arity_1_result e]\n\ngoal (1 subgoal):\n 1. \\tn.\n c_types_agree tn' tn \\ \\ \\ [e] : tn _> tm"} {"_id": "500517", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\2 ^ d\\ *\n \\2 ^ m * enn2real (min (of_nat m) (u x))\\\n \\ \\2 ^ d *\n (2 ^ m * enn2real (min (of_nat m) (u x)))\\"} {"_id": "500518", "text": "proof (prove)\nusing this:\n \\ (a \\ b \\ b \\ c \\ a \\ c)\n\ngoal (1 subgoal):\n 1. \\f. (\\i\\{a, b, c}. f i * i) = x * a + y * b + z * c"} {"_id": "500519", "text": "proof (prove)\nusing this:\n wp \\\n hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id zp\n\ngoal (1 subgoal):\n 1. wp \\ carrier (relative_homology_group (int n) (nsphere n) (upper n))"} {"_id": "500520", "text": "proof (prove)\nusing this:\n \\cxt_to_stmt ?E ?c = ?c'; \\p q. ?c' \\ p ;; q\\\n \\ ?E = [] \\ ?c' = ?c\n\ngoal (1 subgoal):\n 1. cxt_to_stmt E c = c'@[mu] \\ c = c'@[mu] \\ E = []"} {"_id": "500521", "text": "proof (prove)\ngoal (2 subgoals):\n 1. n \\ length D \\ is_word (take j b)\n 2. n \\ length D \\ is_terminal (a ! i)"} {"_id": "500522", "text": "proof (prove)\nusing this:\n set (opaque x) \\ {0.. {0..c. local.bind_state (g c) f)"} {"_id": "500524", "text": "proof (prove)\nusing this:\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom (locals\n (snd s0)) \\In1r\n (Try c1 Catch(C vn) c2)\\ E\n\ngoal (1 subgoal):\n 1. (\\C1 C2.\n \\\\prg = G, cls = accC,\n lcl =\n L\\\\ dom\n (locals (snd s0)) \\In1r c1\\ C1;\n \\prg = G, cls = accC,\n lcl = L(VName vn \\\n Class\n C)\\\\ dom (locals (snd s0)) \\\n{VName vn} \\In1r c2\\ C2\\\n \\ thesis) \\\n thesis"} {"_id": "500525", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (cod Zero = Zero &&& cod One = One) &&& cod j0 = One &&& cod j1 = One"} {"_id": "500526", "text": "proof (prove)\ngoal (1 subgoal):\n 1. - part_circlepath 0 R (- pi / 2) (pi / 2) =\n part_circlepath 0 R (pi / 2) (3 * pi / 2)"} {"_id": "500527", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sig_red sing_reg (=) F p q; rep_list q \\ 0\\\n \\ punit.lt (rep_list q) \\ punit.lt (rep_list p)"} {"_id": "500528", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(\\w.\n (\\x.\n \\xa.\n \\

xa w =\n (\\<^bold>\\xa x) w) \\\n (\\<^bold>\\(\\v.\n \\x.\n existsAt x v \\ (\\xa. \\

xa v = (\\<^bold>\\xa x) v)))\n w)\\"} {"_id": "500529", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\list_all2 (\\) [x] x; list_all2 (\\) x [y]\\\n \\ x \\ (\\x. [x]) ` {x..y}"} {"_id": "500530", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((W ===> M ===> M) ===>\n W ===>\n Monomorphic_Monad.rel_envT R M ===> Monomorphic_Monad.rel_envT R M)\n (\\tell w m. EnvT (\\r. tell w (run_env m r)))\n (\\tell w m. EnvT (\\r. tell w (run_env m r)))"} {"_id": "500531", "text": "proof (state)\nthis:\n finite ({0, 1} \\ S)\n\ngoal (1 subgoal):\n 1. \\x.\n x \\ {0..1} - ({0, 1} \\ S) \\\n f (g x) * vector_derivative g (at x within {0..1}) =\n f (shiftpath (1 - a) (shiftpath a g) x) *\n vector_derivative (shiftpath (1 - a) (shiftpath a g))\n (at x within {0..1})"} {"_id": "500532", "text": "proof (state)\nthis:\n \\m\\j - k. (sdrop k xs !! m) c1 \\ 0\n\ngoal (1 subgoal):\n 1. (xs !! j) c2 \\ (xs !! j) c1"} {"_id": "500533", "text": "proof (prove)\nusing this:\n \\R \\ constant_rules C \\ ?thesis;\n R \\ identity_rules L \\ ?thesis;\n R = top_rule S_Top \\ ?thesis;\n R = nonempty_rule \\ ?thesis\\\n \\ ?thesis\n\ngoal (1 subgoal):\n 1. maintained R G"} {"_id": "500534", "text": "proof (prove)\nusing this:\n [] \\ e \\ v\n FV e = {}\n \\[] \\ ?e \\ ?v; FV ?e = {}\\\n \\ \\v' ob.\n ?e \\* v' \\\n isval v' \\ observe v' ob \\ bs_observe ?v ob\n\ngoal (1 subgoal):\n 1. (\\v' ob.\n \\e \\* v'; observe v' ob;\n bs_observe v ob\\\n \\ thesis) \\\n thesis"} {"_id": "500535", "text": "proof (state)\nthis:\n bval b t1\n\ngoal (1 subgoal):\n 1. \\b s\\<^sub>1 c s\\<^sub>2 s\\<^sub>3 X t.\n \\bval b s\\<^sub>1; (c, s\\<^sub>1) \\ s\\<^sub>2;\n \\X t.\n s\\<^sub>1 = t on L c X \\\n \\t'. (c, t) \\ t' \\ s\\<^sub>2 = t' on X;\n (WHILE b DO c, s\\<^sub>2) \\ s\\<^sub>3;\n \\X t.\n s\\<^sub>2 = t on L (WHILE b DO c) X \\\n \\t'.\n (WHILE b DO c, t) \\ t' \\ s\\<^sub>3 = t' on X;\n s\\<^sub>1 = t on L (WHILE b DO c) X\\\n \\ \\t'.\n (WHILE b DO c, t) \\ t' \\\n s\\<^sub>3 = t' on X"} {"_id": "500536", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subalgebra K (carrier R) R"} {"_id": "500537", "text": "proof (prove)\nusing this:\n Sup A = RES B\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "500538", "text": "proof (state)\ngoal (8 subgoals):\n 1. \\s s' t.\n \\[p] to x in m = App s t; r' = App s' t;\n s \\ s'\\\n \\ SN r\n 2. \\t t' s.\n \\[p] to x in m = App s t; r' = App s t';\n t \\ t'\\\n \\ SN r\n 3. \\s t.\n (\\x \\ [p] to x in m; x \\ r'\\\n \\ [p] to x in m = App (\\ x . t) s \\\n r' = t[x::=s] \\\n x \\ s) \\\n SN r\n 4. \\t t'.\n (\\x \\ [p] to x in m; x \\ r'\\\n \\ [p] to x in m = \\ x . t \\\n r' = \\ x . t' \\\n t \\ t') \\\n SN r\n 5. \\t.\n (\\x \\ [p] to x in m; x \\ r'\\\n \\ [p] to x in m = \\ x . App t (Var x) \\\n r' = t \\ x \\ t) \\\n SN r\n 6. \\s.\n (\\x \\ [p] to x in m; x \\ r'\\\n \\ [p] to x in m = s to x in [Var x] \\\n r' = s \\ x \\ s) \\\n SN r\n 7. \\s u t.\n (\\x \\ z; x \\ [p] to x in m; x \\ r';\n z \\ [p] to x in m; z \\ r'\\\n \\ [p] to x in m = (s to x in t) to z in u \\\n r' = s to x in t to z in u \\\n x \\ (z, s, u) \\\n z \\ (s, t)) \\\n SN r\n 8. \\s s'.\n \\[p] to x in m = [s]; r' = [s']; s \\ s'\\\n \\ SN r"} {"_id": "500539", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x1 x2 i.\n \\\\i.\n (\\\\x2\\\\e =\n \\\\0, 0, 0, 0, 0, 0, 0\\,\n \\quot_dbfm\n (trans_fm [i] \\)\\e\\) =\n (x2 = SyntaxN.Ex i \\);\n \\A.\n (\\\\A\\\\e =\n \\\\\\\\\\e) =\n (A = \\);\n \\c.\n atom c \\ (x2, \\) \\\n atom c \\ (x1, i, x2, \\) \\\n (x1 \\ c) \\ x2 =\n (i \\ c) \\ \\\\\n \\ trans_fm [x1] x2 = trans_fm [i] \\"} {"_id": "500540", "text": "proof (prove)\nusing this:\n \\\\<^sub>F n in sequentially. dist x (f n) \\ a\n\ngoal (1 subgoal):\n 1. dist x y \\ a"} {"_id": "500541", "text": "proof (prove)\nusing this:\n P {||}\n \\\\y. y |\\| ?S \\ y < ?x; P ?S\\\n \\ P (finsert ?x ?S)\n\ngoal (1 subgoal):\n 1. P S"} {"_id": "500542", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cs\\<^bsup>\\'\\<^esup> l' = cs\\<^bsup>\\'\\<^esup> l'"} {"_id": "500543", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier R; a \\ \\\\\n \\ inv (inv a) = a"} {"_id": "500544", "text": "proof (prove)\nusing this:\n pc2 (s 0) = a \\ 0 < sem (s 0)\n\ngoal (1 subgoal):\n 1. (Enabled n2) s"} {"_id": "500545", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lookup (\\b\\B0. u b \\ b) t =\n (\\b\\B0. u b * lookup b t)"} {"_id": "500546", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A * D + B * H = (D * S + H * T) * U"} {"_id": "500547", "text": "proof (prove)\nusing this:\n h measurable_on S\n\ngoal (1 subgoal):\n 1. (\\N g.\n \\N \\ null_sets lebesgue;\n \\n. continuous_on UNIV (g n);\n \\x.\n x \\ N \\\n (\\n. g n x)\n \\ (if x \\ S then h x\n else (0::'b))\\\n \\ thesis) \\\n thesis"} {"_id": "500548", "text": "proof (prove)\nusing this:\n invar1 m\n \\1 m |` {k. (\\v. \\1 m k = Some v) \\ P k} =\n \\1 m |` {k. P k}\n\ngoal (1 subgoal):\n 1. \\2 (restrict (\\(k, uu_). P k) m) =\n \\1 m |` {k. P k}"} {"_id": "500549", "text": "proof (prove)\nusing this:\n \\fst f : fst a \\\\<^sub>1 fst b\\\n \\snd f : snd a \\\\<^sub>2 snd b\\\n\ngoal (1 subgoal):\n 1. \\f : a \\ b\\"} {"_id": "500550", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cauchy_filter F"} {"_id": "500551", "text": "proof (prove)\ngoal (1 subgoal):\n 1. double_dual_iso \\ hom G (Characters (Characters G))"} {"_id": "500552", "text": "proof (chain)\npicking this:\n Opaque x' \\ y"} {"_id": "500553", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x ya.\n poly_rel x ya \\\n list_all2 (list_all2 poly_rel)\n (map (\\s.\n if gcd_poly_i ff_ops x\n (minus_poly_i ff_ops y\n (if s = 0 then []\n else [arith_ops_record.of_int ff_ops\n (int s)])) \\\n [arith_ops_record.one ff_ops]\n then [gcd_poly_i ff_ops x\n (minus_poly_i ff_ops y\n (if s = 0 then []\n else [arith_ops_record.of_int ff_ops (int s)]))]\n else [])\n [0..x.\n if gcd ya (y' - [:of_nat x:]) \\ 1\n then [gcd ya (y' - [:of_int (int x):])] else [])\n [0.. \\\\<^sub>s"} {"_id": "500555", "text": "proof (prove)\nusing this:\n [?a\\<^sub>\\]\\<^sub>\\\n \\\\<^sub>\\ \\\\<^sub>\\ ^\\<^sub>\\ 1\\<^sub>\\\n\ngoal (1 subgoal):\n 1. (cr_vrat ===> cr_vrat ===> (=)) vrat_le' (\\)"} {"_id": "500556", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>fpl \\

\\\\<^esub> = \\ \\\\<^bsub>\\

\\<^esub>"} {"_id": "500557", "text": "proof (prove)\nusing this:\n \\\\<^sub>G (F (f \\\\<^sub>B g), F h) =\n \\\\<^sub>G (C.VV.cod (\\\\<^sub>F (f, g), F h))\n G (\\\\<^sub>F (f, g)) \\\\<^sub>D G (F h) =\n G.H\\<^sub>DoFF.map (\\\\<^sub>F (f, g), F h)\n G.FoH\\<^sub>C.map (\\\\<^sub>F (f, g), F h) =\n G (\\\\<^sub>F (f, g) \\\\<^sub>C F h)\n \\\\<^sub>G (C.VV.dom (\\\\<^sub>F (f, g), F h)) =\n \\\\<^sub>G (F f \\\\<^sub>C F g, F h)\n B.ide f\n B.ide g\n B.ide h\n src\\<^sub>B f = trg\\<^sub>B g\n src\\<^sub>B g = trg\\<^sub>B h\n B.VV.arr ?f =\n (B.arr (fst ?f) \\\n B.arr (snd ?f) \\ src\\<^sub>B (fst ?f) = trg\\<^sub>B (snd ?f))\n C.VV.arr ?f \\\n \\\\<^sub>G (C.VV.cod ?f) \\\\<^sub>D G.H\\<^sub>DoFF.map ?f =\n G.FoH\\<^sub>C.map ?f \\\\<^sub>D \\\\<^sub>G (C.VV.dom ?f)\n\ngoal (1 subgoal):\n 1. G (F \\\\<^sub>B[f, g, h] \\\\<^sub>C\n \\\\<^sub>F (f \\\\<^sub>B g, h)) \\\\<^sub>D\n (\\\\<^sub>G (F (f \\\\<^sub>B g), F h) \\\\<^sub>D\n (G (\\\\<^sub>F (f, g)) \\\\<^sub>D G (F h))) \\\\<^sub>D\n (\\\\<^sub>G (F f, F g) \\\\<^sub>D G (F h)) =\n G (F \\\\<^sub>B[f, g, h] \\\\<^sub>C\n \\\\<^sub>F (f \\\\<^sub>B g, h)) \\\\<^sub>D\n (G (\\\\<^sub>F (f, g) \\\\<^sub>C F h) \\\\<^sub>D\n \\\\<^sub>G (F f \\\\<^sub>C F g, F h)) \\\\<^sub>D\n (\\\\<^sub>G (F f, F g) \\\\<^sub>D G (F h))"} {"_id": "500558", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite I; i \\ I;\n \\j.\n \\j \\ I; j \\ i\\\n \\ degree (p j) < degree (p i)\\\n \\ degree (sum p I) = degree (p i)"} {"_id": "500559", "text": "proof (prove)\nusing this:\n \\?C2 \\ f \\ folding_g.\\;\n folding_g.gallery (?C2 # Bs @ [D])\\\n \\ H \\ set (wall_crossings (?C2 # Bs @ [D]))\n folding_g.gallery (C # (B # Bs) @ [D])\n B \\ f \\ folding_g.\\\n folding_g.gallery (?x # ?xs) \\ folding_g.gallery ?xs\n\ngoal (1 subgoal):\n 1. H \\ set (wall_crossings (C # (B # Bs) @ [D]))"} {"_id": "500560", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom_mult_punit c t (Pm_fmap xs) =\n Pm_fmap\n (if c = (0::'b) then fmempty else shift_map_keys_punit t ((*) c) xs)"} {"_id": "500561", "text": "proof (prove)\ngoal (30 subgoals):\n 1. V0 = id ` V0\nA total of 30 subgoals..."} {"_id": "500562", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dual.check_eqv n \\ \\ \\\n \\.lang\\<^sub>M\\<^sub>2\\<^sub>L n \\ =\n \\.lang\\<^sub>M\\<^sub>2\\<^sub>L n \\"} {"_id": "500563", "text": "proof (prove)\ngoal (1 subgoal):\n 1. RES (fused_resource.fuse core rest) (s_core, s_rest) =\n fuse_converter \\\n RES (callee_of_core core) s_core \\\n RES (callee_of_rest rest) s_rest"} {"_id": "500564", "text": "proof (prove)\ngoal (1 subgoal):\n 1. supp None = {}"} {"_id": "500565", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sublist (xs @ [x]) (ys @ [y]) =\n (x = y \\ suffix xs ys \\ sublist (xs @ [x]) ys)"} {"_id": "500566", "text": "proof (state)\nthis:\n distinct (map ((*\\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))\n (\\i. 1 / 2 * (f i + 1)) \\ UNIV \\ {0<..<1}\n\ngoal (1 subgoal):\n 1. \\e \\ UNIV \\ {- 1<..<1};\n seg\n \\ set (polychain_of (lowest_vertex (fst X, nlex_pdevs (snd X)))\n (map ((*\\<^sub>R) 2) (ccw.selsort 0 (inl (snd X)))));\n length\n (polychain_of (lowest_vertex (fst X, nlex_pdevs (snd X)))\n (map ((*\\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))) \\\n 1\\\n \\ ccw' (fst seg) (snd seg) (aform_val e X)"} {"_id": "500567", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (?c, go_is_done_impl)\n \\ node_rel \\\n ce_rel node_rel \\\\<^sub>r oGSi_rel \\ bool_rel"} {"_id": "500568", "text": "proof (prove)\ngoal (1 subgoal):\n 1. c_rec + c_sig + 2 * n + 2\n \\ 2 * n + 2 + 4 * j + 1 +\n (\\jj = Suc j..M UNIV (\\i. measure_pmf (bernoulli_pmf p)))\n (prod_emb UNIV (\\i. measure_pmf (bernoulli_pmf p)) {n}\n ({n} \\\\<^sub>E {True})) =\n (\\i\\{n}. emeasure (measure_pmf (bernoulli_pmf p)) {True})"} {"_id": "500571", "text": "proof (prove)\nusing this:\n N \\ null_sets M\n null_part M A \\ N\n main_part M A \\ N - A \\ N\n\ngoal (1 subgoal):\n 1. main_part M A \\ N - A \\ null_sets (completion M)"} {"_id": "500572", "text": "proof (prove)\nusing this:\n f analytic_on S\n z \\ S\n\ngoal (1 subgoal):\n 1. (\\a b.\n \\z \\ box a b; f analytic_on box a b\\\n \\ thesis) \\\n thesis"} {"_id": "500573", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a tree, semiring_char_0_class)"} {"_id": "500574", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a.\n Im a = Im (Complex rb ib) \\\n Re a < Re (Complex rb ib) \\\n Im a = Im (Complex rc ic) \\\n Re a \\ Re (Complex rc ic)) \\\n Im (Complex rb ib) = Im (Complex rc ic) \\\n Re (Complex rb ib) \\ Re (Complex rc ic)"} {"_id": "500575", "text": "proof (prove)\nusing this:\n w_vec p qs signs =\n matr_option (dim_row (M_mat signs subsets))\n (mat_inverse (M_mat signs subsets)) *\\<^sub>v\n v_vec p qs subsets\n\ngoal (1 subgoal):\n 1. w_vec p qs signs =\n matr_option (dim_row (M_mat signs subsets))\n (mat_inverse_var (M_mat signs subsets)) *\\<^sub>v\n v_vec p qs subsets"} {"_id": "500576", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\<^bold>\\ (F \\<^bold>\\ G)\n \\ \\ \\\n \\ \\ AX10 \\\\<^sub>H \\<^bold>\\ G"} {"_id": "500577", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lex_ext_unbounded f (a # as) ys =\n ((\\i\n (\\j\n fst (f ((a # as) ! i) (ys ! i))) \\\n (\\i\n length ys < length (a # as),\n (\\i\n (\\j\n fst (f ((a # as) ! i) (ys ! i))) \\\n (\\i\n length ys \\ length (a # as))"} {"_id": "500578", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n. count_it \\ b c s n"} {"_id": "500579", "text": "proof (prove)\ngoal (1 subgoal):\n 1. V = 0 \\ W = 0"} {"_id": "500580", "text": "proof (prove)\ngoal (1 subgoal):\n 1. shows_words xs (r @ s) = shows_words xs r @ s"} {"_id": "500581", "text": "proof (prove)\nusing this:\n \\ lfinite us\n\ngoal (1 subgoal):\n 1. lappend us vs = us"} {"_id": "500582", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A \\\\<^sub>C obsf_oracle callee1(Fault) \\\n obsf_oracle callee2(Fault)"} {"_id": "500583", "text": "proof (prove)\nusing this:\n f \\ X \\\\<^sub>M tree_sigma M\n ?x \\ space X \\ f ?x \\ \\\\\n\ngoal (1 subgoal):\n 1. f \\ X \\\\<^sub>M\n restrict_space (tree_sigma M) (- {\\\\})"} {"_id": "500584", "text": "proof (chain)\npicking this:\n x = \\\\A\\\\e\n atom (decode_Var w) \\ A"} {"_id": "500585", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i\\m. a i * x ^ i) * (\\j\\n. b j * x ^ j) =\n (\\r\\m + n. (\\k\\r. a k * b (r - k)) * x ^ r)"} {"_id": "500586", "text": "proof (prove)\nusing this:\n funToCone F j \\ S.hom S.unity (D j)\n J.arr j\n J.arr ?f \\ U.map ?f = S.unity\n\ngoal (1 subgoal):\n 1. S.dom (funToCone F j) = U.map (J.dom j)"} {"_id": "500587", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal\n (extract_real\n (cexpr_sem (merge V V' (x, \\))\n (e1 *\\<^sub>c\n \\e2\\\\<^sub>c)))\n \\state_measure V \\ =\n \\\\<^sup>+ x. ennreal\n (extract_real\n (cexpr_sem (merge V V' (x, \\)) e1) *\n extract_real\n (cexpr_sem (merge V V' (x, \\))\n (\\e2\\\\<^sub>c)))\n \\state_measure V \\"} {"_id": "500588", "text": "proof (prove)\ngoal (1 subgoal):\n 1. equivalence_of_monoidal_categories (\\\\<^sub>S)\n T\\<^sub>F\\<^sub>S\\<^sub>M\\<^sub>C\n \\\\<^sub>F\\<^sub>S\\<^sub>M\\<^sub>C\n \\\\<^sub>F\\<^sub>S\\<^sub>M\\<^sub>C (\\)\n \\C.T\\<^sub>F\\<^sub>M\\<^sub>C \\C.\\\\<^sub>F\\<^sub>M\\<^sub>C\n \\C.\\\\<^sub>F\\<^sub>M\\<^sub>C \\C.D D.\\ local.map\n \\C.\\ \\C.\\"} {"_id": "500589", "text": "proof (prove)\ngoal (1 subgoal):\n 1. deterministic R \\ w\\ R"} {"_id": "500590", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.quantale (\\) Infs Sup infs (\\) le sups bots tops"} {"_id": "500591", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 \\ root_int_floor_pos p x"} {"_id": "500592", "text": "proof (prove)\nusing this:\n digraph_isomorphism hom\n subgraph H1 G\n subgraph H2 G\n app_iso (inv_iso hom) (app_iso hom H1) =\n app_iso (inv_iso hom) (app_iso hom H2)\n\ngoal (1 subgoal):\n 1. H1 = H2"} {"_id": "500593", "text": "proof (prove)\nusing this:\n (\\m. wf'.obs n (PDGx.PDG_BS S) = {m}) \\\n wf'.obs n (PDGx.PDG_BS S) = {}\n\ngoal (1 subgoal):\n 1. finite (wf'.obs n (PDGx.PDG_BS S))"} {"_id": "500594", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\.\n \\f \\ \\ = \\;\n \\ \\ carrier Zp\\\n \\ thesis) \\\n thesis"} {"_id": "500595", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p =\n (\\x\\keys (mapping_of p).\n MPoly_Type.monom x (MPoly_Type.coeff p x))"} {"_id": "500596", "text": "proof (prove)\nusing this:\n value_flow (residual_network (f_i i))\n (f_aux (Suc i) \\ f_i i) \\\n \\\n \\<^bold>V - {source \\, sink \\} \\ {}\n flow (residual_network (f_i i)) (f_aux (Suc i) \\ f_i i)\n\ngoal (1 subgoal):\n 1. flow_network (residual_network (f_i i)) (f_aux (Suc i) \\ f_i i)"} {"_id": "500597", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\<^sub>\\\n ((dghm_id\n \\ \\\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\n \\)\\ObjMap\\)\n\ngoal (1 subgoal):\n 1. (dghm_id \\ \\\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\n \\)\\ObjMap\\\\a\\ =\n \\\\ObjMap\\\\a\\"} {"_id": "500598", "text": "proof (prove)\ngoal (1 subgoal):\n 1. linearize (takeWhile_queue f q) = takeWhile f (linearize q)"} {"_id": "500599", "text": "proof (state)\nthis:\n b \\ alphabet t\\<^sub>1\n\ngoal (2 subgoals):\n 1. ?P \\\n if a \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then if b \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) a +\n freq (Node w t\\<^sub>1 t\\<^sub>2) b =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>a + w\\<^sub>b\n else weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) a =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>b\n else if b \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) b =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>a\n else weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) =\n weight (Node w t\\<^sub>1 t\\<^sub>2)\n 2. \\ ?P \\\n if a \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then if b \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) a +\n freq (Node w t\\<^sub>1 t\\<^sub>2) b =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>a + w\\<^sub>b\n else weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) a =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>b\n else if b \\ alphabet (Node w t\\<^sub>1 t\\<^sub>2)\n then weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) +\n freq (Node w t\\<^sub>1 t\\<^sub>2) b =\n weight (Node w t\\<^sub>1 t\\<^sub>2) + w\\<^sub>a\n else weight\n (swapLeaves (Node w t\\<^sub>1 t\\<^sub>2) w\\<^sub>a a\n w\\<^sub>b b) =\n weight (Node w t\\<^sub>1 t\\<^sub>2)"} {"_id": "500600", "text": "proof (prove)\nusing this:\n list_all2 cr_dual_ord ?xs ?ys \\ ?xs = map of_dual_ord ?ys\n\ngoal (1 subgoal):\n 1. (list_all2 (pcr_dual_ord (=)) ===> pcr_dual_ord (=)) med med'"} {"_id": "500601", "text": "proof (prove)\ngoal (1 subgoal):\n 1. SN r \\ non_inf r"} {"_id": "500602", "text": "proof (prove)\nusing this:\n P \\ Ts'' [\\] Ts'\n\ngoal (1 subgoal):\n 1. length Ts'' = length Ts'"} {"_id": "500603", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\nat.\n \\steps0\n (Suc 0, Bk # Oc \\ m @ [Oc],\n Bk #\n Oc #\n Bk \\ ln @ Bk # Bk # Oc \\ nat @ Oc # Bk \\ rn)\n t_wcode_adjust n =\n (0, a, b);\n wadjust_stop m nat (a, b); bl_bin () = Suc nat\\\n \\ is_final\n (steps0\n (Suc 0, Bk # Oc # Oc \\ m,\n Bk #\n Oc #\n Bk \\ ln @\n Bk # Bk # Oc \\ nat @ Oc # Bk \\ rn)\n t_wcode_adjust n) \\\n wadjust_stop m\n nat holds_for steps0\n (Suc 0, Bk # Oc # Oc \\ m,\n Bk # Oc # Bk \\ ln @ Bk # Bk # Oc \\ nat @ Oc # Bk \\ rn)\n t_wcode_adjust n"} {"_id": "500604", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\set.\n closed_csubspace set \\\n bounded_clinear (projection set)"} {"_id": "500605", "text": "proof (prove)\nusing this:\n \\[trg f] =\n \\Chn = \\

\\<^sub>1[f.dtrg, f.dtrg],\n Dom =\n \\Leg0 = \\

\\<^sub>1[f.dtrg, f.dtrg],\n Leg1 = \\

\\<^sub>1[f.dtrg, f.dtrg]\\,\n Cod = \\Leg0 = f.dtrg, Leg1 = f.dtrg\\\\\n arrow_of_spans (\\) \\[trg f]\n ide f\n \\[?a] \\\n \\Chn = \\

\\<^sub>0[Chn ?a, Chn ?a], Dom = Dom (?a \\ ?a),\n Cod = Cod ?a\\\n ?\\ \\ ?\\ \\\n if arr ?\\ \\ arr ?\\ \\ src ?\\ = trg ?\\\n then \\Chn = chine_hcomp ?\\ ?\\,\n Dom =\n \\Leg0 =\n Leg0 (Dom ?\\) \\\n \\

\\<^sub>0[Leg0 (Dom ?\\), Leg1 (Dom ?\\)],\n Leg1 =\n Leg1 (Dom ?\\) \\\n \\

\\<^sub>1[Leg0 (Dom ?\\), Leg1 (Dom ?\\)]\\,\n Cod =\n \\Leg0 =\n Leg0 (Cod ?\\) \\\n \\

\\<^sub>0[Leg0 (Cod ?\\), Leg1 (Cod ?\\)],\n Leg1 =\n Leg1 (Cod ?\\) \\\n \\

\\<^sub>1[Leg0\n (Cod ?\\), Leg1\n (Cod ?\\)]\\\\\n else null\n \\arr ?\\; ide ?f; src ?\\ = trg ?f\\\n \\ chine_hcomp ?\\ ?f =\n \\Chn ?\\ \\\n \\

\\<^sub>1[Leg0\n (Dom ?\\), Leg1\n (Dom ?f)] \\Leg0\n (Cod ?\\), Leg1\n (Cod ?f)\\ \\

\\<^sub>0[Leg0\n (Dom ?\\), Leg1 (Dom ?f)]\\\n arrow_of_spans (\\)\n \\Chn = \\

\\<^sub>0[f.dtrg, f.leg1],\n Dom =\n \\Leg0 = f.leg0 \\ \\

\\<^sub>0[f.dtrg, f.leg1],\n Leg1 = \\

\\<^sub>1[f.dtrg, f.leg1]\\,\n Cod = Cod f\\\n arrow_of_spans (\\) (trg f)\n arrow_of_spans (\\) f\n arr ?\\ = arrow_of_spans (\\) ?\\\n \\C.arr ?f; C.cod ?f = ?b\\\n \\ ?b \\ ?f = ?f\n src ?\\ \\\n if arr ?\\ then mkObj (C.cod (Leg0 (Dom ?\\))) else null\n trg ?\\ \\\n if arr ?\\ then mkObj (C.cod (Leg1 (Dom ?\\))) else null\n\ngoal (1 subgoal):\n 1. Chn (\\[trg f] \\ f) =\n \\trg_trg_f.Prj\\<^sub>0\\<^sub>1 \\trg.leg0, f.leg1\\ trg_trg_f.Prj\\<^sub>0\\"} {"_id": "500606", "text": "proof (prove)\ngoal (1 subgoal):\n 1. risk_neutral_prob N = (q = (1 + r - d) / (u - d))"} {"_id": "500607", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\is_rbt lt; is_rbt rt\\\n \\ is_rbt (rbt_union_with f lt rt)"} {"_id": "500608", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_option (map_prod f id) (redT_updT ts ta t) =\n redT_updT (\\t. map_option (map_prod f id) (ts t))\n (convert_new_thread_action f ta) t"} {"_id": "500609", "text": "proof (prove)\ngoal (1 subgoal):\n 1. seqll i (onll \\ P) = onll \\ (seqll i P)"} {"_id": "500610", "text": "proof (prove)\nusing this:\n finite ((\\x. indicat_real \\ x *\\<^sub>R f x) ` space M)\n\ngoal (1 subgoal):\n 1. simple_bochner_integral (restrict_space M \\) f =\n simple_bochner_integral M\n (\\x. indicat_real \\ x *\\<^sub>R f x)"} {"_id": "500611", "text": "proof (prove)\nusing this:\n a \\ carrier R\n b \\ carrier R\n nonsingular a b\n on_curve a b p\\<^sub>1\n on_curve a b p\\<^sub>2\n on_curve a b p\\<^sub>2\n\ngoal (1 subgoal):\n 1. local.add a p\\<^sub>1 (local.add a p\\<^sub>2 p\\<^sub>2) =\n local.add a (local.add a p\\<^sub>1 p\\<^sub>2) p\\<^sub>2"} {"_id": "500612", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lcs x y - y = b - gcs a b"} {"_id": "500613", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. P x) = (\\y z. P (y ||| z))"} {"_id": "500614", "text": "proof (prove)\nusing this:\n dt (Suc t) \\ [] \\ dt (Suc (Suc t)) \\ []\n\ngoal (1 subgoal):\n 1. inf_last_ti dt (t + 2 + k) \\ []"} {"_id": "500615", "text": "proof (prove)\nusing this:\n v = [y, x] @ c @ [x]\n T\\<^sub>p [x, y] (c @ d) (OPT2 (c @ d) [x, y]) =\n length c div 2 + T\\<^sub>p [x, y] d (OPT2 d [x, y]) \\\n length c mod 2 = 0\n\ngoal (1 subgoal):\n 1. 1 + length c div 2 = length v div 2"} {"_id": "500616", "text": "proof (prove)\nusing this:\n \\nofail (step s); rwof m0 cond step s; cond s;\n inres (step s) ?s'\\\n \\ rwof m0 cond step ?s'\n\ngoal (1 subgoal):\n 1. \\nofail (step s); rwof m0 cond step s; cond s\\\n \\ step s \\ SPEC (rwof m0 cond step)"} {"_id": "500617", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pp_pm.deg_shifts (set xs) d (map (Poly_Mapping.map_key PP) fs) =\n map (Poly_Mapping.map_key PP) (deg_shifts_pp xs d fs)"} {"_id": "500618", "text": "proof (prove)\ngoal (2 subgoals):\n 1. (\\n. 0) \\ 0\n 2. \\x.\n 1 \\ x \\\n 0 =\n ((\\rk\\2 * r.\n Fourier_coefficient f k * trigonometric_set k t) /\n real x -\n l) *\n pi -\n (LINT xa|lebesgue_on {0..pi}.\n Fejer_kernel x xa * (f (t + xa) + f (t - xa) - 2 * l))"} {"_id": "500619", "text": "proof (prove)\nusing this:\n supp (\\n. if n = 0 then c else (0::'a)) =\n (if c = (0::'a) then {} else {0})\n\ngoal (1 subgoal):\n 1. bdd_below (supp (\\n. if n = 0 then c else (0::'a))) \\\n (LCM r\\supp (\\n. if n = 0 then c else (0::'a)).\n snd (quotient_of r)) \\\n 0"} {"_id": "500620", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ \\ \\\\ f|` (- domA \\))\n x \\\n ((\\ \\ @ \\ \\\\) f|` (- domA \\))\n x"} {"_id": "500621", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f(x := y), f) \\ oexp"} {"_id": "500622", "text": "proof (prove)\ngoal (1 subgoal):\n 1. invar (deleteMin bq) =\n invar\n (insertList\n (filter (\\t. rank t = 0) (children (getMinTree bq)))\n (meld\n (rev (filter (\\t. 0 < rank t) (children (getMinTree bq))))\n (remove1 (getMinTree bq) bq)))"} {"_id": "500623", "text": "proof (prove)\nusing this:\n mantissa f \\ 0\n m = mantissa f * 2 ^ i\n e = exponent f - int i\n\ngoal (1 subgoal):\n 1. bitlen \\mantissa f\\ + exponent f =\n (if m = 0 then 0 else bitlen \\m\\ + e)"} {"_id": "500624", "text": "proof (prove)\ngoal (1 subgoal):\n 1. atoms\n ((F \\<^bold>\\ G) \\<^bold>\\\n (G \\<^bold>\\ F)) =\n atoms F \\ atoms G"} {"_id": "500625", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x1 t1 t2.\n \\bt_map (f \\ g) t1 = bt_map f (bt_map g t1);\n bt_map (f \\ g) t2 = bt_map f (bt_map g t2)\\\n \\ bt_map (f \\ g) (Br x1 t1 t2) =\n bt_map f (bt_map g (Br x1 t1 t2))"} {"_id": "500626", "text": "proof (prove)\nusing this:\n \\(x, m)\\clkp_set A.\n m \\ real (k x) \\ x \\ X \\ m \\ \\\n A \\ l \\\\<^bsup>g,a,r\\<^esup> l'\n\ngoal (1 subgoal):\n 1. \\(x, m)\\collect_clock_pairs g.\n m \\ real (k x) \\ x \\ X \\ m \\ \\"} {"_id": "500627", "text": "proof (prove)\nusing this:\n \\\\<^sub>F x in at_top. 0 < x powr p'\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in sequentially. 0 < f x"} {"_id": "500628", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\m\\n.\n ( \\ \\ A \\* B, m)\n \\ derivable Ax \\ g3ip*"} {"_id": "500629", "text": "proof (prove)\ngoal (1 subgoal):\n 1. R \\ P \\ (\\_. False) = \\"} {"_id": "500630", "text": "proof (prove)\nusing this:\n {(y, xc'). y \\ cbox (g2 xc) (g1 xc) \\ xc' = xc}\n \\ Dx_pair\n\ngoal (1 subgoal):\n 1. ((\\y.\n partial_vector_derivative (\\a. F a \\ j) i\n (y, xc)) has_integral\n F (g1 xc, xc) \\ j - F (g2 xc, xc) \\ j)\n (cbox (g2 xc) (g1 xc))"} {"_id": "500631", "text": "proof (prove)\nusing this:\n sum_list (replicate n w1) \\ sum_list (replicate n w2)\n w1 \\ w2\n\ngoal (1 subgoal):\n 1. sum_list (replicate (Suc n) w1) \\ sum_list (replicate (Suc n) w2)"} {"_id": "500632", "text": "proof (state)\nthis:\n \\i.\n (case u i of (u, v) \\ \\(u', v'). r u u' OO r' v v')\n (u (Suc i)) (x i) (z i)\n\ngoal (1 subgoal):\n 1. \\x xa.\n \\f.\n (case u f of\n (u, v) \\ \\(u', v'). r u u' OO r' v v')\n (u (Suc f)) (x f) (xa f) \\\n \\ua.\n (\\f. r (fst (u f)) (fst (u (Suc f))) (x f) (ua f)) \\\n (\\f. r' (snd (u f)) (snd (u (Suc f))) (ua f) (xa f))"} {"_id": "500633", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (dom \\') \\ q + card (dom \\)"} {"_id": "500634", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\R module M1; R module M2; R module N; m1 \\ carrier M1;\n m2 \\ carrier M2; r \\ carrier R; bilinear_map f R M1 M2 N;\n aGroup N\\\n \\ r \\\\<^sub>s\\<^bsub>N\\<^esub> f (m1, m2)\n \\ carrier N\n 2. \\R module M1; R module M2; R module N; m1 \\ carrier M1;\n m2 \\ carrier M2; r \\ carrier R; bilinear_map f R M1 M2 N;\n aGroup N\\\n \\ f (r \\\\<^sub>s\\<^bsub>M1\\<^esub> m1, m2) =\n r \\\\<^sub>s\\<^bsub>N\\<^esub> f (m1, m2)"} {"_id": "500635", "text": "proof (prove)\nusing this:\n g \\ carrier G\n U = N #> g\n ?x \\ carrier G \\ inv ?x \\ carrier G\n g \\ h \\ carrier G\n N \\ G\n \\?H \\ ?G; ?x \\ carrier ?G; ?y \\ carrier ?G\\\n \\ ?H #>\\<^bsub>?G\\<^esub> ?x <#>\\<^bsub>?G\\<^esub>\n (?H #>\\<^bsub>?G\\<^esub> ?y) =\n ?H #>\\<^bsub>?G\\<^esub>\n ?x \\\\<^bsub>?G\\<^esub> ?y\n\ngoal (1 subgoal):\n 1. N #> g \\ h <#> (N #> inv g) = N #> g \\ h \\ inv g"} {"_id": "500636", "text": "proof (prove)\ngoal (1 subgoal):\n 1. unit_factor f = f"} {"_id": "500637", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\ set_option (local.MaxR.MaxR_opt X)) =\n (set_option (local.MaxR.MaxR_opt X) = {x})"} {"_id": "500638", "text": "proof (prove)\nusing this:\n \\v.\n fetch_instruction (delayed_pool_write s1) = Inr v \\\n fetch_instruction (delayed_pool_write s2) = Inr v \\\n (\\e. decode_instruction v = Inl e)\n \\e. decode_instruction ?v1.0 = Inl e \\\n \\v2. decode_instruction ?v1.0 = Inr v2\n\ngoal (1 subgoal):\n 1. \\v.\n fetch_instruction (delayed_pool_write s1) = Inr v \\\n fetch_instruction (delayed_pool_write s2) = Inr v \\\n (\\v1. decode_instruction v = Inr v1)"} {"_id": "500639", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^bsup>Suc l\\<^esup>) v = (\\'\\<^bsup>Suc l'\\<^esup>) v"} {"_id": "500640", "text": "proof (prove)\ngoal (1 subgoal):\n 1. span_in_category (\\)\n \\Leg0 = f.leg0 \\ \\

\\<^sub>0[f.dtrg, f.leg1],\n Leg1 = \\

\\<^sub>1[f.dtrg, f.leg1]\\"} {"_id": "500641", "text": "proof (prove)\ngoal (1 subgoal):\n 1. independent S \\ finite S \\ card S \\ DIM('a)"} {"_id": "500642", "text": "proof (prove)\nusing this:\n distincts (xs # xss)\n ?xs \\ set (xs # xss) \\\n list_succ ?xs = perm_restrict f (set ?xs)\n f permutes \\ (sset (xs # xss))\n\ngoal (1 subgoal):\n 1. f = lists_succ (xs # xss)"} {"_id": "500643", "text": "proof (prove)\nusing this:\n 0 < 2 \\\n ((\\x.\n ((1 - x) ^ (2 - 1) / fact (2 - 1)) *\\<^sub>R\n nth_derivative 2 f (X + x *\\<^sub>R H) H) has_integral\n f (X + H) - (\\i<2. (1 / fact i) *\\<^sub>R nth_derivative i f X H))\n {0..1}\n 0 < 2 \\\n f (X + H) =\n (\\i<2. (1 / fact i) *\\<^sub>R nth_derivative i f X H) +\n integral {0..1}\n (\\x.\n ((1 - x) ^ (2 - 1) / fact (2 - 1)) *\\<^sub>R\n nth_derivative 2 f (X + x *\\<^sub>R H) H)\n 0 < 2 \\\n (\\x.\n ((1 - x) ^ (2 - 1) / fact (2 - 1)) *\\<^sub>R\n nth_derivative 2 f (X + x *\\<^sub>R H) H) integrable_on\n {0..1}\n\ngoal (1 subgoal):\n 1. (i has_integral f (X + H) - (f X + nth_derivative 1 f X H)) {0..1} &&&\n f (X + H) = f X + nth_derivative 1 f X H + integral {0..1} i &&&\n i integrable_on {0..1}"} {"_id": "500644", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\gallery Cs; \\ min_gallery Cs;\n {} \\ set (wall_crossings Cs)\\\n \\ \\f g As A B Bs E F Fs.\n (f, g) \\ foldpairs \\\n A \\ f \\ \\ \\\n B \\ g \\ \\ \\\n E \\ g \\ \\ \\\n F \\ f \\ \\ \\\n (Cs = As @ [A, B, F] @ Fs \\\n Cs = As @ [A, B] @ Bs @ [E, F] @ Fs)"} {"_id": "500645", "text": "proof (prove)\nusing this:\n k-smooth_on S f\n 0 < k\n\ngoal (1 subgoal):\n 1. f differentiable_on S"} {"_id": "500646", "text": "proof (prove)\nusing this:\n (\\xs.\n case xs of TNil b \\ bot b\n | TCons x xs' \\ f x xs' xs) `\n Y =\n bot ` terminal ` Y\n local.tSup Y = TNil (flat_lub b (terminal ` Y))\n Complete_Partial_Order.chain (flat_ord b) (terminal ` Y)\n Y \\ {}\n\ngoal (1 subgoal):\n 1. (case local.tSup Y of TNil b \\ bot b\n | TCons x xs' \\ f x xs' (local.tSup Y)) =\n lub ((\\xs.\n case xs of TNil b \\ bot b\n | TCons x xs' \\ f x xs' xs) `\n Y)"} {"_id": "500647", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\H \\ SeqSubstFormP v tm x z s k;\n insert (SeqSubstFormP v tm y z s k) H \\ A\\\n \\ insert (x EQ y) H \\ A"} {"_id": "500648", "text": "proof (prove)\ngoal (1 subgoal):\n 1. R_scalar_mult.R_lin_independent UNIV (\\\\)\n (concat hfvss)"} {"_id": "500649", "text": "proof (state)\nthis:\n det (interchange_rows (bezout_matrix A a b j bezout) a b) = - (1::'a)\n\ngoal (2 subgoals):\n 1. p = (0::'a) \\\n det (bezout_matrix A a b j bezout) = (1::'a)\n 2. p \\ (0::'a) \\\n det (bezout_matrix A a b j bezout) = (1::'a)"} {"_id": "500650", "text": "proof (prove)\nusing this:\n Keys (fst ` set (full_gb (gs @ bs))) = {}\n\ngoal (1 subgoal):\n 1. dgrad_p_set_le d (fst ` set (fst (full_gb (gs @ bs), D)))\n (args_to_set (gs, bs, ps))"} {"_id": "500651", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A \\ set As \\ B \\ set As"} {"_id": "500652", "text": "proof (prove)\nusing this:\n smem_ind s i = Some j\n run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')\n b_e = Grow_memory\n ves = a # list\n a = ConstInt32 x1\n \\smem_ind ?s ?i = Some ?j1; s.mem ?s ! ?j1 = ?m1;\n mem_size ?m1 = ?n1; mem_grow ?m1 (nat_of_int ?c1) = ?mem'1;\n const_list ?cs\\\n \\ \\?s;?vs;?cs @\n [$C ConstInt32 ?c1,\n $Grow_memory]\\ \\_ ?i \\?s\n \\s.mem := (s.mem ?s)\n [?j1 :=\n ?mem'1]\\;?vs;?cs @\n [$C ConstInt32 (int_of_nat ?n1)]\\\n \\smem_ind ?s ?i = Some ?j1; s.mem ?s ! ?j1 = ?m1;\n mem_size ?m1 = ?n1; const_list ?cs\\\n \\ \\?s;?vs;?cs @\n [$C ConstInt32 ?c1,\n $Grow_memory]\\ \\_ ?i \\?s;?vs;?cs @\n [$C ConstInt32 int32_minus_one]\\\n const_list (vs_to_es list)\n\ngoal (1 subgoal):\n 1. \\s;vs;vs_to_es ves @\n [$b_e]\\ \\_ i \\s';vs';es'\\"} {"_id": "500653", "text": "proof (state)\nthis:\n P y\n\ngoal (2 subgoals):\n 1. ?P \\\n \\us vs.\n y # ys = us @ x # vs \\\n (\\u\\set us. \\ P u) \\\n P x \\ xs = filter P vs\n 2. \\ ?P \\\n \\us vs.\n y # ys = us @ x # vs \\\n (\\u\\set us. \\ P u) \\\n P x \\ xs = filter P vs"} {"_id": "500654", "text": "proof (prove)\nusing this:\n \\ nonrec \\\n\ngoal (1 subgoal):\n 1. ccHeap \\ e\\a G|` (- domA \\) \\\n ccExp_syn a (Terms.Let \\ e)"} {"_id": "500655", "text": "proof (prove)\nusing this:\n S' \\ \\<^bold>\\\n set S' \\ \\ (range (extend S f))\n\ngoal (1 subgoal):\n 1. (\\m.\n set S' \\ \\ (extend S f ` {..m}) \\\n thesis) \\\n thesis"} {"_id": "500656", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\G.\n \\F ok G; F \\ G \\ X;\n extend h F ok extend h G \\\n extend h F \\ extend h G\n \\ extend h ` X \\\n extend h F \\ extend h G \\ extend h ` Y\\\n \\ F \\ G \\ Y"} {"_id": "500657", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (A <: B \\ V B \\ V A) \\\n (f1 <:: f2 \\ Vf f2 \\ Vf f1)"} {"_id": "500658", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\valuation K v; s \\ carrier K; t \\ carrier K;\n 0 \\ v s; v t < 0\\\n \\ 0 < v (1\\<^sub>r \\\n -\\<^sub>a t \\\\<^sub>r\n (s \\ t)\\<^bsup>\\ K\\<^esup>)"} {"_id": "500659", "text": "proof (prove)\ngoal (1 subgoal):\n 1. j permutes UNIV \\\n degree (\\i\\UNIV. to_fun A i (j i)) \\ max_perm_degree A"} {"_id": "500660", "text": "proof (prove)\nusing this:\n enat n \\ llength E\n E' n \\ \\ \\\n is_write_seen P (E' n) (ws' n) \\\n thread_start_actions_ok (E' n) \\\n (enat n \\ llength E \\\n enat n \\ llength (E' n)) \\\n ltake (enat (n - Suc 0)) E = ltake (enat (n - Suc 0)) (E' n) \\\n (0 < n \\\n action_tid (E' n) (n - Suc 0) = action_tid E (n - Suc 0) \\\n (if n - Suc 0 \\ read_actions E then sim_action else (=))\n (action_obs (E' n) (n - Suc 0)) (action_obs E (n - Suc 0)) \\\n (\\i read_actions E \\ ws' n i = ws i)) \\\n (\\r\\read_actions (E' n).\n n - Suc 0 \\ r \\\n P,E' n \\ ws' n r \\hb r)\n\ngoal (1 subgoal):\n 1. llength (ltake (enat n) (E' n)) = llength (ltake (enat n) E)"} {"_id": "500661", "text": "proof (prove)\nusing this:\n ((\\t. V (flow0 x t)) has_derivative\n (\\t. t *\\<^sub>R V' (flow0 x s) (f (flow0 x s))))\n (at s)\n\ngoal (1 subgoal):\n 1. ((\\t. V (flow0 x t)) has_derivative\n (*) (V' (flow0 x s) (f (flow0 x s))))\n (at s)"} {"_id": "500662", "text": "proof (prove)\nusing this:\n n cd\\<^bsup>\\\\<^esup>\\ m\n\ngoal (1 subgoal):\n 1. \\ns.\n cs\\<^bsup>\\\\<^esup> n =\n cs\\<^bsup>\\\\<^esup> m @ ns @ [\\ n]"} {"_id": "500663", "text": "proof (prove)\nusing this:\n ?bl \\# filter_mset ((\\) x) \\ \\\n size {#p \\# mset_set (\\ - {x}). p \\ ?bl#} = nat (\\ - 1)\n\ngoal (1 subgoal):\n 1. size\n (\\bl\\#filter_mset ((\\) x)\n \\. {#p \\# mset_set (\\ - {x}).\n p \\ bl#} \\#\n {#bl#}) =\n size (filter_mset ((\\) x) \\) * nat (\\ - 1)"} {"_id": "500664", "text": "proof (state)\nthis:\n (V, V') \\ S\n\ngoal (2 subgoals):\n 1. \\a b aa ba.\n \\(a, b) \\ S; (aa, ba) \\ S\\\n \\ length aa = length ba\n 2. \\a b aa ba i m1 m1' m2 W.\n \\(a, b) \\ S; (aa, ba) \\ S; i < length aa;\n \\aa ! i,m1\\ \\ \\W,m2\\;\n m1 =\\<^bsub>d\\<^esub> m1'\\\n \\ \\W' m2'.\n \\ba ! i,m1'\\ \\\n \\W',m2'\\ \\\n (W, W') \\ S \\ m2 =\\<^bsub>d\\<^esub> m2'"} {"_id": "500665", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A - Coset t = rbt_filter (\\k. k \\ A) t"} {"_id": "500666", "text": "proof (prove)\nusing this:\n a = check\n \\' =\n update\\<^sub>s\\<^sub>t \\\n (\\a: t \\ t'\\\\<^sub>s\\<^sub>t # S)\n \\' = \\ @ [Step \\a: t \\ t'\\\\<^sub>s\\<^sub>t]\n\ngoal (1 subgoal):\n 1. assignment_rhs\\<^sub>e\\<^sub>s\\<^sub>t \\' =\n assignment_rhs\\<^sub>e\\<^sub>s\\<^sub>t \\"} {"_id": "500667", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\c21'.\n \\c1 = c1' @ qs' # c21';\n c2' = c21' @ qs # c2\\\n \\ thesis;\n \\c1' = c1; qs' = qs; c2' = c2\\\n \\ thesis;\n \\c21.\n \\c1' = c1 @ qs # c21; c2 = c21 @ qs' # c2'\\\n \\ thesis;\n c1 = c1' @ us \\ us @ qs # c2 = qs' # c2' \\\n c1 @ us = c1' \\ qs # c2 = us @ qs' # c2'\\\n \\ thesis"} {"_id": "500668", "text": "proof (prove)\nusing this:\n length xl = length al\n distinct xl\n ?i1 < length xl \\ intT (tpOfV (xl ! ?i1)) (al ! ?i1)\n i < length xl\n \\length ?xl = length ?al; distinct ?xl;\n \\i.\n i < length ?xl \\\n intT (tpOfV (?xl ! i)) (?al ! i)\\\n \\ wtE (pickE ?xl ?al) \\\n (\\iy1 - y2 \\ carrier_vec n; y1 \\ carrier_vec n;\n y2 \\ carrier_vec n; x \\ carrier_vec n;\n - y1 \\ carrier_vec n; 0\\<^sub>v n \\ carrier_vec n\\\n \\ y1 \\ y1 - y1 \\ y2 -\n (y2 \\ y1 - y2 \\ y2) =\n (0::'a)"} {"_id": "500670", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>int_group\\<^esub>\n \\ \\{1}\\\\<^bsub>int_group\\<^esub>"} {"_id": "500671", "text": "proof (prove)\nusing this:\n C.ide b\n C.ide a\n ?A \\ S.Univ \\\n S.mkArr ?A ?A (\\x. x) = S.mkIde ?A\n \\C.ide ?b; C.ide ?a\\\n \\ local.set (?b, ?a) \\ S.Univ\n\ngoal (1 subgoal):\n 1. S.mkArr (local.set (b, a)) (local.set (b, a)) (\\h. h) =\n S.mkIde (local.set (b, a))"} {"_id": "500672", "text": "proof (prove)\nusing this:\n {x. A *v x = 0} = {0}\n A *v x = 0\n\ngoal (1 subgoal):\n 1. x = 0"} {"_id": "500673", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hom (\\i. var (f i)) \\ = \\"} {"_id": "500674", "text": "proof (prove)\nusing this:\n ?i \\ agents' \\ finite_total_preorder_on alts' (R' ?i)\n ?i \\ agents' \\ R' ?i = (\\_ _. False)\n pref_profile_wf agents alts R\n agents \\ agents'\n alts \\ alts'\n finite alts'\n R' \\ lift_pref_profile agents alts agents' alts' R\n\ngoal (1 subgoal):\n 1. (agents' \\ {} \\ alts' \\ {}) \\\n (\\i.\n i \\ agents' \\\n finite_total_preorder_on alts' (R' i)) \\\n (\\i x y. i \\ agents' \\ \\ R' i x y)"} {"_id": "500675", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\na.\n \\m.\n \\s1 s2.\n X na s1 s2 \\\n wf_state s1 \\\n wf_state s2 \\\n final s1 = final s2 \\\n (\\x\\set (\\ n).\n \\x' y'.\n Y m x' y' \\\n post x' = post (delta x s1) \\\n post y' = post (delta x s2)) \\\n \\s1 s2.\n Sup (range X) s1 s2 \\\n wf_state s1 \\\n wf_state s2 \\\n final s1 = final s2 \\\n (\\x\\set (\\ n).\n \\x' y'.\n Sup (range Y) x' y' \\\n post x' = post (delta x s1) \\\n post y' = post (delta x s2))"} {"_id": "500676", "text": "proof (prove)\nusing this:\n D.ide ?a \\ \\\\<^sub>D[?a] = ?a\n\ngoal (1 subgoal):\n 1. \\t.\n \\Can t \\ D.ide \\t\\;\n Can \\<^bold>\\\\<^bold>[t\\<^bold>]\\\n \\ D.ide\n \\\\<^bold>\\\\<^bold>[t\\<^bold>]\\"} {"_id": "500677", "text": "proof (prove)\nusing this:\n ?y2 \\ M' \\ ?y2 \\' \\ \\ = ?y2\n\ngoal (1 subgoal):\n 1. \\ \\ = \\'"} {"_id": "500678", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {} \\\n AbstFormP \\Var i\\ (ORD_OF n)\n (quot_dbfm db) (quot_dbfm (abst_dbfm i n db))"} {"_id": "500679", "text": "proof (state)\nthis:\n int (y mod order \\) =\n (int d' - int d) * Number_Theory_Aux.inverse (m - m') (order \\) mod\n int (order \\)\n\ngoal (1 subgoal):\n 1. y =\n nat ((int d' - int d) *\n Number_Theory_Aux.inverse (m - m') (order \\) mod\n int (order \\))"} {"_id": "500680", "text": "proof (prove)\ngoal (1 subgoal):\n 1. comm_group (monoid_vec TYPE('a) n)"} {"_id": "500681", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (mset (map ((!) ss) (\\ (f, length ss))),\n mset (map ((!) ss) (\\ (f, length ss))))\n \\ ns_mul_ext WPO_NS WPO_S"} {"_id": "500682", "text": "proof (prove)\nusing this:\n sanity_wf_ruleset \\\n chain_name \\ set (map fst \\)\n default_action = action.Accept \\ default_action = action.Drop\n matcher_agree_on_exact_matches \\ common_matcher\n unfold_ruleset_CHAIN_safe chain_name default_action (map_of \\) =\n Some rs\n\ngoal (3 subgoals):\n 1. \\m.\n Semantics.matches \\ (optimize_primitive_univ m) p =\n Semantics.matches \\ m p\n 2. unfold_optimize_ruleset_CHAIN optimize_primitive_univ chain_name\n default_action (map_of \\) =\n Some rs\n 3. simple_ruleset rs"} {"_id": "500683", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mon_loc fg (e # w) =\n (case e of LOC a \\ mon_w fg a\n | ENV a \\ {}) \\\n mon_loc fg w"} {"_id": "500684", "text": "proof (prove)\nusing this:\n continuous_on S f \\\n (\\Sa. finite Sa \\ f C1_differentiable_on S - Sa)\n f ` S \\ {0<..}\n\ngoal (1 subgoal):\n 1. continuous_on S (\\x. sqrt (f x)) \\\n (\\Sa.\n finite Sa \\\n (\\x. sqrt (f x)) C1_differentiable_on S - Sa)"} {"_id": "500685", "text": "Step error: Outer syntax error (line 1)\nAt command \"\" (line 1)"} {"_id": "500686", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length [0.. fls_subdegree g\n fls_base_factor g \\ 0\n\ngoal (1 subgoal):\n 1. fls_regpart (f / g) = fls_regpart f / fls_regpart g"} {"_id": "500688", "text": "proof (prove)\nusing this:\n proj2_rep ` (range (($) a) - {a $ i}) =\n proj2_rep ` range (($) a) - {proj2_rep (a $ i)}\n proj2_rep (a $ i) \\ proj2_rep ` range (($) a)\n card (proj2_rep ` range (($) a)) = 3\n\ngoal (1 subgoal):\n 1. card (proj2_rep ` (range (($) a) - {a $ i})) = 2"} {"_id": "500689", "text": "proof (prove)\ngoal (1 subgoal):\n 1. i1 = i2"} {"_id": "500690", "text": "proof (prove)\nusing this:\n \\ \\\n \\\\<^sub>P \\ \\\\x\\Q \\ M'\\N\\ \\ xQ'\n x \\ \\ \\ \\\\<^sub>P\n x \\ M'\n x \\ N\n x \\ xQ'\n\ngoal (1 subgoal):\n 1. \\P'a.\n \\ \\ \\\\x\\(P \\\n Q) \\ \\ \\ P'a \\\n (\\, P'a, \\\\*xvec\\P' \\ xQ') \\ Rel"} {"_id": "500691", "text": "proof (prove)\nusing this:\n NEST a \\\n NEST a \\\n (\\n m. n \\ a \\ m \\ a \\ (n, m) \\ b)\n\ngoal (1 subgoal):\n 1. NEST a"} {"_id": "500692", "text": "proof (prove)\nusing this:\n ?i \\ I \\ {x \\ \\. P ?i x} \\ M\n countable I\n {x \\ \\. \\!i. i \\ I \\ P i x} =\n {x \\ \\.\n \\i\\I.\n P i x \\ (\\j\\I. P j x \\ i = j)}\n\ngoal (1 subgoal):\n 1. {x \\ \\. \\!i. i \\ I \\ P i x} \\ M"} {"_id": "500693", "text": "proof (prove)\ngoal (1 subgoal):\n 1. simple_match_port (sports r) (p_sport p)"} {"_id": "500694", "text": "proof (prove)\nusing this:\n Nom i at j in' H\n ?k \\ A \\\n Nom ?k at ?j in' H \\\n Nom ?i at ?j in' H \\ Nom ?k at ?i in' H\n k \\ A\n Nom j at i in' H\n Nom k at j in' H\n\ngoal (1 subgoal):\n 1. Nom k at i in' H"} {"_id": "500695", "text": "proof (prove)\nusing this:\n :M \\ \\\\\n SNa M\n x:N \\ \\(B)\\\n SNa N\n\ngoal (1 subgoal):\n 1. SNa (Cut .M x.N)"} {"_id": "500696", "text": "proof (prove)\nusing this:\n ?x \\ space M \\\n (\\i. u i ?x) \\ u' ?x\n\ngoal (1 subgoal):\n 1. \\x.\n x \\ space M \\\n liminf (\\n. ereal (u n x)) = ereal (u' x)"} {"_id": "500697", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite (nodes g) \\\n num_of_odd_nodes (g\\edges := {}\\) = 0"} {"_id": "500698", "text": "proof (prove)\nusing this:\n ?ss \\ lists S \\\n flipped_reflections (sum_list ?ss) =\n {t \\ \\w\\W. lconjby w ` S.\n \\ gcd_nat.greater_eq (count_list (lconjseq ?ss) t) 2}\n odd ?m \\ ?m \\ (0::?'b)\n ?x \\ set (lconjseq ss) \\\n count_list (lconjseq ss) ?x = 0\n\ngoal (1 subgoal):\n 1. ss \\ lists S \\\n order.greater_eq (set (lconjseq ss)) (flipped_reflections (sum_list ss))"} {"_id": "500699", "text": "proof (prove)\ngoal (1 subgoal):\n 1. v \\ reachable_nodes g r"} {"_id": "500700", "text": "proof (prove)\nusing this:\n is_cnf \\<^bold>\\l # ls\n\ngoal (1 subgoal):\n 1. is_cnf \\<^bold>\\ls"} {"_id": "500701", "text": "proof (prove)\nusing this:\n R \\ B \\ B\n\ngoal (1 subgoal):\n 1. Domain R \\ Range R \\ B"} {"_id": "500702", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ cball \\ r \\\n (\\i. a i * (x - \\) ^ i) = f x"} {"_id": "500703", "text": "proof (prove)\ngoal (1 subgoal):\n 1. game \\ = game_multi (\\_. \\)"} {"_id": "500704", "text": "proof (prove)\ngoal (1 subgoal):\n 1. indexed_weak_nominal_ts TYPE('idx) (S_action.Act \\)\n (\\\\<^sub>S) (\\\\<^sub>S)"} {"_id": "500705", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k.\n \\k \\ \\; \\k\\ \\ 2 ^ (2 * n)\\\n \\ Measurable.pred (lebesgue_on S)\n (\\x.\n k / 2 ^ n \\ f x \\\n f x < (k + 1) / 2 ^ n)"} {"_id": "500706", "text": "proof (state)\ngoal (1 subgoal):\n 1. AE x in P. \\\\<^sub>F n in sequentially.\n 2 < (\\i\\ zeroring R; M fgover R; submodule R M N; ideal R A;\n A \\ J_rad R;\n carrier (M /\\<^sub>m N) =\n {\\\\<^bsub>M /\\<^sub>m N\\<^esub>}\\\n \\ carrier M = N\n 2. \\\\ zeroring R; M fgover R; submodule R M N; ideal R A;\n A \\ J_rad R;\n carrier M = A \\\\<^bsub>R\\<^esub> M \\ N\\\n \\ A \\\\<^bsub>R\\<^esub> M /\\<^sub>m N =\n carrier (M /\\<^sub>m N)"} {"_id": "500708", "text": "proof (prove)\ngoal (1 subgoal):\n 1. kmp_SPEC s t =\n RETURN (if sublist s t then Some (LEAST i. sublist_at s t i) else None)"} {"_id": "500709", "text": "proof (prove)\nusing this:\n \\?s \\ valid_states;\n \\op\\set ops.\n \\ast_op\\set ast\\. op = abs_ast_operator ast_op;\n sat_preconds_as ?s ops\\\n \\ path_to ?s\n (map (fst \\ the \\ lookup_action) ops)\n (execute_serial_plan_sas_plus ?s ops)\n s \\ valid_states\n \\op\\set (a # ops).\n \\ast_op\\set ast\\. op = abs_ast_operator ast_op\n sat_preconds_as s (a # ops)\n\ngoal (1 subgoal):\n 1. path_to s (map (fst \\ the \\ lookup_action) (a # ops))\n (execute_serial_plan_sas_plus s (a # ops))"} {"_id": "500710", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\n.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n gen_wf_uses gen_cfg_wf n =\n gen_wf_var gen_cfg_wf ` gen_wf.uses' gen_cfg_wf n\n 2. \\n ns m.\n graph_path_base.path2 (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. True) (\\_. gen_wf_inEdges' gen_cfg_wf) () n ns\n m \\\n n \\ set (tl ns) \\\n (\\a aa b.\n (a, aa, b)\n \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) ()\n n \\\n (a, aa, b)\n \\ CFG_SSA_base.allUses\n (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf.uses' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) ()\n m \\\n (\\x.\n x \\ set (tl ns) \\\n (\\ab ac ba.\n (ab, ac, ba)\n \\ CFG_SSA_base.allDefs\n (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) ()\n x \\\n gen_wf_var gen_cfg_wf (ab, ac, ba) \\\n gen_wf_var gen_cfg_wf (a, aa, b))))\n 3. \\n a aa b vs.\n gen_wf.phis' gen_cfg_wf (n, a, aa, b) = Some vs \\\n (\\ab ac ba.\n (ab, ac, ba) \\ set vs \\\n gen_wf_var gen_cfg_wf (ab, ac, ba) =\n gen_wf_var gen_cfg_wf (a, aa, b))\n 4. \\n.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n (\\a aa b.\n (a, aa, b)\n \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) ()\n n \\\n (\\ab ac ba.\n (ab, ac, ba)\n \\ CFG_SSA_base.allDefs\n (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) ()\n n \\\n (a = ab \\\n aa = ac \\ b \\ ba) \\\n gen_wf_var gen_cfg_wf (ab, ac, ba) \\\n gen_wf_var gen_cfg_wf (a, aa, b)))"} {"_id": "500711", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.enum (map abs [0.. \\A\\list_rel"} {"_id": "500714", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y xa ya.\n rel_pred A xa ya \\\n rel_pred (rel_option A) (Some ` xa) (Some ` ya)"} {"_id": "500715", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (MP_Rel ===> (=)) prime_elem_m prime_elem"} {"_id": "500716", "text": "proof (prove)\ngoal (1 subgoal):\n 1. simply_connected (rel_frontier S)"} {"_id": "500717", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mpoly_of_poly i 1 = 1"} {"_id": "500718", "text": "proof (prove)\nusing this:\n exec_meth_d (compP2 P) ([Push v, Store V] @ compE2 e)\n (shift (length [Push v, Store V]) (compxE2 e 0 0)) t h\n (stk, loc, length [Push v, Store V] + pc, xcp) ta h'\n (stk', loc', pc', xcp')\n\ngoal (1 subgoal):\n 1. length [Push v, Store V] \\ pc'"} {"_id": "500719", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sgn z = cis c"} {"_id": "500720", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monotone option.le_fun option_ord B \\\n monotone option.le_fun option_ord\n (\\f. Option.bind (B f) (Some \\ g))"} {"_id": "500721", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite {EqOnUOut p p' u ou |u ou. (u, ou) \\ GL \\ UNIV}"} {"_id": "500722", "text": "proof (prove)\nusing this:\n \\vs. es = map Val vs\n\ngoal (1 subgoal):\n 1. (P,sh \\\\<^sub>b (C\\\\<^sub>sM(es),b) \\) =\n (P,sh \\\\<^sub>b (es,b) \\)"} {"_id": "500723", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y. (f x \\ y) = (x \\ g y) \\\n (\\y. \\x. y = f x) = ((\\x. f (g x)) = id)"} {"_id": "500724", "text": "proof (state)\nthis:\n \\ k \\ \\ (Suc k) 0\n\ngoal (1 subgoal):\n 1. \\ k \\ \\ (Suc k) u"} {"_id": "500725", "text": "proof (prove)\ngoal (6 subgoals):\n 1. \\L N ja.\n \\vals_nonequiv K (Suc n) vv; Ring K; aGroup K;\n valuation K (vv j);\n \\N>L. \\l\\Suc n. an m < vv j (x l) + an N; L < N;\n j \\ Suc n; ja = j; vv j (x j) \\ - \\;\n 1\\<^sub>r \\ carrier K;\n -\\<^sub>a 1\\<^sub>r \\ carrier K\\\n \\ 1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n j^\\<^bsup>K N\\<^esup> \\\n \\\n 2. \\L N ja.\n \\vals_nonequiv K (Suc n) vv; Ring K; aGroup K;\n valuation K (vv j);\n \\N>L. \\l\\Suc n. an m < vv j (x l) + an N; L < N;\n j \\ Suc n; ja = j; vv j (x j) \\ - \\;\n 1\\<^sub>r \\ carrier K;\n -\\<^sub>a 1\\<^sub>r \\ carrier K\\\n \\ an m\n \\ vv j (x j) +\n int N *\\<^sub>a\n vv j\n (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc\n n\\<^esub>)\n j^\\<^bsup>K N\\<^esup>)\n 3. \\L N ja.\n \\vals_nonequiv K (Suc n) vv;\n \\l\\Suc n. x l \\ carrier K; j \\ Suc n; Ring K;\n aGroup K; valuation K (vv j);\n \\N ja.\n ja \\ Suc n \\\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n ja^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n \\N.\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n j^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a 1\\<^sub>r\n \\ carrier K;\n \\ja\\Suc n. vv j (x ja) \\ - \\;\n \\N>L. \\l\\Suc n. an m < vv j (x l) + an N; L < N;\n ja \\ Suc n; ja \\ j\\\n \\ (ja \\ j \\\n an m\n \\ vv j (x ja) +\n vv j\n (1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n ja^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup>)) \\\n (ja = j \\\n an m\n \\ vv j (x j) +\n vv j\n (1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n j^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a 1\\<^sub>r))\n 4. \\vals_nonequiv K (Suc n) vv;\n \\l\\Suc n. x l \\ carrier K; j \\ Suc n; Ring K;\n aGroup K; valuation K (vv j);\n \\N ja.\n ja \\ Suc n \\\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n ja^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n \\N.\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n j^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a 1\\<^sub>r\n \\ carrier K;\n (\\ja\\Suc n.\n (vv j \\ x) ja \\ - \\) \\\n (\\L.\n \\N>L.\n \\l\\Suc n. an m < (vv j \\ x) l + an N)\\\n \\ \\ja\\Suc n.\n (vv j \\ x) ja \\ - \\\n 5. \\vals_nonequiv K (Suc n) vv;\n \\l\\Suc n. x l \\ carrier K; j \\ Suc n; Ring K;\n aGroup K; valuation K (vv j);\n \\N ja.\n ja \\ Suc n \\\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>)\n ja^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup>\n \\ carrier K\\\n \\ \\N.\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc\n n\\<^esub>)\n j^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a 1\\<^sub>r\n \\ carrier K\n 6. \\vals_nonequiv K (Suc n) vv;\n \\l\\Suc n. x l \\ carrier K; j \\ Suc n; Ring K;\n aGroup K; valuation K (vv j)\\\n \\ \\N ja.\n ja \\ Suc n \\\n 1\\<^sub>r \\\n -\\<^sub>a (1\\<^sub>r \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc\n n\\<^esub>)\n ja^\\<^bsup>K N\\<^esup>)^\\<^bsup>K N\\<^esup>\n \\ carrier K"} {"_id": "500726", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fun_eq_on f g (\\ W)"} {"_id": "500727", "text": "proof (chain)\npicking this:\n P (f i) (g i)\n P (g j) (f j)\n \\kk S\n S \\ H <#> S\n\ngoal (1 subgoal):\n 1. g \\ H <#> S"} {"_id": "500730", "text": "proof (prove)\nusing this:\n Norm s0\\\\(G, L)\n \\prg = G, cls = accC,\n lcl = L\\\\In1r (l\\ c)\\T\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom (locals\n (snd (Norm\n s0))) \\In1r (l\\ c)\\ A\n\ngoal (1 subgoal):\n 1. (\\C.\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom (locals\n (snd (Norm\n s0))) \\In1r c\\ C \\\n thesis) \\\n thesis"} {"_id": "500731", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f \\ iscale) ` {- 1..1} \\ cbox (- 1) 1 &&&\n (g \\ iscale) ` {- 1..1} \\ cbox (- 1) 1"} {"_id": "500732", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C.seq\n \\\\_\\\\_\\.Prj\\<^sub>0\\<^sub>1 \\\\\\.leg0, \\.leg1\\ \\_\\\\_\\.Prj\\<^sub>0\\\n (chine_hcomp\n \\Chn = \\\\\\.chine_assoc,\n Dom = Dom ((\\ \\ \\) \\ \\),\n Cod = Cod (\\ \\ \\ \\ \\)\\\n \\)"} {"_id": "500733", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\F Fs.\n \\r \\ upRules; R1 \\ upRules;\n R2 \\ modRules2; R3 \\ modRules2;\n r \\ p_e R2 M1 M2; Ax \\ upRules = {};\n \\M N. Ax \\ p_e modRules2 M N = {};\n Ax \\ modRules2 = {}; upRules \\ modRules2 = {};\n \\M N. upRules \\ p_e modRules2 M N = {};\n snd r = ( \\ \\* \\ Compound F Fs \\) \\\n snd r = ( \\ Compound F Fs \\ \\* \\);\n \\(fst r, snd r) \\ p_e R2 M1 M2;\n R2 \\ modRules2\\\n \\ \\F Fs \\ \\ ps ra.\n (fst r, snd r) =\n extendRule\n ( M1 \\\n \\ \\* M2 \\ \\)\n ra \\\n ra \\ R2 \\\n (ra =\n (ps,\n \\ \\* \\ Modal F\n Fs \\) \\\n ra =\n (ps,\n \\ Modal F\n Fs \\ \\* \\))\\\n \\ r \\ R1"} {"_id": "500734", "text": "proof (prove)\nusing this:\n {k \\ \\. content k \\ 0} \\ \\\n\ngoal (1 subgoal):\n 1. (\\K c d.\n \\K \\ \\; content K = 0; K = cbox c d\\\n \\ thesis) \\\n thesis"} {"_id": "500735", "text": "proof (prove)\ngoal (1 subgoal):\n 1. igAbsRenSTR MOD \\ igAbsCongUSTR MOD"} {"_id": "500736", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\ \\F \\\n \\[F \\ \\F]_v"} {"_id": "500737", "text": "proof (prove)\ngoal (1 subgoal):\n 1. manifold_eucl.tangent_space.dim \\ a\n (range (directional_derivative \\ a)) =\n vs1.dim UNIV"} {"_id": "500738", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gc_invar Ns (enat (Suc i))"} {"_id": "500739", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\finite UNIV; root \\ nil;\n (root, x) \\ next\\<^sup>* \\ x \\ nil;\n x \\ path {root} {root}\\\n \\ x = root"} {"_id": "500740", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cBall C (\\x. f x = g x) \\\n rel_cset (eq_onp (\\x. cin x C)) C C"} {"_id": "500741", "text": "proof (state)\ngoal (1 subgoal):\n 1. G \\ S \\ sccs"} {"_id": "500742", "text": "proof (prove)\ngoal (1 subgoal):\n 1. partial_equivalence A r = partial_equivalence A r'"} {"_id": "500743", "text": "proof (prove)\nusing this:\n \\?x2 \\ A; ?y2 \\ A\\\n \\ (card {i \\ Is. F i = ?y2}\n \\ card {i \\ Is. F i = ?x2}) =\n (card\n {i \\ Is.\n ?y2 \\<^bsub>(single_vote_to_RPR A\n (F i))\\<^esub>\\ ?x2}\n \\ card\n {i \\ Is.\n ?x2 \\<^bsub>(single_vote_to_RPR A\n (F i))\\<^esub>\\ ?y2})\n\ngoal (1 subgoal):\n 1. plurality_rule A Is F =\n MMD_plurality_rule A Is (single_vote_to_RPR A \\ F)"} {"_id": "500744", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\ sinvar G (nP(i := \\))"} {"_id": "500745", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A B OS X D"} {"_id": "500746", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sort_poly_spec p \\ (\\p. RETURN (merge_coeffs p))\n \\ \\ sorted_poly_rel\n (RETURN p' \\\n (\\p'. SPEC (normalize_poly_p\\<^sup>*\\<^sup>* p')))"} {"_id": "500747", "text": "proof (prove)\ngoal (1 subgoal):\n 1. S.seq (Hom_DopxG.map gf) (\\o (DopxC.dom gf))"} {"_id": "500748", "text": "proof (prove)\nusing this:\n AE y in stock_measure t'. 0 \\ eval_cexpr f (case_nat x \\) y\n\ngoal (1 subgoal):\n 1. integrable (stock_measure t') (eval_cexpr f (case_nat x \\))"} {"_id": "500749", "text": "proof (state)\nthis:\n T = []\n P f = []\n\ngoal (2 subgoals):\n 1. \\x1.\n t = Var x1 \\\n \\u.\n term_variants_pred P t u \\ Fun f S = u \\ \\\n 2. \\x21 x22.\n t = Fun x21 x22 \\\n \\u.\n term_variants_pred P t u \\ Fun f S = u \\ \\"} {"_id": "500750", "text": "proof (prove)\ngoal (1 subgoal):\n 1. B.seq (Map \\ \\\\<^sub>B Map \\)\n (B.can (Dom \\ \\<^bold>\\ Dom \\)\n (Dom \\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n Dom \\))"} {"_id": "500751", "text": "proof (prove)\nusing this:\n atom i \\ db\n\ngoal (1 subgoal):\n 1. {} \\\n SubstFormP \\Var i\\ Zero (quot_dbfm db)\n (quot_dbfm db)"} {"_id": "500752", "text": "proof (prove)\nusing this:\n a \\ b\n\ngoal (1 subgoal):\n 1. {..- inverse (real (Suc i))} \\ {a..b} =\n {a..min (- inverse (real (Suc i))) b}"} {"_id": "500753", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P (invoke (a_get_owner_document_tups @ xs) ptr ()) =\n ((known_ptr ptr \\ P (get_owner_document ptr)) \\\n (\\ known_ptr ptr \\ P (invoke xs ptr ())))"} {"_id": "500754", "text": "proof (prove)\nusing this:\n range A \\ M\n\ngoal (1 subgoal):\n 1. (\\i (A ` {..v \\ Null; P,h \\ v :\\ Class C\\\n \\ \\a hT.\n v = Addr a \\\n typeof_addr h a = \\hT\\ \\\n P \\ class_type_of hT \\\\<^sup>* C"} {"_id": "500756", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n.\n EM_remainder' 1\n (\\x.\n (- 1) ^ 1 * pochhammer s 1 *\n complex_of_real (x + a) powr (- s - of_nat 1))\n 0 (real n))\n \\ EM_remainder 1\n (\\x.\n (- 1) ^ 1 * pochhammer s 1 *\n complex_of_real (x + a) powr\n (- s - of_nat 1))\n 0"} {"_id": "500757", "text": "proof (chain)\npicking this:\n g fine {(x, cbox c d)}"} {"_id": "500758", "text": "proof (prove)\nusing this:\n Result.those (map (compile_env_entry \\ \\) []) = Ok ys\n\ngoal (1 subgoal):\n 1. rel_option (\\fd1 fd2. compile_fundef \\ (\\ f) fd1 = Ok fd2)\n (map_of [] f) (map_of ys f)"} {"_id": "500759", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Cop.H f \\ \\\n \\\\<^sup>-\\<^sup>1[f, g, f] \\ Cop.H \\ f =\n \\\\<^sup>-\\<^sup>1[f] \\ \\[f] \\\n Cop.H \\ g \\ \\[g, f, g] \\ Cop.H g \\ =\n \\\\<^sup>-\\<^sup>1[g] \\ \\[g]"} {"_id": "500760", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cinner x y =\n (complex_of_real ((norm (x + y))\\<^sup>2 - (norm (x - y))\\<^sup>2) -\n \\ * complex_of_real ((norm (x + \\ *\\<^sub>C y))\\<^sup>2) +\n \\ * complex_of_real ((norm (x - \\ *\\<^sub>C y))\\<^sup>2)) /\n 4"} {"_id": "500761", "text": "proof (prove)\nusing this:\n Cs = map (\\w. w `\\ C0) (sums ss)\n gallery Cs\n {} \\ set (wall_crossings Cs)\n\ngoal (1 subgoal):\n 1. (\\f g A B E F As Fs Bs.\n \\(f, g) \\ foldpairs; A \\ f \\ \\;\n B \\ g \\ \\; E \\ g \\ \\;\n F \\ f \\ \\;\n Cs = As @ [A, B, F] @ Fs \\\n Cs = As @ [A, B] @ Bs @ [E, F] @ Fs\\\n \\ thesis) \\\n thesis"} {"_id": "500762", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prob_dom PROB1 \\ prob_dom PROB2"} {"_id": "500763", "text": "proof (prove)\nusing this:\n set (insertions A) = set (insertions (remove1 a A)) \\ {(oid, ref)}\n set (insertions (ops @ [a])) = set (insertions ops) \\ {(oid, ref)}\n set (insertions (remove1 a A)) \\ set (insertions ops)\n\ngoal (1 subgoal):\n 1. set (insertions A) \\ set (insertions (ops @ [a]))"} {"_id": "500764", "text": "proof (prove)\ngoal (1 subgoal):\n 1. - \\\n (\\

\n (\\x.\n {(x, y).\n (\\x xa.\n s2p {(x, y). s2p (R `` \\ x) y}\n (x, xa))\\\\\n x y} ``\n \\ x)\n (- UNIV)) =\n UNIV"} {"_id": "500765", "text": "proof (prove)\nusing this:\n other.p x2 = ego.p x2\n 0 < x2\n x2 < other.t_stop\n \\ x2 < ego.t_stop\n\ngoal (1 subgoal):\n 1. other.p x2 \\ ego.p ego.t_stop"} {"_id": "500766", "text": "proof (prove)\nusing this:\n y\\<^sup>+ \\ y\\<^sup>+\\<^sup>T = x\\<^sup>+ \\ - (1::'a)\n\ngoal (1 subgoal):\n 1. transitively_orientable (x\\<^sup>+ \\ - (1::'a))"} {"_id": "500767", "text": "proof (prove)\ngoal (1 subgoal):\n 1. right_gpv (Generative_Probabilistic_Value.Done x) =\n Generative_Probabilistic_Value.Done x"} {"_id": "500768", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ccBinds \\\\(\\, \\) = \\"} {"_id": "500769", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\po'. simple_linux_router_nol12 rt fw pi = Some po'"} {"_id": "500770", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pair_prob_space (K1.T x) (K2.T y)"} {"_id": "500771", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xs ys.\n list = xs @ ys \\ insert_spec list (oid, ref) = xs @ oid # ys"} {"_id": "500772", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s -t\\ta\\ s';\n wset_thread_ok (wset s) (thr s)\\\n \\ wset_thread_ok (wset s') (thr s')"} {"_id": "500773", "text": "proof (prove)\nusing this:\n rk {A, B, C, A', B', C', P, Q} \\ rk {P, Q, R, A', A, B, C, B', C'}\n\ngoal (1 subgoal):\n 1. rk {R, a} = 2"} {"_id": "500774", "text": "proof (prove)\nusing this:\n \\?i < card {i. i < dim_row ?A \\ i \\ ?I};\n ?j < card {j. j < dim_col ?A \\ j \\ ?J}\\\n \\ submatrix ?A ?I ?J $$ (?i, ?j) =\n ?A $$ (pick ?I ?i, pick ?J ?j)\n\ngoal (1 subgoal):\n 1. \\x. submatrix (A x) I J $$ (i, j) = A x $$ (pick I i, pick J j)"} {"_id": "500775", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C = C'"} {"_id": "500776", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. V_of (inv_into X f (f x))) ` X =\n elts (set (V_of ` inv_into X f ` f ` X))"} {"_id": "500777", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AE x in measure_pmf (p N M). AE y in m' N M x. y !! 0 \\ \\"} {"_id": "500778", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj_on (inverse x \\ y)\\<^sub>R (H |\\ x)"} {"_id": "500779", "text": "proof (prove)\nusing this:\n aff_dim T \\ aff_dim (cball a r)\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "500780", "text": "proof (prove)\nusing this:\n (X, Y) \\ swapped\n good X\n\ngoal (1 subgoal):\n 1. skel Y = skel X"} {"_id": "500781", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Big_Step_Unclocked_Single.evaluate env s e (s', res);\n is_cupcake_all_env env; is_cupcake_exp e\\\n \\ cupcake_evaluate_single env e res"} {"_id": "500782", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 220 Amic 284"} {"_id": "500783", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Astack (map Dummy l) = 0"} {"_id": "500784", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\\\<^sub>C[a] \\ C.hom (\\\\<^sub>C \\\\<^sub>C a) a \\\n \\\\<^sub>C \\\\<^sub>C \\\\<^sub>C[a] =\n (\\\\<^sub>C \\\\<^sub>C a) \\\\<^sub>C\n C.assoc' \\\\<^sub>C \\\\<^sub>C a \\\n F \\\\<^sub>C[a]\n \\ D.hom (\\\\<^sub>D \\\\<^sub>D F a) (F a) \\\n \\\\<^sub>D \\\\<^sub>D F \\\\<^sub>C[a] =\n (\\\\<^sub>D \\\\<^sub>D F a) \\\\<^sub>D\n D.assoc' \\\\<^sub>D \\\\<^sub>D (F a)"} {"_id": "500785", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\p\\S.\n (\\(r, v). ((r, p), v)) ` map_graph (\\r. v_hist r p))\n \\ (\\p\\S.\n (\\r. ((r, p), the (v_hist r p))) ` {0.. (Pow G - {{}})))"} {"_id": "500787", "text": "proof (prove)\ngoal (1 subgoal):\n 1. at (a' \\\\<^sub>A a) D j = Da'oDa.map j"} {"_id": "500788", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vpath (vwalk_to_vpath p) G"} {"_id": "500789", "text": "proof (state)\nthis:\n trg (tab\\<^sub>0 r) = r\n\ngoal (1 subgoal):\n 1. tab\\<^sub>0 r \\ r \\ tab\\<^sub>0 r"} {"_id": "500790", "text": "proof (prove)\nusing this:\n subseq (l1 @ l2 @ l3) []\n subseq ?l1.0 [] = (?l1.0 = [])\n\ngoal (1 subgoal):\n 1. subseq (l1 @ l3) []"} {"_id": "500791", "text": "proof (prove)\nusing this:\n set (snd (snd (reduce_system p\n (qs, M_mat [[1], [- 1]] [[], [0]], [[], [0]],\n [[1], [- 1]])))) =\n set (characterize_consistent_signs_at_roots_copr p qs)\n\ngoal (1 subgoal):\n 1. set (snd (snd (reduce_system p\n (qs, mat_of_rows_list 2 [[1, 1], [1, - 1]], [[], [0]],\n [[1], [- 1]])))) =\n set (characterize_consistent_signs_at_roots_copr p qs)"} {"_id": "500792", "text": "proof (prove)\nusing this:\n ov = None\n\ngoal (1 subgoal):\n 1. one_final (subdivFace' g f v n (ov # ovs))"} {"_id": "500793", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x + y = bot \\ x = bot"} {"_id": "500794", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (((S ===> M) ===> A) ===> rel_stateT S M ===> A) case_stateT case_stateT"} {"_id": "500795", "text": "proof (prove)\nusing this:\n edge_succ M a = a\n\ngoal (1 subgoal):\n 1. card (out_arcs G (tail G a)) = 1"} {"_id": "500796", "text": "proof (prove)\ngoal (1 subgoal):\n 1. S $h 0 $h 1 = (0::'a)"} {"_id": "500797", "text": "proof (prove)\nusing this:\n generator g s = Skip ?x2.0 \\\n unstream (filter_trans P g) ?x2.0 = filter P (unstream g ?x2.0)\n generator g s = Yield ?x31.0 ?x32.0 \\\n unstream (filter_trans P g) ?x32.0 = filter P (unstream g ?x32.0)\n\ngoal (1 subgoal):\n 1. unstream (filter_trans P g) s = filter P (unstream g s)"} {"_id": "500798", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x1a \\ SIGT ($f x1a x2a)"} {"_id": "500799", "text": "proof (prove)\ngoal (12 subgoals):\n 1. \\servK \\ symKeys; A \\ bad; B \\ bad\\\n \\ Key authK\n \\ analz (knows Spy []) \\\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set [] \\\n Key servK \\ analz (knows Spy []) \\\n expiredSK Ts []\n 2. \\evsf X Ba.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evsf \\ kerbIV; X \\ synth (analz (knows Spy evsf));\n Key authK \\ analz (knows Spy evsf);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evsf \\\n Key servK \\ analz (knows Spy evsf) \\\n expiredSK Ts evsf\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Spy Ba X # evsf) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Spy Ba X # evsf)) \\\n expiredSK Ts (Says Spy Ba X # evsf)\n 3. \\evs1 Aa.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs1 \\ kerbIV; Key authK \\ analz (knows Spy evs1);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs1 \\\n Key servK \\ analz (knows Spy evs1) \\\n expiredSK Ts evs1\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Aa Kas\n \\Agent Aa, Agent Tgs,\n Number (CT evs1)\\ #\n evs1) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Aa Kas\n \\Agent Aa, Agent Tgs,\nNumber (CT evs1)\\ #\n evs1)) \\\n expiredSK Ts\n (Says Aa Kas\n \\Agent Aa, Agent Tgs,\n Number (CT evs1)\\ #\n evs1)\n 4. \\evs2 authKa A' Aa T1.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs2 \\ kerbIV; Key authKa \\ used evs2;\n authKa \\ symKeys;\n Says A' Kas \\Agent Aa, Agent Tgs, Number T1\\\n \\ set evs2;\n Key authK \\ analz (knows Spy evs2);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs2 \\\n Key servK \\ analz (knows Spy evs2) \\\n expiredSK Ts evs2\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Kas Aa\n (Crypt (shrK Aa)\n \\Key authKa, Agent Tgs, Number (CT evs2),\n Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa,\n Number (CT evs2)\\\\) #\n evs2) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Kas Aa\n (Crypt (shrK Aa)\n\\Key authKa, Agent Tgs, Number (CT evs2),\n Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa,\n Number (CT evs2)\\\\) #\n evs2)) \\\n expiredSK Ts\n (Says Kas Aa\n (Crypt (shrK Aa)\n \\Key authKa, Agent Tgs,\n Number (CT evs2),\n Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa,\n Number (CT evs2)\\\\) #\n evs2)\n 5. \\evs3 Aa T1 Kas' authKa Ta authTicket Ba.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs3 \\ kerbIV;\n Says Aa Kas \\Agent Aa, Agent Tgs, Number T1\\\n \\ set evs3;\n Says Kas' Aa\n (Crypt (shrK Aa)\n \\Key authKa, Agent Tgs, Number Ta,\n Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa,\n Number Ta\\\\)\n \\ set evs3;\n valid Ta wrt T1; Key authK \\ analz (knows Spy evs3);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs3 \\\n Key servK \\ analz (knows Spy evs3) \\\n expiredSK Ts evs3;\n authKa \\ range shrK; authKa \\ symKeys\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Aa Tgs\n \\Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa, Number Ta\\,\n Crypt authKa \\Agent Aa, Number (CT evs3)\\,\n Agent Ba\\ #\n evs3) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Aa Tgs\n \\Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa, Number Ta\\,\nCrypt authKa \\Agent Aa, Number (CT evs3)\\,\nAgent Ba\\ #\n evs3)) \\\n expiredSK Ts\n (Says Aa Tgs\n \\Crypt (shrK Tgs)\n\\Agent Aa, Agent Tgs, Key authKa, Number Ta\\,\n Crypt authKa\n \\Agent Aa, Number (CT evs3)\\,\n Agent Ba\\ #\n evs3)\n 6. \\evs3 Aa T1 Kas' authKa Ta authTicket Ba.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs3 \\ kerbIV;\n Says Aa Kas \\Agent Aa, Agent Tgs, Number T1\\\n \\ set evs3;\n Says Kas' Aa\n (Crypt (shrK Aa)\n \\Key authKa, Agent Tgs, Number Ta, authTicket\\)\n \\ set evs3;\n valid Ta wrt T1; Key authK \\ analz (knows Spy evs3);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs3 \\\n Key servK \\ analz (knows Spy evs3) \\\n expiredSK Ts evs3;\n authTicket \\ analz (knows Spy evs3)\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Aa Tgs\n \\authTicket,\n Crypt authKa \\Agent Aa, Number (CT evs3)\\,\n Agent Ba\\ #\n evs3) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Aa Tgs\n \\authTicket,\nCrypt authKa \\Agent Aa, Number (CT evs3)\\,\nAgent Ba\\ #\n evs3)) \\\n expiredSK Ts\n (Says Aa Tgs\n \\authTicket,\n Crypt authKa\n \\Agent Aa, Number (CT evs3)\\,\n Agent Ba\\ #\n evs3)\n 7. \\evs4 servKa Ba authKa A' Aa Ta T2.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs4 \\ kerbIV; Key servKa \\ used evs4;\n servKa \\ symKeys; Ba \\ Tgs; authKa \\ symKeys;\n Says A' Tgs\n \\Crypt (shrK Tgs)\n \\Agent Aa, Agent Tgs, Key authKa,\n Number Ta\\,\n Crypt authKa \\Agent Aa, Number T2\\,\n Agent Ba\\\n \\ set evs4;\n \\ expiredAK Ta evs4; \\ expiredA T2 evs4;\n servKlife + CT evs4 \\ authKlife + Ta;\n Key authK \\ analz (knows Spy evs4);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs4 \\\n Key servK \\ analz (knows Spy evs4) \\\n expiredSK Ts evs4\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Tgs Aa\n (Crypt authKa\n \\Key servKa, Agent Ba, Number (CT evs4),\n Crypt (shrK Ba)\n \\Agent Aa, Agent Ba, Key servKa,\n Number (CT evs4)\\\\) #\n evs4) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Tgs Aa\n (Crypt authKa\n\\Key servKa, Agent Ba, Number (CT evs4),\n Crypt (shrK Ba)\n \\Agent Aa, Agent Ba, Key servKa,\n Number (CT evs4)\\\\) #\n evs4)) \\\n expiredSK Ts\n (Says Tgs Aa\n (Crypt authKa\n \\Key servKa, Agent Ba,\n Number (CT evs4),\n Crypt (shrK Ba)\n \\Agent Aa, Agent Ba, Key servKa,\n Number (CT evs4)\\\\) #\n evs4)\n 8. \\evs5 authKa servKa Aa authTicket T2 Ba Tgs' Tsa servTicket Aaa.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs5 \\ kerbIV; authKa \\ symKeys; servKa \\ symKeys;\n Says Aa Tgs\n \\authTicket,\n Crypt authKa \\Agent Aa, Number T2\\,\n Agent Ba\\\n \\ set evs5;\n Says Tgs' Aa\n (Crypt authKa\n \\Key servKa, Agent Ba, Number Tsa,\n Crypt (shrK Ba)\n \\Agent Aaa, Agent Ba, Key servKa,\n Number Tsa\\\\)\n \\ set evs5;\n valid Tsa wrt T2; Key authK \\ analz (knows Spy evs5);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs5 \\\n Key servK \\ analz (knows Spy evs5) \\\n expiredSK Ts evs5;\n servKa \\ range shrK\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Aa Ba\n \\Crypt (shrK Ba)\n \\Agent Aaa, Agent Ba, Key servKa, Number Tsa\\,\n Crypt servKa \\Agent Aa, Number (CT evs5)\\\\ #\n evs5) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Aa Ba\n \\Crypt (shrK Ba)\n \\Agent Aaa, Agent Ba, Key servKa, Number Tsa\\,\nCrypt servKa \\Agent Aa, Number (CT evs5)\\\\ #\n evs5)) \\\n expiredSK Ts\n (Says Aa Ba\n \\Crypt (shrK Ba)\n\\Agent Aaa, Agent Ba, Key servKa, Number Tsa\\,\n Crypt servKa\n \\Agent Aa,\n Number (CT evs5)\\\\ #\n evs5)\n 9. \\evs5 authKa servKa Aa authTicket T2 Ba Tgs' Tsa servTicket.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs5 \\ kerbIV; authKa \\ symKeys; servKa \\ symKeys;\n Says Aa Tgs\n \\authTicket,\n Crypt authKa \\Agent Aa, Number T2\\,\n Agent Ba\\\n \\ set evs5;\n Says Tgs' Aa\n (Crypt authKa\n \\Key servKa, Agent Ba, Number Tsa, servTicket\\)\n \\ set evs5;\n valid Tsa wrt T2; Key authK \\ analz (knows Spy evs5);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs5 \\\n Key servK \\ analz (knows Spy evs5) \\\n expiredSK Ts evs5;\n servTicket \\ analz (knows Spy evs5)\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Aa Ba\n \\servTicket,\n Crypt servKa \\Agent Aa, Number (CT evs5)\\\\ #\n evs5) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Aa Ba\n \\servTicket,\nCrypt servKa \\Agent Aa, Number (CT evs5)\\\\ #\n evs5)) \\\n expiredSK Ts\n (Says Aa Ba\n \\servTicket,\n Crypt servKa\n \\Agent Aa,\n Number (CT evs5)\\\\ #\n evs5)\n 10. \\evs6 A' Ba Aa servKa Tsa T3.\n \\servK \\ symKeys; A \\ bad; B \\ bad;\n evs6 \\ kerbIV;\n Says A' Ba\n \\Crypt (shrK Ba)\n \\Agent Aa, Agent Ba, Key servKa,\n Number Tsa\\,\n Crypt servKa \\Agent Aa, Number T3\\\\\n \\ set evs6;\n \\ expiredSK Tsa evs6; \\ expiredA T3 evs6;\n Key authK \\ analz (knows Spy evs6);\n Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set evs6 \\\n Key servK \\ analz (knows Spy evs6) \\\n expiredSK Ts evs6\\\n \\ Says Tgs A\n (Crypt authK\n \\Key servK, Agent B, Number Ts,\n Crypt (shrK B)\n \\Agent A, Agent B, Key servK,\n Number Ts\\\\)\n \\ set (Says Ba Aa (Crypt servKa (Number T3)) #\n evs6) \\\n Key servK\n \\ analz\n (knows Spy\n (Says Ba Aa (Crypt servKa (Number T3)) #\n evs6)) \\\n expiredSK Ts\n (Says Ba Aa (Crypt servKa (Number T3)) # evs6)\nA total of 12 subgoals..."} {"_id": "500800", "text": "proof (prove)\nusing this:\n \\i xs.\n i \\ n \\ set xs \\ {0..0} \\\n (0::'a) \\ len m i i xs\n 0 \\ n\n \\\\i xs.\n i \\ n \\\n set xs \\ {} \\ {0} \\\n (0::'a) \\ len m i i xs;\n canonical_subs n {} m; {} \\ {0..n}; 0 \\ n\\\n \\ \\i xs.\n i \\ n \\\n set xs \\ {} \\ {0} \\\n (0::'a) \\ len (fwi m n 0 n n) i i xs\n \\canonical_subs n {} ?m; {} \\ {0..n};\n (0::'a) \\ ?m ?k ?k; ?k \\ n\\\n \\ canonical_subs n ({} \\ {?k}) (fwi ?m n ?k n n)\n\ngoal (1 subgoal):\n 1. canonical_subs n {0..0} (fw m n 0) \\\n (\\i xs.\n i \\ n \\ set xs \\ {0..0} \\\n (0::'a) \\ len (fw m n 0) i i xs)"} {"_id": "500801", "text": "proof (prove)\nusing this:\n (?m, ?\\, ?m') = \\ ?i \\\n \\p p'. ?m k = Some p \\ ?m' k = Some p'\n limit ?r \\ range ?r\n\ngoal (1 subgoal):\n 1. \\m \\ m'.\n (m, \\, m') \\ limit \\ \\\n \\p p'. m k = Some p \\ m' k = Some p'"} {"_id": "500802", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\it \\ it' \\'.\n \\it \\ S; \\ ` it \\ \\ ` S;\n it' = \\ ` it;\n ((it, \\), \\ ` it, \\') \\ R\\\n \\ c \\ = c' \\'\n 2. \\x it \\ x' it' \\'.\n \\it \\ S; it' \\ \\ ` S;\n x' = \\ x; it' = \\ ` it;\n ((it, \\), it', \\') \\ R; x \\ it; x' \\ it';\n c \\; c' \\'\\\n \\ f x \\\n \\ \\\n {(\\, \\').\n ((it - {x}, \\), it' - {x'},\n \\')\n \\ R}\n (f' x' \\')\n 3. \\\\ \\'.\n (({}, \\), {}, \\') \\ R \\\n (\\, \\') \\ R'\n 4. \\it \\ it' \\'.\n \\it \\ S; it' \\ \\ ` S;\n it' = \\ ` it; ((it, \\), it', \\') \\ R;\n it \\ {}; it' \\ {}; \\ c \\;\n \\ c' \\'\\\n \\ (\\, \\') \\ R'"} {"_id": "500803", "text": "proof (prove)\nusing this:\n isrlfm p\n\ngoal (1 subgoal):\n 1. Ifm (x # bs) (\\ p (t, n)) =\n Ifm (Inum (x # bs) t / real_of_int n # bs) p \\\n bound0 (\\ p (t, n))"} {"_id": "500804", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P \\ ty_of_htype U \\ Class (class_type_of U)"} {"_id": "500805", "text": "proof (prove)\nusing this:\n (HF.Inl ?q \\ ST.nextl {HF.Inl (dfa.init MS)} u) =\n (?q = dfa.nextl MS (dfa.init MS) u)\n\ngoal (1 subgoal):\n 1. (HF.Inl q \\ ST.nextl {HF.Inl (dfa.init MS)} (u @ [x])) =\n (q = dfa.nextl MS (dfa.init MS) (u @ [x]))"} {"_id": "500806", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (CASE 0 \\ c) &&& CASE (eSuc n) \\ d n"} {"_id": "500807", "text": "proof (prove)\ngoal (1 subgoal):\n 1. !x \\ !y \\ t y \\ t x"} {"_id": "500808", "text": "proof (prove)\nusing this:\n w \\ x\n w \\ y\n w \\ V\n w \\ pverts G - V\n G.gen_iapath V x q y\n w \\ set (pawalk_verts x q)\n in_degree (with_proj G) w \\ 2\n\ngoal (1 subgoal):\n 1. (\\e.\n \\e \\ set q; snd e = w\\\n \\ thesis) \\\n thesis"} {"_id": "500809", "text": "proof (prove)\ngoal (1 subgoal):\n 1. findIndices\\P\\(a : xs) =\n If P\\a\n then 0 : map\\(+1)\\(findIndices\\P\\xs)\n else map\\(+1)\\(findIndices\\P\\xs)"} {"_id": "500810", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [\\<^bold>\\\\<^bold>\\\\x,F\\ \\<^bold>\\\n \\<^bold>\\\\<^bold>\\\\x,F\\ in v]"} {"_id": "500811", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ R (\\ S M) = \\ (R O S) M"} {"_id": "500812", "text": "proof (state)\nthis:\n \\\\.\n \\,\\\n \\\\<^sub>t\\<^bsub>/F\\<^esub> ({\\} \\\n I \\\n b)\n c ({t. (t, \\) \\ V} \\ I),A\n\ngoal (1 subgoal):\n 1. \\,\\\n \\\\<^sub>t\\<^bsub>/F\\<^esub> P whileAnno b I V c Q,A"} {"_id": "500813", "text": "proof (state)\nthis:\n \\x\\{0..1}.\n onorm (\\i. f' (a + x *\\<^sub>R (b - a)) i - f' x0 i) \\ B\n\ngoal (1 subgoal):\n 1. norm (f b - f a - f' x0 (b - a)) \\ norm (b - a) * B"} {"_id": "500814", "text": "proof (prove)\nusing this:\n (u, mon_w fg w, P) \\ S_cs fg k\n mon_e fg e = {}\n (u, ?M, ?P) \\ S_cs fg ?k \\\n (v, ?M, ?P) \\ S_cs fg ?k\n\ngoal (1 subgoal):\n 1. (v, mon_w fg (w @ [e]), P) \\ S_cs fg k"} {"_id": "500815", "text": "proof (state)\nthis:\n ((\\t. v t $ i) has_vderiv_on (\\t. 0)) T\n\ngoal (1 subgoal):\n 1. v t $ i = x0"} {"_id": "500816", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\mcont lub ord Sup (\\) f;\n mcont lub ord Sup (\\) g\\\n \\ mcont lub ord Sup (\\) (\\x. f x + g x)"} {"_id": "500817", "text": "proof (prove)\nusing this:\n \\as'.\n exec_plan s as = exec_plan s as' \\\n subseq as' as \\ length as' \\ problem_plan_bound PROB\n\ngoal (1 subgoal):\n 1. \\as'.\n exec_plan s as = exec_plan s as' \\\n subseq as' as \\ length as' < problem_plan_bound PROB + 1"} {"_id": "500818", "text": "proof (prove)\ngoal (1 subgoal):\n 1. count cs (shd w) k < length cs"} {"_id": "500819", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wellformed_policy2Pr (x # xs) \\ wellformed_policy2Pr xs"} {"_id": "500820", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map vec\\<^sub>h (factorization_lattice u k m) =\n factorization_lattice (map_poly hom u) k (hom m)"} {"_id": "500821", "text": "proof (prove)\nusing this:\n porder_on (actions E) (happens_before P E)\n \\ P,E \\ w' \\hb ws r'\n ws r' \\ write_actions E\n w' = ws r'\n\ngoal (1 subgoal):\n 1. False"} {"_id": "500822", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Dag p low high pret; no \\ Null;\n \\not. Dag no low high not \\ thesis;\n no \\ set_of pret \\ var no = n\\\n \\ thesis"} {"_id": "500823", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a fract, euclidean_ring_gcd_class)"} {"_id": "500824", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coeff\n (\\X\\Pow A.\n monom ((- (1::'b)) ^ card X * prod f X) (card (A - X)))\n k =\n (\\X | X \\ A \\ card X = card A - k.\n (- (1::'b)) ^ (card A - k) * prod f X)"} {"_id": "500825", "text": "proof (prove)\ngoal (1 subgoal):\n 1. component_of_term ` Keys B\n \\ component_of_term ` Keys (phull B)"} {"_id": "500826", "text": "proof (prove)\nusing this:\n 0 < a\n 0 < b\n \\\\<^sub>F x in at_top. 0 < ln x / x powr (b / a)\n\ngoal (1 subgoal):\n 1. ((\\x. (ln x / x powr (b / a)) powr a) \\ 0)\n at_top"} {"_id": "500827", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ DBSpec \\ FullSpec"} {"_id": "500828", "text": "proof (chain)\npicking this:\n n \\ m"} {"_id": "500829", "text": "proof (prove)\ngoal (1 subgoal):\n 1. abss (l, u) = (l, [u]\\<^sub>\\)"} {"_id": "500830", "text": "proof (prove)\nusing this:\n LeftDerivation \\ (take (ladder_n L' (Suc index)) D')\n (ladder_\\ \\ D' L' (Suc index))\n ladder_\\ (\\ @ \\) D L (Suc index) =\n \\ @ \\' \\\n LeftDerivationIntro (ladder_\\ \\ D' L' (Suc index))\n (ladder_i L (Suc index) - length \\) (snd e)\n (ladder_ix L (Suc index)) (derivation_shift E (length \\) 0)\n (ladder_j L (Suc index) - length \\) \\'\n D' = derivation_shift D (length \\) 0\n Derivation ?a ?D ?b \\ Derive ?a ?D = ?b\n L' = ladder_cut_prefix (length \\) L\n LeftDerivationIntro ?\\ ?i ?r ?ix ?D ?j ?\\ =\n (\\\\.\n LeftDerives1 ?\\ ?i ?r \\ \\\n ?ix < length (snd ?r) \\\n snd ?r ! ?ix = ?\\ ! ?j \\\n LeftDerivationFix \\ (?i + ?ix) ?D ?j ?\\)\n n = ladder_n L (Suc index - 1)\n m = ladder_n L (Suc index)\n e = D ! n\n E = drop (Suc n) (take m D)\n LeftDerivation \\ (take (ladder_n L' (Suc index)) D')\n (ladder_\\ \\ D' L' (Suc index))\n LeftDerivation ?a ?D ?b \\ Derivation ?a ?D ?b\n ladder_\\ ?a ?D ?L ?index = Derive ?a (take (ladder_n ?L ?index) ?D)\n \\?index < length ?L; 0 < length ?L\\\n \\ ladder_n (ladder_cut_prefix ?d ?L) ?index =\n ladder_n ?L ?index\n take ?n (derivation_shift ?D ?left ?right) =\n derivation_shift (take ?n ?D) ?left ?right\n\ngoal (1 subgoal):\n 1. LeftDerivation \\ (take (ladder_n L' (Suc index)) D')\n (ladder_\\ \\ D' L' (Suc index)) \\\n ladder_\\ (\\ @ \\) D L (Suc index) =\n \\ @ ladder_\\ \\ D' L' (Suc index) \\\n ladder_\\ (\\ @ \\) D L (Suc index) =\n \\ @ ladder_\\ \\ D' L' (Suc index)"} {"_id": "500831", "text": "proof (prove)\nusing this:\n Right_a.arr \\ \\ arr \\ \\ \\ \\ a \\ null\n H\\<^sub>R ?f \\ \\\\. \\ \\ ?f\n Right_a.arr ?f \\ Right_a.arr (H\\<^sub>R a' ?f)\n\ngoal (1 subgoal):\n 1. Right_a.arr (\\ \\ a')"} {"_id": "500832", "text": "proof (prove)\nusing this:\n m1 \\ m2\n m2 \\ m1\n\ngoal (1 subgoal):\n 1. m1 = m2"} {"_id": "500833", "text": "proof (prove)\ngoal (1 subgoal):\n 1. k \\ (GREATEST k. P k)"} {"_id": "500834", "text": "proof (prove)\ngoal (1 subgoal):\n 1. smap f (of_seq g) = of_seq (f \\ g)"} {"_id": "500835", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AE x in lborel. x \\ s m ` cbox a b"} {"_id": "500836", "text": "proof (prove)\nusing this:\n infs (even \\ f)\n (gtrace (sdrop n (fromN 1 ||| r))\n (gtarget (stake n (fromN 1 ||| r)) (0, p)))\n\ngoal (1 subgoal):\n 1. infs even\n (smap f\n (gtrace (sdrop n (fromN 1 ||| r))\n (gtarget (stake n (fromN 1 ||| r)) (0, p))))"} {"_id": "500837", "text": "proof (prove)\nusing this:\n a \\ a\n\ngoal (1 subgoal):\n 1. False"} {"_id": "500838", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\m.\n real (card {(n, d). n \\ {0<..m} \\ d dvd n}) /\n real (card {0<..m}) -\n (ln (real m) + 2 * euler_mascheroni - 1))\n \\ \\(\\m.\n (sum_upto (\\n. real (divisor_count n))\n (real m) -\n (real m * ln (real m) +\n (2 * euler_mascheroni - 1) * real m)) /\n real m)"} {"_id": "500839", "text": "proof (prove)\ngoal (1 subgoal):\n 1. similar_mat_wit (four_block_mat A1 A2 A0 A3)\n (four_block_mat A1 (A2 * P') A0 B)\n (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 (n - 1))\n (0\\<^sub>m (n - 1) 1) P')\n (four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 (n - 1))\n (0\\<^sub>m (n - 1) 1) Q')"} {"_id": "500840", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fls_shift k (of_nat a) $$ n = (if n = - k then of_nat a else (0::'a))"} {"_id": "500841", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_bset R (bset_of_option x1) (bset_of_option x2) = rel_option R x1 x2"} {"_id": "500842", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B\\X.\n \\a\\X.\n a \\ set_option\n (local.MaxR.MaxR_opt\n (B \\ {a})) \\\n set_option (local.MaxR.MaxR_opt (B \\ {a})) =\n set_option (local.MaxR.MaxR_opt B)"} {"_id": "500843", "text": "proof (prove)\nusing this:\n \\\\ (\\ ?s); ?x\\<^sub>i \\ lvars (\\ ?s);\n ?x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ ?s) ?x\\<^sub>i)\\\n \\ \\ (\\ (pivot ?x\\<^sub>i ?x\\<^sub>j ?s))\n \\\\ (\\ ?s); ?x\\<^sub>i \\ lvars (\\ ?s);\n ?x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ ?s) ?x\\<^sub>i)\\\n \\ ?x\\<^sub>i\n \\ lvars (\\ (pivot ?x\\<^sub>i ?x\\<^sub>j ?s))\n\ngoal (1 subgoal):\n 1. PivotAndUpdate eq_idx_for_lvar pivot_and_update"} {"_id": "500844", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (path \\, k) \\ scop \\ (path \\', k') \\ scp"} {"_id": "500845", "text": "proof (prove)\ngoal (1 subgoal):\n 1. blinfun_apply -` UNIV \\ UNIV = UNIV"} {"_id": "500846", "text": "proof (prove)\nusing this:\n igSwapAbsIPresIGWlsAbs MOD\n igWlsAbsDisj MOD\n igWlsAbsIsInBar MOD\n\ngoal (1 subgoal):\n 1. igSwapAbsIPresIGWlsAbsSTR (errMOD MOD)"} {"_id": "500847", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vector_space = vector_space_with (+) (0::'b) (-) uminus"} {"_id": "500848", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\stp ln rn.\n steps0 (Suc 0, Bk # Bk # ires, Oc \\ Suc rs @ Bk \\ n)\n t_fourtimes stp =\n (Suc t_fourtimes_len, Bk \\ ln @ Bk # Bk # ires,\n Oc \\ Suc (4 * rs) @ Bk \\ rn) \\\n \\stp ln rn.\n steps0\n (Suc 0 +\n length\n (t_wcode_main_first_part @\n shift t_twice (length t_wcode_main_first_part div 2) @\n [(L, 1), (L, 1)]) div\n 2,\n Bk # Bk # ires, Oc \\ Suc rs @ Bk \\ n)\n ((t_wcode_main_first_part @\n shift t_twice (length t_wcode_main_first_part div 2) @\n [(L, 1), (L, 1)]) @\n shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)])\n stp =\n (Suc t_fourtimes_len +\n length\n (t_wcode_main_first_part @\n shift t_twice (length t_wcode_main_first_part div 2) @\n [(L, 1), (L, 1)]) div\n 2,\n Bk \\ ln @ Bk # Bk # ires,\n Oc \\ Suc (4 * rs) @ Bk \\ rn)"} {"_id": "500849", "text": "proof (prove)\ngoal (1 subgoal):\n 1. det (mat_of_rows n (map (row A) [0..T) =\n gso.Gramian_determinant (map (row A) [0..x. x \\ X)\n adm (\\x. x \\ Y)\n \\i. fst (A i) \\ X\n \\i. snd (A i) \\ Y\n\ngoal (1 subgoal):\n 1. Lub A \\ X \\ Y"} {"_id": "500851", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ B \\ [x \\ 0] (mod int p)"} {"_id": "500852", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\n f a b.\n \\Ring R;\n surjec\\<^bsub>R,(r\\\\<^bsub>{j. j \\ Suc n}\\<^esub>\n B)\\<^esub> A_to_prodag R {j. j \\ Suc n} S B;\n \\k\\Suc (Suc n). ideal R (J k);\n \\k\\Suc (Suc n). B k = R /\\<^sub>r J k;\n \\k\\Suc (Suc n). S k = pj R (J k);\n \\i\\Suc (Suc n).\n \\j\\Suc (Suc n).\n i \\ j \\ coprime_ideals R (J i) (J j);\n \\k\\Suc (Suc n). Ring (R /\\<^sub>r J k);\n A_to_prodag R {i. i \\ Suc (Suc n)} S B\n \\ rHom R (r\\\\<^bsub>{i. i \\ Suc (Suc n)}\\<^esub> B);\n A_to_prodag R {j. j \\ Suc (Suc n)} S B\n \\ carrier R \\\n carrier (r\\\\<^bsub>{i. i \\ Suc (Suc n)}\\<^esub> B);\n f \\ carrier (r\\\\<^bsub>{j. j \\ Suc (Suc n)}\\<^esub> B);\n ideal R (i\\\\<^bsub>R,Suc n\\<^esub> J);\n a \\ i\\\\<^bsub>R,Suc n\\<^esub> J; b \\ J (Suc (Suc n));\n a \\ b = 1\\<^sub>r\\\n \\ \\a\\carrier R.\n A_to_prodag R {j. j \\ Suc (Suc n)} S B a = f\n 2. \\n.\n \\Ring R;\n (\\k\\Suc n. ideal R (J k)) \\\n (\\k\\Suc n. B k = R /\\<^sub>r J k) \\\n (\\k\\Suc n. S k = pj R (J k)) \\\n (\\i\\Suc n.\n \\j\\Suc n.\n i \\ j \\\n coprime_ideals R (J i) (J j)) \\\n surjec\\<^bsub>R,(r\\\\<^bsub>{j. j \\ Suc n}\\<^esub>\n B)\\<^esub> A_to_prodag R {j. j \\ Suc n} S B;\n \\k\\Suc (Suc n). ideal R (J k);\n \\k\\Suc (Suc n). B k = R /\\<^sub>r J k;\n \\k\\Suc (Suc n). S k = pj R (J k);\n \\i\\Suc (Suc n).\n \\j\\Suc (Suc n).\n i \\ j \\ coprime_ideals R (J i) (J j);\n \\j.\n j \\ {j. j \\ Suc n} \\\n j \\ {j. j \\ Suc (Suc n)}\\\n \\ \\k\\{i. i \\ Suc (Suc n)}. Ring (B k)"} {"_id": "500853", "text": "proof (prove)\nusing this:\n sym theta\n theta \\ RetrT (theta \\ bis RetrT)\n Bisim.mono Sretr\n Bisim.mono ZOretr\n Bisim.mono ZOretrT\n Bisim.mono Wretr\n Bisim.mono WretrT\n Bisim.mono RetrT\n \\Bisim.mono ?Retr; sym ?theta;\n ?theta \\ ?Retr (?theta \\ bis ?Retr)\\\n \\ ?theta \\ bis ?Retr\n\ngoal (1 subgoal):\n 1. theta \\ bis RetrT"} {"_id": "500854", "text": "proof (prove)\nusing this:\n convex (Epigraph UNIV f)\n\ngoal (1 subgoal):\n 1. convex (domain f)"} {"_id": "500855", "text": "proof (prove)\nusing this:\n (l # ls) ! i = (p, Read volatile a t # is, \\, sb, \\, \\)\n i = 0\n\ngoal (1 subgoal):\n 1. l = (p, Read volatile a t # is, \\, sb, \\, \\)"} {"_id": "500856", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lm.ball m P = (\\x\\set (assoc_list.impl_of m). P x)"} {"_id": "500857", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f1 x / f2 x) \\ L F (\\x. g1 x / g2 x)) =\n (f1 \\ L F g1)"} {"_id": "500858", "text": "proof (prove)\nusing this:\n Suc n \\ N\n 1 \\ n\n n < m\n a \\ b\n finite Y\n\ngoal (1 subgoal):\n 1. ((- 1) ^ n * bernoulli (Suc n) / fact (Suc n)) *\\<^sub>R\n (fs n (real_of_int b) - fs n (real_of_int a)) +\n EM_remainder' n (fs n) (real_of_int a) (real_of_int b) =\n EM_remainder' (Suc n) (fs (Suc n)) (real_of_int a) (real_of_int b)"} {"_id": "500859", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (A - B) $$ i = A $$ i - B $$ i"} {"_id": "500860", "text": "proof (state)\nthis:\n n = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\\n \\m y.\n odd m \\\n 0 < m \\\n m < 2 ^ Suc n \\\n y \\ {a (real m / 2 ^ Suc n)..b (real m / 2 ^ Suc n)} \\\n f y = f x\n 2. n \\ 0 \\\n \\m y.\n odd m \\\n 0 < m \\\n m < 2 ^ Suc n \\\n y \\ {a (real m / 2 ^ Suc n)..b (real m / 2 ^ Suc n)} \\\n f y = f x"} {"_id": "500861", "text": "proof (prove)\ngoal (1 subgoal):\n 1. SN (inv_image\n (lex_two\n {(fss, gts).\n (case gts of (f, ss) \\ weight (Fun f ss))\n < (case fss of (f, ss) \\ weight (Fun f ss))}\n {(fss, gts).\n (case gts of (f, ss) \\ weight (Fun f ss))\n \\ (case fss of (f, ss) \\ weight (Fun f ss))}\n {(fss, gts).\n pr_strict (case fss of (f, ss) \\ (f, length ss))\n (case gts of (f, ss) \\ (f, length ss))})\n (\\x. (x, x)))"} {"_id": "500862", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pmf (N \\ K) i \\ pmf N i"} {"_id": "500863", "text": "proof (prove)\nusing this:\n u ?n \\ real_cond_exp M F f x\n\ngoal (1 subgoal):\n 1. c \\ real_cond_exp M F f x"} {"_id": "500864", "text": "proof (prove)\nusing this:\n op\\<^sub>2 \\ set ops\n op\\<^sub>1 \\ set ((\\\\ \\ \\ t) ! k)\n op\\<^sub>2 \\ set ((\\\\ \\ \\ t) ! k)\n index (\\\\<^sub>\\) op\\<^sub>1 \\\n index (\\\\<^sub>\\) op\\<^sub>2\n\ngoal (1 subgoal):\n 1. \\ are_operators_interfering op\\<^sub>1 op\\<^sub>2"} {"_id": "500865", "text": "proof (state)\ngoal (1 subgoal):\n 1. T \\ u exposed_face_of S"} {"_id": "500866", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\0 < n; distinct_pds K n P; ideal (O\\<^bsub>K P n\\<^esub>) I;\n I \\ {\\\\<^bsub>O\\<^bsub>K P n\\<^esub>\\<^esub>};\n I \\ carrier (O\\<^bsub>K P n\\<^esub>); j \\ n; Ring K;\n aGroup K; j \\ n\\\n \\ (\\\\<^bsub>K P j\\<^esub>)\n (\\\\<^sub>e K (\\k.\n Zl_mI K P I k \\\\<^sub>r\n mprod_exp K (K_gamma k) (Kb\\<^bsub>K n P\\<^esub>)\n n\\<^bsub>K\\<^esub>\\<^bsup>m_zmax_pdsI K n P I\\<^esup>) n) =\n LI K (\\\\<^bsub>K P j\\<^esub>) I"} {"_id": "500867", "text": "proof (prove)\ngoal (1 subgoal):\n 1. timpl_closure_set M TI \\\\<^sub>c u"} {"_id": "500868", "text": "proof (prove)\nusing this:\n valid_edge ax\n sourcenode ax = sourcenode a'\n kind ax = (\\cf. False)\\<^sub>\\\n \\a''.\n valid_edge a'' \\\n sourcenode a'' = sourcenode a' \\\n intra_kind (kind a'') \\\n a'' = a\n\ngoal (1 subgoal):\n 1. ax = a"} {"_id": "500869", "text": "proof (state)\nthis:\n order (G\\carrier := P\\) = p ^ a * k\n\ngoal (1 subgoal):\n 1. snd_sylow (G\\carrier := P\\) p a (card P div p ^ a)"} {"_id": "500870", "text": "proof (prove)\nusing this:\n qGood (qOp delta inp binp)\n liftAll (\\a. qGood a \\ qGood a #[[rho]]) inp\n liftAll (\\a. qGoodAbs a \\ qGoodAbs a $[[rho]])\n binp\n\ngoal (1 subgoal):\n 1. liftAll (\\x. qGood x #[[rho]]) inp \\\n liftAll (\\x. qGoodAbs x $[[rho]]) binp"} {"_id": "500871", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isDERIV j (fa ^\\<^sub>e n) xs =\n (if n = 0 then True else isDERIV j fa xs)"} {"_id": "500872", "text": "proof (prove)\nusing this:\n a = 0\n\ngoal (1 subgoal):\n 1. \\x. 1 \\ x \\ Im (Ln (Complex a x)) = pi / 2"} {"_id": "500873", "text": "proof (prove)\nusing this:\n \\\\ : w \\\\<^sub>B\n w \\ w\\\n ?g \\ ?f =\n (if arr ?f \\ arr ?g \\ trg\\<^sub>B ?f = src\\<^sub>B ?g\n then ?g \\\\<^sub>B ?f else null)\n arr ?f = (B.arr ?f \\ src\\<^sub>B ?f = a \\ trg\\<^sub>B ?f = a)\n\ngoal (1 subgoal):\n 1. \\\\ : fst (cod \\, cod \\) \\\n snd (cod \\,\n cod \\) \\ cod\n \\\\"} {"_id": "500874", "text": "proof (prove)\ngoal (1 subgoal):\n 1. X #= X''"} {"_id": "500875", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\complete t; sorted1 (inorder t)\\\n \\ inorder (treeD (Tree23_Map.del x t)) =\n AList_Upd_Del.del_list x (inorder t)"} {"_id": "500876", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i.\n (sterm_sem I (if i = vid1 then Var vid1 else Const 0) has_derivative\n frechet I (if i = vid1 then Var vid1 else Const 0) x)\n (at x)"} {"_id": "500877", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ANR S \\ ANR (connected_component_set S x)"} {"_id": "500878", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom 1 1 [^]\\<^bsub>mult_of R\\<^esub> CARD('a) ^ degree f = monom 1 1"} {"_id": "500879", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xa \\# dom_m N \\\n fmrestrict_set (insert xa l1) N = fmrestrict_set l1 N"} {"_id": "500880", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mersenne_mod k n < 2 ^ n + 2 ^ (2 * n + 2 - n)"} {"_id": "500881", "text": "proof (prove)\ngoal (1 subgoal):\n 1. PRE (\\s.\n True) (IF (\\s.\n s ''y''\n \\ s ''x'') THEN (''z'' ::= (\\s.\n s ''x'')) ELSE (''z'' ::= (\\s.\n s ''y'')) FI) POST (\\s.\ns ''z'' = max (s ''x'') (s ''y''))"} {"_id": "500882", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_poly complex_of_real (map_poly Im r) = 0 \\\n map_poly complex_of_real (map_poly Re r) +\n smult \\ (map_poly complex_of_real (map_poly Im r)) =\n map_poly complex_of_real (map_poly Re r)"} {"_id": "500883", "text": "proof (prove)\nusing this:\n P t\n\ngoal (1 subgoal):\n 1. \\k. k \\ set K \\ P k"} {"_id": "500884", "text": "proof (prove)\nusing this:\n pt TYPE('b) TYPE('a)\n pt TYPE('c) TYPE('a)\n fs TYPE('b) TYPE('a)\n fs TYPE('c) TYPE('a)\n at TYPE('a)\n finite Xs\n finite Ys\n Xs \\ {}\n Ys \\ {}\n\ngoal (1 subgoal):\n 1. x \\ Xs \\ Ys = (x \\ Xs \\ x \\ Ys)"} {"_id": "500885", "text": "proof (prove)\nusing this:\n (neg A IMP neg A) IMP (neg (neg A) IMP neg (neg A)) IMP A IMP neg (neg A)\n \\ boolean_axioms\n\ngoal (1 subgoal):\n 1. H \\ A \\ H \\ neg (neg A)"} {"_id": "500886", "text": "proof (prove)\nusing this:\n sorted (map fst auxlist)\n ?t1 \\ set (map fst auxlist) \\ ?t1 \\ nt\n\ngoal (1 subgoal):\n 1. sorted\n (map fst\n (map (\\(t, a1, a2).\n (t, if pos then join a1 True rel1 else a1 \\ rel1,\n if mem (nt - t) I then a2 \\ join rel2 pos a1\n else a2))\n auxlist @\n [(nt, rel1, if left I = 0 then rel2 else empty_table)]))"} {"_id": "500887", "text": "proof (prove)\nusing this:\n continuous_on ({0..1} \\ {0..1}) h\n h ` ({0..1} \\ {0..1}) \\ s\n ?x \\ {0..1} \\ h (0, ?x) = p ?x\n ?x \\ {0..1} \\ h (1, ?x) = q ?x\n ?x \\ {0..1} \\\n pathstart (h \\ Pair ?x) = pathstart p\n ?x \\ {0..1} \\\n pathfinish (h \\ Pair ?x) = pathfinish p\n\ngoal (1 subgoal):\n 1. homotopic_paths s p q"} {"_id": "500888", "text": "proof (prove)\ngoal (1 subgoal):\n 1. CU.Model wtFsym wtPsym (length \\ arOf) (length \\ parOf)\n \\ (intT any) intF intP"} {"_id": "500889", "text": "proof (prove)\nusing this:\n outstanding_refs is_volatile_Read\\<^sub>s\\<^sub>b (r # sb) \\ A\n\ngoal (1 subgoal):\n 1. outstanding_refs is_volatile_Read\\<^sub>s\\<^sub>b sb \\ A"} {"_id": "500890", "text": "proof (prove)\ngoal (1 subgoal):\n 1. uniform_limit (cball s \\)\n (\\x s. EM_remainder' n (f s) (real_of_int a) (real x))\n (\\s. EM_remainder n (f s) a) sequentially"} {"_id": "500891", "text": "proof (prove)\nusing this:\n match (t\\<^sub>1 $ t\\<^sub>2) u = Some env\n\ngoal (1 subgoal):\n 1. (\\u\\<^sub>1 u\\<^sub>2 env\\<^sub>1 env\\<^sub>2.\n \\u = app u\\<^sub>1 u\\<^sub>2;\n match t\\<^sub>1 u\\<^sub>1 = Some env\\<^sub>1;\n match t\\<^sub>2 u\\<^sub>2 = Some env\\<^sub>2;\n env = env\\<^sub>1 ++\\<^sub>f env\\<^sub>2\\\n \\ thesis) \\\n thesis"} {"_id": "500892", "text": "proof (prove)\ngoal (1 subgoal):\n 1. correctCompositionDiffLevelsSYSTEM"} {"_id": "500893", "text": "proof (prove)\ngoal (1 subgoal):\n 1. expectation_gpv' (c response) \\ 1"} {"_id": "500894", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\Ma. M = Ma \\ (\\n. count Ma n < base);\n inj_on f (set_mset M)\\\n \\ \\Ma.\n image_mset f M = Ma \\\n (\\n. count Ma n < base)"} {"_id": "500895", "text": "proof (prove)\nusing this:\n g \\ Cn 1 r_ifeq_else_diverg [f, Z, Z]\n\ngoal (1 subgoal):\n 1. eval g [x] =\n (if x \\ {x. eval r_phi [x, x] \\} then Some 0 else None)"} {"_id": "500896", "text": "proof (prove)\nusing this:\n 2 \\ ?x \\ \\ ?x < 443 / 139 * (?x / ln ?x)\n\ngoal (1 subgoal):\n 1. 0 < n \\\n 139 / 443 * (real n * ln (real n)) \\ real (nth_prime (n - 1))"} {"_id": "500897", "text": "proof (prove)\nusing this:\n prim_invar2_init Q \\ \\ prim_invar2_ctd Q \\\n prim_invar2_init Q ?\\ \\\n prim_invar2_ctd (Q' Q ?\\ u) (\\' Q ?\\ u)\n prim_invar2_init Q ?\\ \\\n T_measure2 (Q' Q ?\\ u) (\\' Q ?\\ u) < T_measure2 Q ?\\\n prim_invar2_ctd Q ?\\ \\\n prim_invar2_ctd (Q' Q ?\\ u) (\\' Q ?\\ u)\n prim_invar2_ctd Q ?\\ \\\n T_measure2 (Q' Q ?\\ u) (\\' Q ?\\ u) < T_measure2 Q ?\\\n\ngoal (1 subgoal):\n 1. prim_invar2_init (Q' Q \\ u) (\\' Q \\ u) \\\n prim_invar2_ctd (Q' Q \\ u) (\\' Q \\ u) &&&\n T_measure2 (Q' Q \\ u) (\\' Q \\ u) < T_measure2 Q \\"} {"_id": "500898", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Ord i \\ Goedel_I.HF i = W i"} {"_id": "500899", "text": "proof (prove)\nusing this:\n set (coeffs (\\(x, i)\\xis. [:- x, 1:] ^ Suc i))\n \\ \\\n\ngoal (1 subgoal):\n 1. map_poly Re (\\(x, i)\\xis. [:- x, 1:] ^ Suc i) =\n (\\(q,\n i)\\complex_roots_to_real_factorization xis. q ^ i) \\\n ((q, j)\n \\ set (complex_roots_to_real_factorization xis) \\\n irreducible q \\\n j \\ 0 \\ monic q \\ degree q \\ {1, 2})"} {"_id": "500900", "text": "proof (prove)\nusing this:\n trg g = src f\n src g = trg f\n trg g' = src f\n src g' = trg f\n \\\\' : src f \\ src f\\\n \\\\' : src f \\ g' \\\n f\\\n \\\\' : trg f \\ trg f\\\n \\\\' : f \\\n g' \\ trg f\\\n \\arr ?f; cod ?f = ?b\\ \\ ?b \\ ?f = ?f\n\ngoal (1 subgoal):\n 1. (g' \\ \\') \\\n ((g' \\ f) \\ g') \\ (\\' \\ g') =\n (g' \\ \\') \\ (\\' \\ g')"} {"_id": "500901", "text": "proof (state)\nthis:\n e \\ funcset UNIV {- 1..1}\n x = aform_val e X\n y = aform_val e Y\n\ngoal (1 subgoal):\n 1. f x y \\ Affine (F d X Y)"} {"_id": "500902", "text": "proof (prove)\nusing this:\n \\x = ?\\';\n xa = ?e1.0 \\?bop\\ ?e2.0; xb = High;\n ?\\' \\ ?e1.0 : High;\n ?\\' \\ ?e2.0 : High\\\n \\ thesis\n \\x = ?\\';\n xa = ?e1.0 \\?bop\\ ?e2.0; xb = High;\n ?\\' \\ ?e1.0 : Low;\n ?\\' \\ ?e2.0 : High\\\n \\ thesis\n x = \\\n xa = e1 \\bop\\ e2\n xb = High\n \\ \\ e1 : lev\n \\ \\ e2 : High\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "500903", "text": "proof (chain)\npicking this:\n At = Eq p"} {"_id": "500904", "text": "proof (prove)\nusing this:\n \\i u.\n k + k' \\ i \\\n (P \\ Q) (stake i \\ @- u) \\\n (\\m\\k.\n Q (\\(m \\ i - m) @- u) \\\n (\\p(p \\ i - p) @- u)))\n\ngoal (1 subgoal):\n 1. \\i.\n \\n\\i.\n \\m\\k.\n \\u.\n Q (\\(m \\ n - m) @- u) \\\n (\\p(p \\ n - p) @- u))"} {"_id": "500905", "text": "proof (prove)\nusing this:\n t \\ SPR.jkbpC\n x \\ rel_ext\n (\\uu_.\n \\t'.\n uu_ = (tFirst t', tLast t') \\\n t' \\ SPR.jkbpC \\ spr_jview a t' = spr_jview a t)\n y \\ rel_ext\n (\\uu_.\n \\t'.\n uu_ = (tFirst t', tLast t') \\\n t' \\ SPR.jkbpC \\ spr_jview a t' = spr_jview a t)\n\ngoal (1 subgoal):\n 1. (x, y)\n \\ rel_ext\n (\\((u, v), u', v').\n envObs a u = envObs a u' \\ envObs a v = envObs a v')"} {"_id": "500906", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (R \\ R') `` X \\ X \\\n SN_on (R \\ R') X =\n (SN_on (R'\\<^sup>* O R O R'\\<^sup>*) X \\ SN_on R' X)"} {"_id": "500907", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Set.is_empty (DList_set dxs) =\n (case ID CEQ('a) of\n None \\\n Code.abort STR ''is_empty DList_set: ceq = None''\n (\\_. Set.is_empty (DList_set dxs))\n | Some x \\ DList_Set.null dxs)"} {"_id": "500908", "text": "proof (prove)\nusing this:\n evaluate True env s e' (s', r)\n\ngoal (1 subgoal):\n 1. \\s' r. evaluate True env s (Raise e') (s', r)"} {"_id": "500909", "text": "proof (chain)\npicking this:\n w \\ T"} {"_id": "500910", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i.\n (f (Suc (j_of ((j_of ^^ i) 0))), f (Suc ((j_of ^^ i) 0))) \\ R"} {"_id": "500911", "text": "proof (prove)\nusing this:\n \\ys \\ set (subseqs xs); distinct xs\\\n \\ sorted_on (linord_of_list (rev xs)) ys\n y # ys \\ set (subseqs xs)\n distinct xs\n \\distinct (rev xs);\n Field (linord_of_list (rev xs)) = set (rev xs)\\\n \\ Linear_order (linord_of_list (rev xs))\n\ngoal (1 subgoal):\n 1. sorted_on (linord_of_list (rev xs)) (y # ys)"} {"_id": "500912", "text": "proof (prove)\nusing this:\n \\x\\A. x \\ t\n t \\ A\n\ngoal (1 subgoal):\n 1. extremed A (\\)"} {"_id": "500913", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\mem.\n \\\\mem.\n locally_sound_mode_use\n \\(x \\ e) \\\n annos, mds\\<^sub>2, mem\\;\n \\Stop, mds\\<^sub>2', mem'\\\n \\ loc_reach\n \\(x \\ e) \\\n annos, mds\\<^sub>2, mem\\\\\n \\ mds\\<^sub>2' \\ mds \\ annos"} {"_id": "500914", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(c, s)\\set (uset p).\n \\c\\\\<^sub>p\\<^bsup>vs\\<^esup> < (0::'a) \\\n - Itm vs (x # bs) s\n \\ \\c\\\\<^sub>p\\<^bsup>vs\\<^esup> * x \\\n (0::'a) < \\c\\\\<^sub>p\\<^bsup>vs\\<^esup> \\\n \\c\\\\<^sub>p\\<^bsup>vs\\<^esup> * x\n \\ - Itm vs (x # bs) s"} {"_id": "500915", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\l r n.\n \\k. (l, r) = (Bk \\ k, <0>) \\\n \\ is_final (steps0 (1, l, r) dither n)"} {"_id": "500916", "text": "proof (prove)\ngoal (1 subgoal):\n 1. z \\ \\\\<^sub>\\\\<^sub>0 \\\n Gamma_series z \\ (0::'a)"} {"_id": "500917", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ (if \\ = [] then e else Let \\ e) =\n (if \\ \\ \\ = [] then \\ \\ e\n else Let (\\ \\ \\) (\\ \\ e))"} {"_id": "500918", "text": "proof (state)\ngoal (1 subgoal):\n 1. [A,e,B]\\<^bsub>T\\<^esub> = [A,d,B]\\<^bsub>T\\<^esub>"} {"_id": "500919", "text": "proof (prove)\nusing this:\n x\\<^sub>C \\ (mdss ! i) AsmNoWrite\n priv_is_asm_priv mdss \\\n priv_is_guar_priv mdss \\\n new_asms_only_for_priv mdss \\\n (\\i - range var\\<^sub>C_of = {})\n x\\<^sub>C \\ - range var\\<^sub>C_of\n i < length (conc.restrict_modes mdss (- range var\\<^sub>C_of))\n length (conc.restrict_modes ?mdss ?X) = length ?mdss\n length mdss = length cms\n\ngoal (1 subgoal):\n 1. False"} {"_id": "500920", "text": "proof (prove)\nusing this:\n d' = 0 \\ e \\ 0 \\\n \\(a', b', c')\\set b.\n (a' = 0 \\ b' \\ 0) \\\n (\\(d, e, f)\\set a.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f = 0) \\\n (\\(d, e, f)\\set b.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f < 0) \\\n (\\(d, e, f)\\set c.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f \\ 0) \\\n (\\(d, e, f)\\set d.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f \\ 0)\n \\ (\\(a', b', c')\\set b.\n (a' = 0 \\ b' \\ 0) \\\n (\\(d, e, f)\\set a.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f = 0) \\\n (\\(d, e, f)\\set b.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f < 0) \\\n (\\(d, e, f)\\set c.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f \\ 0) \\\n (\\(d, e, f)\\set d.\n \\y'>- c' / b'.\n \\x\\{- c' / b'<..y'}.\n d * x\\<^sup>2 + e * x + f \\ 0))\n\ngoal (1 subgoal):\n 1. d' = 0 \\ e \\ 0 \\ False"} {"_id": "500921", "text": "proof (prove)\nusing this:\n lead_coeff\n (map2 (\\) p1 (replicate (length p1 - length p2) \\ @ p2)) =\n lead_coeff p1 \\\n lead_coeff (replicate (length p1 - length p2) \\ @ p2)\n lead_coeff p1 \\ carrier R\n length p2 < length p1\n\ngoal (1 subgoal):\n 1. lead_coeff\n (map2 (\\) p1 (replicate (length p1 - length p2) \\ @ p2)) =\n lead_coeff p1"} {"_id": "500922", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F \\ G \\ UNIV \\ A \\ B"} {"_id": "500923", "text": "proof (state)\ngoal (1 subgoal):\n 1. P"} {"_id": "500924", "text": "proof (prove)\ngoal (1 subgoal):\n 1. even (sum_list (map f xs)) = even (sum_list (map g xs))"} {"_id": "500925", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set {\\0\\<^sub>\\, a\\,\n \\1\\<^sub>\\, b\\}\n \\\\<^sub>\\ vproduct (set {[]\\<^sub>\\, 1\\<^sub>\\})\n f \\\n \\a, b\\\n \\\\<^sub>\\ f (0\\<^sub>\\) \\\\<^sub>\\\n f (1\\<^sub>\\)"} {"_id": "500926", "text": "proof (chain)\npicking this:\n SecurityInvariant_preliminaries sinvar_spec"} {"_id": "500927", "text": "proof (state)\nthis:\n C = D @ [(l, u)]\n D \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t \\ = B\n [(l, u)] \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t \\ = [(l, s')]\n\ngoal (1 subgoal):\n 1. \\l B s'.\n (l, s) =\n dual\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p\n ((l, s') \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p\n \\) \\\n prefix\n ((B \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t \\) @\n [(l, s') \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p \\])\n (A \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t \\)"} {"_id": "500928", "text": "proof (prove)\nusing this:\n example.is_valid_state_id (PeerA, 3)\n example.is_valid_state_id (PeerB, 3)\n set (example.received_messages (PeerA, 3)) =\n set (example.received_messages (PeerB, 3))\n\ngoal (1 subgoal):\n 1. example.state (PeerA, 3) = example.state (PeerB, 3)"} {"_id": "500929", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Fvars (psubst \\ txs) =\n Fvars \\ - snd ` set txs \\\n \\\n {if x \\ Fvars \\ then FvarsT t else {} |t x.\n (t, x) \\ set txs}"} {"_id": "500930", "text": "proof (prove)\nusing this:\n p = p1 @ (a, w) # (w, b) # p2\n (a, b) = (u, v) \\ (a, b) = (v, u)\n\ngoal (1 subgoal):\n 1. co_path (u, v) w ((a, w) # (w, b) # p2) = (a, b) # co_path (u, v) w p2"} {"_id": "500931", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (LEAST n. A $ i $ n \\ (0::'a))\n < (LEAST n. A $ (i + (1::'rows)) $ n \\ (0::'a))"} {"_id": "500932", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (any, replicate n False) \\ (set \\ \\ \\) n"} {"_id": "500933", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cmod (f z) \\ exp (cmod (complex_of_real pi * h z))"} {"_id": "500934", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a\\set as.\n hd_coeff a \\ 0 \\ divisor a \\ 0 \\\n Z.normal ((qe_pres\\<^sub>1 \\ hd_coeffs1) as)"} {"_id": "500935", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_of_lazy_sequence hit_bound = [None]"} {"_id": "500936", "text": "proof (prove)\ngoal (3 subgoals):\n 1. hs_dfs_\\ ` wa_initial (det_wa_wa (hs_dfs_dwa \\i post))\n \\ wa_initial (dfs_algo (hs.\\ \\i) (hs_R post))\n 2. hs_dfs_\\ ` wa_invar (det_wa_wa (hs_dfs_dwa \\i post))\n \\ wa_invar (dfs_algo (hs.\\ \\i) (hs_R post))\n 3. \\s.\n s \\ wa_invar (det_wa_wa (hs_dfs_dwa \\i post)) \\\n hs_dfs_\\ s\n \\ wa_cond\n (dfs_algo (hs.\\ \\i)\n (hs_R post)) \\\n s \\ wa_cond (det_wa_wa (hs_dfs_dwa \\i post))"} {"_id": "500937", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\t x.\n thr s t = \\(x, no_wait_locks)\\ \\\n wset s t = None \\ \\diverge t (x, shr s)"} {"_id": "500938", "text": "proof (prove)\nusing this:\n \\ lfinite P\n \\\\ lfinite ?P; lset ?P \\ S; valid_path ?P;\n path_conforms_with_strategy p ?P well_ordered_strategy\\\n \\ \\n.\n \\m\\n.\n path_strategies ?P $ n = path_strategies ?P $ m\n lset P \\ S\n valid_path P\n path_conforms_with_strategy p P well_ordered_strategy\n\ngoal (1 subgoal):\n 1. (\\n.\n (\\m.\n n \\ m \\\n path_strategies P $ n = path_strategies P $ m) \\\n thesis) \\\n thesis"} {"_id": "500939", "text": "proof (prove)\ngoal (1 subgoal):\n 1. foldr (\\xs. max (length xs)) xss 0 =\n foldr (\\x. max (length x))\n (filter (\\ys. ys \\ []) xss) 0"} {"_id": "500940", "text": "proof (prove)\ngoal (1 subgoal):\n 1. kbo_std ground_heads_var (>\\<^sub>s) \\ 0 extf\n (\\_. \\) (\\_. \\) wt_sym"} {"_id": "500941", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\<^bold>g [^] (n * int (order G)) = (\\<^bold>g [^] order G) [^] n"} {"_id": "500942", "text": "proof (prove)\nusing this:\n qGood qX\n (qX, qY) \\ {(qY, qY') |qY qY'. (asTerm qY, asTerm qY') \\ rel}\n good (asTerm ?qX) = qGood ?qX\n qGood ?qX \\ pick (asTerm ?qX) #= ?qX\n \\good ?X90; (?X90, ?Y90) \\ rel\\\n \\ good ?Y90 \\ skel ?Y90 = skel ?X90\n qGood ?qX \\ skel (asTerm ?qX) = qSkel ?qX\n\ngoal (1 subgoal):\n 1. qGood qY \\ qSkel qY = qSkel qX"} {"_id": "500943", "text": "proof (prove)\nusing this:\n p \\ q\n a \\ p\n a \\ q\n proj2_Col p q a\n\ngoal (1 subgoal):\n 1. cross_ratio p q a b = 1"} {"_id": "500944", "text": "proof (prove)\ngoal (1 subgoal):\n 1. partial_function_definitions Heap_ord Heap_lub &&&\n Heap_lub {} \\ Heap Map.empty"} {"_id": "500945", "text": "proof (prove)\nusing this:\n list_all (eval e f g) G \\\n eval (e\\0:z\\) f g a\n G \\ Exists a\n \\e f g.\n list_all (eval e f g) G \\ eval e f g (Exists a)\n a[App n []/0] # G \\ b\n \\e f g.\n list_all (eval e f g) (a[App n []/0] # G) \\\n eval e f g b\n list_all (\\p. n \\ params p) G\n n \\ params a\n n \\ params b\n\ngoal (1 subgoal):\n 1. list_all (eval e (f(n := \\x. z)) g) G \\\n eval e (f(n := \\x. z)) g b"} {"_id": "500946", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x = 0"} {"_id": "500947", "text": "proof (chain)\npicking this:\n j = i \\ i \\ set js \\ i \\ j \\ i \\ set js"} {"_id": "500948", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (?fi1, cp_array)\n \\ (array_assn\n nat_assn)\\<^sup>k \\\\<^sub>a array_assn nat_assn"} {"_id": "500949", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\Obj\\\n\ngoal (1 subgoal):\n 1. (\\ \\\\<^sub>T\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\\<^sub>-\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\n dghm_id \\)\\NTMap\\\\a\\ =\n \\\\NTMap\\\\a\\"} {"_id": "500950", "text": "proof (prove)\nusing this:\n te q (i + 1) \\ t\n correct p t\n correct q t\n \\?s \\ ?t; correct ?p ?t; correct ?q ?t\\\n \\ \\IC ?p ?i ?t - IC ?q ?j ?t\\\n \\ \\IC ?p ?i ?s - IC ?q ?j ?s\\ +\n 2 * \\ * (?t - ?s)\n\ngoal (1 subgoal):\n 1. \\IC p i t - IC q (i + 1) t\\\n \\ \\IC p i (te q (i + 1)) - IC q (i + 1) (te q (i + 1))\\ +\n 2 * \\ * (t - te q (i + 1))"} {"_id": "500951", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F' = hensel_fpxs1 * hensel_fpxs2"} {"_id": "500952", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (uncurry2 full_checker_l_impl,\n uncurry2 (\\spec A _. PAC_checker_specification spec A))\n \\ full_poly_assn\\<^sup>k *\\<^sub>a\n full_poly_input_assn\\<^sup>d *\\<^sub>a\n fully_pac_assn\\<^sup>k \\\\<^sub>a hr_comp\n (code_status_assn \\\\<^sub>a\n full_vars_assn \\\\<^sub>a\n hr_comp polys_assn\n (\\nat_rel,\n sorted_poly_rel O mset_poly_rel\\fmap_rel))\n {((st, G), st', G').\n st = st' \\\n (st \\ FAILED \\\n (G, G') \\ Id \\\\<^sub>r polys_rel)}"} {"_id": "500953", "text": "proof (prove)\ngoal (1 subgoal):\n 1. enc_respects_binary_pred Pred = rel_respects_binary_pred indRelR Pred"} {"_id": "500954", "text": "proof (chain)\npicking this:\n p = smult c (\\(x, i)\\xis. [:- x, 1:] ^ Suc i)"} {"_id": "500955", "text": "proof (prove)\nusing this:\n (?x = ?y) =\n (Ree ?x = Ree ?y \\\n Im1 ?x = Im1 ?y \\\n Im2 ?x = Im2 ?y \\\n Im3 ?x = Im3 ?y \\\n Im4 ?x = Im4 ?y \\\n Im5 ?x = Im5 ?y \\ Im6 ?x = Im6 ?y \\ Im7 ?x = Im7 ?y)\n (\\) = (*)\n ln (1::?'a) = (0::?'a)\n of_real ?x \\ (1::?'a) = ?x\n\ngoal (1 subgoal):\n 1. Octonions.cnj (r *\\<^sub>R x) = r *\\<^sub>R Octonions.cnj x"} {"_id": "500956", "text": "proof (chain)\npicking this:\n dim_col x = dim_row x\n \\x * ?B = 1\\<^sub>m (dim_row x);\n ?B * x = 1\\<^sub>m (dim_row ?B)\\\n \\ \\A'.\n y ** A' = mat (1::'a) \\ A' ** y = mat (1::'a)\n \\y ** ?A' = mat (1::'a); ?A' ** y = mat (1::'a)\\\n \\ \\B.\n x * B = 1\\<^sub>m (dim_row x) \\\n B * x = 1\\<^sub>m (dim_row B)"} {"_id": "500957", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\1 \\\\<^sub>\\ \\2,\n \\1' \\\\<^sub>\\ \\2' \\\\<^sub>C\n map_converter id id (map_sum f1 f2) (map_sum g1 g2) 1\\<^sub>C \\"} {"_id": "500958", "text": "proof (prove)\ngoal (1 subgoal):\n 1. B abs_summable_on I"} {"_id": "500959", "text": "proof (prove)\nusing this:\n cmod (croot n x) ^ n = cmod y ^ n\n n \\ 0\n\ngoal (1 subgoal):\n 1. cmod (croot n x) = cmod y"} {"_id": "500960", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (sfilter P s = x ## s') =\n (P x \\\n ((\\xs. \\ holds P xs) suntil\n (\\xs.\n HLD {x} xs \\ nxt (\\s. sfilter P s = s') xs))\n s)"} {"_id": "500961", "text": "proof (prove)\ngoal (1 subgoal):\n 1. refl Le"} {"_id": "500962", "text": "proof (prove)\nusing this:\n \\r k (1 $ 0) ^ k = 1 $ 0; r k (a $ 0) ^ k = a $ 0;\n 1 $ 0 \\ (0::'a); a $ 0 \\ (0::'a)\\\n \\ (r k (1 $ 0 / a $ 0) = r k (1 $ 0) / r k (a $ 0)) =\n (fps_radical r k (1 / a) =\n fps_radical r k 1 / fps_radical r k a)\n r k (a $ 0) ^ k = a $ 0\n r k (1::'a) ^ k = (1::'a)\n a $ 0 \\ (0::'a)\n\ngoal (1 subgoal):\n 1. (r k (inverse (a $ 0)) = r k (1::'a) / r k (a $ 0)) =\n (fps_radical r k (inverse a) = fps_radical r k 1 / fps_radical r k a)"} {"_id": "500963", "text": "proof (prove)\nusing this:\n card {1, 2} = 2\n\ngoal (1 subgoal):\n 1. 2 \\ card (Units (residue_ring (p ^ n)))"} {"_id": "500964", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.iso (\\ (local.dom \\t\\))"} {"_id": "500965", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 2 * spmf (P1_game_alt m1 m2 D) True - 1 =\n spmf\n (R1 m1 m2 \\\n (\\rview.\n D rview \\ (\\b'. return_spmf (True = b'))))\n True -\n (1 -\n spmf\n (funct m1 m2 \\\n (\\(out1, out2).\n S1 m1 out1 \\\n (\\sview.\n D sview \\ (\\b'. return_spmf (False = b')))))\n True)"} {"_id": "500966", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fUnionM ((\\x. ask (\\r. run_nondet (f x r))) |`| A) =\n ask (\\r. fUnionM ((\\x. run_nondet (f x r)) |`| A))"} {"_id": "500967", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\StableNoDecomp.stable_pair_on UNIV (XD, XH);\n XD = {Xd1, Xd1', Xd2, Xd2'} \\\n XD = {Xd1, Xd1', Xd2} \\\n XD = {Xd1, Xd1', Xd2'} \\\n XD = {Xd1, Xd1'} \\\n XD = {Xd1, Xd2, Xd2'} \\\n XD = {Xd1, Xd2} \\\n XD = {Xd1, Xd2'} \\\n XD = {Xd1} \\\n XD = {Xd1', Xd2, Xd2'} \\\n XD = {Xd1', Xd2} \\\n XD = {Xd1', Xd2'} \\\n XD = {Xd1'} \\\n XD = {Xd2, Xd2'} \\ XD = {Xd2} \\ XD = {Xd2'} \\ XD = {};\n XH = {Xd1, Xd1', Xd2, Xd2'} \\\n XH = {Xd1, Xd1', Xd2} \\\n XH = {Xd1, Xd1', Xd2'} \\\n XH = {Xd1, Xd1'} \\\n XH = {Xd1, Xd2, Xd2'} \\\n XH = {Xd1, Xd2} \\\n XH = {Xd1, Xd2'} \\\n XH = {Xd1} \\\n XH = {Xd1', Xd2, Xd2'} \\\n XH = {Xd1', Xd2} \\\n XH = {Xd1', Xd2'} \\\n XH = {Xd1'} \\\n XH = {Xd2, Xd2'} \\\n XH = {Xd2} \\ XH = {Xd2'} \\ XH = {}\\\n \\ XD = {} \\ XH = {Xd1, Xd1', Xd2, Xd2'}"} {"_id": "500968", "text": "proof (prove)\nusing this:\n x \\ orbit f a\n f x \\ orbit f b\n y \\ segment f b (f x)\n\ngoal (1 subgoal):\n 1. x \\ orbit f b"} {"_id": "500969", "text": "proof (prove)\nusing this:\n f = Insert x\n\ngoal (1 subgoal):\n 1. t\\<^sub>s f s + \\\\<^sub>d (nxt f s) - \\\\<^sub>d s\n \\ (case s of\n (t, uu_) \\\n case f of\n Insert x \\ (6 * e + 2) * real (height t + 1) + 1\n | Delete x \\\n (6 * e + 1) * real (height t) + 4 / cd + 4)"} {"_id": "500970", "text": "proof (prove)\nusing this:\n \\p2\\carrier.\n \\\\.\n \\ \\ atlas \\\n codomain \\ = ball (0::'b) 3 \\\n (\\c\\I. domain \\ \\ \\ c) \\\n (\\j.\n p2 \\ V j \\\n domain \\ \\ V j) \\\n p2 \\ domain \\ \\ apply_chart \\ p2 = (0::'b)\n\ngoal (1 subgoal):\n 1. (\\\\.\n \\\\p.\n p \\ carrier \\\n codomain (\\ p) = ball (0::'b) 3;\n \\p.\n p \\ carrier \\\n \\c\\I. domain (\\ p) \\ \\ c;\n \\p j.\n p \\ V j \\ domain (\\ p) \\ V j;\n \\p j.\n p \\ carrier \\ p \\ domain (\\ p);\n \\p.\n p \\ carrier \\\n apply_chart (\\ p) p = (0::'b);\n \\p.\n p \\ carrier \\ \\ p \\ atlas\\\n \\ thesis) \\\n thesis"} {"_id": "500971", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\C'. A' B' Perp C' B' \\ thesis) \\\n thesis"} {"_id": "500972", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a\\#A. image_mset (Pair a) (B a + C a)) =\n (\\a\\#A. image_mset (Pair a) (B a)) +\n (\\a\\#A. image_mset (Pair a) (C a))"} {"_id": "500973", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i\\elts n.\n (xs #\\<^sub>\\ x)\\i\\ \\\\<^sub>\\ A i"} {"_id": "500974", "text": "proof (state)\nthis:\n Well_order s\n Well_order t\n embedS s t f\n\ngoal (1 subgoal):\n 1. r *o s x \\ F; y \\ F\\\n \\ (if x \\ y \\ F then top else bot) =\n (if x \\ F then top else bot) \\\n (if y \\ F then top else bot)"} {"_id": "500976", "text": "proof (prove)\ngoal (1 subgoal):\n 1. insert (u IN Eats t z) H \\ C"} {"_id": "500977", "text": "proof (prove)\nusing this:\n type_wf h2\n \\\\h h' w.\n \\w \\ set_val_locs new_character_data_ptr;\n h \\ w \\\\<^sub>h h'\\\n \\ type_wf h \\ type_wf h';\n reflp (\\h h'. type_wf h \\ type_wf h');\n transp (\\h h'. type_wf h \\ type_wf h')\\\n \\ type_wf h2 \\ type_wf h3\n\ngoal (1 subgoal):\n 1. type_wf h3"} {"_id": "500978", "text": "proof (prove)\nusing this:\n v \\ V\n polymap (pCons a p) v =\n lincomb' (coeffs (pCons a p))\n (v #\n map (\\S. S v)\n (map ((\\) T) (map endpow [0..S. S v)\n (id \\ V # map ((\\) T) (map endpow [0..r args. 0 < rec_exec r args)\n [Cn 3 rec_disj\n [Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 0) [recf.id 3 0]],\n Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 1) [recf.id 3 0]]],\n Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 2) [recf.id 3 0]],\n Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 3) [recf.id 3 0]],\n Cn 3 rec_less [Cn 3 (constn 3) [recf.id 3 0], recf.id 3 2]]))\n [p, r, a]"} {"_id": "500980", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cost\n (if i = 0 then ((True, Suc i, inc_state state), 0)\n else let (state', cost_add) = (state', c_add); di = d_state state' i;\n dsi = d_state state' (Suc i); dim1 = d_state state' (i - 1);\n (num, denom) = quotient_of \\;\n cond = di * di * denom \\ num * dim1 * dsi; local_cost = 5\n in if cond\n then ((True, Suc i, inc_state state'), local_cost + cost_add)\n else case basis_reduction_swap_cost i state' of\n (res, cost_swap) \\\n (res, local_cost + cost_swap + cost_add))\n \\ body_cost"} {"_id": "500981", "text": "proof (prove)\nusing this:\n \\n.\n Bernstein_Poly x p c d = Bernstein_Poly n p c d \\ n \\ p\n\ngoal (1 subgoal):\n 1. Bernstein_Poly x p c d\n \\ poly_vs.span {Bernstein_Poly n p c d |n. n \\ p}"} {"_id": "500982", "text": "proof (state)\nthis:\n \\A\\set (map wordinterval_to_set V).\n \\a1\\A. \\a2\\A. same_fw_behaviour_one a1 a2 c rs\n\ngoal (1 subgoal):\n 1. V \\ set (groupWIs c rs) \\\n \\w1\\set V.\n \\w2\\set V.\n \\a1\\wordinterval_to_set w1.\n \\a2\\wordinterval_to_set w2.\n same_fw_behaviour_one a1 a2 c rs"} {"_id": "500983", "text": "proof (prove)\ngoal (2 subgoals):\n 1. interfree_aux (Some Mutator, \\False\\, Some Collector)\n 2. OG_Hoare.post (Some Mutator, \\False\\) \\\n (OG_Hoare.post (Some Collector, \\False\\) \\\n \\\\i.\n False \\\n \\(\\)\n (OG_Hoare.post ((Collector\n \\False\\\n \\\n Mutator\n \\False\\) !\n i))\\)\n \\ \\False\\"} {"_id": "500984", "text": "proof (prove)\nusing this:\n Arr v1 \\\n \\<^bold>\\C.dom (un_Prim t1)\\<^bold>\\ = Cod v1\n t1 = \\<^bold>\\un_Prim t1\\<^bold>\\ \\\n C.arr (un_Prim t1) \\ Diag t1\n Diag t2 \\ t2 \\ \\<^bold>\\\n v1 = \\<^bold>\\un_Prim v1\\<^bold>\\ \\\n C.arr (un_Prim v1) \\ Diag v1\n Diag v2 \\ v2 \\ \\<^bold>\\\n Diag u\n Diag w\n Seq u w\n u \\ \\<^bold>\\\n \\Diag ?t; Diag ?u; Dom ?t = Cod ?u\\\n \\ Diag\n (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u)\n \\Diag ?t; Diag ?u; Dom ?t = Cod ?u\\\n \\ Dom (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u) =\n Dom ?u\n \\Diag ?t; Diag ?u; Dom ?t = Cod ?u\\\n \\ Cod (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u) =\n Cod ?t\n \\Diag ?t; Diag ?u\\\n \\ Diag\n (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u)\n \\Diag ?t; Diag ?u\\\n \\ Dom (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u) =\n Dom ?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n Dom ?u\n \\Diag ?t; Diag ?u\\\n \\ Cod (?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?u) =\n Cod ?t \\<^bold>\\\\<^bold>\\\\<^bold>\\\n Cod ?u\n ?t \\ \\<^bold>\\ \\\n \\<^bold>\\?f\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n ?t =\n \\<^bold>\\?f\\<^bold>\\ \\<^bold>\\ ?t\n\ngoal (1 subgoal):\n 1. u \\<^bold>\\\\<^bold>\\\\<^bold>\\ w \\\n \\<^bold>\\"} {"_id": "500985", "text": "proof (prove)\nusing this:\n \\,\\,p\\\\<^sub>g \\r #\n rs, s\\ \\ t\n\ngoal (1 subgoal):\n 1. \\\\ti.\n \\\\,\\,p\\\\<^sub>g \\[r], s\\ \\ ti;\n \\,\\,p\\\\<^sub>g \\rs, ti\\ \\ t;\n no_matching_Goto \\ p [r]\\\n \\ thesis;\n \\\\,\\,p\\\\<^sub>g \\[r], s\\ \\ t;\n \\ no_matching_Goto \\ p [r]\\\n \\ thesis\\\n \\ thesis"} {"_id": "500986", "text": "proof (prove)\nusing this:\n incidence_system (?f ` \\) (blocks_image \\ ?f)\n finite \\\n\ngoal (1 subgoal):\n 1. finite_incidence_system (f ` \\) (blocks_image \\ f)"} {"_id": "500987", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p \\ projr f = projr (p \\ f)"} {"_id": "500988", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Gromov_hyperbolic_subset delta A"} {"_id": "500989", "text": "proof (prove)\nusing this:\n gen infs cs (u @- a ## v)\n\ngoal (1 subgoal):\n 1. gen infs cs v"} {"_id": "500990", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\p. p \\ carrier (K [X]) \\ h p = local.eval p x"} {"_id": "500991", "text": "proof (prove)\ngoal (1 subgoal):\n 1. while_option (uninst_code.cond uses phis g)\n (uninst.step (usesOf \\ uses)\n (\\g. Mapping.lookup (phis g)) g)\n ((usesOf \\ uses) g, Mapping.lookup (phis g)) =\n map_option (\\a. map_prod usesOf Mapping.lookup (fst a))\n (while_option\n (\\((u, p), triv_phis, nodes_of_uses, nodes_of_phis).\n \\ Set.is_empty triv_phis)\n (\\((u, p), triv_phis, nodes_of_uses, nodes_of_phis).\n case uninst_code.step_codem (\\_. u) (\\_. p) g\n (Max triv_phis)\n (uninst_code.substitution_code (\\g. p) g\n (Max triv_phis))\n nodes_of_uses nodes_of_phis of\n (u', p') \\\n ((u', p'),\n uninst_code.triv_phis' (\\_. p') g (Max triv_phis)\n triv_phis nodes_of_phis,\n uninst_code.nodes_of_uses' g (Max triv_phis)\n (uninst_code.substitution_code (\\g. p) g\n (Max triv_phis))\n (Mapping.keys (uninst_code.ssa.phidefNodes phis g))\n nodes_of_uses,\n uninst_code.nodes_of_phis' g (Max triv_phis)\n (uninst_code.substitution_code (\\g. p) g\n (Max triv_phis))\n nodes_of_phis))\n ((uses g, phis g), ssa.trivial_phis g,\n uninst_code.ssa.useNodes_of uses g,\n uninst_code.ssa.phiNodes_of phis g))"} {"_id": "500992", "text": "proof (prove)\nusing this:\n arr \\\n arr \\\n src \\ = trg \\\n arr \\'\n arr \\'\n src \\' = trg \\'\n seq \\' \\\n seq \\' \\\n Dom \\' = Cod \\ \\ Dom \\' = Cod \\\n Dom (\\' \\ \\) = Dom \\ \\\n Dom (\\' \\ \\) = Dom \\ \\\n Cod (\\' \\ \\) = Cod \\' \\\n Cod (\\' \\ \\) = Cod \\'\n C.span\n (Chn (\\' \\ \\) \\\n \\

\\<^sub>1[\\.dom.leg0, \\.dom.leg1])\n (Chn (\\' \\ \\) \\\n \\

\\<^sub>0[\\.dom.leg0, \\.dom.leg1])\n ?\\ \\ ?\\ \\\n if arrow_of_spans (\\) ?\\ \\\n arrow_of_spans (\\) ?\\ \\ Dom ?\\ = Cod ?\\\n then \\Chn = Chn ?\\ \\ Chn ?\\, Dom = Dom ?\\,\n Cod = Cod ?\\\\\n else Null\n seq ?\\ ?\\ =\n (arrow_of_spans (\\) ?\\ \\\n arrow_of_spans (\\) ?\\ \\ Dom ?\\ = Cod ?\\)\n arr ?\\ = arrow_of_spans (\\) ?\\\n \\arr ?\\; arr ?\\; src ?\\ = trg ?\\\\\n \\ \\chine_hcomp ?\\\n ?\\ : Leg0\n (Dom ?\\) \\\\\n Leg1\n (Dom ?\\) \\\\<^sub>C Leg0 (Cod ?\\) \\\\\n Leg1 (Cod ?\\)\\\n \\arr ?\\; arr ?\\; src ?\\ = trg ?\\\\\n \\ C.commutative_square (Leg0 (Cod ?\\))\n (Leg1 (Cod ?\\))\n (Chn ?\\ \\\n \\

\\<^sub>1[Leg0 (Dom ?\\), Leg1 (Dom ?\\)])\n (Chn ?\\ \\\n \\

\\<^sub>0[Leg0 (Dom ?\\), Leg1 (Dom ?\\)])\n seq ?\\ ?\\ \\\n src (?\\ \\ ?\\) = src ?\\ \\ src ?\\\n seq ?\\ ?\\ \\\n trg (?\\ \\ ?\\) = trg ?\\ \\ trg ?\\\n\ngoal (1 subgoal):\n 1. C.dom \\'.cod.leg0 =\n C.cod\n (Chn (\\' \\ \\) \\\n \\

\\<^sub>1[\\.dom.leg0, \\.dom.leg1])"} {"_id": "500993", "text": "proof (prove)\ngoal (1 subgoal):\n 1. insert (OrdP (Var i)) H \\ A"} {"_id": "500994", "text": "proof (prove)\ngoal (1 subgoal):\n 1. alw (state_eq None) j"} {"_id": "500995", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (option2list k) = option2set k"} {"_id": "500996", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cxt_to_stmt E c = Stop \\ c = Stop \\ E = []"} {"_id": "500997", "text": "proof (prove)\nusing this:\n rank y n = Some i\n rank x n = Some j\n stable_rank x j\n\ngoal (1 subgoal):\n 1. i \\ j \\ stable_rank y i"} {"_id": "500998", "text": "proof (prove)\ngoal (1 subgoal):\n 1. FF (C.VV.comp \\\\ \\\\) =\n (F (fst \\\\) \\\\<^sub>D F (fst \\\\),\n F (snd \\\\) \\\\<^sub>D F (snd \\\\))"} {"_id": "500999", "text": "proof (prove)\nusing this:\n \\i. G'.L (r i) (w i)\n\ngoal (1 subgoal):\n 1. \\r. is_acc_run r \\ (\\i. L (r i) (w i))"} {"_id": "501000", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ns'' ms'' z.\n \\old.pathsConverge g n ns'' m ms'' z;\n prefix ns'' ((ns @ tl ns') @ [defNode g p]);\n prefix ms'' ((ms @ tl ms') @ [defNode g p])\\\n \\ thesis) \\\n thesis"} {"_id": "501001", "text": "proof (prove)\nusing this:\n \\i. 0 < \\ i\n\ngoal (1 subgoal):\n 1. (\\i. \\i i)))\n \\ eexp (\\n. - ereal (\\ n))"} {"_id": "501002", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x y.\n wordinterval_CIDR_split1 rs = (Some x, y) \\\n thesis) \\\n thesis"} {"_id": "501003", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Max_by fst (ran (process_mru vs |` Q)) = Max_by fst (vote_set vs Q)"} {"_id": "501004", "text": "proof (prove)\ngoal (1 subgoal):\n 1. in_language\n (subst\n \\init = (),\n prod = \\_. {|[], [\\, \\, S, S, \\]|}\\\n {|[Inr (init\n \\init = (),\n prod =\n \\_.\n {|[], [\\, \\, S, S, \\]|}\\)]|})\n []"} {"_id": "501005", "text": "proof (prove)\ngoal (1 subgoal):\n 1. uint (w >>> n) = take_bit LENGTH('a) (sint w div 2 ^ n)"} {"_id": "501006", "text": "proof (prove)\nusing this:\n \\j\\2 * d. lose (t + j + (3 + k)) = [False]\n\ngoal (1 subgoal):\n 1. \\j\\2 * d. lose (t + 3 + k + j) = [False]"} {"_id": "501007", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ u \\ x \\ monad u"} {"_id": "501008", "text": "proof (prove)\ngoal (1 subgoal):\n 1. diameter (ball a r) = 2 * r"} {"_id": "501009", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 \\ X \\\n (x \\ (\\x. x - Suc 0) ` X) = (Suc x \\ X)"} {"_id": "501010", "text": "proof (prove)\nusing this:\n z \\ (\\u\\A\\<^sup>\\.\n \\v\\A\\<^sup>\\. {u @@ v})\n\ngoal (1 subgoal):\n 1. (\\u v.\n \\u \\ A\\<^sup>\\; v \\ A\\<^sup>\\;\n z = u @@ v\\\n \\ thesis) \\\n thesis"} {"_id": "501011", "text": "proof (chain)\npicking this:\n C =m\n \\\\<^sub># (factors_of_factor_tree l) *\n \\\\<^sub># (factors_of_factor_tree r)"} {"_id": "501012", "text": "proof (prove)\ngoal (1 subgoal):\n 1. punit.monom_mult (lookup q 0) 0\n (subst_pp\n (\\x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0) 0) +\n (\\t\\keys q - {0}.\n monomial (lookup q t) 0 *\n subst_pp\n (\\x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0)\n t) =\n monomial (lookup q 0) 0 +\n (\\t\\keys q - {0}.\n monomial (lookup q t) 0 *\n subst_pp\n (\\x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0) t)"} {"_id": "501013", "text": "proof (prove)\ngoal (1 subgoal):\n 1. faccs \\ t = {q. accs \\ t q}"} {"_id": "501014", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P.events =\n sigma_sets UNIV\n {Pi\\<^sub>E UNIV X |X.\n (\\i. X i \\ sets borel) \\\n finite {i. X i \\ UNIV}}"} {"_id": "501015", "text": "proof (prove)\nusing this:\n evalUni F1 x =\n (\\l\\set (map (list_conj_Uni \\ map AtomUni) (DNFUni F1)).\n evalUni l x)\n evalUni F2 x =\n (\\l\\set (map (list_conj_Uni \\ map AtomUni) (DNFUni F2)).\n evalUni l x)\n\ngoal (1 subgoal):\n 1. (evalUni F1 x \\ evalUni F2 x) =\n (\\l\\set (map (list_conj_Uni \\ map AtomUni)\n (DNFUni F1)) \\\n set (map (list_conj_Uni \\ map AtomUni)\n (DNFUni F2)).\n evalUni l x)"} {"_id": "501016", "text": "proof (prove)\nusing this:\n class P C = \\(D, fs, ms)\\\n\ngoal (1 subgoal):\n 1. (\\FDTs'.\n FDTs = map (\\(F, b, T). ((F, C), b, T)) fs @ FDTs' \\\n map_of FDTs' (F, C) = None \\\n thesis) \\\n thesis"} {"_id": "501017", "text": "proof (prove)\nusing this:\n set xs = A\n distinct xs\n Linorder_Relations.sorted_wrt R xs\n length xs = card A\n\ngoal (1 subgoal):\n 1. bij_betw (linorder_rank R A) A {.. paths t \\ x # xs \\ paths t'\n\ngoal (1 subgoal):\n 1. x # xs \\ paths (t \\\\ t')"} {"_id": "501019", "text": "proof (prove)\nusing this:\n ?i \\ I \\ y ?i \\ S ?i\n x \\ Pi\\<^sub>E J S\n J \\ I\n J \\ (I - J) = {} \\\n (merge J (I - J) (x, y) \\ Pi\\<^sub>E (J \\ (I - J)) S) =\n (x \\ Pi J S \\ y \\ Pi (I - J) S)\n x \\ A\n\ngoal (1 subgoal):\n 1. merge J (I - J) (x, y)\n \\ (\\x. restrict x J) -` A \\ Pi\\<^sub>E I S"} {"_id": "501020", "text": "proof (prove)\nusing this:\n n < length ias\n\ngoal (1 subgoal):\n 1. interrupt_action_ok' (redT_updIs (redT_updI is ia) (take n ias))\n (ias ! n)"} {"_id": "501021", "text": "proof (state)\ngoal (1 subgoal):\n 1. set_iterator (set_iterator_union it_a it_b) (S_a \\ S_b)"} {"_id": "501022", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f ^^ k) x = x \\ (f ^^ l) x = (f ^^ (l mod k)) x"} {"_id": "501023", "text": "proof (prove)\nusing this:\n \\N.\n \\m\\N.\n \\n\\N. dist (X m $ ?i) (X n $ ?i) < r / real CARD('n)\n\ngoal (1 subgoal):\n 1. \\i.\n \\m\\N i.\n \\n\\N i. dist (X m $ i) (X n $ i) < r / real CARD('n)"} {"_id": "501024", "text": "proof (state)\nthis:\n \\x\\E - E'.\n \\ (forest\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = insert x E'\\ \\\n subgraph\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = insert x E'\\\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = E\\)\n\ngoal (1 subgoal):\n 1. spanning_forest\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = E'\\\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = E\\ \\\n s.SpanningForest E'"} {"_id": "501025", "text": "proof (prove)\nusing this:\n proj.quorum (I - S)\n proj.quorum U\n\ngoal (1 subgoal):\n 1. False"} {"_id": "501026", "text": "proof (prove)\nusing this:\n \\ isVal e\n heap_upds_ok (\\, S)\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501027", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(f ` R `` {x}) \\ \\{f y |y. (x, y) \\ R}"} {"_id": "501028", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homeomorphism (Collect (chart_basis_domainP b)) UNIV (chart_basis b)\n local.chart_basis_inv"} {"_id": "501029", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F \\ \\ (range f) \\\n \\n. F \\ f n"} {"_id": "501030", "text": "proof (prove)\nusing this:\n \\n'. \\' n' \\ local.return\n ?x \\ local.nodes \\\n \\\\ n.\n is_path \\ \\ \\ 0 = ?x \\ \\ n = local.return\n is_path \\\n is_path ?\\ \\ ?\\ ?k \\ local.nodes\n\ngoal (1 subgoal):\n 1. (\\\\l n.\n \\is_path \\l; \\ k = \\l 0;\n \\l n = local.return\\\n \\ thesis) \\\n thesis"} {"_id": "501031", "text": "proof (prove)\nusing this:\n f \\ h holomorphic_on S\n \\z\\S. (f \\ h) z \\ 0\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501032", "text": "proof (prove)\nusing this:\n a \\ c\n b \\ c\n\ngoal (1 subgoal):\n 1. - inf (- a) (- b) \\ c"} {"_id": "501033", "text": "proof (state)\nthis:\n unit p = (fst (unit p), fst (snd (unit p)), snd (snd (unit p)))\n\ngoal (1 subgoal):\n 1. (\\x. Ifm (real_of_int x # bs) p) = Ifm bs (cooper p) \\\n qfree (cooper p)"} {"_id": "501034", "text": "proof (prove)\nusing this:\n \\p \\ b\\ S \\p\\\\<^sub>u\n\ngoal (1 subgoal):\n 1. \\p\\ while\\<^sup>\\ b do S od\n \\\\ b \\ p\\\\<^sub>u"} {"_id": "501035", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum power2 {..n} = (2 * n ^ 3 + 3 * n\\<^sup>2 + n) div 6"} {"_id": "501036", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\l s.\n pre C1 Q l s \\\n preList upds (IF b THEN C1 ELSE C2) l s \\\n bval b s \\\n pre C1 Q l s \\ preList upds C1 l s\n 2. \\\\<^sub>1 {\\l s.\n pre C1 Q l s \\ preList upds C1 l s}\n strip C1\n { time\n C1 \\ \\l s.\n Q l s \\ postList upds l s}"} {"_id": "501037", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (init_fin_descend_thr ts t = None) = (ts t = None)"} {"_id": "501038", "text": "proof (prove)\nusing this:\n s \\ ball z \\\n t \\ {real_of_int a..real x}\n 0 < \\\n ball z \\ \\ U\n\ngoal (1 subgoal):\n 1. ((\\z. f z t) has_field_derivative f' s t)\n (at s within ball z \\)"} {"_id": "501039", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finprod G f A \\ carrier G"} {"_id": "501040", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\g' n v vs v'.\n \\p_g g' (n, v) = Some vs; v' \\ set vs; g' = g\\\n \\ var g' v' = var g' v\n 2. \\g' n v vs v'.\n \\p_g g' (n, v) = Some vs; v' \\ set vs;\n g' \\ g\\\n \\ var g' v' = var g' v\n 3. \\n g v v'.\n \\n \\ set (\\n g); v \\ step.allDefs g n;\n v' \\ step.allDefs g n; v \\ v'\\\n \\ var g v' \\ var g v\n 4. \\u p g.\n \\CFG_SSA_Transformed \\e \\n invar inEdges' Entry\n oldDefs oldUses defs u p var;\n CFG_SSA_wf_base.redundant \\n inEdges' defs u p g\\\n \\ chooseNext_all (u g) (p g) g \\ dom (p g) \\\n CFG_SSA_wf_base.trivial \\n inEdges' defs u p\n g (snd (chooseNext_all (u g) (p g) g))"} {"_id": "501041", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A $ from_nat c $ from_nat c dvd\n diagonal_to_Smith_i [i + 1..1 ===> B\\<^sub>1 ===> B\\<^sub>1) ===>\n B\\<^sub>1 ===>\n (B\\<^sub>1 ===> B\\<^sub>1 ===> B\\<^sub>1) ===>\n (B\\<^sub>1 ===> B\\<^sub>1) ===>\n ((=) ===> B\\<^sub>1 ===> B\\<^sub>1) ===>\n (B\\<^sub>2 ===> B\\<^sub>2 ===> B\\<^sub>2) ===>\n B\\<^sub>2 ===>\n (B\\<^sub>2 ===> B\\<^sub>2 ===> B\\<^sub>2) ===>\n (B\\<^sub>2 ===> B\\<^sub>2) ===>\n ((=) ===> B\\<^sub>2 ===> B\\<^sub>2) ===> (=))\n (\\plus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n zero\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n minus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n uminus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1 scale\\<^sub>1.\n vector_space_pair_ow (Collect (Domainp B\\<^sub>1))\n plus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n zero\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n minus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1\n uminus\\<^sub>V\\<^sub>S\\<^sub>_\\<^sub>1 scale\\<^sub>1\n (Collect (Domainp B\\<^sub>2)))\n vector_space_pair_with"} {"_id": "501043", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sv \\ atoms\n \\<^bold>\\map\n(\\(x, y).\n encode_negative_transition_frame_axiom (\\ prob_with_noop abs_prob )\n x y)\n(List.product [0.. prob_with_noop abs_prob )\\<^sub>\\)) \\\n atoms\n \\<^bold>\\map\n(\\(x, y).\n encode_positive_transition_frame_axiom (\\ prob_with_noop abs_prob )\n x y)\n(List.product [0.. prob_with_noop abs_prob )\\<^sub>\\));\n t < Suc h\\\n \\ valid_state_var sv"} {"_id": "501044", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (uncurry2 ?f3, uncurry2 remap_polys_l2)\n \\ poly_assn\\<^sup>k *\\<^sub>a\n (hs.assn string_assn)\\<^sup>d *\\<^sub>a\n polys_assn_input\\<^sup>d \\\\<^sub>a status_assn\n raw_string_assn \\\\<^sub>a\n hs.assn string_assn \\\\<^sub>a polys_assn"} {"_id": "501045", "text": "proof (prove)\nusing this:\n B.internal_equivalence g f (B.inv \\') (B.inv \\)\n B.equivalence_map ?f \\ B.is_left_adjoint ?f\n B.equivalence_map ?f \\\n \\g \\ \\.\n B.internal_equivalence ?f g \\ \\\n \\g : local.Cod\n F \\\\<^sub>B local.Dom F\\\n\ngoal (1 subgoal):\n 1. \\g : local.Cod\n F \\\\<^sub>B local.Dom\n F\\ \\\n B.is_left_adjoint g \\\n \\g\\\\<^sub>B = \\g\\\\<^sub>B"} {"_id": "501046", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a_kernel R (image_ring f R) f = I"} {"_id": "501047", "text": "proof (prove)\ngoal (1 subgoal):\n 1. insert (SeqWRP s k' y) H \\\n SyntaxN.Ex y' (SeqWRP s k (Var y') AND y EQ Q_Succ (Var y'))"} {"_id": "501048", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Group.group (add_monoid (R\\carrier := H\\)) &&&\n Group.monoid (R\\carrier := H\\)"} {"_id": "501049", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\l. \\n. poly_mapping.lookup p l * g l *\n (poly_mapping.lookup q n * g n)) =\n (\\m. poly_mapping.lookup p m * g m) *\n (\\m. poly_mapping.lookup q m * g m)"} {"_id": "501050", "text": "proof (prove)\nusing this:\n ss = [t]\n bst t\n bst t \\\n real (T_insert a t) + \\ (Splay_Tree.insert a t) - \\ t\n \\ 4 * \\ t + 3\n\ngoal (1 subgoal):\n 1. real (cost f ss) + \\ (exec f ss) - sum_list (map \\ ss)\n \\ U f ss"} {"_id": "501051", "text": "proof (state)\nthis:\n continuous_on UNIV (\\x. Blinfun (FunctionFrechet I ?f5 x))\n\ngoal (3 subgoals):\n 1. \\args i.\n \\\\i. dfree (args i);\n \\i.\n continuous_on UNIV\n (blin_frechet (good_interp I) (simple_term (args i)))\\\n \\ continuous_on UNIV\n (blin_frechet (good_interp I)\n (simple_term ($f i args)))\n 2. \\\\\\<^sub>1 \\\\<^sub>2.\n \\dfree \\\\<^sub>1;\n continuous_on UNIV\n (blin_frechet (good_interp I) (simple_term \\\\<^sub>1));\n dfree \\\\<^sub>2;\n continuous_on UNIV\n (blin_frechet (good_interp I)\n (simple_term \\\\<^sub>2))\\\n \\ continuous_on UNIV\n (blin_frechet (good_interp I)\n (simple_term\n (Plus \\\\<^sub>1 \\\\<^sub>2)))\n 3. \\\\\\<^sub>1 \\\\<^sub>2.\n \\dfree \\\\<^sub>1;\n continuous_on UNIV\n (blin_frechet (good_interp I) (simple_term \\\\<^sub>1));\n dfree \\\\<^sub>2;\n continuous_on UNIV\n (blin_frechet (good_interp I)\n (simple_term \\\\<^sub>2))\\\n \\ continuous_on UNIV\n (blin_frechet (good_interp I)\n (simple_term\n (Times \\\\<^sub>1 \\\\<^sub>2)))"} {"_id": "501052", "text": "proof (prove)\ngoal (1 subgoal):\n 1. legendre_aux (real k) 2 = multiplicity 2 (fact k)"} {"_id": "501053", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\nat prod.\n prod \\ {(x, y). x \\ y} \\\n (case prod of\n (lx, ux) \\ (lb_arctan nat lx, ub_arctan nat ux))\n \\ {(x, y). x \\ y}"} {"_id": "501054", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.ev_blocks_part m A"} {"_id": "501055", "text": "proof (state)\nthis:\n \\ \\N \\\n \\P \\\n \\\\A\\_v \\\n \\Q\n\ngoal (1 subgoal):\n 1. \\ \\N \\\n \\(\\P \\\n \\\\A\\_v) \\\n P \\ Q"} {"_id": "501056", "text": "proof (prove)\nusing this:\n x \\ y\n z \\ y\n\ngoal (1 subgoal):\n 1. x \\ z \\ y"} {"_id": "501057", "text": "proof (prove)\nusing this:\n interaction_bound' (Generative_Probabilistic_Value.Done None)\n \\ interaction_bound consider\n (Generative_Probabilistic_Value.Done None)\n\ngoal (1 subgoal):\n 1. interaction_bound' (Generative_Probabilistic_Value.Done None) = 0"} {"_id": "501058", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LLL_impl_inv state' i fs' \\\n local.basis_reduction_add_rows_loop i fs (Suc j) = fs'"} {"_id": "501059", "text": "proof (prove)\nusing this:\n qGoodBinp qbinp\n asBinp qbinp = asBinp qbinp'\n\ngoal (1 subgoal):\n 1. (qbinp, qbinp')\n \\ {(qbinp, qbinp').\n (\\i. (qbinp i = None) = (qbinp' i = None)) \\\n (\\i v1 v2.\n qbinp i = Some v1 \\ qbinp' i = Some v2 \\\n v1 $= v2)}"} {"_id": "501060", "text": "proof (state)\nthis:\n \\f.\n f \\ A \\ nofail (f x) \\

c\n <\\r. hn_rel Ry (f x) r * F>\n\ngoal (1 subgoal):\n 1.

c <\\r. hn_rel Ry (INF f\\A. f x) r * F>"} {"_id": "501061", "text": "proof (prove)\nusing this:\n [\\mmlholed'. Lfilled 0 lholed' [$Return] (cs1 @ cs2 @ ($* es))\n\ngoal (1 subgoal):\n 1. False"} {"_id": "501064", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mat_kernel A \\ mat_kernel (B * A)"} {"_id": "501065", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite A; finite B; A \\ B = {}\\\n \\ mset_set (A \\ B) = mset_set A + mset_set B"} {"_id": "501066", "text": "proof (prove)\nusing this:\n f = x # xs\n st_inv (App (Suc ok') f r)\n\ngoal (1 subgoal):\n 1. st_inv (exec (App (Suc ok') f r))"} {"_id": "501067", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\etp.OT_12.adv_P2 n m1 m2 D\\ = etp.OT_12.adv_P2 n m1 m2 D"} {"_id": "501068", "text": "proof (prove)\ngoal (1 subgoal):\n 1. compact (\\ (F ` {..n}))"} {"_id": "501069", "text": "proof (prove)\nusing this:\n 0 \\ pmf (sds R36) a\n 0 \\ pmf (sds R36) b\n 0 \\ pmf (sds R36) c\n 0 \\ pmf (sds R36) d\n pmf (sds R36) a + pmf (sds R36) b + pmf (sds R36) c + pmf (sds R36) d = 1\n\ngoal (1 subgoal):\n 1. pmf (sds R36) d = 1 / 2 - pmf (sds R36) c"} {"_id": "501070", "text": "proof (state)\nthis:\n Diag b \\ Ide b\n\ngoal (1 subgoal):\n 1. Can ((a \\<^bold>\\ b) \\<^bold>\\ c) \\\n Arr ((a \\<^bold>\\ b) \\<^bold>\\ c) \\\n Dom ((a \\<^bold>\\ b) \\<^bold>\\ c) =\n (a \\<^bold>\\ b) \\<^bold>\\ c \\\n Cod ((a \\<^bold>\\ b) \\<^bold>\\ c) =\n (\\<^bold>\\a\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n \\<^bold>\\b\\<^bold>\\) \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c"} {"_id": "501071", "text": "proof (prove)\ngoal (1 subgoal):\n 1. exp (ln_Gamma x) < exp (ln_Gamma y)"} {"_id": "501072", "text": "proof (prove)\nusing this:\n x1a_ ::= x2_ {x3_} \\ {c'. strip c' = i}\n \\a\\A. x1a_ ::= x2_ {x3_} \\ a\n\ngoal (1 subgoal):\n 1. x1a_ ::= x2_ {x3_} \\ lift \\ i A"} {"_id": "501073", "text": "proof (prove)\nusing this:\n ?q \\ accessible \\\n path_to ?q \\ {u. nextl (init M) u = ?q}\n\ngoal (1 subgoal):\n 1. q \\ accessible \\ nextl (init M) (path_to q) = q"} {"_id": "501074", "text": "proof (prove)\nusing this:\n (k, \\)\n \\ {(1, \\x.\n case (x, 0) of\n (x, y) \\\n ((x - 1 / 2) * d,\n (2 * y - 1) * d *\n sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2)))),\n (- 1,\n \\x.\n case (x, 1) of\n (x, y) \\\n ((x - 1 / 2) * d,\n (2 * y - 1) * d *\n sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2))))} \\\n {(- 1,\n \\y.\n case (0, y) of\n (x, y) \\\n ((x - 1 / 2) * d,\n (2 * y - 1) * d *\n sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2)))),\n (1, \\y.\n case (1, y) of\n (x, y) \\\n ((x - 1 / 2) * d,\n (2 * y - 1) * d *\n sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2))))}\n\ngoal (1 subgoal):\n 1. continuous_on {0..1} \\ \\\n (\\S. finite S \\ \\ C1_differentiable_on {0..1} - S)"} {"_id": "501075", "text": "proof (prove)\nusing this:\n \\Ts'. P,E,h \\ es' [:] Ts' \\ conformable Ts' Ts''\n es = e' # es'\n P,E,h \\ e' : T'\n T' = T'' \\ (\\C. T' = NT \\ T'' = Class C)\n\ngoal (1 subgoal):\n 1. \\Ts'.\n P,E,h \\ es [:] Ts' \\ conformable Ts' (T'' # Ts'')"} {"_id": "501076", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (P \\ P) = true"} {"_id": "501077", "text": "proof (prove)\nusing this:\n (0::'a)\n \\ (\\x\\T. \\y\\S. {x - y}) \\\n \\a.\n a \\ (0::'a) \\\n (\\x\\\\x\\T. \\y\\S. {x - y}.\n 0 \\ a \\ x)\n\ngoal (1 subgoal):\n 1. (\\a.\n \\a \\ (0::'a);\n \\x\\{x - y |x y. x \\ T \\ y \\ S}.\n 0 \\ a \\ x\\\n \\ thesis) \\\n thesis"} {"_id": "501078", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ss \\ list n A;\n \\(q, t)\\set qs. q < n \\ t \\ A; p \\ w;\n semilat (A, r, f); ss = merges f qs ss\\\n \\ {q. \\t.\n (q, t) \\ set qs \\\n t \\\\<^bsub>f\\<^esub> ss ! q \\\n ss ! q} \\\n (w - {p})\n \\ w"} {"_id": "501079", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distinct (zip xs ys)"} {"_id": "501080", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xs.\n eval (AllQ (fm.Atom x)) xs = eval (push_forall (AllQ (fm.Atom x))) xs"} {"_id": "501081", "text": "proof (prove)\nusing this:\n \\cnv.\n \\\\.\n wiring (\\_ideal \\) (\\_real \\) (cnv \\)\n (w \\) \\\n wiring (\\_ideal \\) (\\_ideal \\)\n (cnv \\ \\ sim \\) (id, id)\n\ngoal (1 subgoal):\n 1. (\\cnv.\n \\\\\\.\n wiring (\\_ideal \\) (\\_real \\)\n (cnv \\) (w \\);\n \\\\.\n wiring (\\_ideal \\) (\\_ideal \\)\n (cnv \\ \\ sim \\) (id, id)\\\n \\ thesis) \\\n thesis"} {"_id": "501082", "text": "proof (prove)\ngoal (1 subgoal):\n 1. factors_of_factor_tree (create_factor_tree xs) = mset xs"} {"_id": "501083", "text": "proof (prove)\ngoal (3 subgoals):\n 1. finite {..x\\{..o (annos C2 ! x) (vars (strip C2))\n \\ m\\<^sub>o (annos C1 ! x) (vars (strip C2))\n 3. \\a\\{..o (annos C2 ! a) (vars (strip C2))\n < m\\<^sub>o (annos C1 ! a) (vars (strip C2))"} {"_id": "501084", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite UNIV;\n \\x. (if atom0 x then some else none) \\ label0 x\\\n \\ False"} {"_id": "501085", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\q.\n \\0 < k; 0 < q;\n [ r, mod q * k ] = [ r div k, mod q ] \\ k \\ r mod k;\n [ r div k, mod q ] \\ k \\ r mod k \\ k =\n [ r div k, mod q ] \\ k \\ k\\\n \\ [ r div k, mod q ] \\ k \\ k =\n [ r div k, mod q ]"} {"_id": "501086", "text": "proof (prove)\nusing this:\n F' \\ X\n F' \\ E\n E \\ X = {}\n {} \\ F'\n\ngoal (1 subgoal):\n 1. False"} {"_id": "501087", "text": "proof (prove)\ngoal (1 subgoal):\n 1. u \\ w = w \\ u"} {"_id": "501088", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\y.\n \\x(a, b, c)\\set c. a * x\\<^sup>2 + b * x + c \\ 0"} {"_id": "501089", "text": "proof (chain)\npicking this:\n \\t\\\\ M i s. M, t \\ p"} {"_id": "501090", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fragmentsLS s' \\ fragmentsLS s"} {"_id": "501091", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i \\ Suc n;\n P \\ {j. j \\ Suc n} \\ Collect (prime_ideal R);\n skip i \\ {i. i \\ n} \\ {i. i \\ Suc n}\\\n \\ compose {j. j \\ n} P (skip i) ` {j. j \\ n} =\n P ` ({j. j \\ Suc n} - {i})"} {"_id": "501092", "text": "proof (prove)\ngoal (1 subgoal):\n 1. transp (ko.lt (key_order_of_nat_term_order_inv ox))"} {"_id": "501093", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\c d.\n \\c \\ 0; d \\ 0;\n (\\x.\n real x powr p *\n (1 +\n integral {a..real x}\n (\\u. u powr p * ln u powr p' / u powr (p + 1))))\n \\ \\(\\x.\n c * real x powr p +\n d * real x powr p *\n ln (real x) powr (p' + 1))\\\n \\ thesis) \\\n thesis"} {"_id": "501094", "text": "proof (prove)\nusing this:\n gAbsCongS MOD\n\ngoal (1 subgoal):\n 1. igAbsCongS (fromMOD MOD)"} {"_id": "501095", "text": "proof (prove)\nusing this:\n \\inv\\<^bsub>C\\<^esub> \\' \\ carrier C;\n \\ \\ carrier C\\\n \\ inv\\<^bsub>C\\<^esub> (inv\\<^bsub>C\\<^esub> \\') =\n \\\n character G \\\n character G \\'\n\ngoal (1 subgoal):\n 1. \\ = \\'"} {"_id": "501096", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k.\n \\\\j. j < D * E \\ isDERIV i (A ! j) xs;\n \\j. j < E \\ isDERIV i (B ! j) xs;\n k < E\\\n \\ isDERIV i (A ! (k + E * j)) xs"} {"_id": "501097", "text": "proof (prove)\ngoal (1 subgoal):\n 1. circuit_in carrier C\\<^sub>1 &&& circuit_in carrier C\\<^sub>2"} {"_id": "501098", "text": "proof (prove)\nusing this:\n summable g'\n (\\\\<^sup>+ x. ennreal (g' x)) = integral\\<^sup>N (count_space B) g\n suminf g' = enn2real (\\\\<^sup>+ x. ennreal (g' x))\n\ngoal (1 subgoal):\n 1. g' sums s"} {"_id": "501099", "text": "proof (prove)\nusing this:\n Gramian_matrix fs l \\ carrier_mat l l\n\ngoal (1 subgoal):\n 1. \\ i l t * det (Gramian_matrix fs l) =\n det (replace_col (Gramian_matrix fs l)\n (Gramian_matrix fs l *\\<^sub>v vec l (\\ i l)) t)"} {"_id": "501100", "text": "proof (prove)\nusing this:\n (a, b) \\ symcl Rel\n\ngoal (1 subgoal):\n 1. (a, b) \\ Rel \\ (b, a) \\ Rel"} {"_id": "501101", "text": "proof (state)\nthis:\n while_option (\\A. f A \\ A) f a = Some P\n\ngoal (1 subgoal):\n 1. fixp_above a = the (while_option (\\A. f A \\ A) f a)"} {"_id": "501102", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\eP\\<^sub>3 eV\\<^sub>3 \\'.\n \\{$$} \\ e\\<^sub>3, eP\\<^sub>3, eV\\<^sub>3 : \\;\n \\\\\\<^sub>P.\n \\\\\\<^sub>P, eP\\<^sub>2\\ P\\ \\\\\\<^sub>P @\n \\', eP\\<^sub>3\\;\n \\\\''.\n \\\\' @\n \\'', eV\\<^sub>2\\ V\\ \\\\'', eV\\<^sub>3\\\\\n \\ thesis) \\\n thesis"} {"_id": "501103", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i < length (filter P xs);\n \\i'.\n \\i' < length xs; P (xs ! i')\\\n \\ Q (xs ! i')\\\n \\ Q (filter P xs ! i)"} {"_id": "501104", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(sdim, card_UNIV_class)"} {"_id": "501105", "text": "proof (prove)\nusing this:\n prod_list ds dvd a div d\n prod_list ds dvd ?a \\\n digit_encode ds ?a = replicate (length ds) 0\n prod_list (d # ds) dvd a\n ?a dvd ?b \\ ?b mod ?a = (0::?'a)\n ?a * ?b dvd ?c \\ ?a dvd ?c\n prod_list (?x # ?xs) = ?x * prod_list ?xs\n\ngoal (1 subgoal):\n 1. a mod d # digit_encode ds (a div d) = 0 # replicate (length ds) 0"} {"_id": "501106", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (INF p\\{}. ereal (awalk_cost f p)) = \\"} {"_id": "501107", "text": "proof (prove)\nusing this:\n pes = init @ (p, e) # rest\n cupcake_pmatch cenv p v0 [] = Match env'\n\ngoal (1 subgoal):\n 1. (p, e) \\ set pes"} {"_id": "501108", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \\ degree v < degree u}\\<^esub>v\\B. a\n v \\\\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \\ degree v < degree u}\\<^esub>\n v) =\n (\\v\\B. Polynomial.smult (a v) v)"} {"_id": "501109", "text": "proof (chain)\npicking this:\n EH\\<^sub>N n =\n (\\\\<^sub>ai\\{1..}. measure_pmf.prob (H\\<^sub>N n) {i..})\n integrable (measure_pmf (H\\<^sub>N n)) real"} {"_id": "501110", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P \\ carrier_mat m m \\\n A \\ carrier_mat m n \\ Q \\ carrier_mat n n"} {"_id": "501111", "text": "proof (prove)\nusing this:\n A \\ Univ\n |A| \n arr (mkIde ?A) = (?A \\ Univ \\ |?A| )\n arr (mkArr ?A ?B ?F) =\n (?A \\ Univ \\\n |?A| \\\n ?B \\ Univ \\\n |?B| \\ ?F \\ ?A \\ ?B)\n \\?A \\ Univ; |?A| \\\n \\ local.dom (mkIde ?A) = mkIde ?A\n \\?A \\ Univ; |?A| \\\n \\ cod (mkIde ?A) = mkIde ?A\n \\?A \\ Univ; |?A| \\\n \\ Fun (mkIde ?A) = (\\x\\?A. x)\n\ngoal (1 subgoal):\n 1. mkArr A A (\\x. x) = mkIde A"} {"_id": "501112", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\P'' p.\n \\\\ \\ P' = \\' \\ P'';\n set p\n \\ set (bn \\') \\ set (bn \\);\n PQ' = P'' \\ (p \\ Q)\\\n \\ thesis;\n bn \\ \\* \\';\n \\' \\ PQ' = \\ \\ P' \\ Q\\\n \\ thesis"} {"_id": "501113", "text": "proof (prove)\nusing this:\n mC_C theta c c1 s s1 P (lift P1 F1)\n\ngoal (1 subgoal):\n 1. mC_C theta c1 c s1 s (lift P1 F1 ` P) (inv_into P (lift P1 F1))"} {"_id": "501114", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (case fmlookup xs s of None \\ 0::'a\n | Some v \\ v) =\n (case fmlookup\n (fmfilter (\\k. fmlookup xs k \\ Some (0::'a)) xs)\n s of\n None \\ 0::'a | Some v \\ v)"} {"_id": "501115", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\1 / 2 - ln (ln 2) - meissel_mertens\\ \\ 2 / ln 2 + 1 / 2"} {"_id": "501116", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n\\n\\<^sub>0'. f n \\ C\\<^sub>4"} {"_id": "501117", "text": "proof (prove)\nusing this:\n hconf h\n preallocated h\n\ngoal (1 subgoal):\n 1. bisim1_list1 t h (Throw a', loc) [] \\a'\\ []"} {"_id": "501118", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\s_repr d_repr s_range d_range.\n (s_repr, d_repr) \\ set E \\\n map_of V s_repr = Some s_range \\\n p_src p \\ wordinterval_to_set s_range \\\n map_of V d_repr = Some d_range \\\n p_dst p \\ wordinterval_to_set d_range"} {"_id": "501119", "text": "proof (prove)\ngoal (1 subgoal):\n 1. no_trailing P (x # xs) =\n (no_trailing P xs \\ (xs = [] \\ \\ P x))"} {"_id": "501120", "text": "proof (prove)\ngoal (1 subgoal):\n 1. close_eq s (Eq a b) = (a = b)"} {"_id": "501121", "text": "proof (prove)\nusing this:\n P \\ subgroups_of_size p\n P \\ conjP.stabilizer P\n\ngoal (1 subgoal):\n 1. P \\ conjP.fixed_points"} {"_id": "501122", "text": "proof (state)\nthis:\n norm (f y - f x) / norm (y - x) \\ Nf y + kF\n\ngoal (2 subgoals):\n 1. norm (g' (f y - f x - f' (y - x))) / norm (y - x) \\ Nf y * kG\n 2. 0 \\ Ng y"} {"_id": "501123", "text": "proof (prove)\nusing this:\n \\?j1.0 < ?j2.0; ?j2.0 < length zs\\\n \\ zs ! ?j1.0 < zs ! ?j2.0\n ?x \\ set zs \\ ?x < Suc i\n j1 < j2\n j2 < length (snd (mk_eqcl xs zs (Suc i) T))\n\ngoal (1 subgoal):\n 1. snd (mk_eqcl xs zs (Suc i) T) ! j1 < snd (mk_eqcl xs zs (Suc i) T) ! j2"} {"_id": "501124", "text": "proof (state)\nthis:\n group_hom G2 G3 g2\n\ngoal (1 subgoal):\n 1. solvable G2 \\ solvable G1 \\ solvable G3"} {"_id": "501125", "text": "proof (prove)\ngoal (1 subgoal):\n 1. drlex_pp 0 s"} {"_id": "501126", "text": "proof (prove)\nusing this:\n run c \\ \\' r\n run c \\ \\ s\n \\ is_exn \\\n\ngoal (1 subgoal):\n 1. \\\\is_exn \\'; \\' = \\\\\n \\ thesis;\n \\\\ is_exn \\'; \\' = \\; r = s\\\n \\ thesis\\\n \\ thesis"} {"_id": "501127", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i.\n of_int_complex_interval a +\n C i *\n ipoly_complex_interval p (C i)) \\\\<^sub>c\n of_int a + c * ipoly p c"} {"_id": "501128", "text": "proof (prove)\nusing this:\n x\\<^sup>T ; r ; x\\<^sup>T \\ - r\\<^sup>T \\ x\\<^sup>\\\n\ngoal (1 subgoal):\n 1. x\\<^sup>T ; r ; x\\<^sup>T \\ - r\\<^sup>T +\n x\\<^sup>T ; r ; x\\<^sup>T \\ r\\<^sup>T\n \\ x\\<^sup>\\ + x\\<^sup>T\\<^sup>\\"} {"_id": "501129", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\0 < n; d dvd n\\\n \\ card {k \\ {0<..n}. gcd k n = d} =\n card (totatives (n div d))"} {"_id": "501130", "text": "proof (prove)\nusing this:\n \\ f\\<^sub>1 \\ \\ f\\<^sub>2 = {}\n \\?f'2 \\ \\ g; ?f'2 \\ f\\\n \\ (\\ f\\<^sub>1 \\ \\ f\\<^sub>2) \\\n \\ ?f'2 =\n {}\n edges_disj g\n\ngoal (1 subgoal):\n 1. edges_disj g'"} {"_id": "501131", "text": "proof (prove)\nusing this:\n instance_of\\<^sub>l\\<^sub>s Cg C\\<^sub>1 \\ falsifies\\<^sub>g G Cg\n instance_of\\<^sub>l\\<^sub>s Cg\n (C\\<^sub>1 \\\\<^sub>l\\<^sub>s (\\x. \\ (''1'' @ x)))\n\ngoal (1 subgoal):\n 1. falsifies\\<^sub>c G\n (C\\<^sub>1 \\\\<^sub>l\\<^sub>s\n (\\x. \\ (''1'' @ x)))"} {"_id": "501132", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P. invertible P \\ P ** A = A"} {"_id": "501133", "text": "proof (prove)\nusing this:\n dim_col x = dim_row x\n \\x * ?B = 1\\<^sub>m (dim_row x);\n ?B * x = 1\\<^sub>m (dim_row ?B)\\\n \\ \\A'.\n y ** A' = mat (1::'a) \\ A' ** y = mat (1::'a)\n \\y ** ?A' = mat (1::'a); ?A' ** y = mat (1::'a)\\\n \\ \\B.\n x * B = 1\\<^sub>m (dim_row x) \\\n B * x = 1\\<^sub>m (dim_row B)\n\ngoal (1 subgoal):\n 1. (square_mat x \\\n (\\B.\n x * B = 1\\<^sub>m (dim_row x) \\\n B * x = 1\\<^sub>m (dim_row B))) =\n (\\A'. y ** A' = mat (1::'a) \\ A' ** y = mat (1::'a))"} {"_id": "501134", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a \\ G \\ quotient.invertible (Class a)"} {"_id": "501135", "text": "proof (prove)\nusing this:\n (\\x. ennreal (Px x)) \\ borel_measurable MX\n emeasure MX {x \\ space MX. ennreal (Px x) \\ 0} \\ 0\n\ngoal (1 subgoal):\n 1. log b \\

(x in MX. Px x \\ 0)\n \\ log b (Sigma_Algebra.measure MX (space MX))"} {"_id": "501136", "text": "proof (prove)\ngoal (1 subgoal):\n 1. extended_ord_pm_powerprod (\\s t. ord (PP s) (PP t))\n (\\s t. ord_strict (PP s) (PP t))"} {"_id": "501137", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k\\n. Ring (B k) \\\n Ring\n \\carrier = ac_fProd_Rg n B, pop = prod_pOp {i. i \\ n} B,\n mop = prod_mOp {i. i \\ n} B, zero = prod_zero {i. i \\ n} B,\n tp = prod_tOp {i. i \\ n} B,\n un = prod_one {i. i \\ n} B\\"} {"_id": "501138", "text": "proof (prove)\ngoal (1 subgoal):\n 1. UCAST('a \\ 'b) w !! n =\n (w !! n \\ n < min LENGTH('a) LENGTH('b))"} {"_id": "501139", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (uncurry2\n (uncurry\n (\\x.\n return \\\\\\ check_mult_l_mult_err_impl x)),\n uncurry2 (uncurry check_mult_l_mult_err))\n \\ poly_assn\\<^sup>k *\\<^sub>a poly_assn\\<^sup>k *\\<^sub>a\n poly_assn\\<^sup>k *\\<^sub>a\n poly_assn\\<^sup>k \\\\<^sub>a raw_string_assn"} {"_id": "501140", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s x.\n \\stable pre rely; stable post rely;\n \\V.\n \\ P sat [pre \\ b \\\n {V}, {(s, t).\n s =\n t}, UNIV, {s.\n (V, s) \\ guar} \\\n post] \\\n (\\s.\n {l. l ! 0 = (Some P, s) \\ l \\ cptn} \\\n assum (pre \\ b \\ {V}, {(x, y). x = y})\n \\ {c. fst (last c) = None \\\n (V, snd (last c)) \\ guar \\\n snd (last c) \\ post});\n x \\ assum (pre, rely); fst (last x) = None;\n x ! 0 = (Some (Await b P), s); x \\ cptn\\\n \\ snd (last x) \\ post"} {"_id": "501141", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. \\ P (dual x)) = (\\x'. \\ P x')"} {"_id": "501142", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\string_unop_type op \\ \\\\<^sub>1;\n string_unop_type op \\ \\\\<^sub>2\\\n \\ \\\\<^sub>1 = \\\\<^sub>2"} {"_id": "501143", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (rel_set A ===> (A ===> A ===> A) ===> (=)) semigroup_add_ow\n semigroup_add_ow"} {"_id": "501144", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Big_Step_Unclocked_Single.evaluate v s e bv \\\n Big_Step_Unclocked.evaluate v s e bv"} {"_id": "501145", "text": "proof (prove)\nusing this:\n \\x\\one_chain1.\n case x of\n (k, \\) \\\n real_of_int k * line_integral F basis \\ =\n (case f x of\n (k, \\) \\\n real_of_int k * line_integral F basis \\) \\\n line_integral_exists F basis \\\n\ngoal (1 subgoal):\n 1. one_chain_line_integral F basis one_chain1 =\n one_chain_line_integral F basis one_chain2"} {"_id": "501146", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {R23s \\\n UNIV \\\n (l3_inv5 \\ l3_inv6 \\ l3_inv7 \\ l3_inv8 \\\n l3_inv9)} TS.trans l2, TS.trans l3 {> R23s}"} {"_id": "501147", "text": "proof (prove)\ngoal (1 subgoal):\n 1. k_dfs = KBPAlg.dfs k_empt (k_frontier a)"} {"_id": "501148", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {B. B \\\\<^sub>c A \\ cfinite B} =\n acset ` {B. B \\ rcset A \\ finite B}"} {"_id": "501149", "text": "proof (prove)\ngoal (1 subgoal):\n 1. common_sudiv_exists (two_chain_vertical_boundary {rot_diamond_cube})\n (boundary diamond_cube) \\\n common_reparam_exists (boundary diamond_cube)\n (two_chain_vertical_boundary {rot_diamond_cube})"} {"_id": "501150", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B.\n lower_asymptotic_density B = 1 \\\n (\\n.\n Sigma_Algebra.measure P (space M \\ (T ^^ n) -` A n) *\n indicat_real B n)\n \\ 0"} {"_id": "501151", "text": "proof (prove)\nusing this:\n \\ is_empty q\n\ngoal (1 subgoal):\n 1. the (priority q (PQ.min q)) = snd (hd (alist_of q))"} {"_id": "501152", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inversions_between (permute_list \\1 xs) (permute_list \\2 ys) =\n map_prod (inv \\1) (inv \\2) ` inversions_between xs ys"} {"_id": "501153", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Node\\a\\l\\r = Node\\a'\\l'\\r') =\n (a = a' \\\n (a \\ \\ \\ l = l' \\ r = r'))"} {"_id": "501154", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 2 - 2 ^ (LENGTH('e) - Suc 0) - int LENGTH('f)\n \\ denormal_exponent TYPE(('e, 'f) IEEE.float) + int i"} {"_id": "501155", "text": "proof (prove)\nusing this:\n \\u\\U. 0 < w u \\ w u \\ real c\n u \\ V\n \\\n (P\\<^sub>1 \\ wrap B\\<^sub>1 \\ P\\<^sub>2 \\\n wrap B\\<^sub>2 \\\n {{v} |v. v \\ V}) =\n U\n\ngoal (1 subgoal):\n 1. 0 < w u"} {"_id": "501156", "text": "proof (prove)\ngoal (1 subgoal):\n 1. tw_cmpser G m f \\ {h. red_ch_cd G f m h} \\ {}"} {"_id": "501157", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dlist_eq \\ equivclp double"} {"_id": "501158", "text": "proof (prove)\nusing this:\n locally path_connected T\n S retract_of T\n\ngoal (1 subgoal):\n 1. locally path_connected S"} {"_id": "501159", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (P\\<^sub>1 \\ wrap B\\<^sub>1 \\ {{v} |v. v \\ L}) +\n card (P\\<^sub>2 \\ wrap B\\<^sub>2)\n \\ card P + card (P\\<^sub>2 \\ wrap B\\<^sub>2)"} {"_id": "501160", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ff.\n \\I \\ {}; \\j\\I. aGroup (A j); aGroup S;\n \\j\\I. g j \\ aHom S (A j);\n aGroup (a\\\\<^bsub>I\\<^esub> A);\n aGroup (Ag_ind (a\\\\<^bsub>I\\<^esub> A) ff);\n ff \\ carrier (a\\\\<^bsub>I\\<^esub> A) \\ B;\n bij_to ff (carrier (a\\\\<^bsub>I\\<^esub> A)) B;\n aGroup (Ag_ind (a\\\\<^bsub>I\\<^esub> A) ff);\n \\j\\I.\n g j \\ aHom S (A j) \\\n (\\!f.\n f \\ aHom S S \\\n (\\j\\I. compos S (g j) f = g j));\n \\j\\I.\n ProjInd I A ff j\n \\ aHom (Ag_ind (a\\\\<^bsub>I\\<^esub> A) ff)\n (A j) \\\n (\\!f.\n f \\ aHom (Ag_ind (a\\\\<^bsub>I\\<^esub> A) ff) S \\\n (\\j\\I.\n compos (Ag_ind (a\\\\<^bsub>I\\<^esub> A) ff) (g j) f =\n ProjInd I A ff j))\\\n \\ \\h.\n bijec\\<^bsub>(a\\\\<^bsub>I\\<^esub> A),S\\<^esub> h"} {"_id": "501161", "text": "proof (prove)\nusing this:\n connected (sphere a r)\n 0 < r\n sphere a r = {x, y}\n x \\ y\n finite (sphere a r)\n connected ?S \\\n finite ?S = (?S = {} \\ (\\a. ?S = {a}))\n\ngoal (1 subgoal):\n 1. False"} {"_id": "501162", "text": "proof (prove)\ngoal (1 subgoal):\n 1. App t u \\ RED \\"} {"_id": "501163", "text": "proof (prove)\nusing this:\n b |\\| c \\ {b |\\| c |b c. b \\ B \\ c \\ C}\n a \\ A\n\ngoal (1 subgoal):\n 1. \\a' bc.\n a |\\| b |\\| c = a' |\\| bc \\\n a' \\ A \\\n bc \\ {b |\\| c |b c. b \\ B \\ c \\ C}"} {"_id": "501164", "text": "proof (prove)\nusing this:\n naturally_isomorphic (\\) (\\) (\\x. x \\ b)\n (\\x. CCC.prod x b)\n left_adjoint_functor (\\) (\\) (\\x. x \\ b)\n\ngoal (1 subgoal):\n 1. left_adjoint_functor (\\) (\\) (\\x. CCC.prod x b)"} {"_id": "501165", "text": "proof (prove)\nusing this:\n 0 < x \\ lg (2 ^ x) = x\n 0 < Suc x\n\ngoal (1 subgoal):\n 1. lg (2 ^ Suc x) = Suc x"} {"_id": "501166", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\y.\n \\mem_context_val_w32 9\n (ucast (ptp OR ucast (ucast (va >> 18)))) (mem s1) =\n Some y;\n mem_context_val_w32 9 (ucast (ptp OR ucast (ucast (va >> 18))))\n ((mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))) =\n Some y\\\n \\ (let et_val = y AND 3\n in if et_val = 0 then None\n else if et_val = 1\n then let ptp = y AND 4294967292\n in ptd_lookup va ptp (mem s1) (2 + 1)\n else if et_val = 2\n then let ppn = ucast (y >> 8); va_offset = ucast (ucast va)\n in Some ((ucast ppn << 12) OR va_offset, ucast y)\n else None) =\n (let et_val = y AND 3\n in if et_val = 0 then None\n else if et_val = 1\n then let ptp = y AND 4294967292\n in ptd_lookup va ptp\n ((mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None)))\n (2 + 1)\n else if et_val = 2\n then let ppn = ucast (y >> 8); va_offset = ucast (ucast va)\n in Some ((ucast ppn << 12) OR va_offset, ucast y)\n else None)"} {"_id": "501167", "text": "proof (prove)\nusing this:\n v = combine Ks Us \\ u \\ \\\n combine Ks Us \\ F\n Span F [u] = E\n (?u \\ line_extension ?K ?a ?E) =\n (\\k\\?K. \\v\\?E. ?u = k \\ ?a \\ v)\n\ngoal (1 subgoal):\n 1. v \\ E"} {"_id": "501168", "text": "proof (prove)\nusing this:\n isInBar (xs, s)\n wls s X\n wlsSEM SEM\n wlsSEM SEM \\\n termFSwSbImorph (semInt SEM) (semIntAbs SEM) (asIMOD SEM)\n\ngoal (1 subgoal):\n 1. semIntAbs SEM (Abs xs x X) = igAbs (asIMOD SEM) xs x (semInt SEM X)"} {"_id": "501169", "text": "proof (prove)\nusing this:\n C.ide a'\n \\ \\' \\ Cones.SET a'\n C.ide ?a \\ Cones.map ?a = S.mkIde (\\ ` D.cones ?a)\n Cop.in_hom a' a' a' \\\n \\\\.map\n a' : Cones.map\n a' \\\\<^sub>S Y a a'\\\n\ngoal (1 subgoal):\n 1. \\.FUN a' (\\.FUN a' (\\ \\')) =\n compose (\\.DOM a') (\\.FUN a') (\\.FUN a')\n (\\ \\')"} {"_id": "501170", "text": "proof (state)\ngoal (1 subgoal):\n 1. wa_precise_refine WC WA (\\2 \\ \\1)"} {"_id": "501171", "text": "proof (prove)\nusing this:\n \\x y z.\n x \\ y \\\n x \\ z \\\n y \\ z \\ x \\ A \\ y \\ A \\ z \\ A\n\ngoal (1 subgoal):\n 1. card A = 2"} {"_id": "501172", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pathVertices u p = u # map snd p"} {"_id": "501173", "text": "proof (state)\nthis:\n qi \\ s\n qi \\ Qi\n\ngoal (2 subgoals):\n 1. \\x q.\n \\q \\ Q \\ q \\ Qi \\ {};\n accs \\ss x q\\\n \\ \\q\\Qi. accs \\ x q\n 2. \\x q.\n \\q \\ Qi; accs \\ x q\\\n \\ \\xa\\Q.\n xa \\ Qi \\ {} \\\n accs \\ss x xa"} {"_id": "501174", "text": "proof (prove)\nusing this:\n indicator {x \\ space St. P x}\n \\ borel_measurable St \\\n E_inf s (indicator {x \\ space St. P x}) =\n (\\D\\K s.\n \\\\<^sup>+ t. E_inf t\n (\\\\.\n indicator {x \\ space St. P x}\n (t ## \\))\n \\measure_pmf D)\n\ngoal (1 subgoal):\n 1. P_inf s P =\n (\\D\\K s.\n \\\\<^sup>+ t. P_inf t\n (\\\\. P (t ## \\))\n \\measure_pmf D)"} {"_id": "501175", "text": "proof (prove)\ngoal (1 subgoal):\n 1. S homeomorphic T \\ locally compact S = locally compact T"} {"_id": "501176", "text": "proof (prove)\nusing this:\n distinct (map fst xs)\n fst x \\ set (map fst xs)\n\ngoal (1 subgoal):\n 1. distinct (map fst (xs @ [x]))"} {"_id": "501177", "text": "proof (prove)\nusing this:\n p \\ carrier P\n q \\ carrier P\n deg R q < n\n P \\ UP R\n \\UP_ring ?R; deg ?R ?p < ?m; ?p \\ carrier (UP ?R)\\\n \\ coeff (UP ?R) ?p ?m = \\\\<^bsub>?R\\<^esub>\n UP_ring R\n \\?p \\ carrier P; ?q \\ carrier P\\\n \\ (?p \\\\<^bsub>P\\<^esub> ?q) ?n =\n ?p ?n \\ ?q ?n\n ?f \\ carrier P \\ coeff (UP R) ?f = ?f\n ?f \\ carrier P \\ ?f ?n \\ carrier R\n \\deg R ?p < ?m; ?p \\ carrier P\\\n \\ coeff P ?p ?m = \\\n\ngoal (1 subgoal):\n 1. (p \\\\<^bsub>P\\<^esub> q) n = p n"} {"_id": "501178", "text": "proof (prove)\ngoal (1 subgoal):\n 1. degree (nlex_pdevs x) = degree x"} {"_id": "501179", "text": "proof (prove)\ngoal (1 subgoal):\n 1. tarski_absolute real_hyp2_C real_hyp2_B"} {"_id": "501180", "text": "proof (prove)\ngoal (1 subgoal):\n 1. G,A\\{Normal\n ((\\Y' s' s. s' = s \\ fst s = None) \\.\n G\\init\\n \\.\n (\\s. s\\\\(G, L)) \\.\n (\\s.\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom\n (locals\n (snd s)) \\In1r\n (c1 Finally c2)\\ C))}\n .c1 Finally c2.\n {\\Y s' s.\n Y = \\ \\\n G\\s \\c1 Finally c2\\ s'}"} {"_id": "501181", "text": "proof (prove)\nusing this:\n x \\ carrier\n \\carrier =\n Pi I (carrier \\ (f \\ Gs)) \\\n {f. \\x.\n x \\ I \\ f x = undefined},\n monoid.mult =\n \\x y.\n \\i\\I.\n x i \\\\<^bsub>(f \\ Gs) i\\<^esub> y i,\n one =\n \\i\\I.\n \\\\<^bsub>(f \\ Gs) i\\<^esub>\\\n i \\ I\n\ngoal (1 subgoal):\n 1. (if i \\ I then J i else (\\_. undefined)) (K i (x i)) = x i"} {"_id": "501182", "text": "proof (prove)\nusing this:\n \\\\\\.\n \\ \\ carrier_mat d d \\\n partial_density_operator \\ \\\n partial_density_operator (?DS \\);\n \\\\.\n \\ \\ carrier_mat d d \\\n partial_density_operator \\ \\\n ?DS \\ \\ carrier_mat d d\\\n \\ partial_density_operator\n (denote_while_n_iter ?M0.0 M1 ?DS ?n \\)\n \\?\\ \\ carrier_mat d d;\n partial_density_operator ?\\\\\n \\ DS ?\\ \\ carrier_mat d d \\\n partial_density_operator (DS ?\\)\n M0 \\ carrier_mat d d\n M1 \\ carrier_mat d d\n\ngoal (1 subgoal):\n 1. partial_density_operator (denote_while_n_iter M0 M1 DS k \\)"} {"_id": "501183", "text": "proof (prove)\nusing this:\n ennreal (pmf p x) = 1\n\ngoal (1 subgoal):\n 1. pmf p i = pmf (return_pmf x) i"} {"_id": "501184", "text": "proof (prove)\ngoal (1 subgoal):\n 1. enat n < llength (ltl xs) \\ enat (Suc n) < llength xs"} {"_id": "501185", "text": "proof (prove)\nusing this:\n n = 0\n natToVertexList v f es ! n = Some v'\n nths (take (Suc (es ! (n - Suc 0))) (verticesFrom f v))\n (set (take n es)) =\n removeNones (take n (natToVertexList v f es))\n distinct (vertices f)\n v \\ \\ f\n vs = verticesFrom f v\n incrIndexList es |es| |verticesFrom f v|\n Suc n \\ |es|\n |vs| = |vertices f|\n nths (verticesFrom f v) {0} = [v]\n vs \\ []\n es ! 0 = 0\n\ngoal (1 subgoal):\n 1. nths (take (Suc (es ! (Suc n - 1))) vs) (set (take (Suc n) es)) =\n removeNones (take (Suc n) (natToVertexList v f es))"} {"_id": "501186", "text": "proof (state)\nthis:\n i = 0\n\ngoal (1 subgoal):\n 1. i = 0 \\ False"} {"_id": "501187", "text": "proof (prove)\ngoal (1 subgoal):\n 1. higher_differentiable_on S (\\x. (f x, g x)) n"} {"_id": "501188", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n \\Idomain R; PolynRg A B Y; h \\ rHom S B; Ring S;\n Subring R S; Idomain S; Subring A B; Ring A; Ring B; aGroup R;\n aGroup A; x \\ carrier R; y \\ carrier R\\\n \\ erH R S X A B Y h (x \\\\<^sub>r y) =\n erH R S X A B Y h\n x \\\\<^sub>r\\<^bsub>A\\<^esub>\n erH R S X A B Y h y"} {"_id": "501189", "text": "proof (prove)\nusing this:\n f \\ F\n F \\ FINFUNC\n x0 \\ {f z |f. f \\ F \\ z = s.maxim (SUPP f)} \\\n (\\a\\{f z |f. f \\ F \\ z = s.maxim (SUPP f)}.\n (x0, a) \\ r)\n s.maxim (SUPP f) = z \\ f z = x0\n (f0(z := r.zero, z := x0), f(z := r.zero, z := x0)) \\ oexp\n\ngoal (1 subgoal):\n 1. (f(z := x0), f) \\ oexp"} {"_id": "501190", "text": "proof (prove)\ngoal (1 subgoal):\n 1. M *v axis i 1 = f i - x0"} {"_id": "501191", "text": "proof (state)\nthis:\n True\n\ngoal (1 subgoal):\n 1. (\\f.\n \\orthogonal_transformation f; det (matrix f) = 1;\n f a = b\\\n \\ thesis) \\\n thesis"} {"_id": "501192", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\bst_wrt (\\) t; Splay_Heap.partition p t = (l', r')\\\n \\ set_tree t = set_tree l' \\ set_tree r'"} {"_id": "501193", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dens_ctxt_\\ (vs, vs', \\, \\) \\\\<^sub>d\n e \\\n (\\\\ x. ennreal (eval_cexpr f \\ x)) &&&\n is_density_expr (vs, vs', \\, \\) REAL f"} {"_id": "501194", "text": "proof (prove)\nusing this:\n convergent (\\n. sum f {..n. C + sum g {.. b \\ p \\ 0\n then sign_changes (sturm_squarefree p) a -\n sign_changes (sturm_squarefree p) b\n else 0) =\n card {x. a < x \\ x \\ b \\ poly p' x = 0}"} {"_id": "501196", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fcard A - fcard B \\ fcard (A |-| B)"} {"_id": "501197", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b aa ba.\n \\P \\ \\e,s\\ \\*\n \\a,b\\;\n P \\ \\a,b\\ \\\n \\aa,ba\\\\\n \\ P \\ \\V:=a,\n b\\ \\\n \\V:=aa,ba\\"} {"_id": "501198", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\infinite C; |A| \\o |C|; |B| \\o |C|\\\n \\ |A \\ B| \\o |C|"} {"_id": "501199", "text": "proof (prove)\nusing this:\n t2 \\ keys\n (punit.monom_mult (1::'a) (l - i.lpp (focus X h))\n (flatten (i.punit.tail (focus X h))))\n\ngoal (1 subgoal):\n 1. (\\t3.\n \\t3 \\ keys (flatten (i.punit.tail (focus X h)));\n t2 = l - i.lpp (focus X h) + t3\\\n \\ thesis) \\\n thesis"} {"_id": "501200", "text": "proof (prove)\nusing this:\n open U\n topological_basis K\n\ngoal (1 subgoal):\n 1. (\\B.\n \\B \\ K; U = \\ B\\\n \\ thesis) \\\n thesis"} {"_id": "501201", "text": "proof (prove)\nusing this:\n \\n. (nxt ^^ n) \\ xs\n\ngoal (1 subgoal):\n 1. (LEAST n. (nxt ^^ n) \\ xs) \\ n"} {"_id": "501202", "text": "proof (prove)\nusing this:\n P[<(rev p \\ s)>] \\ Q[<(rev p \\ s)>]\n\ngoal (1 subgoal):\n 1. (p \\ P)[] \\ (p \\ Q)[]"} {"_id": "501203", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\.\n \\\\ : f \\\\<^sub>C\n g \\\\<^sub>C src\\<^sub>C\n g\\ \\\n F \\ =\n unit (src\\<^sub>C g) \\\\<^sub>D\n \\ \\\\<^sub>D\n (D.inv (F f) \\\\<^sub>D g') \\\\<^sub>D\n (F f \\\\<^sub>D D.inv \\) \\\\<^sub>D\n D.inv (\\ (f, g)) \\\n thesis) \\\n thesis"} {"_id": "501204", "text": "proof (prove)\ngoal (1 subgoal):\n 1. upper (A + B) = upper A + upper B"} {"_id": "501205", "text": "proof (state)\nthis:\n \\ \\P \\\n \\\\A\\_v \\\n \\Q\n\ngoal (1 subgoal):\n 1. \\ \\(\\P \\\n \\\\A\\_v) \\\n P \\ Q"} {"_id": "501206", "text": "proof (prove)\ngoal (1 subgoal):\n 1. complex_of_real\n ((LBINT t=ereal 0..ereal T. 2 * (sin (t * (x - a)) / t)) -\n (LBINT t=ereal 0..ereal T. 2 * (sin (t * (x - b)) / t))) =\n complex_of_real\n (2 *\n (sgn (x - a) * Si (T * \\x - a\\) -\n sgn (x - b) * Si (T * \\x - b\\)))"} {"_id": "501207", "text": "proof (prove)\nusing this:\n wfactors G cs c\n c \\ Units G\n\ngoal (1 subgoal):\n 1. 0 < length cs"} {"_id": "501208", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\x1 x1a x1b x1c x1d x1e a b r x1f x2.\n \\I s; (ii, i) \\ R s; (ti, t) \\ R s;\n (ei, e) \\ R s; (x1, restrict_top i lv True) \\ R s;\n (x1a, restrict_top t lv True) \\ R s;\n (x1b, restrict_top e lv True) \\ R s;\n (x1c, restrict_top i lv False) \\ R s;\n (x1d, restrict_top t lv False) \\ R s;\n (x1e, restrict_top e lv False) \\ R s;\n (a, case lowest_tops\n [restrict_top i lv True, restrict_top t lv True,\n restrict_top e lv True] of\n None \\\n case restrict_top i lv True of\n Trueif \\ restrict_top t lv True\n | Falseif \\ restrict_top e lv True\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv True) x True)\n (restrict_top (restrict_top t lv True) x True)\n (restrict_top (restrict_top e lv True) x True))\n (ifex_ite (restrict_top (restrict_top i lv True) x False)\n (restrict_top (restrict_top t lv True) x False)\n (restrict_top (restrict_top e lv True) x False)))\n \\ R b;\n I b; les s b; r = (x1f, x2);\n (x1f,\n case lowest_tops\n [restrict_top i lv False, restrict_top t lv False,\n restrict_top e lv False] of\n None \\\n case restrict_top i lv False of\n Trueif \\ restrict_top t lv False\n | Falseif \\ restrict_top e lv False\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv False) x True)\n (restrict_top (restrict_top t lv False) x True)\n (restrict_top (restrict_top e lv False) x True))\n (ifex_ite (restrict_top (restrict_top i lv False) x False)\n (restrict_top (restrict_top t lv False) x False)\n (restrict_top (restrict_top e lv False) x False)))\n \\ R x2 \\\n I x2 \\ les b x2\\\n \\ (a, ?n1.243 x1 x1a x1b x1c x1d x1e a b r x1f x2)\n \\ R x2\n 2. \\x1 x1a x1b x1c x1d x1e a b r x1f x2.\n \\I s; (ii, i) \\ R s; (ti, t) \\ R s;\n (ei, e) \\ R s; (x1, restrict_top i lv True) \\ R s;\n (x1a, restrict_top t lv True) \\ R s;\n (x1b, restrict_top e lv True) \\ R s;\n (x1c, restrict_top i lv False) \\ R s;\n (x1d, restrict_top t lv False) \\ R s;\n (x1e, restrict_top e lv False) \\ R s;\n (a, case lowest_tops\n [restrict_top i lv True, restrict_top t lv True,\n restrict_top e lv True] of\n None \\\n case restrict_top i lv True of\n Trueif \\ restrict_top t lv True\n | Falseif \\ restrict_top e lv True\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv True) x True)\n (restrict_top (restrict_top t lv True) x True)\n (restrict_top (restrict_top e lv True) x True))\n (ifex_ite (restrict_top (restrict_top i lv True) x False)\n (restrict_top (restrict_top t lv True) x False)\n (restrict_top (restrict_top e lv True) x False)))\n \\ R b;\n I b; les s b; r = (x1f, x2);\n (x1f,\n case lowest_tops\n [restrict_top i lv False, restrict_top t lv False,\n restrict_top e lv False] of\n None \\\n case restrict_top i lv False of\n Trueif \\ restrict_top t lv False\n | Falseif \\ restrict_top e lv False\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv False) x True)\n (restrict_top (restrict_top t lv False) x True)\n (restrict_top (restrict_top e lv False) x True))\n (ifex_ite (restrict_top (restrict_top i lv False) x False)\n (restrict_top (restrict_top t lv False) x False)\n (restrict_top (restrict_top e lv False) x False)))\n \\ R x2 \\\n I x2 \\ les b x2\\\n \\ (x1f, ?n2.243 x1 x1a x1b x1c x1d x1e a b r x1f x2)\n \\ R x2\n 3. \\x1 x1a x1b x1c x1d x1e a b r x1f x2 ra.\n \\I s; (ii, i) \\ R s; (ti, t) \\ R s;\n (ei, e) \\ R s; (x1, restrict_top i lv True) \\ R s;\n (x1a, restrict_top t lv True) \\ R s;\n (x1b, restrict_top e lv True) \\ R s;\n (x1c, restrict_top i lv False) \\ R s;\n (x1d, restrict_top t lv False) \\ R s;\n (x1e, restrict_top e lv False) \\ R s;\n (a, case lowest_tops\n [restrict_top i lv True, restrict_top t lv True,\n restrict_top e lv True] of\n None \\\n case restrict_top i lv True of\n Trueif \\ restrict_top t lv True\n | Falseif \\ restrict_top e lv True\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv True) x True)\n (restrict_top (restrict_top t lv True) x True)\n (restrict_top (restrict_top e lv True) x True))\n (ifex_ite (restrict_top (restrict_top i lv True) x False)\n (restrict_top (restrict_top t lv True) x False)\n (restrict_top (restrict_top e lv True) x False)))\n \\ R b;\n I b; les s b; r = (x1f, x2);\n (x1f,\n case lowest_tops\n [restrict_top i lv False, restrict_top t lv False,\n restrict_top e lv False] of\n None \\\n case restrict_top i lv False of\n Trueif \\ restrict_top t lv False\n | Falseif \\ restrict_top e lv False\n | Some x \\\n IFC x\n (ifex_ite (restrict_top (restrict_top i lv False) x True)\n (restrict_top (restrict_top t lv False) x True)\n (restrict_top (restrict_top e lv False) x True))\n (ifex_ite (restrict_top (restrict_top i lv False) x False)\n (restrict_top (restrict_top t lv False) x False)\n (restrict_top (restrict_top e lv False) x False)))\n \\ R x2 \\\n I x2 \\ les b x2;\n case ra of\n (ni, s') \\\n (ni,\n IFC lv (?n1.243 x1 x1a x1b x1c x1d x1e a b r x1f x2)\n (?n2.243 x1 x1a x1b x1c x1d x1e a b r x1f x2))\n \\ R s' \\\n I s' \\ les x2 s'\\\n \\ case ra of\n (r, s') \\\n (r, IFC lv\n (case lowest_tops\n [restrict_top i lv True, restrict_top t lv True, restrict_top e lv True] of\n None \\\n case restrict_top i lv True of\n Trueif \\\n restrict_top t lv True\n | Falseif \\\n restrict_top e lv True\n | Some x \\\n IFC x\n(ifex_ite (restrict_top (restrict_top i lv True) x True)\n (restrict_top (restrict_top t lv True) x True)\n (restrict_top (restrict_top e lv True) x True))\n(ifex_ite (restrict_top (restrict_top i lv True) x False)\n (restrict_top (restrict_top t lv True) x False)\n (restrict_top (restrict_top e lv True) x False)))\n (case lowest_tops\n [restrict_top i lv False, restrict_top t lv False,\n restrict_top e lv False] of\n None \\\n case restrict_top i lv False of\n Trueif \\\n restrict_top t lv False\n | Falseif \\\n restrict_top e lv False\n | Some x \\\n IFC x\n(ifex_ite (restrict_top (restrict_top i lv False) x True)\n (restrict_top (restrict_top t lv False) x True)\n (restrict_top (restrict_top e lv False) x True))\n(ifex_ite (restrict_top (restrict_top i lv False) x False)\n (restrict_top (restrict_top t lv False) x False)\n (restrict_top (restrict_top e lv False) x False))))\n \\ R s' \\\n I s' \\ les s s'"} {"_id": "501209", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n.\n \\\\(shiftr1 ^^ n) u \\ (shiftr1 ^^ n) v;\n (shiftr1 ^^ n) u \\ (shiftr1 ^^ n) v\\\n \\ u \\ v;\n shiftr1 ((shiftr1 ^^ n) u) \\ shiftr1 ((shiftr1 ^^ n) v);\n shiftr1 ((shiftr1 ^^ n) u) \\ shiftr1 ((shiftr1 ^^ n) v);\n (shiftr1 ^^ n) u \\ (shiftr1 ^^ n) v\\\n \\ u \\ v"} {"_id": "501210", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\x C1.\n strip C1 = strip (SKIP {x}) \\\n length (annos C1) = length (annos (SKIP {x}))\n 2. \\x1a x2 x3 C1.\n strip C1 = strip (x1a ::= x2 {x3}) \\\n length (annos C1) = length (annos (x1a ::= x2 {x3}))\n 3. \\C21 C22 C1.\n \\\\C1.\n strip C1 = strip C21 \\\n length (annos C1) = length (annos C21);\n \\C1.\n strip C1 = strip C22 \\\n length (annos C1) = length (annos C22);\n strip C1 = strip (C21;;\n C22)\\\n \\ length (annos C1) = length (annos (C21;;\n C22))\n 4. \\x1a C21 C22 x4 C1.\n \\\\C1.\n strip C1 = strip C21 \\\n length (annos C1) = length (annos C21);\n \\C1.\n strip C1 = strip C22 \\\n length (annos C1) = length (annos C22);\n strip C1 = strip (IF x1a THEN C21 ELSE C22\n {x4})\\\n \\ length (annos C1) =\n length (annos (IF x1a THEN C21 ELSE C22\n {x4}))\n 5. \\x1a x2 C2 x4 C1.\n \\\\C1.\n strip C1 = strip C2 \\\n length (annos C1) = length (annos C2);\n strip C1 = strip ({x1a}\n WHILE x2 DO C2\n {x4})\\\n \\ length (annos C1) = length (annos ({x1a}\n WHILE x2 DO C2\n {x4}))"} {"_id": "501211", "text": "proof (prove)\nusing this:\n x \\ (\\' i\\domain y. ball ((y)\\<^sub>F i) (e y))\n ?x \\ S \\ 0 < e ?x\n y \\ S\n\ngoal (1 subgoal):\n 1. dist x y < e y"} {"_id": "501212", "text": "proof (prove)\nusing this:\n rel_sum boa bob x y\n rel_sum boa bob y x\n x \\ {x. setl x \\ A \\ setr x \\ B}\n\ngoal (1 subgoal):\n 1. x = y"} {"_id": "501213", "text": "proof (prove)\nusing this:\n c \\ 0 \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. integrable (count_space A) (\\x. c *\\<^sub>R f x)"} {"_id": "501214", "text": "proof (prove)\ngoal (1 subgoal):\n 1. argmax\n (sum (summedBidVector\n (pseudoAllocation\n (randomEl\n (takeAll\n (\\x.\n x \\ maximalStrictAllocations N (set \\) b)\n (allAllocationsAlg ({seller} \\ N) \\))\n r) <|\n (({seller} \\ N) \\ finestpart (set \\)))\n ({seller} \\ N) (set \\)))\n (maximalStrictAllocations N (set \\) b) =\n {randomEl\n (takeAll\n (\\x. x \\ maximalStrictAllocations N (set \\) b)\n (allAllocationsAlg ({seller} \\ N) \\))\n r}"} {"_id": "501215", "text": "proof (prove)\nusing this:\n v |\\| vertices\n\ngoal (1 subgoal):\n 1. \\\\c.\n \\c \\ set conclusions; v = (c, [])\\\n \\ thesis;\n \\c is.\n \\c \\ set conclusions; is \\ it_paths (it' c);\n v = (c, 0 # is)\\\n \\ thesis\\\n \\ thesis"} {"_id": "501216", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\b_ag A \\ H ; x \\ H; Group (b_ag A)\\\n \\ x \\ carrier A"} {"_id": "501217", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Ord i \\ WR (succ i) (q_Eats y y) = WR i y"} {"_id": "501218", "text": "proof (prove)\nusing this:\n arr ?f = ide ?f\n\ngoal (1 subgoal):\n 1. seq g f = (arr f \\ f = g)"} {"_id": "501219", "text": "proof (prove)\nusing this:\n ide ?a = \\?a : ?a \\ ?a\\\n C1.ide ?a =\n \\?a : ?a \\\\<^sub>1 ?a\\\n C2.ide ?a =\n \\?a : ?a \\\\<^sub>2 ?a\\\n\ngoal (1 subgoal):\n 1. ide f = (C1.ide (fst f) \\ C2.ide (snd f))"} {"_id": "501220", "text": "proof (prove)\nusing this:\n (h, as) \\\n P1 a ap * P2 b bp * F \\\\<^sub>A P1 a' ap * P2 b' bp * F'\n\ngoal (1 subgoal):\n 1. (h, as) \\\n P2 b bp * (P1 a ap * F) \\\\<^sub>A P2 b' bp * (P1 a' ap * F')"} {"_id": "501221", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {. \\X .} = \\(assert ` X)"} {"_id": "501222", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bij (put \\)"} {"_id": "501223", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ACI_nullable s_ = ([] \\ ACI_lang s_)"} {"_id": "501224", "text": "proof (prove)\ngoal (1 subgoal):\n 1. k = m + n"} {"_id": "501225", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prod f (A - B') / f x = prod f A / prod f B' / f x"} {"_id": "501226", "text": "proof (prove)\nusing this:\n a < 2 ^ n + i \\ b < 2 ^ n + j \\\n 2 ^ n + i \\ a \\ a < 2 ^ (n + 1) + i \\ b < 2 ^ n + j \\\n a < 2 ^ n + i \\ 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j \\\n 2 ^ n + i \\ a \\\n a < 2 ^ (n + 1) + i \\ 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j\n a < 2 ^ n + i \\ b < 2 ^ n + j \\\n square2 (n + 1) i j - {(a, b)} - L0 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ tiling Ls\n 2 ^ n + i \\ a \\\n a < 2 ^ (n + 1) + i \\ b < 2 ^ n + j \\\n square2 (n + 1) i j - {(a, b)} - L1 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ tiling Ls\n a < 2 ^ n + i \\\n 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j \\\n square2 (n + 1) i j - {(a, b)} - L3 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ tiling Ls\n 2 ^ n + i \\ a \\\n a < 2 ^ (n + 1) + i \\\n 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j \\\n square2 (n + 1) i j - {(a, b)} - L2 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ tiling Ls\n a < 2 ^ n + i \\ b < 2 ^ n + j \\\n L0 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ square2 (n + 1) i j - {(a, b)}\n 2 ^ n + i \\ a \\\n a < 2 ^ (n + 1) + i \\ b < 2 ^ n + j \\\n L1 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ square2 (n + 1) i j - {(a, b)}\n a < 2 ^ n + i \\\n 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j \\\n L3 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ square2 (n + 1) i j - {(a, b)}\n 2 ^ n + i \\ a \\\n a < 2 ^ (n + 1) + i \\\n 2 ^ n + j \\ b \\ b < 2 ^ (n + 1) + j \\\n L2 (2 ^ n + i - 1) (2 ^ n + j - 1)\n \\ square2 (n + 1) i j - {(a, b)}\n\ngoal (1 subgoal):\n 1. square2 (Suc n) i j - {(a, b)} \\ tiling Ls"} {"_id": "501227", "text": "proof (prove)\nusing this:\n c \\ c \\ (c - x) \\ (c - x)\n 0 < x \\ b\n\ngoal (1 subgoal):\n 1. c \\ b < x \\ b"} {"_id": "501228", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pred_pmf P (p \\ f) = pred_pmf (pred_pmf P \\ f) p"} {"_id": "501229", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\determ (wp a); determ (wp b); well_def b\\\n \\ determ (wp (a ;; b))"} {"_id": "501230", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"} {"_id": "501231", "text": "proof (chain)\npicking this:\n sum_bot_ord (C ?y' f) (C ?y' g)"} {"_id": "501232", "text": "proof (prove)\nusing this:\n (inv\\<^bsub>DirProds G I\\<^esub> x \\\\<^bsub>DirProds G I\\<^esub>\n x)\n i =\n \\\\<^bsub>G i\\<^esub>\n (inv\\<^bsub>DirProds G I\\<^esub> x \\\\<^bsub>DirProds G I\\<^esub>\n x)\n i =\n (inv\\<^bsub>DirProds G I\\<^esub> x) i \\\\<^bsub>G i\\<^esub> x i\n x \\ carrier (DirProds G I)\n inv\\<^bsub>DirProds G I\\<^esub> x \\ carrier (DirProds G I)\n\ngoal (1 subgoal):\n 1. (inv\\<^bsub>DirProds G I\\<^esub> x) i = inv\\<^bsub>G i\\<^esub> x i"} {"_id": "501233", "text": "proof (prove)\ngoal (1 subgoal):\n 1. integral {real_of_int a..real_of_int b}\n (\\t. pbernpoly n t *\\<^sub>R f t) =\n (bernoulli (Suc n) / real (Suc n)) *\\<^sub>R\n (f (real_of_int b) - f (real_of_int a)) -\n integral {real_of_int a..real_of_int b}\n (\\t. pbernpoly (Suc n) t *\\<^sub>R f' t) /\\<^sub>R\n real (Suc n)"} {"_id": "501234", "text": "proof (prove)\nusing this:\n set (r # rs) \\ R\n set (s # ss) \\ R\n\ngoal (1 subgoal):\n 1. set (map2 (-) (r # rs) (s # ss)) \\ R"} {"_id": "501235", "text": "proof (prove)\nusing this:\n reachable_state sys s\n \\\\.\n prerun sys \\ \\\n (\\\\I\\) \\\n\ngoal (1 subgoal):\n 1. I s"} {"_id": "501236", "text": "proof (prove)\nusing this:\n w # ws \\ carrier \\\\<^bsub>I\\<^esub>\n x \\ X\n snd w \\ I\n g \\ I \\ carrier G\n g (snd w) \\ carrier G\n\ngoal (1 subgoal):\n 1. ws \\ carrier \\\\<^bsub>I\\<^esub>"} {"_id": "501237", "text": "proof (prove)\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501238", "text": "proof (prove)\nusing this:\n map (\\i. (i, pdevs_apply X i))\n (filter\n (\\x. ((\\) (0::'a) \\ snd) (x, pdevs_apply X x))\n [0..) (0::'a) \\ snd)\n (map (\\i. (i, pdevs_apply X i)) [0..) (0::'a) \\ snd)\n (map (\\i. (i, pdevs_apply X i)) [0..Runs_table s2 stlsss2\n\ngoal (1 subgoal):\n 1. s2 \\2 tmap fst id stlsss2"} {"_id": "501240", "text": "proof (prove)\nusing this:\n g = t\n\ngoal (1 subgoal):\n 1. (THE fa. \\i. f v $ i = (\\x\\UNIV. fa x * Y $ x $ i)) i =\n (\\x\\UNIV.\n (THE f. \\i. v $ i = (\\x\\UNIV. f x * X $ x $ i)) x *\n (THE fa.\n \\i. f (X $ x) $ i = (\\x\\UNIV. fa x * Y $ x $ i))\n i)"} {"_id": "501241", "text": "proof (prove)\nusing this:\n \\is. P (T ! i) s \\ S ! i = s \\ \\\n length T = length S\n\ngoal (1 subgoal):\n 1. (\\U.\n \\length T = length U;\n \\i\\<^sub>l\\<^sub>i\\<^sub>s\\<^sub>t \\\\\n \\ thesis) \\\n thesis"} {"_id": "501242", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\A\\\\<^sub>o\\<^sub>p = 0) = (A = 0)"} {"_id": "501243", "text": "proof (prove)\ngoal (1 subgoal):\n 1. possible_allocations_rel G N\n \\ injectionsUniverse \\\n {a. Range a \\ N \\\n Domain a \\ all_partitions G}"} {"_id": "501244", "text": "proof (prove)\nusing this:\n \\\\c. 0 < v c; v c \\ n; 0 < v c; v c \\ n;\n \\u. \\ u \\\\<^bsub>v,n\\<^esub> M\\\n \\ \\ u \\\\<^bsub>v,n\\<^esub> reset M n (v c)\n d\n M' \\ reset M n (v c) d\n \\k\\n. 0 < k \\ (\\c. v c = k)\n \\c.\n 0 < v c \\\n (\\x y.\n v x \\ n \\ v y \\ n \\ v x = v y \\\n x = y)\n v c \\ n\n u \\ {u. u \\\\<^bsub>v,n\\<^esub> M'}\n {u. u \\\\<^bsub>v,n\\<^esub> M} = {}\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501245", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b. a \\ 0 \\ M = moebius_similarity a b"} {"_id": "501246", "text": "proof (prove)\nusing this:\n literal_ordering A L\n L \\ C\n A \\ C - {L} \\\n (\\B.\n B \\ C - {L} \\ A \\ B \\\n literal_ordering B A)\n \\literal_ordering ?A ?B; literal_ordering ?B ?C\\\n \\ literal_ordering ?A ?C\n\ngoal (1 subgoal):\n 1. L \\ C \\\n (\\B.\n B \\ C \\ L \\ B \\\n literal_ordering B L)"} {"_id": "501247", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P,h' \\ l' (:\\) E(V \\ T)"} {"_id": "501248", "text": "proof (prove)\nusing this:\n integrable (lborel \\\\<^sub>M lborel) f\n integrable (lborel \\\\<^sub>M lborel) ?f \\\n LBINT x. LBINT y. ?f (x, y) =\n integral\\<^sup>L (lborel \\\\<^sub>M lborel) ?f\n\ngoal (1 subgoal):\n 1. LBINT x. LBINT y. f (x, y) =\n integral\\<^sup>L (lborel \\\\<^sub>M lborel) f"} {"_id": "501249", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_antirange_kleene_algebra.bdia (v ::= e) \\P\\ =\n \\\\s.\n \\w. s v = e (s(v := w)) \\ P (s(v := w))\\"} {"_id": "501250", "text": "proof (prove)\ngoal (1 subgoal):\n 1. relative_homology_group p (subtopology X S) T \\\n relative_homology_group p X T"} {"_id": "501251", "text": "proof (prove)\nusing this:\n ss = [h]\n is_root h\n f = Insert x\n is_root ?hp \\\n \\ (Pairing_Heap_Tree.insert ?x ?hp) - \\ ?hp\n \\ log 2 (real (size ?hp + 1))\n \\s\\set ss. is_root s\n length ss = arity f\n\ngoal (1 subgoal):\n 1. real (cost f ss) + \\ (exec f ss) - sum_list (map \\ ss)\n \\ U f ss"} {"_id": "501252", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\heap.\n P heap \\\n (case run_state (State (\\heap. the (execute b heap))) heap of\n (v1, heap1) \\\n case execute b heap of None \\ False\n | Some (v2, heap2) \\\n v1 = v2 \\ heap1 = heap2 \\ P heap2)"} {"_id": "501253", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\S.\n fst SK = Some S \\\n Crypt (Auth_ShaKey n) (PriKey S) \\ used s) \\\n fst SK = None"} {"_id": "501254", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (LEAST i. seq (Suc \\ + Suc \\ + i) = seq (Suc \\))\n < (LEAST i. seq (\\ + \\ + i) = seq \\)"} {"_id": "501255", "text": "proof (state)\nthis:\n (b / a)\\<^sup>2 - 4 * (c / a) = (b\\<^sup>2 - 4 * a * c) / a\\<^sup>2\n\ngoal (4 subgoals):\n 1. 0 \\ (b / a)\\<^sup>2 - 4 * (c / a)\n 2. \\1\\<^sup>2 + b / a * \\1 + c / a = 0\n 3. \\2\\<^sup>2 + b / a * \\2 + c / a = 0\n 4. \\1 \\ \\2"} {"_id": "501256", "text": "proof (prove)\nusing this:\n \\ j < i\n\ngoal (1 subgoal):\n 1. opt_bst_wpl i j = (local.opt_bst i j, local.wpl i j (local.opt_bst i j))"} {"_id": "501257", "text": "proof (prove)\nusing this:\n m = 0\n\ngoal (1 subgoal):\n 1. bit (set_bit (Suc n) a) m =\n bit (a mod (2::'a) + (2::'a) * set_bit n (a div (2::'a))) m"} {"_id": "501258", "text": "proof (prove)\nusing this:\n r p \\ r i\n r i \\ ad t \\ r q\n \\x| (d t \\ r i) \\ r i\n\ngoal (1 subgoal):\n 1. \\while t inv i do x od| i \\ r i \\ ad t"} {"_id": "501259", "text": "proof (prove)\nusing this:\n internal p ?T \\ treesize (delete p ?T) < treesize ?T\n internal (a # p) T\n\ngoal (1 subgoal):\n 1. \\T1 T2. T = Branching T1 T2"} {"_id": "501260", "text": "proof (prove)\ngoal (1 subgoal):\n 1. arg_max_list f l \\ set l"} {"_id": "501261", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x = closest_point S a"} {"_id": "501262", "text": "proof (prove)\nusing this:\n cmod (contour_integral (part_circlepath c r a b) f)\n \\ integral {0..1} (\\x. B * 1 * (r * \\b - a\\) * 1)\n\ngoal (1 subgoal):\n 1. cmod (contour_integral (part_circlepath c r a b) f)\n \\ B * r * \\b - a\\"} {"_id": "501263", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i. P i) = (P 1 \\ P 2)"} {"_id": "501264", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ terminal subs (f n); f (Suc n) \\ subs (f n);\n x \\ set (atoms (f n))\\\n \\ x \\ set (atoms (f (Suc n)))"} {"_id": "501265", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monotone (rel_prod orda ordb) ordc f"} {"_id": "501266", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bit a n = (drop_bit n a AND (1::'a) = (1::'a))"} {"_id": "501267", "text": "proof (prove)\nusing this:\n \\y. ?x \\ y\n\ngoal (1 subgoal):\n 1. (\\z.\n ord.max (\\) z1 z2 \\ z \\\n thesis) \\\n thesis"} {"_id": "501268", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\n. real_of_ereal (enn2ereal (u n))) \\ l)\n F"} {"_id": "501269", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k::nat. \\(i::nat) j::nat. f (i + j) \\ f k\nvariables:\n f :: nat \\ 'c\ntype variables:\n 'c :: complete_lattice"} {"_id": "501270", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bl \\# \\ \\ del_block_sys.add_block bl bl = \\"} {"_id": "501271", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.basis_reduction_swap i ((f1, f2), dmus, ds) =\n (let di = ds !! i; dsi = ds !! Suc i; im1 = i - 1; dim1 = ds !! im1;\n fi = hd f2; fim1 = hd f1; dmu_i_im1 = dmus !! i !! im1; fi' = fim1;\n fim1' = fi\n in (False, im1, (tl f1, fim1' # fi' # tl f2),\n local.swap_mu dmus i dmu_i_im1 dim1 di dsi,\n iarray_update ds i ((dsi * dim1 + dmu_i_im1 * dmu_i_im1) div di)))"} {"_id": "501272", "text": "proof (prove)\nusing this:\n local.square_free_factorization p (c, bs)\n\ngoal (1 subgoal):\n 1. (p = smult c (\\(a, i)\\set bs. a ^ Suc i) &&&\n ((a, i) \\ set bs \\\n square_free a \\ degree a \\ 0)) &&&\n (\\(a, i) \\ set bs; (b, j) \\ set bs;\n (a, i) \\ (b, j)\\\n \\ coprime a b) &&&\n (p = 0 \\ c = (0::'a) \\ bs = []) &&& distinct bs"} {"_id": "501273", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\eqa\\set t.\n lhs eqa = y \\\n Mapping.lookup\n (foldl\n (\\v' e.\n Mapping.update (lhs e) (rhs_eq_val (\\ s) x v e) v')\n (\\ s) t)\n y =\n Some (rhs_eq_val (\\ s) x v eqa)"} {"_id": "501274", "text": "proof (prove)\ngoal (1 subgoal):\n 1. G,A\\{=:n} In1l e\\ {G\\} \\\n G,A\\{Normal\n ((\\Y' s' s. s' = s \\ normal s) \\.\n G\\init\\n)}\n .Expr e.\n {\\Y s' s.\n G\\s \\In1r (Expr e)\\\\ (Y, s')}"} {"_id": "501275", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inside (path_image (p +++ linepath a b)) \\\n outside (path_image (p +++ linepath a b)) =\n {} &&&\n frontier (inside (path_image (p +++ linepath a b))) =\n path_image (p +++ linepath a b) &&&\n frontier (outside (path_image (p +++ linepath a b))) =\n path_image (p +++ linepath a b)"} {"_id": "501276", "text": "proof (prove)\nusing this:\n p \\ {0<..1}\n\ngoal (1 subgoal):\n 1. measure_pmf.prob (geometric_pmf p) (UNIV - {..!x. x \\ U \\ P x\n then Some (THE x. x \\ U \\ P x) else None) =\n None"} {"_id": "501279", "text": "proof (prove)\ngoal (1 subgoal):\n 1. norm (f t x) \\ B"} {"_id": "501280", "text": "proof (prove)\nusing this:\n t5 \\ keys (lookup (i.punit.tail (focus X h)) t4)\n\ngoal (1 subgoal):\n 1. t5 \\ keys (lookup (focus X h) t4)"} {"_id": "501281", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\(s1', t1)|\\|possible_steps e1 s1 r1 l i.\n \\(s2', t2)|\\|possible_steps e2 s2 r2 l i.\n evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \\\n executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2'\n (evaluate_updates t2 i r2) es) \\\n (\\(s2', t2)|\\|possible_steps e2 s2 r2 l i.\n \\(s1', t1)|\\|possible_steps e1 s1 r1 l i.\n evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \\\n executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2'\n (evaluate_updates t2 i r2) es) \\\n executionally_equivalent e1 s1 r1 e2 s2 r2 ((l, i) # es)"} {"_id": "501282", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bprv (neg (PPf (encPf prf) \\\\\\))"} {"_id": "501283", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inside (inside S) \\ S"} {"_id": "501284", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\infinite I; t0 \\ I\\\n \\ iNextStrong t0 I P = iNextWeak t0 I P"} {"_id": "501285", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa.\n x \\ xa \\\n (\\va u.\n move ts ts' v=va\\u \\\n True \\\n (\\v ua.\n u=v\\ua \\\n (\\va u.\n v=va--u \\\n True \\\n (\\v ua.\n u=v--ua \\\n ((0 < \\ext v\\ \\\n len v ts' x = ext v \\\n restrict v (res ts') x = lan v \\\n |lan v| = 1) \\\n 0 < \\ext v\\ \\\n len v ts' xa = ext v \\\n restrict v (res ts') xa = lan v \\\n |lan v| = 1) \\\n True)) \\\n True)) \\\n (\\ts'.\n ts' \\<^bold>\\ ts' \\\n (\\va u.\n move ts' ts' (move ts ts' v)=va\\u \\\n True \\\n (\\v ua.\n u=v\\ua \\\n (\\va u.\n v=va--u \\\n True \\\n (\\v ua.\n u=v--ua \\\n ((0 < \\ext v\\ \\\n len v ts' x = ext v \\\n restrict v (res ts') x = lan v \\\n |lan v| = 1) \\\n 0 < \\ext v\\ \\\n len v ts' xa = ext v \\\n restrict v (res ts') xa = lan v \\\n |lan v| = 1) \\\n True)) \\\n True)))"} {"_id": "501286", "text": "proof (prove)\nusing this:\n ran (process_mru vs |` Q) \\ {}\n vote_set vs Q \\ {}\n\ngoal (1 subgoal):\n 1. Q \\ voters vs \\ {}"} {"_id": "501287", "text": "proof (state)\ngoal (1 subgoal):\n 1. P (iti c f \\0)"} {"_id": "501288", "text": "proof (prove)\nusing this:\n x1 \\ b \\\n (\\u>0. u < 1 \\ x2 = (1 - u) *\\<^sub>R x1 + u *\\<^sub>R b)\n\ngoal (1 subgoal):\n 1. (\\v.\n \\0 < v; v < 1;\n x2 = (1 - v) *\\<^sub>R x1 + v *\\<^sub>R b\\\n \\ thesis) \\\n thesis"} {"_id": "501289", "text": "proof (prove)\ngoal (1 subgoal):\n 1. degree_m (Polynomial.smult (lead_coeff f) (prod_list hs)) =\n degree (Polynomial.smult (lead_coeff f) (prod_list hs))"} {"_id": "501290", "text": "proof (prove)\nusing this:\n \\\\\\_\\_\\.Prj\\<^sub>1\\<^sub>1 : HHH\\\\\\\\.chine \\\\<^sub>C \\\\.apex\\\n \\ \\ \\ \\\n if arr \\ \\ arr \\ \\ src \\ = trg \\\n then \\Chn = chine_hcomp \\ \\,\n Dom =\n \\Leg0 = \\.leg0 \\ \\\\.prj\\<^sub>0,\n Leg1 = \\.leg1 \\ \\\\.prj\\<^sub>1\\,\n Cod =\n \\Leg0 =\n \\.cod.leg0 \\\n \\

\\<^sub>0[\\.cod.leg0, \\.cod.leg1],\n Leg1 =\n \\.cod.leg1 \\\n \\

\\<^sub>1[\\.cod.leg0, \\.cod.leg1]\\\\\n else null\n \\ide ?g; ide ?f; src ?g = trg ?f\\\n \\ chine_hcomp ?g ?f =\n Leg0 (Dom ?g) \\\\ Leg1 (Dom ?f)\n \\\\.cod.apex \\ C.dom \\\\.cod.leg0\n src \\ = trg \\\n C.cospan \\.leg0 \\.leg1\n \\C.ide ?b; C.seq ?b ?f\\\n \\ ?b \\ ?f = ?f\n\ngoal (1 subgoal):\n 1. \\.chine \\\n \\\\.prj\\<^sub>1 \\\n \\\\_\\_\\.Prj\\<^sub>1\\<^sub>1 =\n \\\\.prj\\<^sub>1 \\\n \\\\_\\_\\.Prj\\<^sub>1\\<^sub>1"} {"_id": "501291", "text": "proof (prove)\ngoal (1 subgoal):\n 1. S' retract_of UNIV"} {"_id": "501292", "text": "proof (prove)\nusing this:\n (\\ lnull ?xs) = (\\x xs'. ?xs = x % xs')\n \\ lnull w\n\ngoal (1 subgoal):\n 1. (\\a v. w = $ v \\ thesis) \\\n thesis"} {"_id": "501293", "text": "proof (prove)\nusing this:\n size \\1[M]\\<^sub>\\\\<^sub>2 \\ 2 ^ (size \\1 + 1)\n size \\2[M]\\<^sub>\\\\<^sub>2 \\ 2 ^ (size \\2 + 1)\n\ngoal (1 subgoal):\n 1. size \\1 U\\<^sub>n \\2[M]\\<^sub>\\\\<^sub>2\n \\ 2 ^ (size \\1 + 1) + 2 ^ (size \\2 + 1) + 1"} {"_id": "501294", "text": "proof (prove)\nusing this:\n (THE xa.\n xa \\ carrier (K [X]) \\\n pirreducible K xa \\\n local.eval xa x = \\ \\ lead_coeff xa = \\)\n \\ carrier (K [X]) \\\n pirreducible K\n (THE xa.\n xa \\ carrier (K [X]) \\\n pirreducible K xa \\\n local.eval xa x = \\ \\ lead_coeff xa = \\) \\\n local.eval\n (THE xa.\n xa \\ carrier (K [X]) \\\n pirreducible K xa \\\n local.eval xa x = \\ \\ lead_coeff xa = \\)\n x =\n \\ \\\n lead_coeff\n (THE xa.\n xa \\ carrier (K [X]) \\\n pirreducible K xa \\\n local.eval xa x = \\ \\ lead_coeff xa = \\) =\n \\\n\ngoal (1 subgoal):\n 1. (Irr K x \\ carrier (K [X]) &&& pirreducible K (Irr K x)) &&&\n lead_coeff (Irr K x) = \\ &&& local.eval (Irr K x) x = \\"} {"_id": "501295", "text": "proof (prove)\nusing this:\n vcard xs = ZFC_in_HOL.succ (vcard xs')\n vfinite xs'\n \\\\<^sub>\\ xs = vcard xs\n\ngoal (1 subgoal):\n 1. xs' = xs \\\\<^sup>l\\<^sub>\\ vcard xs'"} {"_id": "501296", "text": "proof (prove)\nusing this:\n set (map of_int_hom.vec_hom fs_init) \\ Rn \\\n distinct (map of_int_hom.vec_hom fs_init) \\\n gs.lin_indpt (set (map of_int_hom.vec_hom fs_init))\n set (rows B) = set (map of_int_hom.vec_hom fs_init)\n\ngoal (1 subgoal):\n 1. gs.lin_indpt (set (rows B))"} {"_id": "501297", "text": "proof (prove)\nusing this:\n O x y\n \\ O x z\n\ngoal (1 subgoal):\n 1. P (x \\ (y \\ z)) (x \\ y)"} {"_id": "501298", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (list_R1 F = l) = (F = unR l)"} {"_id": "501299", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_nan x \\ \\ is_finite x"} {"_id": "501300", "text": "proof (prove)\nusing this:\n C \\ \\\n (\\(k,\n v)\\{0..} \\ set (\\\\<^sub>\\).\n {{{(State k (index (\\\\<^sub>\\) v))\\<^sup>+,\n (State (Suc k)\n (index (\\\\<^sub>\\) v))\\} \\\n {(Operator k (index (\\\\<^sub>\\) op))\\<^sup>+ |op.\n op \\ set (\\\\<^sub>\\) \\\n v \\ set (add_effects_of op)}}})\n\ngoal (1 subgoal):\n 1. clause_semantics \\ C"} {"_id": "501301", "text": "proof (prove)\ngoal (13 subgoals):\n 1. mapl_G id id = id\n 2. \\f1 f2 g1 g2.\n mapl_G (g1 \\ f1) (g2 \\ f2) =\n mapl_G g1 g2 \\ mapl_G f1 f2\n 3. \\x f1 f2 g1 g2.\n \\\\z1. z1 \\ set1_G x \\ f1 z1 = g1 z1;\n \\z2. z2 \\ set2_G x \\ f2 z2 = g2 z2\\\n \\ mapl_G f1 f2 x = mapl_G g1 g2 x\n 4. \\f1 f2. set1_G \\ mapl_G f1 f2 = (`) f1 \\ set1_G\n 5. \\f1 f2. set2_G \\ mapl_G f1 f2 = (`) f2 \\ set2_G\n 6. card_order bd_G\n 7. cinfinite bd_G\n 8. \\x. |set1_G x| \\o bd_G\n 9. \\x. |set2_G x| \\o bd_G\n 10. \\R1 R2 S1 S2.\n rell_G R1 R2 OO rell_G S1 S2 \\ rell_G (R1 OO S1) (R2 OO S2)\nA total of 13 subgoals..."} {"_id": "501302", "text": "proof (prove)\nusing this:\n exec_step_input P C M pc ics = StepC C' Cs\n ics = Called Cs\n\ngoal (1 subgoal):\n 1. ics = Calling C' Cs"} {"_id": "501303", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lookup (delete k al) = (lookup al)(k := None)"} {"_id": "501304", "text": "proof (prove)\nusing this:\n Lam [a].t \\\\<^sub>1 t'\n a \\ t'\n\ngoal (1 subgoal):\n 1. \\t''. t' = Lam [a].t'' \\ t \\\\<^sub>1 t''"} {"_id": "501305", "text": "proof (prove)\nusing this:\n x \\ \\ (fv_trm ` set ts)\n match ts (map (MFOTL.eval_trm (map the v)) ts) = Some f\n v[x := None] = tabulate f 0 n\n\ngoal (1 subgoal):\n 1. f x = None &&& length v = n"} {"_id": "501306", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa.\n \\\\x. f x \\ \\{g xa |xa. (xa, x) \\ R};\n xa \\ {f y |y. (x, y) \\ R}\\\n \\ xa \\ g x"} {"_id": "501307", "text": "proof (prove)\nusing this:\n var_infinite undefined\n var_regular undefined\n varSortAsSort_inj Delta\n arityOf_lt_var undefined Delta\n barityOf_lt_var undefined Delta\n sort_lt_var undefined undefined\n\ngoal (1 subgoal):\n 1. FixVars TYPE('var) TYPE('varSort)"} {"_id": "501308", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Good c \\ events"} {"_id": "501309", "text": "proof (prove)\ngoal (1 subgoal):\n 1. c d \\ c d"} {"_id": "501310", "text": "proof (prove)\ngoal (1 subgoal):\n 1. alw (holds P) xs = pred_stream P xs"} {"_id": "501311", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gfp = complete_lattice.lfp Sup (\\x y. y \\ x)"} {"_id": "501312", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf'\\<^sub>s\\<^sub>s\\<^sub>t V S"} {"_id": "501313", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A B CTSP D E'"} {"_id": "501314", "text": "proof (prove)\nusing this:\n folding.gallery (B # C # Cs @ [D]) \\\n adjacentchain (B # C # Cs @ [D])\n \\B \\ f \\ folding.\\;\n C \\ folding.\\ - f \\ folding.\\; B \\ C\\\n \\ f ` C = B\n folding.gallery (B # C # Cs @ [D]) \\\n folding.gallery (f \\ (B # C # Cs @ [D]))\n folding.gallery (B # B # f \\ Cs @ [D]) \\\n folding.gallery (B # f \\ Cs @ [D])\n \\is_arg_min length (\\Ds. folding.gallery (B # Ds @ [D]))\n ?x;\n folding.gallery (B # f \\ Cs @ [D])\\\n \\ \\ order.greater (length ?x)\n (length (f \\ Cs))\n min_gallery (B # (C # Cs) @ [D]) =\n (B \\ D \\\n is_arg_min length (\\zs. folding.gallery (B # zs @ [D])) (C # Cs))\n\ngoal (1 subgoal):\n 1. \\B \\ f \\ folding.\\;\n C \\ folding.\\ - f \\ folding.\\;\n D \\ f \\ folding.\\;\n folding.gallery (B # C # Cs @ [D])\\\n \\ \\ min_gallery (B # C # Cs @ [D])"} {"_id": "501315", "text": "proof (prove)\ngoal (1 subgoal):\n 1. aform_val e (truncate_aform p z) =\n aform_val e z + aform_val e (truncate_error_aform p z)"} {"_id": "501316", "text": "proof (prove)\ngoal (1 subgoal):\n 1. aprimedivisor n ^ multiplicity (aprimedivisor n) n = n"} {"_id": "501317", "text": "proof (prove)\ngoal (9 subgoals):\n 1. (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ true\\<^sub>r,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ true\\<^sub>r)\n \\ Rv\n 2. (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ false\\<^sub>r,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ false\\<^sub>r)\n \\ Rv\n 3. \\ai a'i.\n (ai, a'i) \\ R \\\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ prop\\<^sub>r(ai),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_\n prop\\<^sub>r(a'i))\n \\ Rv\n 4. \\ai a'i.\n (ai, a'i) \\ R \\\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ nprop\\<^sub>r(ai),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_\n nprop\\<^sub>r(a'i))\n \\ Rv\n 5. \\ai a'i b b'.\n \\(ai, a'i) \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ ai,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ a'i)\n \\ Rv;\n (b, b') \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ b,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ b')\n \\ Rv\\\n \\ (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_\n (ai and\\<^sub>r b),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_\n a'h_ (a'i and\\<^sub>r b'))\n \\ Rv\n 6. \\ai a'i b b'.\n \\(ai, a'i) \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ ai,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ a'i)\n \\ Rv;\n (b, b') \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ b,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ b')\n \\ Rv\\\n \\ (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_\n (ai or\\<^sub>r b),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_\n a'h_ (a'i or\\<^sub>r b'))\n \\ Rv\n 7. \\ai a'i.\n \\(ai, a'i) \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ ai,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ a'i)\n \\ Rv\\\n \\ (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_\n (X\\<^sub>r ai),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_\n a'h_ (X\\<^sub>r a'i))\n \\ Rv\n 8. \\ai a'i b b'.\n \\(ai, a'i) \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ ai,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ a'i)\n \\ Rv;\n (b, b') \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ b,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ b')\n \\ Rv\\\n \\ (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_\n (ai U\\<^sub>r b),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_\n a'h_ (a'i U\\<^sub>r b'))\n \\ Rv\n 9. \\ai a'i b b'.\n \\(ai, a'i) \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ ai,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ a'i)\n \\ Rv;\n (b, b') \\ \\R\\ltlr_rel;\n (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_ b,\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_ a'h_ b')\n \\ Rv\\\n \\ (rec_ltlr a_ aa_ ab_ ac_ ad_ ae_ af_ ag_ ah_\n (ai R\\<^sub>r b),\n rec_ltlr a'_ a'a_ a'b_ a'c_ a'd_ a'e_ a'f_ a'g_\n a'h_ (a'i R\\<^sub>r b'))\n \\ Rv"} {"_id": "501318", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AboveS r (f ` T) \\ {}"} {"_id": "501319", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\bij \\; bij \\\\\n \\ (\\\\<^sub>E (inv \\) (inv \\) e' = e) =\n (\\\\<^sub>E \\ \\ e = e')"} {"_id": "501320", "text": "proof (prove)\nusing this:\n p x \\ 0\n p x \\ 1\n finite (supp p)\n \\x\\supp (dist_remove p x). healthy (wp (a x))\n\ngoal (1 subgoal):\n 1. bd_cts (wp (SetPC a (\\_. p)))"} {"_id": "501321", "text": "proof (prove)\nusing this:\n rewrite_negated_primitives (disc, sel) C ?negate_f m = m\n\ngoal (1 subgoal):\n 1. rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m = m"} {"_id": "501322", "text": "proof (chain)\npicking this:\n card (local.blues_seen w' p) = 0\n local.valid w'"} {"_id": "501323", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fetch (tp1 @ ci ly (start_of ly as) ins @ tp2) s b =\n fetch (ci ly (start_of ly as) ins) (s - length tp1 div 2) b"} {"_id": "501324", "text": "proof (prove)\nusing this:\n ((\\ 0 \\ gba.V0 \\ ipath gba.E \\) \\\n gba.is_acc \\) \\\n (\\i. local.gba.L (\\ i) (\\ i))\n\ngoal (1 subgoal):\n 1. \\ 0 \\ gba.V0"} {"_id": "501325", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P\\<^sub>1 \\\\<^bsub>M\\<^esub> Q \\\n P\\<^sub>2 \\\\<^bsub>M\\<^esub> Q"} {"_id": "501326", "text": "proof (prove)\nusing this:\n a = a' - length E'\n action_obs E'' a' = NormalAction (ReadMem ad al v)\n lnth E'' a' = (t, NormalAction (ReadMem ad al v))\n length E' \\ a'\n\ngoal (1 subgoal):\n 1. action_obs E a = NormalAction (ReadMem ad al v) &&&\n lnth E a = (t, NormalAction (ReadMem ad al v))"} {"_id": "501327", "text": "proof (prove)\nusing this:\n lalternate xs = LCons x21 x22\n eSuc (enat ?n) = enat (Suc ?n)\n lalternate ?xs =\n (case ?xs of LNil \\ LNil\n | LCons x xs \\ LCons x (lalternate (ltl xs)))\n lnull ?xs \\ lalternate ?xs = LNil\n \\ lnull ?llist \\\n LCons (lhd ?llist) (ltl ?llist) = ?llist\n lhd (LCons ?x21.0 ?x22.0) = ?x21.0\n\ngoal (1 subgoal):\n 1. ltake (enat (Suc n)) (lalternate xs) =\n lalternate (ltake (enat (Suc (Suc (2 * n)))) xs)"} {"_id": "501328", "text": "proof (prove)\nusing this:\n {X \\ \\. x \\ X} \\ {}\n\ngoal (1 subgoal):\n 1. (\\Y.\n \\Y \\ {X \\ \\. x \\ X};\n \\Y'.\n (Y', Y)\n \\ {(K, L).\n K \\ \\ \\\n L \\ \\ \\ L \\ K} \\\n Y' \\ {X \\ \\. x \\ X}\\\n \\ thesis) \\\n thesis"} {"_id": "501329", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom 1 b [^]\\<^bsub>R\\<^esub> CARD('a) ^ m' = monom 1 b"} {"_id": "501330", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mono \\"} {"_id": "501331", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (TP, TQ) \\ TRel"} {"_id": "501332", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\e.\n \\\\a'.\n \\((fh, lh), a', f', l') \\ pr_algo_lts';\n a = project_operation a'\\\n \\ thesis;\n a = PUSH; f' = push_effect fh e; l' = lh;\n push_precond fh lh e\\\n \\ thesis\n 2. \\u.\n \\\\a'.\n \\((fh, lh), a', f', l') \\ pr_algo_lts';\n a = project_operation a'\\\n \\ thesis;\n a = RELABEL; f' = fh; l' = relabel_effect fh lh u;\n relabel_precond fh lh u\\\n \\ thesis"} {"_id": "501333", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\A'0 C'0 D'0 F'0.\n B Out A'0 A' \\\n B Out C'0 C' \\\n E Out D'0 D' \\\n E Out F'0 F' \\\n Cong B A'0 E D'0 \\ Cong B C'0 E F'0 \\\n Cong A'0 C'0 D'0 F'0"} {"_id": "501334", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (case SUCCEED of FAILi \\ FAIL\n | RES X \\ Sup (f ` X)) =\n SUCCEED"} {"_id": "501335", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj \\"} {"_id": "501336", "text": "proof (prove)\nusing this:\n ?q \\ accessible \\ nextl (dfa.init M) (path_to ?q) = ?q\n minimal\n\ngoal (1 subgoal):\n 1. (path_to (dfa.init M), []) \\ eq_app_right language"} {"_id": "501337", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (flow0 x ` (existence_ivl0 x \\ {..0}) \\ K) =\n (rev.flow0 x ` (rev.existence_ivl0 x \\ {0..}) \\ K)"} {"_id": "501338", "text": "proof (prove)\nusing this:\n empty_store_buffers ts\\<^sub>s\\<^sub>b'\n\ngoal (1 subgoal):\n 1. \\i p is xs sb \\ \\ \\.\n i < length ts\\<^sub>s\\<^sub>b' \\\n ts\\<^sub>s\\<^sub>b' ! i =\n (p, is, xs, sb, \\, \\, \\) \\\n sb = []"} {"_id": "501339", "text": "proof (prove)\nusing this:\n xs prefix of i\n \\?es prefix of ?i;\n ?m \\ set (node_deliver_messages ?es)\\\n \\ Deliver ?m \\ set ?es\n\ngoal (1 subgoal):\n 1. \\x51 x52 x53 x51a x52a x53a.\n \\(a, Store x51 x52 x53) \\ set (node_deliver_messages xs);\n (x, Store x51a x52a x53a) \\ set (node_deliver_messages xs);\n \\ hb (a, Store x51 x52 x53) (x, Store x51a x52a x53a) \\\n \\ hb (x, Store x51a x52a x53a) (a, Store x51 x52 x53);\n b = Store x51 x52 x53; y = Store x51a x52a x53a\\\n \\ \\Store x51 x52 x53\\ \\\n \\Store x51a x52a x53a\\ =\n \\Store x51a x52a x53a\\ \\\n \\Store x51 x52 x53\\"} {"_id": "501340", "text": "proof (prove)\nusing this:\n d dvd b\n d dvd a\n\ngoal (1 subgoal):\n 1. (\\u v.\n \\d * u = - b; d * v = a\\\n \\ thesis) \\\n thesis"} {"_id": "501341", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ts1'.\n \\mRed1 (ls1, (ts1, m1), ws1, is1)\n (ls1, (ts1', m1), ws1, is1) \\\n (\\t.\n no_\\moves2 s2 t \\\n no_\\moves1 (ls1, (ts1', m1), ws1, is1) t) \\\n (ls1, (ts1', m1), ws1, is1) \\m s2"} {"_id": "501342", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\i. sup (X i) (Y i)) \\ sup x y) net"} {"_id": "501343", "text": "proof (state)\nthis:\n continuous_on (s - B) f\n\ngoal (1 subgoal):\n 1. \\x B.\n \\x \\ s; open B; x \\ B\\\n \\ \\A.\n open A \\\n f x \\ A \\\n (\\y\\s.\n f y \\ A \\ y \\ B)"} {"_id": "501344", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Order E; Order F; ord_isom D E f; ord_isom E F g\\\n \\ ord_isom D F (compose (carrier D) g f)"} {"_id": "501345", "text": "proof (prove)\nusing this:\n \\ (derive_right x) i1 < \\ (derive_right x) i2\n snd x \\ \\Final i2\\ \\\n snd x \\ \\Final i1\\\n is_interval x\n\ngoal (1 subgoal):\n 1. \\ x i1 < \\ x i2"} {"_id": "501346", "text": "proof (state)\nthis:\n take k (replicate (Suc k) (2::?'a3)) = replicate k (2::?'a3)\n\ngoal (1 subgoal):\n 1. partial_state.encode1 (replicate (Suc k) 2) {0..a l h.\n (l < h \\\n mset (lran a (l + 1) h) = mset_ran a {l + 1..\n mset (if l < h then a l # lran a (l + 1) h else []) =\n mset_ran a {l..l (flit_of_hlit l)"} {"_id": "501349", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rcvd (rho n) p\n \\ SHOmsgVectors A r p (coarse_run rho r) (HOs r p) (SHOs r p)"} {"_id": "501350", "text": "proof (prove)\nusing this:\n d_IN j x' \\ \\\n \\ < \\\n d_IN (\\e. (if e \\ set E then \\ else 0) + j e) x' =\n \\ * of_nat (card (set (filter (\\(y, x''). x'' = x') E))) +\n d_IN j x'\n\ngoal (1 subgoal):\n 1. d_IN (\\e. (if e \\ set E then \\ else 0) + j e)\n x' \\\n \\"} {"_id": "501351", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b c.\n ideal_generated {a, b, c} = ideal_generated {1::'a} \\\n (\\p q.\n ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"} {"_id": "501352", "text": "proof (prove)\nusing this:\n (nfa_startnode A, q) \\ (succsr (subset_succs A))\\<^sup>*\n p \\ set (subset_succs A q)\n\ngoal (1 subgoal):\n 1. (nfa_startnode A, p) \\ (succsr (subset_succs A))\\<^sup>*"} {"_id": "501353", "text": "proof (prove)\nusing this:\n atom s1 \\ (v, i, x, x')\n atom k1 \\ (v, i, x, x', s1)\n atom s \\ (v, i, x, x', k1, s1)\n atom k \\ (v, i, x, x', s, k1, s1)\n\ngoal (1 subgoal):\n 1. {SubstFormP v i x x'} \\ SubstFormP v i (Q_Neg x) (Q_Neg x')"} {"_id": "501354", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier A; 0 \\ m; n < 0; m * n \\ 0;\n m \\ 0\\\n \\ aSum A (nat m) (aSum A (nat (- n)) (-\\<^sub>a a)) =\n aSum A (nat (- (m * n))) (-\\<^sub>a a)"} {"_id": "501355", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\l p.\n l \\ [] \\\n cs_pop\n p <\\r.\n case r of\n (r, p') \\\n cs_list (tl l) p' * \\ (r = hd l)>\\<^sub>t"} {"_id": "501356", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cf_sol R S X X c \\\n snd c 0 = \\\\<^bsub>S\\<^esub> \\\n snd c (Suc 0) = 1\\<^sub>r\\<^bsub>S\\<^esub>"} {"_id": "501357", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\f. sum' f \\)\n \\ Group.iso\n (subgroup_generated\n (sum_group \\ (\\T. chain_group p (subtopology X T)))\n (carrier\n (sum_group \\\n (\\T. chain_group p (subtopology X T))) \\\n (\\\\<^sub>E T\\\\.\n singular_relcycle_set p (subtopology X T) {})))\n (subgroup_generated (chain_group p X)\n (singular_relcycle_set p X {}))"} {"_id": "501358", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ga.accept = local.sa.accept"} {"_id": "501359", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a vec, show_class)"} {"_id": "501360", "text": "proof (prove)\ngoal (1 subgoal):\n 1. small\n {b. \\a\\elts A. [a, b]\\<^sub>\\ \\\\<^sub>\\ r}"} {"_id": "501361", "text": "proof (state)\nthis:\n \\\\ : F f \\\\<^sub>D\n g' \\\\<^sub>D trg\\<^sub>D\n (F f)\\\n\ngoal (1 subgoal):\n 1. \\unit\n (src\\<^sub>C\n g) : trg\\<^sub>D\n (F f) \\\\<^sub>D F\n (trg\\<^sub>C f)\\"} {"_id": "501362", "text": "proof (prove)\ngoal (1 subgoal):\n 1. unstables r step start =\n sorted_list_of_set {p. succs p \\ {} \\ p < length start}"} {"_id": "501363", "text": "proof (prove)\nusing this:\n lossless_spmf p\n\ngoal (1 subgoal):\n 1. P (GPV p)"} {"_id": "501364", "text": "proof (prove)\nusing this:\n thr'_invar (active_threads s)\n\ngoal (1 subgoal):\n 1. \\.active_threads (state_\\ s) = {}"} {"_id": "501365", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ #c$ = #c &&&\n (\\f. \\ f$ = f) &&&\n (\\f. \\ f$ = f)) &&&\n (\\f. \\ f$ = f) &&&\n \\ (\\x. P x)$ = (\\x. P x$) &&&\n \\ (\\x. P x)$ = (\\x. P x$)"} {"_id": "501366", "text": "proof (prove)\nusing this:\n False\n\ngoal (1 subgoal):\n 1. r \\ RED \\"} {"_id": "501367", "text": "proof (prove)\nusing this:\n mat\\<^sub>h C * mat\\<^sub>h D = mat\\<^sub>h (1\\<^sub>m n)\n mat\\<^sub>h D * mat\\<^sub>h C = mat\\<^sub>h (1\\<^sub>m n)\n\ngoal (1 subgoal):\n 1. {mat\\<^sub>h (C * B * D), mat\\<^sub>h B, mat\\<^sub>h C, mat\\<^sub>h D}\n \\ carrier_mat (dim_row (mat\\<^sub>h (C * B * D)))\n (dim_row (mat\\<^sub>h (C * B * D))) \\\n mat\\<^sub>h C * mat\\<^sub>h D =\n 1\\<^sub>m (dim_row (mat\\<^sub>h (C * B * D))) \\\n mat\\<^sub>h D * mat\\<^sub>h C =\n 1\\<^sub>m (dim_row (mat\\<^sub>h (C * B * D))) \\\n mat\\<^sub>h (C * B * D) = mat\\<^sub>h C * mat\\<^sub>h B * mat\\<^sub>h D"} {"_id": "501368", "text": "proof (prove)\ngoal (1 subgoal):\n 1. field (image_ring f R)"} {"_id": "501369", "text": "proof (prove)\nusing this:\n length x2 \\ length tss\n length ts' \\ length x2\n\ngoal (1 subgoal):\n 1. length ts' < length (a # tss)"} {"_id": "501370", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monad_altc3 local.return_env local.bind_env local.altc_env"} {"_id": "501371", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pderiv p = q * pderiv (p div q) + p div q * pderiv q"} {"_id": "501372", "text": "proof (state)\nthis:\n coprime q1 q2\n coprime q3 q4\n\ngoal (1 subgoal):\n 1. \\0 < a; 0 < b; 0 < c; 0 < d; coprime a c; coprime b d\\\n \\ gcd (a * b) (c * d) = gcd a d * gcd b c"} {"_id": "501373", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf\\<^sub>s\\<^sub>t X A'"} {"_id": "501374", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\T' A'.\n lb T' A' (Suc j) \\\n real\n (makespan (T(min\\<^sub>k T m := T (min\\<^sub>k T m) + t (Suc j))))\n \\ 3 / 2 * real (makespan T')"} {"_id": "501375", "text": "proof (state)\nthis:\n (\\n. moebius_mu n / (real n)\\<^sup>2) = 6 / pi\\<^sup>2\n\ngoal (1 subgoal):\n 1. (\\n. moebius_mu n / real (n\\<^sup>2)) sums (6 / pi\\<^sup>2)"} {"_id": "501376", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum fi L = 1"} {"_id": "501377", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\y. \\x. y = mset x"} {"_id": "501378", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at x0 within S.\n return_time p2 (poincare_map p1 x) + return_time p1 x =\n return_time p2 x"} {"_id": "501379", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [a].x \\ ABS_set"} {"_id": "501380", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (f ` set xs) = fold ((\\) \\ f) xs \\"} {"_id": "501381", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ([], Y) \\ futures Q []"} {"_id": "501382", "text": "proof (prove)\nusing this:\n sim_det_def.perfect_sec_P2 R2_OT12 S2_OT12 ?funct_OT12.0 (?m0.0, ?m1.0)\n ?\\\n OT_12_sim.perfect_sec_P2 ?m1.0 ?m2.0 \\\n R2_OT12 ?m1.0 ?m2.0 =\n funct_OT_12 ?m1.0 ?m2.0 \\ (\\(s1, s2). S2_OT12 ?m2.0 s2)\n\ngoal (1 subgoal):\n 1. R2_OT12 (m0, m1) \\ =\n funct_OT_12 (m0, m1) \\ \\\n (\\(out1, out2). S2_OT12 \\ out2)"} {"_id": "501383", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ eval\n (And (fm.Atom (Leq (A * B)))\n (fm.Atom (Eq (A\\<^sup>2 - B\\<^sup>2 * c))))\n (L[var := sqrt Cv])) =\n eval\n (Or (fm.Atom (Less (- A * B)))\n (fm.Atom (Neq (A\\<^sup>2 - B\\<^sup>2 * c))))\n (L[var := sqrt Cv])"} {"_id": "501384", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((if hasLR urp1_alice 1 \\0 then \\allow ()\\\n else \\deny ()\\) =\n \\) =\n False"} {"_id": "501385", "text": "proof (prove)\ngoal (1 subgoal):\n 1. u \\ gureachable A w v"} {"_id": "501386", "text": "proof (prove)\nusing this:\n nprv F (subst \\ t x)\n\ngoal (1 subgoal):\n 1. nprv F \\"} {"_id": "501387", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map f {} {}"} {"_id": "501388", "text": "proof (prove)\nusing this:\n \\W.\n W \\ I \\\n W \\ U \\\n W \\ U' \\\n \\ U W \\\\<^bsub>U\\<^esub> =\n \\ U' W \\\\<^bsub>U'\\<^esub>\n \\\\<^bsub>U\\<^esub> \\ \\ U\n \\\\<^bsub>U'\\<^esub> \\ \\ U'\n U \\ I\n U' \\ I\n\ngoal (1 subgoal):\n 1. (U, \\\\<^bsub>U\\<^esub>) \\ (U', \\\\<^bsub>U'\\<^esub>)"} {"_id": "501389", "text": "proof (prove)\nusing this:\n x \\ auto G\n y \\ auto G\n\ngoal (1 subgoal):\n 1. x \\\\<^bsub>BijGroup (carrier G)\\<^esub> y \\ auto G"} {"_id": "501390", "text": "proof (prove)\ngoal (12 subgoals):\n 1. \\a b c.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n b \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n c \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ a \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n b \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n c =\n a \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n (b \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n c)\n 2. \\a b.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n b \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ a \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n b =\n b \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n a\n 3. \\k\\I. Ring (A k) \\\n mop (r\\\\<^bsub>I\\<^esub> A)\n \\ carrier (r\\\\<^bsub>I\\<^esub> A) \\\n carrier (r\\\\<^bsub>I\\<^esub> A)\n 4. \\a.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ -\\<^sub>a\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub> a \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n a =\n \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n 5. \\k\\I. Ring (A k) \\\n \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n \\ carrier (r\\\\<^bsub>I\\<^esub> A)\n 6. \\a.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub> \\\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n a =\n a\n 7. \\k\\I. Ring (A k) \\\n (\\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>)\n \\ carrier (r\\\\<^bsub>I\\<^esub> A) \\\n carrier (r\\\\<^bsub>I\\<^esub> A) \\\n carrier (r\\\\<^bsub>I\\<^esub> A)\n 8. \\a b c.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n b \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n c \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ a \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n b \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n c =\n a \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n (b \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n c)\n 9. \\a b.\n \\\\k\\I. Ring (A k);\n a \\ carrier (r\\\\<^bsub>I\\<^esub> A);\n b \\ carrier (r\\\\<^bsub>I\\<^esub> A)\\\n \\ a \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n b =\n b \\\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n a\n 10. \\k\\I. Ring (A k) \\\n 1\\<^sub>r\\<^bsub>r\\\\<^bsub>I\\<^esub> A\\<^esub>\n \\ carrier (r\\\\<^bsub>I\\<^esub> A)\nA total of 12 subgoals..."} {"_id": "501391", "text": "proof (prove)\nusing this:\n ct_suffix ts' (map TSome ts)\n \\cs csa.\n \\ ct_suffix cs csa \\\n csa = ccs cs csa @ ccsa cs csa \\ ct_list_eq (ccsa cs csa) cs\n\ngoal (1 subgoal):\n 1. take (length (map TSome ts) - length (ccsa ts' (map TSome ts)))\n (map TSome ts) @\n ccsa ts' (map TSome ts) =\n map TSome ts"} {"_id": "501392", "text": "proof (prove)\nusing this:\n wtA at\n I.wtE (kE \\)\n\ngoal (1 subgoal):\n 1. I.satA (kE \\) at = F.satA \\ at"} {"_id": "501393", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ring\n (S\\carrier :=\n (\\X. the_elem (h ` X)) `\n carrier (R Quot a_kernel R S h),\n zero :=\n the_elem\n (h ` \\\\<^bsub>R Quot a_kernel R S h\\<^esub>)\\)"} {"_id": "501394", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\m'. m = fmmap f m' \\ thesis) \\\n fmmap id m = fmmap (f \\ inv f) m"} {"_id": "501395", "text": "proof (state)\nthis:\n t obs = t obs'\n\ngoal (1 subgoal):\n 1. \\( fst ; OB ) = \\( fst ; t \\ OB )"} {"_id": "501396", "text": "proof (prove)\nusing this:\n \\[k\\<^sup>*, h, u] \\\n ((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\ (tab \\ w) =\n \\[k\\<^sup>*, h, u] \\\n ((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w') \\ \\\n seq \\[k\\<^sup>*, h, u]\n (((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\ (tab \\ w))\n seq \\[k\\<^sup>*, h, u]\n ((((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w')) \\\n \\)\n ide u\n ide v\n src p\\<^sub>0 = trg w\n ide w\n ide w'\n \\\\ : p\\<^sub>1 \\\n w \\ p\\<^sub>1 \\\n w'\\\n trg k\\<^sup>* = src k\n src k\\<^sup>* = trg k\n trg h = trg k\n \\ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\\\n \\ local.iso \\[?f, ?g, ?h]\n local.iso ?f \\ section ?f\n section ?g \\ local.mono ?g\n \\local.mono \\[k\\<^sup>*, h, u];\n seq \\[k\\<^sup>*, h, u]\n (((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\ (tab \\ w));\n seq \\[k\\<^sup>*, h, u]\n ((((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w')) \\\n \\);\n \\[k\\<^sup>*, h, u] \\\n ((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\ (tab \\ w) =\n \\[k\\<^sup>*, h, u] \\\n (((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w')) \\\n \\\\\n \\ (((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w')) \\\n \\ =\n ((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\\n (tab \\ w)\n (?h \\ ?g) \\ ?f = ?h \\ ?g \\ ?f\n\ngoal (1 subgoal):\n 1. ((k\\<^sup>* \\ h) \\ \\) \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w] \\ (tab \\ w) =\n (((k\\<^sup>* \\ h) \\ \\') \\\n \\[k\\<^sup>* \\ h, p\\<^sub>0, w'] \\\n (tab \\ w')) \\\n \\"} {"_id": "501397", "text": "proof (prove)\nusing this:\n \\ \\ a1 : t\n \\ \\ a2 : t\n \\ \\ s\n taval a1 s v1\n taval a2 s v2\n v1 = Rv x2_\n\ngoal (1 subgoal):\n 1. \\a. tbval (Less a1 a2) s a"} {"_id": "501398", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\0 < ab_; ab_ \\ aa_; aa_ \\ length b_;\n aa_ - Suc 0 < length b_; ab_ - Suc 0 < length b_\\\n \\ last (take ab_ b_) # drop aa_ b_ =\n drop (aa_ - Suc 0)\n (b_[aa_ - Suc 0 := b_ ! (ab_ - Suc 0)])"} {"_id": "501399", "text": "proof (prove)\ngoal (1 subgoal):\n 1. constructive_security\n (\\\\.\n 1\\<^sub>C |\\<^sub>=\n (CIPHER.enc \\ |\\<^sub>= CIPHER.dec \\) \\\n parallel_wiring \\\n parallel_resource1_wiring \\\n CIPHER.KEY.res \\ \\\n (1\\<^sub>C |\\<^sub>= MAC.enm \\ |\\<^sub>= MAC.dem \\ \\\n 1\\<^sub>C |\\<^sub>= parallel_wiring \\\n parallel_resource1_wiring \\\n MAC.RO.res \\ \\ INSEC.res))\n (\\_. SEC.res)\n (\\\\.\n CNV Message_Authentication_Code.sim (Inl None) \\\n CNV One_Time_Pad.sim None)\n (\\\\.\n \\_uniform (Set.Collect (valid_mac_query \\))\n (insert None\n (Some ` (nlists UNIV \\ \\ nlists UNIV \\))))\n (\\\\. \\_uniform UNIV {None, Some \\})\n (\\\\.\n \\_uniform (nlists UNIV \\) UNIV \\\\<^sub>\\\n \\_uniform UNIV (insert None (Some ` nlists UNIV \\)))\n (\\_. enat q) True\n (\\\\.\n (id, map_option length) \\\\<^sub>w\n (insec_query_of, map_option snd))"} {"_id": "501400", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (D x, D y) \\ I"} {"_id": "501401", "text": "proof (prove)\nusing this:\n Tpows_v = map (\\S. S v) Tpows\n Tpows = map (VectorSpaceEnd.endpow V T) [0.. V\n\ngoal (1 subgoal):\n 1. \\ R_lin_independent Tpows_v"} {"_id": "501402", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isCont cmod a"} {"_id": "501403", "text": "proof (prove)\nusing this:\n e = []\n\ngoal (1 subgoal):\n 1. \\\\\\ \\ e : ts _> ts'"} {"_id": "501404", "text": "proof (prove)\ngoal (1 subgoal):\n 1. the_elem\n (Outside' {seller} `\n (argmax \\ sum)\n (summedBidVector\n (pseudoAllocation\n (randomEl\n (takeAll\n (\\x.\n winningAllocationRel (N \\ {seller}) (set \\)\n ((\\) x) b)\n (allAllocationsAlg (N \\ {seller}) \\))\n r) <|\n ((N \\ {seller}) \\ finestpart (set \\)))\n (N \\ {seller}) (set \\))\n ((argmax \\ sum) b\n (allAllocations (N \\ {seller}) (set \\)))) =\n Outside' {seller}\n (the_elem\n ((argmax \\ sum)\n (summedBidVector\n (pseudoAllocation\n (randomEl\n (takeAll\n (\\x.\n winningAllocationRel (N \\ {seller})\n (set \\) ((\\) x) b)\n (allAllocationsAlg (N \\ {seller}) \\))\n r) <|\n ((N \\ {seller}) \\ finestpart (set \\)))\n (N \\ {seller}) (set \\))\n ((argmax \\ sum) b\n (allAllocations ({seller} \\ N) (set \\)))))"} {"_id": "501405", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (SOME xi.\n xi \\ Gromov_boundary \\\n Gromov_extension f xi = xi \\ additive_strength f xi < 0)\n \\ Gromov_boundary &&&\n Gromov_extension f\n (SOME xi.\n xi \\ Gromov_boundary \\\n Gromov_extension f xi = xi \\ additive_strength f xi < 0) =\n (SOME xi.\n xi \\ Gromov_boundary \\\n Gromov_extension f xi = xi \\ additive_strength f xi < 0) &&&\n additive_strength f\n (SOME xi.\n xi \\ Gromov_boundary \\\n Gromov_extension f xi = xi \\ additive_strength f xi < 0)\n < 0"} {"_id": "501406", "text": "proof (state)\nthis:\n (C * (A @\\<^sub>r B))\\<^sup>T = (A @\\<^sub>r B)\\<^sup>T * C\\<^sup>T\n\ngoal (1 subgoal):\n 1. C * (A @\\<^sub>r B) = A @\\<^sub>r D * B"} {"_id": "501407", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (?c, gcollapse_impl)\n \\ node_rel \\\n \\\\nat_rel\\bs_set_rel\\as_rel \\\\<^sub>r\n \\node_rel\\as_rel \\\\<^sub>r\n \\nat_rel\\as_rel \\\\<^sub>r\n oGSi_rel \\\\<^sub>r\n \\nat_rel \\\\<^sub>r\n \\node_rel\\list_set_rel\\as_rel \\\n \\\\\\nat_rel\\bs_set_rel\\as_rel \\\\<^sub>r\n \\node_rel\\as_rel \\\\<^sub>r\n \\nat_rel\\as_rel \\\\<^sub>r\n oGSi_rel \\\\<^sub>r\n \\nat_rel \\\\<^sub>r\n \\node_rel\\list_set_rel\\as_rel\\nres_rel"} {"_id": "501408", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pairwise orthogonal (insert (a - proj_onto a C) C)"} {"_id": "501409", "text": "proof (prove)\nusing this:\n \\x y.\n x \\ g y = bot \\ f x \\ y = bot\n\ngoal (1 subgoal):\n 1. conjugate f g"} {"_id": "501410", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s (trms\\<^sub>s\\<^sub>t (to_st A))"} {"_id": "501411", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ContractsWithSubstitutesAndIRC X3d X3h PX3d CX3h"} {"_id": "501412", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s.\n s \\ P \\\n (\\Z.\n s \\ I Z \\\n I Z \\ - b \\ Q \\ A \\ A)"} {"_id": "501413", "text": "proof (prove)\nusing this:\n depth P = 0\n depth Q = 0\n uhnf P\n uhnf Q\n\ngoal (1 subgoal):\n 1. P = \\ &&& Q = \\"} {"_id": "501414", "text": "proof (prove)\nusing this:\n s \\ sset ss\n x \\ sset s\n\ngoal (1 subgoal):\n 1. x \\ sset (smerge ss)"} {"_id": "501415", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\wf_prog wf_md P; (C, D) \\ (subcls1 P)\\<^sup>+\\\n \\ C \\ D"} {"_id": "501416", "text": "proof (prove)\ngoal (1 subgoal):\n 1. GDERIV (\\x. k) x :> (0::'a)"} {"_id": "501417", "text": "proof (prove)\nusing this:\n xs \\ set (map (\\x. [x]) l)\n\ngoal (1 subgoal):\n 1. (\\x. xs = [x] \\ thesis) \\ thesis"} {"_id": "501418", "text": "proof (prove)\nusing this:\n charts_eucl \\ manifold_eucl.atlas ?k\n\ngoal (1 subgoal):\n 1. chart_eucl \\ manifold_eucl.atlas k"} {"_id": "501419", "text": "proof (prove)\nusing this:\n bound0 p\n\ngoal (1 subgoal):\n 1. Ifm vs (x # bs) p = Ifm vs bs (decr0 p)"} {"_id": "501420", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (''z'' ::= (\\s. s ''x'')) ; (''x'' ::= (\\s. s ''y'')) ;\n (''y'' ::= (\\s. s ''z''))\n \\ rel_R\n \\\\s. s ''x'' = a \\ s ''y'' = b\\\n \\\\s. s ''x'' = b \\ s ''y'' = a\\"} {"_id": "501421", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (supto\\(MkI\\i)\\(MkI\\j) = [::]) = (j < i) &&&\n ([::] = supto\\(MkI\\i)\\(MkI\\j)) = (j < i)"} {"_id": "501422", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n d e.\n Poly_Mapping.keys d\n \\ singular_simplex_set p (subtopology X (S - U)) \\\n Poly_Mapping.keys e\n \\ singular_simplex_set p (subtopology X T) \\\n (singular_subdivision p ^^ n) c = d + e"} {"_id": "501423", "text": "proof (prove)\ngoal (1 subgoal):\n 1. FEval mo e\n (subst\n (\\v. case v of 0 \\ u | Suc n \\ n)\n A) =\n FEval mo (case_nat (e u) e) A"} {"_id": "501424", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fun_eq_on g f (\\ X) \\\n ChamberComplexMorphism X Y g"} {"_id": "501425", "text": "proof (prove)\ngoal (1 subgoal):\n 1. quotient_map (prod_topology X Y) Y snd =\n (topspace X = {} \\ topspace Y = {})"} {"_id": "501426", "text": "proof (prove)\nusing this:\n ?P Perp2 ?A ?B ?C ?D \\\n \\X Y. Col ?P X Y \\ X Y Perp ?A ?B \\ X Y Perp ?C ?D\n \\?A1.0 ?A2.0 Perp ?C1.0 ?C2.0;\n ?B1.0 ?B2.0 Perp ?C1.0 ?C2.0\\\n \\ ?A1.0 ?A2.0 Par ?B1.0 ?B2.0\n ?A ?B Perp ?C ?D \\\n ?A ?B Perp ?C ?D \\\n ?B ?A Perp ?C ?D \\\n ?A ?B Perp ?D ?C \\\n ?B ?A Perp ?D ?C \\\n ?C ?D Perp ?A ?B \\\n ?C ?D Perp ?B ?A \\ ?D ?C Perp ?A ?B \\ ?D ?C Perp ?B ?A\n PO Perp2 A B C D\n\ngoal (1 subgoal):\n 1. A B Par C D"} {"_id": "501427", "text": "proof (state)\nthis:\n cmp (a, a) =\n G (\\\\<^sub>F (a, a)) \\\\<^sub>D \\\\<^sub>G (F a, F a)\n\ngoal (1 subgoal):\n 1. (G (F.unit a) \\\\<^sub>D G.unit (F.map\\<^sub>0 a)) \\\\<^sub>D\n \\\\<^sub>D[map\\<^sub>0 a] =\n (local.map \\\\<^sub>B[a] \\\\<^sub>D cmp (a, a)) \\\\<^sub>D\n (G (F.unit a) \\\\<^sub>D G.unit (F.map\\<^sub>0 a) \\\\<^sub>D\n G (F.unit a) \\\\<^sub>D G.unit (F.map\\<^sub>0 a))"} {"_id": "501428", "text": "proof (prove)\nusing this:\n geq_arg max\n\ngoal (1 subgoal):\n 1. problem_plan_bound (snapshot PROB (fmrestrict_set vs s))\n \\ wlp (\\x y. y \\ state_successors (prob_proj PROB vs) x)\n (\\s. problem_plan_bound (snapshot PROB s)) max\n (\\x y. x + y + 1) (fmrestrict_set vs s) lss"} {"_id": "501429", "text": "proof (state)\ngoal (1 subgoal):\n 1. simple_fw rs (p\\p_src := s1\\) =\n simple_fw rs (p\\p_src := s2\\) \\\n simple_fw rs (p\\p_dst := s1\\) =\n simple_fw rs (p\\p_dst := s2\\)"} {"_id": "501430", "text": "proof (prove)\ngoal (1 subgoal):\n 1. right (Nonce N) = Nonce N"} {"_id": "501431", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Bag xs + Bag ys = Bag (join (\\x (n1, n2). n1 + n2) xs ys)"} {"_id": "501432", "text": "proof (prove)\nusing this:\n strict_mono r\n monoseq (X \\ r)\n\ngoal (1 subgoal):\n 1. (\\n m.\n m \\ n \\\n (X \\ r) m \\ (X \\ r) n) \\\n (\\n m.\n m \\ n \\ (X \\ r) n \\ (X \\ r) m)"} {"_id": "501433", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\y \\ x; z \\ 0\\\n \\ x * z \\ y * z"} {"_id": "501434", "text": "proof (prove)\nusing this:\n \\\n \\ {uu_.\n \\CAs DA AAs As \\ E.\n uu_ = Infer (CAs @ [DA]) E \\\n ord_resolve_rename S CAs DA AAs As \\ E}\n\ngoal (1 subgoal):\n 1. (\\CAs AAs As \\.\n \\ord_resolve_rename S CAs (main_prem_of \\) AAs As\n \\ (concl_of \\);\n CAs = side_prems_of \\\\\n \\ thesis) \\\n thesis"} {"_id": "501435", "text": "proof (prove)\ngoal (1 subgoal):\n 1. functor (\\) C S.map"} {"_id": "501436", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x1a x2a A t m n S1 t1 m1 S2.\n \\W x2a (gen ($ S1 A) t1 # $ S1 A) m1 = Some (S2, t, m);\n W x1a A n = Some (S1, t1, m1); new_tv n A;\n new_tv m1 (gen ($ S1 A) t1 # $ S1 A) \\\n Some (S2, t, m) =\n W x2a (gen ($ S1 A) t1 # $ S1 A) m1 \\\n $ S2 (gen ($ S1 A) t1 # $ S1 A) |- x2a :: t;\n new_tv m1 S1 \\ new_tv m1 t1\\\n \\ $ S2 (gen ($ S1 A) t1 # $ S1 A) |- x2a :: t\n 2. \\x1a x2a A t m n S1 t1 m1 S2.\n \\\\A S t m n.\n new_tv n A \\\n Some (S, t, m) = W x1a A n \\\n $ S A |- x1a :: t;\n \\A S t m n.\n new_tv n A \\\n Some (S, t, m) = W x2a A n \\ $ S A |- x2a :: t;\n W x2a (gen ($ S1 A) t1 # $ S1 A) m1 = Some (S2, t, m);\n W x1a A n = Some (S1, t1, m1); new_tv n A\\\n \\ free_tv S2 \\\n (free_tv t1 - free_tv ($ S1 A)) =\n {}"} {"_id": "501437", "text": "proof (prove)\nusing this:\n \\\\\\\\<^bold>G (G \\).\n G \\ \\ \\ \\\n (\\\\.\n G \\ \\ \\ \\\n accept\\<^sub>R\n (semi_mojmir_def.step \\ \\af\\<^sub>G (Abs \\),\n semi_mojmir_def.initial (Abs \\),\n {(mojmir_to_rabin_def.fail\\<^sub>R \\ \\af\\<^sub>G\n (Abs \\) {q. \\ \\\\<^sub>P Rep q} \\\n mojmir_to_rabin_def.merge\\<^sub>R \\af\\<^sub>G (Abs \\)\n {q. \\ \\\\<^sub>P Rep q} i,\n mojmir_to_rabin_def.succeed\\<^sub>R \\af\\<^sub>G\n (Abs \\) {q. \\ \\\\<^sub>P Rep q} i) |\n i. i < semi_mojmir_def.max_rank \\ \\af\\<^sub>G\n (Abs \\)})\n w)\n\ngoal (1 subgoal):\n 1. (\\\\.\n \\\\ \\ \\<^bold>G (G \\); G \\ \\ \\;\n \\\\.\n G \\ \\ \\ \\\n accept\\<^sub>R\n (semi_mojmir_def.step \\ \\af\\<^sub>G (Abs \\),\n semi_mojmir_def.initial (Abs \\),\n {(mojmir_to_rabin_def.fail\\<^sub>R \\ \\af\\<^sub>G\n (Abs \\) {q. \\ \\\\<^sub>P Rep q} \\\n mojmir_to_rabin_def.merge\\<^sub>R \\af\\<^sub>G\n (Abs \\) {q. \\ \\\\<^sub>P Rep q} i,\n mojmir_to_rabin_def.succeed\\<^sub>R \\af\\<^sub>G\n (Abs \\) {q. \\ \\\\<^sub>P Rep q} i) |\n i. i < semi_mojmir_def.max_rank \\ \\af\\<^sub>G\n (Abs \\)})\n w\\\n \\ thesis) \\\n thesis"} {"_id": "501438", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.linorder le lt"} {"_id": "501439", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x y z.\n (x, y) \\ Id_on A \\ (x, z) \\ Id_on A \\\n y = z) \\\n (\\x y z.\n (x, z) \\ Id_on A \\ (y, z) \\ Id_on A \\\n x = y)"} {"_id": "501440", "text": "proof (prove)\nusing this:\n primitive (Polynomial.monom 1 1) \\\n monic (Polynomial.monom 1 1) \\ degree (Polynomial.monom 1 1) = 1\n (?fi, ?i) \\ set hs \\\n content ?fi = 1 \\ 0 < lead_coeff ?fi\n\ngoal (1 subgoal):\n 1. (fi, i)\n \\ set (if n = 0 then hs\n else (Polynomial.monom 1 1, n - 1) # hs) \\\n primitive fi \\ 0 < lead_coeff fi"} {"_id": "501441", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a - b \\ I"} {"_id": "501442", "text": "proof (prove)\nusing this:\n k \\ m\n cond_B x\n\ngoal (1 subgoal):\n 1. take k a \\ take k x\n \\ b \\ map (max_y (take k x)) [0.. (0::'b) \\\n has_bochner_integral M f x) \\\n has_bochner_integral M (\\x. f x / c) (x / c)"} {"_id": "501444", "text": "proof (prove)\nusing this:\n finite X\n top_on\\<^sub>o (Some x_) (- X)\n top_on\\<^sub>o (Some y_) (- X)\n Some x_ \\ Some y_\n\ngoal (1 subgoal):\n 1. m\\<^sub>o (Some y_) X \\ m\\<^sub>o (Some x_) X"} {"_id": "501445", "text": "proof (prove)\nusing this:\n atom i \\ t\n\ngoal (1 subgoal):\n 1. H \\ t SUBS t"} {"_id": "501446", "text": "proof (prove)\ngoal (1 subgoal):\n 1. unit_ball_vol (real (2 * n)) = pi ^ n / fact n"} {"_id": "501447", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {ys'.\n \\ys.\n xs @ ys \\ traces P \\\n ys' = filter (\\y. L y = Low) ys}\n \\ {ys'.\n \\ys.\n xs @ x # ys \\ traces P \\\n ys' = filter (\\y. L y = Low) ys}"} {"_id": "501448", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fls_X_intpow i / fls_X_intpow j = fls_X_intpow (i - j)"} {"_id": "501449", "text": "proof (prove)\ngoal (1 subgoal):\n 1. find_fund_sol D = (if is_square D then (0, 0) else pell.fund_sol D)"} {"_id": "501450", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sample_uniform q \\\n (\\b.\n sample_uniform q \\\n (\\r.\n sample_uniform q \\\n (\\e2.\n let s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q;\n s2 = (x * y + (q - s1)) mod q;\n d2 = ((e2 + q - x) mod q * b + (q - r)) mod q\n in return_spmf ((y, b, d2, e2), s1, s2)))) =\n sample_uniform q \\\n (\\b.\n sample_uniform q \\\n (\\r.\n sample_uniform q \\\n (\\e2.\n let s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q;\n s2 = (x * y + (q - s1)) mod q;\n d2 = (e2 * b + (q - s2)) mod q\n in return_spmf ((y, b, d2, e2), s1, s2))))"} {"_id": "501451", "text": "proof (prove)\nusing this:\n \\f. f 0 \\ X \\ (\\i. (f i, f (Suc i)) \\ r)\n WCR_on r\n {x. \\f.\n f 0 \\ {x} \\ (\\i. (f i, f (Suc i)) \\ r)}\n x \\ X\n\ngoal (1 subgoal):\n 1. \\f. f 0 \\ {x} \\ (\\i. (f i, f (Suc i)) \\ r)"} {"_id": "501452", "text": "proof (prove)\nusing this:\n ts'' = ts @ [typeof v']\n\ngoal (1 subgoal):\n 1. typeof v = typeof v' &&& ts' = ts"} {"_id": "501453", "text": "proof (prove)\nusing this:\n ?p2 \\ RED \\ \\\n SN ([?p2] \\ [x]t\\K)\n\ngoal (1 subgoal):\n 1. [x]t\\K \\ SRED \\"} {"_id": "501454", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f x \\ f y"} {"_id": "501455", "text": "proof (prove)\ngoal (1 subgoal):\n 1. compat r r id"} {"_id": "501456", "text": "proof (prove)\nusing this:\n qReachable cfg1 Q cfg2\n p \\ Q\n\ngoal (1 subgoal):\n 1. states cfg1 p = states cfg2 p"} {"_id": "501457", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homotopic_with_canon (\\f. True) T U (g \\ id)\n (g \\ (\\x. b))"} {"_id": "501458", "text": "proof (prove)\nusing this:\n inner_aforms' p XS (map pdevs_of_real rs) = Some R\n\ngoal (1 subgoal):\n 1. approx_slp_outer p (Suc 0)\n [inner_floatariths (map floatarith.Var [0..s. (s, b)) xs)\n (map (\\s. (s, b)) ys) =\n map (\\s. (s, b)) xs @ map (\\s. (s, b)) ys"} {"_id": "501460", "text": "proof (prove)\nusing this:\n term_ok \\ (mk_eq (Abs \\ (t $ Bv 0)) (decr 0 t))\n typ_of (mk_eq (Abs \\ (t $ Bv 0)) (decr 0 t)) = Some propT\n wf_term (sig \\) t\n \\\\<^sub>\\ t : \\ \\ \\'\n \\\\<^sub>\\ ?t : ?ty \\ is_closed ?t\n term_ok ?\\ ?t \\ subst_bv ?u ?t = ?t\n is_closed ?t \\ subst_bv ?u ?t = ?t\n\ngoal (1 subgoal):\n 1. typ_of (mk_eq (Abs \\ (t $ Bv 0)) t) = Some propT \\\n term_ok \\ (mk_eq (Abs \\ (t $ Bv 0)) t)"} {"_id": "501461", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P a\\<^sub>1 \\ P a\\<^sub>2"} {"_id": "501462", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\C.\n \\countable C; C \\ K;\n pairwise\n (\\i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C;\n negligible (S - (\\i\\C. cball (a i) (r i)))\\\n \\ thesis) \\\n thesis"} {"_id": "501463", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homotopic_loops S g h"} {"_id": "501464", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (partGen P) = \\ P"} {"_id": "501465", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A $$ (if i' < i then i' else Suc i', if j' < j then j' else Suc j') =\n mat (dim_row A - 1) (dim_col A - 1)\n (\\(i', j').\n A $$\n (if i' < i then i' else Suc i', if j' < j then j' else Suc j')) $$\n (i', j')"} {"_id": "501466", "text": "proof (prove)\nusing this:\n RETURN (dfs_impl3' cmpk succi v0i vdi)\n \\ SPEC (\\r. r = ((src, tgt) \\ E\\<^sup>*))\n\ngoal (1 subgoal):\n 1. dfs_impl3' cmpk succi v0i vdi = ((src, tgt) \\ E\\<^sup>*)"} {"_id": "501467", "text": "proof (prove)\nusing this:\n word_of_int (- (sint a div - sint b)) * b +\n word_of_int (sint a mod - sint b) =\n a\n\ngoal (1 subgoal):\n 1. word_of_int (- (- sint a div sint b)) * b +\n word_of_int (- (- sint a mod sint b)) =\n a"} {"_id": "501468", "text": "proof (prove)\nusing this:\n f \\\\<^sub>A\\<^bsub>\\\\<^esub> f = f\n\ngoal (1 subgoal):\n 1. a \\\\<^sub>\\ \\\\<^sub>\\\n (\\\\Comp\\) \\\n \\\\Comp\\\\a\\ =\n ZFC_in_HOL.set\n {\\[f, f]\\<^sub>\\, f\\}\\a\\"} {"_id": "501469", "text": "proof (prove)\nusing this:\n out_arcs G v = out_arcs H v\n v \\ verts G - verts H\n\ngoal (1 subgoal):\n 1. del_vert_props G GM v"} {"_id": "501470", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cs\\<^bsup>\\\\<^esup> k =\n map \\\n (sorted_list_of_set {i. k cd\\<^bsup>\\\\<^esup>\\ i}) @\n [\\ k]"} {"_id": "501471", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f \\ 0"} {"_id": "501472", "text": "proof (prove)\ngoal (1 subgoal):\n 1. typeof v3 = t"} {"_id": "501473", "text": "proof (prove)\nusing this:\n (M, N) \\ ns_mul_ext Id R\n\ngoal (1 subgoal):\n 1. (\\X Z Y.\n \\M = X + Z; N = Y + Z;\n \\y\\#Y. \\x\\#X. (x, y) \\ R\\\n \\ thesis) \\\n thesis"} {"_id": "501474", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\N.\n (N = 0 \\ 1 = fac N) \\\n (N \\ 0 \\\n N - 1 < N \\\n (\\R.\n R = fac (N - 1) \\ N * fac (N - 1) = fac N))"} {"_id": "501475", "text": "proof (prove)\nusing this:\n x \\ X\n\ngoal (1 subgoal):\n 1. ind_mor_btw_stalks_axioms X x"} {"_id": "501476", "text": "proof (prove)\nusing this:\n indRelRPO \\ UNIV \\ UNIV \\\n (\\x\\UNIV. x \\\\\\\\R x)\n\ngoal (1 subgoal):\n 1. P \\\\\\\\R P"} {"_id": "501477", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f \\ linepath a b = linepath (f a) (f b)"} {"_id": "501478", "text": "proof (prove)\nusing this:\n c < prod_list (nths (dims (dematricize rmodes A ds)) (- rmodes))\n\ngoal (1 subgoal):\n 1. mat (prod_list (nths (dims (dematricize rmodes A ds)) rmodes))\n (prod_list (nths (dims (dematricize rmodes A ds)) (- rmodes)))\n (\\(r, c).\n lookup (dematricize rmodes A ds)\n (weave rmodes\n (digit_encode (nths (dims (dematricize rmodes A ds)) rmodes) r)\n (digit_encode (nths (dims (dematricize rmodes A ds)) (- rmodes))\n c))) $$\n (r, c) =\n A $$ (r, c)"} {"_id": "501479", "text": "proof (prove)\nusing this:\n AbstAtomic v i x x'\n\ngoal (1 subgoal):\n 1. \\A.\n x = \\quot_dbfm A\\e \\\n x' =\n \\quot_dbfm\n (abst_dbfm (decode_Var v) (nat_of_ord i) A)\\e"} {"_id": "501480", "text": "proof (prove)\nusing this:\n finite {(u y - u x) / (f x - f y) |x y. x \\[le] y \\ f y < f x}\n\ngoal (1 subgoal):\n 1. 0 < Min (insert 1\n {(u y - u x) / (f x - f y) |x y.\n x \\[le] y \\ f y < f x})"} {"_id": "501481", "text": "proof (prove)\nusing this:\n t \\\\<^sub>\\ t'\n \\ \\ t : T\n\ngoal (1 subgoal):\n 1. \\ \\ t' : T"} {"_id": "501482", "text": "proof (prove)\nusing this:\n P (subtopology (prod_topology X Y) ({a} \\ topspace Y))\n P (subtopology (prod_topology X Y) (topspace X \\ {b}))\n X homeomorphic_space prod_topology X (subtopology Y {b})\n Y homeomorphic_space prod_topology (subtopology X {a}) Y\n\ngoal (1 subgoal):\n 1. Q X \\ R Y"} {"_id": "501483", "text": "proof (state)\nthis:\n \\[F2](f) \\ \\[F1](f)\n\ngoal (14 subgoals):\n 1. \\f. \\[bot](f) = UNIV\n 2. \\f' h' F' g'.\n (f' \\ \\[filtermap h' F'](g')) =\n ((\\x. f' (h' x)) \\ \\[F'](\\x. g' (h' x)))\n 3. \\f' F'' g' h'.\n f' \\ \\[F''](g') \\\n (\\x. f' (h' x))\n \\ \\[filtercomap h' F''](\\x. g' (h' x))\n 4. \\f F1 g F2.\n \\f \\ \\[F1](g); f \\ \\[F2](g)\\\n \\ f \\ \\[sup F1 F2](g)\n 5. \\f g F h.\n \\\\<^sub>F x in F. f x = g x \\\n (f \\ \\[F](h)) = (g \\ \\[F](h))\n 6. \\f g F.\n \\\\<^sub>F x in F. f x = g x \\\n \\[F](f) = \\[F](g)\n 7. \\c F f.\n c \\ (0::'b) \\\n \\[F](\\x. c * f x) = \\[F](f)\n 8. \\c f F g.\n c \\ (0::'b) \\\n ((\\x. c * f x) \\ \\[F](g)) =\n (f \\ \\[F](g))\n 9. \\f F g h.\n f \\ \\[F](g) \\\n (\\x. h x * f x) \\ \\[F](\\x. h x * g x)\n 10. \\f F g.\n \\\\\\<^sub>F x in F. f x \\ (0::'b);\n \\\\<^sub>F x in F. g x \\ (0::'b);\n f \\ \\[F](g)\\\n \\ (\\x. inverse (g x))\n \\ \\[F](\\x. inverse (f x))\nA total of 14 subgoals..."} {"_id": "501484", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f.\n f \\ UNIV \\\n carrier (residue_ring (p ^ k)) \\\n maps_to_n (int (nat (p ^ k - 1))) f"} {"_id": "501485", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k. i \\ k \\ f (i, k) = bot"} {"_id": "501486", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pi \\ (x - y) = pi \\ x - pi \\ y"} {"_id": "501487", "text": "proof (prove)\ngoal (1 subgoal):\n 1. module_pair_on U\\<^sub>M\\<^sub>_\\<^sub>1 U\\<^sub>M\\<^sub>_\\<^sub>2\n (*s\\<^sub>1) (*s\\<^sub>2)"} {"_id": "501488", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i < ar T; j \\ ar (T !- i)\\\n \\ P T i j"} {"_id": "501489", "text": "proof (prove)\nusing this:\n p = 2\n\ngoal (1 subgoal):\n 1. p = sum4sq_int (1, 1, 0, 0)"} {"_id": "501490", "text": "proof (prove)\ngoal (1 subgoal):\n 1. det (addcol (monom (1::'a) (Suc i)) (n - 1) (n - 1 - Suc i)\n (mk_poly_sub A (n - 1) i)) =\n det A"} {"_id": "501491", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite A \\\n sorted_list_of_set A = sorted_list_of_set_alt A"} {"_id": "501492", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bitlen (b * 2 ^ c) = bitlen b + int c"} {"_id": "501493", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a mod_ring, comm_ring_class)"} {"_id": "501494", "text": "proof (prove)\ngoal (1 subgoal):\n 1. h i \\ {- 1<..<1}"} {"_id": "501495", "text": "proof (prove)\nusing this:\n \\ \\ induced_automorph f g\n\ngoal (1 subgoal):\n 1. chamber (insert v z) &&& chamber (insert v' z')"} {"_id": "501496", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Suc 0 \\ length xs;\n map ((\\a. rec_exec a (m # xs)) \\\n recf.id (Suc (length xs)))\n [Suc 0..i. xs ! (i - Suc 0)) [Suc 0..\n (m # xs) ! length xs = xs ! (length xs - Suc 0)\\\n \\ map (\\i. xs ! (i - Suc 0))\n [Suc 0..x. length x = \\) ?s8 \\\n \\_ideal \\ \\c S_CHAN.sec_oracle ?s8 \\\n\ngoal (1 subgoal):\n 1. \\_ideal \\ \\\\<^sub>\\ \\_common \\ \n \\res S_CHAN.res \\"} {"_id": "501498", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C' P D B IFSC D' P' C B"} {"_id": "501499", "text": "proof (prove)\nusing this:\n i \\ CBasis\n j \\ CBasis\n\ngoal (1 subgoal):\n 1. (\\k\\CBasis. if k = j then cnj (g k) else 0) = cnj (g j)"} {"_id": "501500", "text": "proof (prove)\ngoal (1 subgoal):\n 1. perzeta (1 / 2) s =\n (if s = 1 then - Ln 2 else (2 powr (1 - s) - 1) / (s - 1)) *\n ((s - 1) * pre_zeta 1 s + 1)"} {"_id": "501501", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((array_of_list [], 0), [])\n \\ {(c, a).\n a = rev (take (snd c) (list_of_array (fst c))) \\\n snd c \\ array_length (fst c)} O\n \\R\\list_rel"} {"_id": "501502", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eq_imp\n (\\r' s.\n (sys_fM s, map_option obj_fields (sys_heap s r'),\n map_option obj_mark (sys_heap s r'),\n filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)))\n marked_deletions"} {"_id": "501503", "text": "proof (prove)\nusing this:\n \\e n c.\n \\p pa s sa m.\n \\ca cb.\n (\\ c \\ e \\ ca \\\n msgs ca n \\ [] \\\n hd (msgs c n) = Marker \\\n msgs c n = [] \\ Recv n p pa s sa m = e) \\\n (\\ c \\ e \\ cb \\\n hd (msgs c n) = Marker \\\n hd (msgs cb n) = hd (msgs c n) \\\n msgs c n = [] \\ Recv n p pa s sa m = e)\n RecvMarker i' p' q' = ev \\\n states c p' = s'' \\\n channel i = Some (q', p') \\\n recv i p' q' s'' s''' m' \\\n 0 < length (msgs c i) \\ hd (msgs c i) = Msg m'\n states c p = s \\\n channel i = Some (q, p) \\\n recv i p q s s' m \\\n 0 < length (msgs c i) \\ hd (msgs c i) = Msg m\n\ngoal (1 subgoal):\n 1. occurs_on ev = p"} {"_id": "501504", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (J \\s clss_of_interp I) = (J = I)"} {"_id": "501505", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. \\ x) \\ m \\\\<^sub>n SPEC \\"} {"_id": "501506", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ A \\\n extreme_bound A (\\) {x} y = (x = y)"} {"_id": "501507", "text": "proof (prove)\ngoal (1 subgoal):\n 1. traces_prefixclosed (induceES SES)"} {"_id": "501508", "text": "proof (prove)\nusing this:\n embed r' p f\n Well_order r \\ Well_order r' \\ Well_order p\n Field r = {0.. Field p = {0.. {0..CI T\\<^esub> \\ Vx : A \\ e : B'\n Sig iS\\<^bsub>CI T\\<^esub> \\ Vx : B' \\ f E@ Vx : B\n\ngoal (1 subgoal):\n 1. [A,e,B']\\<^bsub>T\\<^esub> ;;\\<^bsub>iC\\<^bsub>CI T\\<^esub>\\<^esub>\n [B',f E@ Vx,B]\\<^bsub>T\\<^esub> =\n [A,sub e in f E@ Vx,B]\\<^bsub>T\\<^esub>"} {"_id": "501510", "text": "proof (chain)\npicking this:\n n' \\ set ns"} {"_id": "501511", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\Source Trigger Guard Action Update Target.\n \\Source \\ States A; (S, Source) \\ ChiRel (HA ST);\n TS \\ HPT ST; S \\ StepConf (HA ST) (Conf ST) TS;\n A \\ the (CompFun (HA ST) S); S \\ Conf ST;\n S \\ ChiStar (HA ST) `` {y. \\x\\TS. y = source x};\n (Source, (Trigger, Guard, Action, Update), Target) \\ TS;\n Target \\ States A\\\n \\ \\x\\TS. S = target x\n 2. \\T U.\n \\TS \\ HPT ST; S \\ StepConf (HA ST) (Conf ST) TS;\n S \\ Target TS; A \\ the (CompFun (HA ST) S);\n S \\ Conf ST; S \\ ChiStar (HA ST) `` Source TS;\n \\x.\n x \\ States A \\\n x \\ StepConf (HA ST) (Conf ST) TS;\n T \\ States A; (U, T) \\ ChiStar (HA ST); U \\ Source TS;\n (S, T) \\ ChiRel (HA ST); U \\ T\\\n \\ False"} {"_id": "501512", "text": "proof (prove)\nusing this:\n bst (splay a t)\n f = Insert a\n ss = [t]\n bst t\n\ngoal (1 subgoal):\n 1. bst (exec f ss)"} {"_id": "501513", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\new_shadow_root_ptr = the (cast x);\n h' = put (the (cast x)) (create_shadow_root_obj Open []) h;\n shadow_root_ptr.Ref\n (Suc (fMax (shadow_root_ptr.the_ref |`| shadow_root_ptrs h))) =\n the (cast x);\n x |\\| document_ptr_kinds h; is_shadow_root_ptr_kind x\\\n \\ False"} {"_id": "501514", "text": "proof (prove)\nusing this:\n if a \\ externals (ioa.asig A) then tr = [a] else tr = []\n trace (ioa.asig A) (append_exec e_A' e) =\n append (trace (ioa.asig A) e) (trace (ioa.asig A) e_A')\n\ngoal (1 subgoal):\n 1. trace (ioa.asig A) e_A =\n (if a \\ externals (ioa.asig A) then Cons a (trace (ioa.asig A) e_A')\n else trace (ioa.asig A) e_A')"} {"_id": "501515", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\x. f x \\ A; \\x. g x \\ A\\\n \\ prod_fun f g x \\ A"} {"_id": "501516", "text": "proof (prove)\nusing this:\n numbers_of_marks (Suc (Suc n)) (sieve (Suc n) (mark_out n bs)) =\n {q \\ numbers_of_marks (Suc (Suc n)) (mark_out n bs).\n \\m\\numbers_of_marks (Suc (Suc n)) (mark_out n bs).\n \\ m dvd_strict q}\n\ngoal (1 subgoal):\n 1. numbers_of_marks (Suc n) (sieve n (True # bs)) =\n {q \\ numbers_of_marks (Suc n) (True # bs).\n \\m\\numbers_of_marks (Suc n) (True # bs).\n \\ m dvd_strict q}"} {"_id": "501517", "text": "proof (prove)\nusing this:\n X = Y\n\ngoal (1 subgoal):\n 1. Y Out P Q"} {"_id": "501518", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_formula n \\ \\\n local.wf n (rexp.Inter (rexp_of_alt n \\) (ENC n (FOV \\)))"} {"_id": "501519", "text": "proof (prove)\nusing this:\n (\\\\ the (map_of \\\n x) \\\\<^bsub>\\\\<^sub>1\\<^esub>)\\\n r' =\n (\\\\ the (map_of \\\n x) \\\\<^bsub>\\\\<^sub>2\\<^esub>)\\\n r'\n x \\ domA \\\n\ngoal (1 subgoal):\n 1. (\\ ++\\<^bsub>domA\n \\\\<^esub> \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>\\\\<^sub>1\\<^esub>)\n x\\\n r' =\n (\\|\\<^sup>\\\\<^bsub>r\\<^esub> ++\\<^bsub>domA\n \\\\<^esub> \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>\\\\<^sub>2\\<^esub>)\n x\\\n r'"} {"_id": "501520", "text": "proof (prove)\ngoal (1 subgoal):\n 1. weak_reduction_coupled_simulation (TRel\\<^sup>+) Target"} {"_id": "501521", "text": "proof (prove)\ngoal (6 subgoals):\n 1. \\h shadow_root_ptr shadow_root_mode.\n \\ShadowRootClass.type_wf h;\n shadow_root_ptr |\\| shadow_root_ptr_kinds h\\\n \\ h \\ ok set_mode shadow_root_ptr\n shadow_root_mode\n 2. \\h shadow_root_ptr shadow_root_mode.\n h \\ ok set_mode shadow_root_ptr\n shadow_root_mode \\\n shadow_root_ptr |\\| shadow_root_ptr_kinds h\n 3. \\w shadow_root_ptr h h' x.\n \\w \\ set_mode_locs shadow_root_ptr;\n h \\ w \\\\<^sub>h h';\n x |\\| object_ptr_kinds h\\\n \\ x |\\| object_ptr_kinds h'\n 4. \\w shadow_root_ptr h h' x.\n \\w \\ set_mode_locs shadow_root_ptr;\n h \\ w \\\\<^sub>h h';\n x |\\| object_ptr_kinds h'\\\n \\ x |\\| object_ptr_kinds h\n 5. \\w shadow_root_ptr h h'.\n \\w \\ set_mode_locs shadow_root_ptr;\n h \\ w \\\\<^sub>h h';\n ShadowRootClass.type_wf h\\\n \\ ShadowRootClass.type_wf h'\n 6. \\w shadow_root_ptr h h'.\n \\w \\ set_mode_locs shadow_root_ptr;\n h \\ w \\\\<^sub>h h';\n ShadowRootClass.type_wf h'\\\n \\ ShadowRootClass.type_wf h"} {"_id": "501522", "text": "proof (prove)\ngoal (1 subgoal):\n 1. |Field r \\ Field p| \\o |Field r' \\ Field p'|"} {"_id": "501523", "text": "proof (chain)\npicking this:\n set_mset A \\ {a}"} {"_id": "501524", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\h ptr' sc thesis ptr.\n \\\\owner_document.\n h \\ Shadow_DOM.get_owner_document ptr'\n \\\\<^sub>r owner_document \\\n cast\\<^sub>d\\<^sub>o\\<^sub>c\\<^sub>u\\<^sub>m\\<^sub>e\\<^sub>n\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n owner_document\n \\ set sc \\\n thesis;\n Shadow_DOM.heap_is_wellformed h; ShadowRootClass.type_wf h;\n ShadowRootClass.known_ptrs h;\n h \\ Shadow_DOM_SC_DOM_Components.get_scdom_component ptr\n \\\\<^sub>r sc;\n ptr' \\ set sc\\\n \\ thesis\n 2. \\h ptr owner_document ptr' owner_document' x.\n \\Shadow_DOM.heap_is_wellformed h; ShadowRootClass.type_wf h;\n ShadowRootClass.known_ptrs h;\n h \\ Shadow_DOM.get_owner_document ptr\n \\\\<^sub>r owner_document;\n h \\ Shadow_DOM.get_owner_document ptr'\n \\\\<^sub>r owner_document';\n owner_document \\ owner_document';\n x \\ set |h \\ Shadow_DOM_SC_DOM_Components.get_scdom_component\n ptr|\\<^sub>r;\n x \\ set |h \\ Shadow_DOM_SC_DOM_Components.get_scdom_component\n ptr'|\\<^sub>r\\\n \\ False"} {"_id": "501525", "text": "proof (prove)\nusing this:\n discrete ?X \\ card {f. CK_nf_pos f} = 1\n \\part ?X; \\ discrete ?X\\\n \\ card {f. CK_nf_pos f} = 3\n \\ed_ou ?X; \\ discrete ?X\\\n \\ card {f. CK_nf_pos f} = 4\n \\open_unresolvable ?X; \\ ed_ou ?X\\\n \\ card {f. CK_nf_pos f} = 5\n \\extremally_disconnected ?X; \\ part ?X;\n \\ ed_ou ?X\\\n \\ card {f. CK_nf_pos f} = 5\n kuratowski ?X \\ card {f. CK_nf_pos f} = 7\n\ngoal (1 subgoal):\n 1. card {f. CK_nf_pos f} \\ {1, 3, 4, 5, 7}"} {"_id": "501526", "text": "proof (prove)\nusing this:\n P o\\<^sub>C\\<^sub>L P = P\n P = P*\n P = A o\\<^sub>C\\<^sub>L Proj S o\\<^sub>C\\<^sub>L A*\n \\?P o\\<^sub>C\\<^sub>L ?P = ?P; ?P = ?P*\\\n \\ Proj (?P *\\<^sub>S \\) = ?P\n\ngoal (1 subgoal):\n 1. \\M.\n P = Proj M \\\n space_as_set M =\n range ((*\\<^sub>V) (A o\\<^sub>C\\<^sub>L Proj S o\\<^sub>C\\<^sub>L A*))"} {"_id": "501527", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_fun id MPoly Const\\<^sub>0 (0::'a) * P =\n map_fun id MPoly Const\\<^sub>0 (0::'a)"} {"_id": "501528", "text": "proof (prove)\nusing this:\n \\ triangle (Suc x\\<^sub>0) \\ x\\<^sub>1\n\ngoal (1 subgoal):\n 1. (if triangle (Suc x\\<^sub>0) \\ x\\<^sub>1 then Suc x\\<^sub>0\n else GREATEST y. y \\ x\\<^sub>0 \\ triangle y \\ x\\<^sub>1) =\n (GREATEST y. y \\ Suc x\\<^sub>0 \\ triangle y \\ x\\<^sub>1)"} {"_id": "501529", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (rel_pmf R ===> (R ===> (=)) ===> (=)) local.sample_state\n local.sample_state"} {"_id": "501530", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\x. L x \\ K (x, a); L a \\ L b \\ L c;\n \\x. L x \\ x = a \\ x = b \\ x = c;\n \\y x. K (x, y) \\ H (x, y);\n \\x y. K (x, y) \\ \\ R (x, y);\n \\x. H (a, x) \\ \\ H (c, x);\n \\x. x \\ b \\ H (a, x);\n \\x. \\ R (x, a) \\ H (b, x);\n \\x. H (a, x) \\ H (b, x);\n \\x. \\y. \\ H (x, y); a \\ b\\\n \\ K (a, a)"} {"_id": "501531", "text": "proof (prove)\ngoal (4 subgoals):\n 1. Group.comm_monoid S\n 2. inv\\<^bsub>S\\<^esub> h a \\ carrier S\n 3. h a \\ carrier S\n 4. inv\\<^bsub>S\\<^esub> h a \\\\<^bsub>S\\<^esub> h a =\n \\\\<^bsub>S\\<^esub>"} {"_id": "501532", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f ` I \\ {X. directed X r \\ X \\ {}}"} {"_id": "501533", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length (filter (\\y. y < x) xs)\n \\ nat \\7 / 10 * real n + 3\\"} {"_id": "501534", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nths (weave A is1 is2) A = is1 &&& nths (weave A is1 is2) (- A) = is2"} {"_id": "501535", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\' \\ \\ is \\' : T"} {"_id": "501536", "text": "proof (prove)\nusing this:\n finite (supp {P. P \\ x})\n \\?P9 \\ {P. P \\ x};\n ?P9 \\\\\\<^sub>S ?Q9\\\n \\ ?Q9 \\ {P. P \\ x}\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501537", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P y"} {"_id": "501538", "text": "proof (state)\ngoal (3 subgoals):\n 1. Ball P finite\n 2. \\A\\P.\n \\B\\P. A \\ B \\ A \\ B = {}\n 3. sum f {.. P)"} {"_id": "501539", "text": "proof (state)\nthis:\n \\i. \\\\ g i \\\\<^sup>+ g (i + 1)\n\ngoal (1 subgoal):\n 1. False"} {"_id": "501540", "text": "proof (prove)\nusing this:\n \\x. y \\ x \\ ! y \\ x = x\n\ngoal (1 subgoal):\n 1. (y \\ ! (d x)) = (y \\ x \\ bot)"} {"_id": "501541", "text": "proof (state)\ngoal (7 subgoals):\n 1. \\a volatile t.\n \\is = Read volatile a t # is';\n \\' = \\(t \\ m a); sb' = sb; m' = m;\n \\' = \\; \\' = \\; \\' = \\; \\' = \\;\n buffered_val sb a = None\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 2. \\a D f A L R W.\n \\is = Write False a (D, f) A L R W # is';\n \\' = \\;\n sb' =\n sb @ [Write\\<^sub>s\\<^sub>b False a (D, f) (f \\) A L R W];\n m' = m; \\' = \\; \\' = \\; \\' = \\;\n \\' = \\\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 3. \\a D f A L R W.\n \\is = Write True a (D, f) A L R W # is';\n \\' = \\;\n sb' =\n sb @ [Write\\<^sub>s\\<^sub>b True a (D, f) (f \\) A L R W];\n m' = m; \\' = \\; \\' = \\; \\' = \\;\n \\' = \\\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 4. \\is = Fence # is'; sb = []; \\' = \\; sb' = [];\n m' = m; \\' = \\; \\' = \\; \\' = \\; \\' = \\\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 5. \\cond t a D f ret A L R W.\n \\is = RMW a t (D, f) cond ret A L R W # is'; sb = [];\n \\' = \\(t \\ m a); sb' = []; m' = m;\n \\' = \\; \\' = \\; \\' = \\; \\' = \\;\n \\ cond (\\(t \\ m a))\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 6. \\cond t a D f ret A L R W.\n \\is = RMW a t (D, f) cond ret A L R W # is'; sb = [];\n \\' = \\(t \\\n ret (m a) (f (\\(t \\ m a))));\n sb' = []; m' = m(a := f (\\(t \\ m a))); \\' = \\;\n \\' = \\; \\' = \\; \\' = \\;\n cond (\\(t \\ m a))\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 7. \\A L R W.\n \\is = Ghost A L R W # is'; \\' = \\; sb' = sb;\n m' = m; \\' = \\; \\' = \\; \\' = \\;\n \\' = \\\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' "} {"_id": "501542", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((=) ===> cyclic ===> cyclic) map map"} {"_id": "501543", "text": "proof (prove)\nusing this:\n A \\ \\\n SimplicialComplex.min_maxsimpchain A Cs\n Cs = C # Ds @ [D]\n\ngoal (1 subgoal):\n 1. min_gallery (C # Ds @ [D])"} {"_id": "501544", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\k. birkhoff_sum f (1 + k) x) ` {0..n} =\n (\\k. f x + birkhoff_sum f k (T x)) ` {0..n}"} {"_id": "501545", "text": "proof (prove)\ngoal (1 subgoal):\n 1. symp R = (R = R\\\\)"} {"_id": "501546", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A $ a $ b = (0::'a)"} {"_id": "501547", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_val xs ys \\ xs = map Val ys"} {"_id": "501548", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\\n xs \\ k \\\\<^sub> [ k - 1, mod k, n ] =\n xs \\

k \\\\<^sub> [\\n]"} {"_id": "501549", "text": "proof (prove)\nusing this:\n h \\ \\\n\ngoal (1 subgoal):\n 1. deriv f \\ \\ c \\ h = \\"} {"_id": "501550", "text": "proof (prove)\nusing this:\n invar_trie (Trie vo ts)\n (?k'2, ?t'2) \\ set (AList.delete_aux k ts) \\\n \\ is_empty_trie ?t'2 \\ invar_trie ?t'2\n\ngoal (1 subgoal):\n 1. invar_trie (Trie vo (AList.delete_aux k ts))"} {"_id": "501551", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ({Inv}\n WHILE b DO c\n {P} \\\n w) =\n (\\Inv' c' P'.\n w = {Inv'}\n WHILE b DO c'\n {P'} \\\n c \\ c' \\\n Inv \\ Inv' \\ P \\ P')"} {"_id": "501552", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dom (EmptyMap S) = S"} {"_id": "501553", "text": "proof (prove)\nusing this:\n orthogonal (vector A B) (vector C B)\n\ngoal (1 subgoal):\n 1. vecsqnorm (vector C A) = vecsqnorm (vector B A) + vecsqnorm (vector C B)"} {"_id": "501554", "text": "proof (prove)\nusing this:\n \\ (merge h1 h2) - \\ h1 - \\ h2\n \\ log 2 (real (size_hp h1 + size_hp h2 + 1)) + 1\n\ngoal (1 subgoal):\n 1. real (cost f ss) + \\ (exec f ss) - sum_list (map \\ ss)\n \\ U f ss"} {"_id": "501555", "text": "proof (prove)\ngoal (1 subgoal):\n 1. drefines \\ G (repeat 0 a) (repeat 0 b)"} {"_id": "501556", "text": "proof (prove)\nusing this:\n \\w' : src\\<^sub>C\n \\ \\\\<^sub>C src\\<^sub>C\n f\\ \\\n C.ide w' \\ D.isomorphic (F w') w\n D.iso ?f \\ D.inv ?f \\\\<^sub>D ?f = D.dom ?f\n src\\<^sub>D (F g) = trg\\<^sub>D w\n \\D.arr ?f; D.cod ?f = ?b\\\n \\ ?b \\\\<^sub>D ?f = ?f\n\ngoal (1 subgoal):\n 1. (D.inv (\\ (r \\\\<^sub>C f, w')) \\\\<^sub>D\n \\ (r \\\\<^sub>C f, w')) \\\\<^sub>D\n (F \\ \\\\<^sub>D F w') =\n F \\ \\\\<^sub>D F w'"} {"_id": "501557", "text": "proof (prove)\ngoal (1 subgoal):\n 1. equivalence_functor (\\\\<^sub>S) (\\\\<^sub>S) R"} {"_id": "501558", "text": "proof (prove)\nusing this:\n (pmf (sds R10) a < pmf (sds R5) d \\\n pmf (sds R10) a + (pmf (sds R10) c + pmf (sds R10) d)\n < pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)) \\\n pmf (sds R10) a = pmf (sds R5) d \\\n pmf (sds R10) a + (pmf (sds R10) c + pmf (sds R10) d) =\n pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)\n pmf (sds R5) b = 0\n 0 \\ pmf (sds R5) a\n 0 \\ pmf (sds R5) b\n 0 \\ pmf (sds R5) c\n 0 \\ pmf (sds R5) d\n pmf (sds R5) a + pmf (sds R5) b + pmf (sds R5) c + pmf (sds R5) d = 1\n\ngoal (1 subgoal):\n 1. 1 / 2 \\ pmf (sds R5) d"} {"_id": "501559", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [] \\ V"} {"_id": "501560", "text": "proof (chain)\npicking this:\n fv (throw a) \\ W"} {"_id": "501561", "text": "proof (prove)\nusing this:\n 2 ^ s1 * word_of_nat (mq - 2 ^ sq * nq) < 2 ^ s2\n\ngoal (1 subgoal):\n 1. word_of_nat (mq - 2 ^ sq * nq) < 2 ^ (s2 - s1)"} {"_id": "501562", "text": "proof (prove)\ngoal (1 subgoal):\n 1. run_option (get f) = get (\\s. run_option (f s))"} {"_id": "501563", "text": "proof (prove)\ngoal (1 subgoal):\n 1. atU U {#s#} = atU_s U s"} {"_id": "501564", "text": "proof (prove)\nusing this:\n \\ permutes alts\n finite agents\n\ngoal (1 subgoal):\n 1. {#weak_ranking (map_relation (inv \\) (R x))\n . x \\# mset_set agents#} =\n {#map ((`) \\) (weak_ranking (R x)). x \\# mset_set agents#}"} {"_id": "501565", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\f x.\n Generative_Probabilistic_Value.Done f \\\n Generative_Probabilistic_Value.Done x =\n Generative_Probabilistic_Value.Done (f x)\n 2. \\g f x.\n Generative_Probabilistic_Value.Done\n (\\g f x. g (f x)) \\\n g \\\n f \\\n x =\n g \\ (f \\ x)\n 3. \\x.\n Generative_Probabilistic_Value.Done (\\x. x) \\ x =\n x\n 4. \\f x.\n f \\ Generative_Probabilistic_Value.Done x =\n Generative_Probabilistic_Value.Done (\\f. f x) \\ f"} {"_id": "501566", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set ls' \\ set xs &&& set rs' \\ set xs"} {"_id": "501567", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj bd\\<^sub>\\"} {"_id": "501568", "text": "proof (prove)\nusing this:\n i \\ C\n f pmax = Max (reduce f PR)\n f pmax \\ f i\n\ngoal (1 subgoal):\n 1. \\i\\C. Max (reduce f PR) \\ f i"} {"_id": "501569", "text": "proof (prove)\nusing this:\n 2 < p\n prime p\n\ngoal (1 subgoal):\n 1. card ((*) 2 ` {0<..(p - 1) div 2} \\ {(p - 1) div 2<..}) =\n nat ((p - 1) div 2) -\n card ((*) 2 ` {0<..(p - 1) div 2} - {(p - 1) div 2<..})"} {"_id": "501570", "text": "proof (prove)\nusing this:\n DIM('a) = 1\n \\bounded ?S; connected (- ?S); DIM(?'a) = 1\\\n \\ ?S = {}\n compact ?U \\ bounded ?U\n (?S homotopy_eqv {}) = (?S = {})\n ({} homotopy_eqv ?S) = (?S = {})\n S homotopy_eqv T\n compact S\n compact T\n\ngoal (1 subgoal):\n 1. connected (- S) = connected (- T)"} {"_id": "501571", "text": "proof (prove)\ngoal (1 subgoal):\n 1. min (fpxs_val f) (fpxs_val g) \\ fpxs_val (f - g)"} {"_id": "501572", "text": "proof (prove)\nusing this:\n \\v\\V. \\t\\V\\<^bsub>T\\<^esub>. v \\ bag t\n ?t \\ V\\<^bsub>T\\<^esub> \\ finite (bag ?t)\n (0 < card ?A) = (?A \\ {} \\ finite ?A)\n ?a \\ {} \\ ?P\n (?x < Max bag_cards) = (\\a\\bag_cards. ?x < a)\n\ngoal (1 subgoal):\n 1. 0 < Max bag_cards"} {"_id": "501573", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\l.\n \\b = a \\ l; l \\ L\\\n \\ thesis) \\\n thesis"} {"_id": "501574", "text": "proof (prove)\nusing this:\n \\\\ \\redex\n c1_,s\\ \\ Stuck \\\n \\\\ \\c1_,s\\ \\ Stuck\n \\\\ \\redex\n c2_,s\\ \\ Stuck \\\n \\\\ \\c2_,s\\ \\ Stuck\n \\\\ \\redex\n (Catch c1_\n c2_),s\\ \\ Stuck\n\ngoal (1 subgoal):\n 1. \\\\ \\Catch c1_\n c2_,s\\ \\ Stuck"} {"_id": "501575", "text": "proof (prove)\nusing this:\n multiplicative_function (\\d. h d / f d)\n\ngoal (1 subgoal):\n 1. (\\d | d dvd a \\ coprime N d. moebius_mu d * (h d / f d)) =\n (\\p\\{p. p \\# prime_factorization a \\ \\ p dvd N}.\n 1 - h p / f p)"} {"_id": "501576", "text": "proof (state)\nthis:\n B \\ nhds y\n\ngoal (1 subgoal):\n 1. \\x y.\n \\x \\ carrier; y \\ carrier; x \\ y\\\n \\ \\u\\nhds x.\n \\v\\nhds y. u \\ v = {}"} {"_id": "501577", "text": "proof (prove)\nusing this:\n P \\ (0, s, stk) \\^k1 (size P, s'', stk'')\n P @ P' \\ (size P, s'', stk'') \\^k2 (j, s', stk')\n j = size P + j0\n\ngoal (1 subgoal):\n 1. \\k1 k2 s'' stk''.\n P \\ (0, s, stk) \\^k1\n (size P, s'', stk'') \\\n P' \\ (0, s'', stk'') \\^k2\n (j - size P, s', stk')"} {"_id": "501578", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fst i = arith_type UMUL \\\n fst i = arith_type SMUL \\ fst i = arith_type SMULcc;\n get_trap_set s = {}\\\n \\ (fst i = arith_type SUB \\\n \\ snd (sub_instr i s)) \\\n (fst i = arith_type SUBcc \\\n \\ snd (sub_instr i s)) \\\n (fst i = arith_type SUBX \\\n \\ snd (sub_instr i s)) \\\n (fst i \\ arith_type SUB \\\n fst i \\ arith_type SUBcc \\\n fst i \\ arith_type SUBX \\\n (fst i = arith_type ADD \\\n \\ snd (add_instr i s)) \\\n (fst i = arith_type ADDcc \\\n \\ snd (add_instr i s)) \\\n (fst i = arith_type ADDX \\\n \\ snd (add_instr i s)) \\\n (fst i \\ arith_type ADD \\\n fst i \\ arith_type ADDcc \\\n fst i \\ arith_type ADDX \\\n (fst i = shift_type SLL \\\n \\ snd (shift_instr i s)) \\\n (fst i = shift_type SRL \\\n \\ snd (shift_instr i s)) \\\n (fst i = shift_type SRA \\\n \\ snd (shift_instr i s)) \\\n (fst i \\ shift_type SLL \\\n fst i \\ shift_type SRL \\\n fst i \\ shift_type SRA \\\n (fst i = logic_type ANDs \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ANDcc \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ANDN \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ANDNcc \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ORs \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ORcc \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type ORN \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type XORs \\\n \\ snd (logical_instr i s)) \\\n (fst i = logic_type XNOR \\\n \\ snd (logical_instr i s)) \\\n (fst i \\ logic_type ANDs \\\n fst i \\ logic_type ANDcc \\\n fst i \\ logic_type ANDN \\\n fst i \\ logic_type ANDNcc \\\n fst i \\ logic_type ORs \\\n fst i \\ logic_type ORcc \\\n fst i \\ logic_type ORN \\\n fst i \\ logic_type XORs \\\n fst i \\ logic_type XNOR \\\n (fst i = nop_type NOP \\\n \\ snd (nop_instr i s)) \\\n (fst i \\ nop_type NOP \\\n (fst i = sethi_type SETHI \\\n \\ snd (sethi_instr i s)) \\\n (fst i \\\n sethi_type SETHI \\\n (fst i = load_store_type STB \\\n \\ snd (store_instr i s)) \\\n (fst i = load_store_type STH \\\n \\ snd (store_instr i s)) \\\n (fst i = load_store_type ST \\\n \\ snd (store_instr i s)) \\\n (fst i = load_store_type STA \\\n \\ snd (store_instr i s)) \\\n (fst i = load_store_type STD \\\n \\ snd (store_instr i s)) \\\n (fst i \\ load_store_type STB \\\n fst i \\ load_store_type STH \\\n fst i \\ load_store_type ST \\\n fst i \\ load_store_type STA \\\n fst i \\\n load_store_type STD \\\n (fst i = load_store_type LDSB \\\n \\ snd (load_instr i s)) \\\n (fst i = load_store_type LDUB \\\n \\ snd (load_instr i s)) \\\n (fst i =\n load_store_type LDUBA \\\n \\ snd (load_instr i s)) \\\n (fst i = load_store_type LDUH \\\n \\ snd (load_instr i s)) \\\n (fst i = load_store_type LD \\\n \\ snd (load_instr i s)) \\\n (fst i = load_store_type LDA \\\n \\ snd (load_instr i s)) \\\n (fst i = load_store_type LDD \\\n \\ snd (load_instr i s)) \\\n (fst i \\ load_store_type LDSB \\\n fst i \\ load_store_type LDUB \\\n fst i \\ load_store_type LDUBA \\\n fst i \\ load_store_type LDUH \\\n fst i \\ load_store_type LD \\\n fst i \\ load_store_type LDA \\\n fst i \\\n load_store_type LDD \\\n \\ snd (mul_instr i s)))))))))"} {"_id": "501579", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ carrier R; p \\ carrier P\\\n \\ a \\\\<^bsub>P\\<^esub> p \\ carrier P"} {"_id": "501580", "text": "proof (prove)\ngoal (1 subgoal):\n 1. multiplicity p (n choose k) = 0"} {"_id": "501581", "text": "proof (state)\ngoal (2 subgoals):\n 1. arr \\[f] \\\n src f \\ sources \\[f] \\ trg f \\ targets \\[f]\n 2. \\\\[f] : f \\\n src f \\ f\\"} {"_id": "501582", "text": "proof (prove)\nusing this:\n asymp_powser h F f cs\n\ngoal (1 subgoal):\n 1. asymp_powser (h \\ g) G (f \\ g) cs"} {"_id": "501583", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\U' V g g'.\n \\open U'; U' \\ {w. - pi < Im w \\ Im w < pi};\n Ln z \\ U'; open V; z \\ V; homeomorphism U' V exp g;\n \\y.\n y \\ V \\ (g has_derivative g' y) (at y);\n \\y. y \\ V \\ g' y = inv ((*) (exp (g y)));\n \\y. y \\ V \\ bij ((*) (exp (g y)))\\\n \\ thesis) \\\n thesis"} {"_id": "501584", "text": "proof (prove)\nusing this:\n Enc Y K' \\ HashSet G \\ analz H\n\ngoal (1 subgoal):\n 1. Enc Y K' \\ analz H"} {"_id": "501585", "text": "proof (state)\nthis:\n (\\y\\set (Abs_State.dom S2) - set (Abs_State.dom S1) \\\n set (Abs_State.dom S2) \\ set (Abs_State.dom S1).\n m_ivl (fun S2 y)) =\n (\\y\\set (Abs_State.dom S2) - set (Abs_State.dom S1).\n m_ivl (fun S2 y)) +\n (\\y\\set (Abs_State.dom S2) \\ set (Abs_State.dom S1).\n m_ivl (fun S2 y))\n\ngoal (1 subgoal):\n 1. \\x\\set (Abs_State.dom S2).\n (x \\ set (Abs_State.dom S1) \\\n fun S1 x \\ fun S2 x) \\\n (x \\ set (Abs_State.dom S1) \\\n \\ \\ fun S2 x) \\\n (\\x\\set (Abs_State.dom S2). m_ivl (fun S2 x))\n \\ (\\x\\set (Abs_State.dom S1). m_ivl (fun S1 x))"} {"_id": "501586", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Field r \\ {} \\ o.zero = const"} {"_id": "501587", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n rel_bset (=\\<^sub>\\) x y = rel_bset (=\\<^sub>\\) y x"} {"_id": "501588", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_zariski_open Spec"} {"_id": "501589", "text": "proof (prove)\nusing this:\n ?x \\ results_gpv \\' gpv' \\ f ?x = g ?x\n\ngoal (1 subgoal):\n 1. expectation_gpv fail \\ f gpv = expectation_gpv fail' \\' g gpv'"} {"_id": "501590", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s \\ reach \\\n \\t ik.\n initk = ik \\\n hotel t \\\n state.cards s = Trace.cards t \\\n state.isin s = Trace.isin t \\\n state.roomk s = Trace.roomk t \\\n state.owns s = Trace.owns t \\\n state.currk s = Trace.currk t \\\n state.issued s = Trace.issued t \\\n state.safe s =\n (\\r. Trace.safe t r \\ Trace.owns t r = None)"} {"_id": "501591", "text": "proof (prove)\nusing this:\n i < k\n bs ! i \\ set bs \\ 0 < bs ! i\n min 1 (min ((bs ! i / 2) powr (p + 1)) ((bs ! i * 3 / 2) powr (p + 1)))\n \\ min 1 ((bs ! i + (hs ! i) x / x) powr (p + 1))\n\ngoal (1 subgoal):\n 1. c2 / min 1 ((bs ! i + (hs ! i) x / x) powr (p + 1))\n \\ c2 /\n min 1\n (min ((bs ! i / 2) powr (p + 1)) ((bs ! i * 3 / 2) powr (p + 1)))"} {"_id": "501592", "text": "proof (prove)\nusing this:\n inj S.UP\n inj_on Inr ?A\n \\inj ?f; inj ?g\\ \\ inj (?f \\ ?g)\n\ngoal (1 subgoal):\n 1. inj (S.UP \\ Inr)"} {"_id": "501593", "text": "proof (prove)\nusing this:\n continuous_on T g\n g ` T = S\n\ngoal (1 subgoal):\n 1. closedin (top_of_set T) (f ` U)"} {"_id": "501594", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite\n {i. \\ (\\jt lt (seq i) \\\n punit.lt (rep_list (seq j)) adds\n punit.lt (rep_list (seq i)))}"} {"_id": "501595", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\d\\Basis.\n \\l r.\n strict_mono r \\\n (\\e>0.\n \\\\<^sub>F n in sequentially.\n \\i\\d.\n dist (f (r n) \\ i) (l \\ i) < e)"} {"_id": "501596", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card\n {x. regular x \\\n x \\ - - h \\\n x \\ - choose_arc h \\ x \\ - choose_arc h\\<^sup>T}\n < card {x. regular x \\ x \\ - - h}"} {"_id": "501597", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (invKey K \\ symKeys) = (K \\ symKeys)"} {"_id": "501598", "text": "proof (prove)\nusing this:\n tmbound0 (snd (split0 ?t))\n\ngoal (1 subgoal):\n 1. islin\n (let (c, s) = split0 (simptm t)\n in if c = 0\\<^sub>p then neq s else NEq (CNP 0 c s))"} {"_id": "501599", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f absolutely_integrable_on S"} {"_id": "501600", "text": "proof (prove)\ngoal (6 subgoals):\n 1. \\c v.\n \\ v : deriv c Zero \\\n \\ injval Zero c v : Zero\n 2. \\c v.\n \\ v : deriv c One \\\n \\ injval One c v : One\n 3. \\x c v.\n \\ v : deriv c (Atom x) \\\n \\ injval (Atom x) c v : Atom x\n 4. \\x1 x2 c v.\n \\\\c v.\n \\ v : deriv c x1 \\\n \\ injval x1 c v : x1;\n \\c v.\n \\ v : deriv c x2 \\\n \\ injval x2 c v : x2;\n \\ v : deriv c (Plus x1 x2)\\\n \\ \\ injval (Plus x1 x2) c v : Plus x1 x2\n 5. \\x1 x2 c v.\n \\\\c v.\n \\ v : deriv c x1 \\\n \\ injval x1 c v : x1;\n \\c v.\n \\ v : deriv c x2 \\\n \\ injval x2 c v : x2;\n \\ v : deriv c (Times x1 x2)\\\n \\ \\ injval (Times x1 x2) c v : Times x1 x2\n 6. \\x c v.\n \\\\c v.\n \\ v : deriv c x \\\n \\ injval x c v : x;\n \\ v : deriv c (Star x)\\\n \\ \\ injval (Star x) c v : Star x"} {"_id": "501601", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\{} \\ \\ IMP \\; ground_fm \\;\n ground_fm \\\\\n \\ {PfP \\\\\\} \\\n PfP \\\\\\"} {"_id": "501602", "text": "proof (prove)\nusing this:\n goodEnv rho\n freshEnv zs z rho\n zs \\ ys \\ z \\ {y, y1}\n\ngoal (1 subgoal):\n 1. freshEnv zs z (rho &[Var ys y1 / y]_ys)"} {"_id": "501603", "text": "proof (prove)\nusing this:\n a = (sfs, i)\n\ngoal (1 subgoal):\n 1. exec_step_ind (exec_step_input P C M pc ics) P h stk loc C M pc ics frs\n sh (xp', h', frs', sh')"} {"_id": "501604", "text": "proof (chain)\npicking this:\n \\r\\#prime_factorization x.\n [a ^ ((p - 1) div r) \\ 1] (mod p)"} {"_id": "501605", "text": "proof (prove)\ngoal (1 subgoal):\n 1. success (shiftr_f a i) h"} {"_id": "501606", "text": "proof (prove)\nusing this:\n ?e \\ insert tick (range ev)\n\ngoal (1 subgoal):\n 1. \\a0 \\ tick;\n {x. ev x = a0 \\ ev x \\ set al} \\ A;\n {x. ev x \\ set al} \\ A \\\n (al, b) \\ F (CHAOS A);\n tickFree al\\\n \\ a0 \\ ev ` A \\\n (\\a. ev a = a0) \\\n (al, b) \\ F (CHAOS A)"} {"_id": "501607", "text": "proof (prove)\ngoal (1 subgoal):\n 1. empt_ord.Red_F_\\ = Red_F_\\"} {"_id": "501608", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(char, cenum_class)"} {"_id": "501609", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Av + Bv * sqrt Cv = 0 \\ Av\\<^sup>2 = Bv\\<^sup>2 * Cv"} {"_id": "501610", "text": "proof (prove)\nusing this:\n \\ finite_list xs \\ last\\xs = \\\n \\ finite_list (a : xs)\n\ngoal (1 subgoal):\n 1. last\\(a : xs) = \\"} {"_id": "501611", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a xs.\n (\\ i \\ dom\\<^sub>l xs \\ f (xs ! i)) =\n foldr (\\) (map f xs) true \\\n (\\ i \\ dom\\<^sub>l\n (a # xs) \\ f ((a # xs) ! i)) =\n foldr (\\) (map f (a # xs)) true"} {"_id": "501612", "text": "proof (prove)\nusing this:\n shd ?a \\ sset ?a\n\ngoal (1 subgoal):\n 1. sset (gtrace r v) \\ {}"} {"_id": "501613", "text": "proof (prove)\ngoal (1 subgoal):\n 1. i \\ \\_rep p \\\n - i \\ \\_rep (uminus_rep p)"} {"_id": "501614", "text": "proof (prove)\nusing this:\n (monom 1 CARD('a) - monom 1 1) \\\\<^sub>p v =\n (\\x\\UNIV. [:0, 1:] - [:x:]) \\\\<^sub>p v\n\ngoal (1 subgoal):\n 1. v ^ CARD('a) - v = (\\x\\UNIV. v - [:x:])"} {"_id": "501615", "text": "proof (prove)\ngoal (1 subgoal):\n 1. adjuster n w us = a \\\n a \\ carrier_vec n \\\n wits' = map (\\ i) [0..\n a =\n sumlist (map (\\j. - \\ i j \\\\<^sub>v gso j) [0..c. 0 < v c\n \\x y.\n v x \\ n \\ v y \\ n \\ v x = v y \\ x = y\n \\c.\n v c \\ n \\\n dbm_entry_val x None (Some c) (M 0 (v c)) \\\n dbm_entry_val x (Some c) None (M (v c) 0)\n \\c1 c2.\n v c1 \\ n \\ v c2 \\ n \\\n dbm_entry_val x (Some c1) (Some c2) (M (v c1) (v c2))\n \\c.\n v c \\ n \\\n dbm_entry_val xa None (Some c)\n (if v c = 0 then \\ else M 0 (v c)) \\\n dbm_entry_val xa (Some c) None (if v c = 0 then \\ else M (v c) 0)\n \\c1 c2.\n v c1 \\ n \\ v c2 \\ n \\\n dbm_entry_val xa (Some c1) (Some c2)\n (if v c1 = v c2 then \\ else M (v c1) (v c2))\n v c1 \\ n\n v c2 \\ n\n\ngoal (1 subgoal):\n 1. \\c1 c2.\n \\\\c. 0 < v c;\n \\x y.\n v x \\ n \\ v y \\ n \\ v x = v y \\\n x = y;\n \\c.\n v c \\ n \\\n dbm_entry_val x None (Some c) (M 0 (v c)) \\\n dbm_entry_val x (Some c) None (M (v c) 0);\n \\c1 c2.\n v c1 \\ n \\ v c2 \\ n \\\n dbm_entry_val x (Some c1) (Some c2) (M (v c1) (v c2));\n \\c.\n v c \\ n \\\n dbm_entry_val xa None (Some c)\n (if v c = 0 then \\ else M 0 (v c)) \\\n dbm_entry_val xa (Some c) None\n (if v c = 0 then \\ else M (v c) 0);\n \\c1 c2.\n v c1 \\ n \\ v c2 \\ n \\\n dbm_entry_val xa (Some c1) (Some c2)\n (if v c1 = v c2 then \\ else M (v c1) (v c2));\n v c1 \\ n; v c2 \\ n\\\n \\ dbm_entry_val xa (Some c1) (Some c2)\n (M (v c1) (v c2))"} {"_id": "501617", "text": "proof (prove)\ngoal (1 subgoal):\n 1. closed_csubspace (orthogonal_complement A)"} {"_id": "501618", "text": "proof (state)\nthis:\n f = Polynomial.smult c' (\\(a, i)\\set fs'. a ^ Suc i)\n\ngoal (4 subgoals):\n 1. f = Polynomial.smult c (\\(a, i)\\set fs. a ^ Suc i)\n 2. \\a i. (a, i) \\ set fs \\ square_free a\n 3. \\a i. (a, i) \\ set fs \\ 0 < degree a\n 4. \\a i b j.\n \\(a, i) \\ set fs; (b, j) \\ set fs;\n (a, i) \\ (b, j)\\\n \\ algebraic_semidom_class.coprime a b"} {"_id": "501619", "text": "proof (prove)\nusing this:\n d = CARD('n)\n (xi, x) \\ aforms_rel\n (si, s) \\ \\rl_rel\\sctn_rel\n length xi = d\n length (normal si) = d\n\ngoal (1 subgoal):\n 1. (inter_aform_plane optns (eucl_of_list_aform xi)\n (Sctn (eucl_of_list (normal si)) (pstn si)),\n inter_sctn_spec (Affine (eucl_of_list_aform xi))\n (Sctn (eucl_of_list (normal si)) (pstn si)))\n \\ \\br Affine top\\nres_rel"} {"_id": "501620", "text": "proof (prove)\ngoal (9 subgoals):\n 1. \\C I J.\n \\G\\C\\I;\n G\\I\\I J\\\n \\ G\\C\\J\n 2. \\I J T Ja.\n \\G\\I\\I J;\n G\\Iface J\\Iface Ja\\\n \\ G\\Iface I\\Iface Ja\n 3. \\I T D.\n G\\Class Object\\Class D \\\n G\\Iface I\\Class D\n 4. \\I T Ia.\n G\\Class Object\\Iface Ia \\\n G\\Iface I\\Iface Ia\n 5. \\C D T Da.\n \\G\\C\\\\<^sub>C D;\n G\\Class D\\Class Da\\\n \\ G\\Class C\\Class Da\n 6. \\C D T I.\n \\G\\C\\\\<^sub>C D;\n G\\Class D\\Iface I\\\n \\ G\\Class C\\Iface I\n 7. \\T Ta D.\n G\\Class Object\\Class D \\\n G\\T.[]\\Class D\n 8. \\T Ta I.\n G\\Class Object\\Iface I \\\n G\\T.[]\\Iface I\n 9. \\S T Ta T'.\n \\G\\RefT T\\T';\n G\\RefT S\\RefT T;\n \\Ta.\n G\\RefT T\\Ta \\\n G\\RefT S\\Ta;\n G\\RefT T.[]\\T'.[];\n \\t. T' = RefT t\\\n \\ G\\RefT S.[]\\T'.[]"} {"_id": "501621", "text": "proof (prove)\ngoal (4 subgoals):\n 1. eqvt HRP_graph_aux\n 2. \\x y. HRP_graph x y \\ True\n 3. \\P x.\n (\\s xa x' k.\n \\atom s \\ (xa, x', k); atom k \\ (xa, x');\n x = (xa, x')\\\n \\ P) \\\n P\n 4. \\s x x' k sa xa x'a ka.\n \\atom s \\ (x, x', k); atom k \\ (x, x');\n atom sa \\ (xa, x'a, ka); atom ka \\ (xa, x'a);\n (x, x') = (xa, x'a)\\\n \\ SyntaxN.Ex s\n (SyntaxN.Ex k (SeqHRP x x' (Var s) (Var k))) =\n SyntaxN.Ex sa\n (SyntaxN.Ex ka (SeqHRP xa x'a (Var sa) (Var ka)))"} {"_id": "501622", "text": "proof (prove)\nusing this:\n xvec \\* C\n \\ \\ M \\ K\n distinct xvec\n set xvec \\ supp N\n xvec \\* Tvec\n length xvec = length Tvec\n xvec \\* \\\n xvec \\* M\n xvec \\* K\n K\\N[xvec::=Tvec]\\ \\ P[xvec::=Tvec] =\n M\\N\\ \\ P'\n\ngoal (1 subgoal):\n 1. Prop C \\ (M\\\\*xvec N\\.P) M N P'"} {"_id": "501623", "text": "proof (prove)\nusing this:\n is_open\\<^sub>Y ?U \\\n is_open\\<^sub>X (f \\<^sup>\\ X ?U)\n \\is_open\\<^sub>Y ?V; ?q \\ \\\\<^sub>Y ?V\\\n \\ \\\\<^sub>f ?V ?q\n \\ \\\\<^sub>X (f \\<^sup>\\ X ?V)\n x \\ X\n is_open\\<^sub>X (f \\<^sup>\\ X (U \\ U'))\n\ngoal (1 subgoal):\n 1. stx.mult_rel\n (stx.class_of (f \\<^sup>\\ X U) (\\\\<^sub>f U q))\n (stx.class_of (f \\<^sup>\\ X U') (\\\\<^sub>f U' q')) =\n stx.class_of (f \\<^sup>\\ X (U \\ U'))\n (mult_str\\<^sub>X (f \\<^sup>\\ X (U \\ U'))\n (\\\\<^sub>X (f \\<^sup>\\ X U)\n (f \\<^sup>\\ X (U \\ U')) (\\\\<^sub>f U q))\n (\\\\<^sub>X (f \\<^sup>\\ X U')\n (f \\<^sup>\\ X (U \\ U')) (\\\\<^sub>f U' q')))"} {"_id": "501624", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n ((Some x, y) \\ K2.acc) =\n (\\z. y = Some z \\ (x, z) \\ acc)"} {"_id": "501625", "text": "proof (prove)\nusing this:\n \\fv ?t \\ subst_domain \\;\n ground (subst_range \\)\\\n \\ fv (?t \\ \\) = {}\n\ngoal (1 subgoal):\n 1. interpretation\\<^sub>s\\<^sub>u\\<^sub>b\\<^sub>s\\<^sub>t\n \\ \\\n ground (M \\\\<^sub>s\\<^sub>e\\<^sub>t \\)"} {"_id": "501626", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [\\k = 2..n.\n of_nat (n choose k) * p ^ k * 1 ^ (n - k) = 0] (mod p\\<^sup>2)"} {"_id": "501627", "text": "proof (prove)\nusing this:\n 2 \\ card (f ` {0.. length ys"} {"_id": "501628", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.semiring = semiring_ow UNIV"} {"_id": "501629", "text": "proof (prove)\ngoal (1 subgoal):\n 1. range to_metric_completion\n \\ closure (to_metric_completion ` A)"} {"_id": "501630", "text": "proof (prove)\nusing this:\n set_integrable (count_space UNIV) A f\n set_integrable (count_space UNIV) B f\n\ngoal (1 subgoal):\n 1. set_integrable (count_space UNIV) (A \\ B) f"} {"_id": "501631", "text": "proof (prove)\nusing this:\n N \\ m\n m < n\n\ngoal (1 subgoal):\n 1. - (t * Ln (of_nat n)) - - (t * Ln (of_nat m)) =\n (\\k = Suc m..n. t * Ln (of_nat (k - 1)) - t * Ln (of_nat k))"} {"_id": "501632", "text": "proof (prove)\ngoal (18 subgoals):\n 1. eqvt trans_fm_graph_aux\n 2. \\x y.\n trans_fm_graph x y \\\n (case x of (xs, uu_) \\ fresh_star (atom ` set xs)) y\n 3. \\P x.\n \\\\e t u. x = (e, t IN u) \\ P;\n \\e t u. x = (e, t EQ u) \\ P;\n \\e A B. x = (e, A OR B) \\ P;\n \\e A. x = (e, Neg A) \\ P;\n \\k e A.\n \\atom k \\ e; x = (e, SyntaxN.Ex k A)\\\n \\ P\\\n \\ P\n 4. \\e t u ea ta ua.\n (e, t IN u) = (ea, ta IN ua) \\\n DBMem (trans_tm e t) (trans_tm e u) =\n DBMem (trans_tm ea ta) (trans_tm ea ua)\n 5. \\e t u ea ta ua.\n (e, t IN u) = (ea, ta EQ ua) \\\n DBMem (trans_tm e t) (trans_tm e u) =\n DBEq (trans_tm ea ta) (trans_tm ea ua)\n 6. \\e t u ea A B.\n \\eqvt_at trans_fm_sumC (ea, A);\n eqvt_at trans_fm_sumC (ea, B);\n (case (ea, A) of (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (ea, A));\n (case (ea, B) of (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (ea, B));\n (e, t IN u) = (ea, A OR B)\\\n \\ DBMem (trans_tm e t) (trans_tm e u) =\n DBDisj (trans_fm_sumC (ea, A))\n (trans_fm_sumC (ea, B))\n 7. \\e t u ea A.\n \\eqvt_at trans_fm_sumC (ea, A);\n (case (ea, A) of (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (ea, A));\n (e, t IN u) = (ea, Neg A)\\\n \\ DBMem (trans_tm e t) (trans_tm e u) =\n DBNeg (trans_fm_sumC (ea, A))\n 8. \\e t u k ea A.\n \\eqvt_at trans_fm_sumC (k # ea, A);\n (case (k # ea, A) of\n (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (k # ea, A));\n atom k \\ ea; (e, t IN u) = (ea, SyntaxN.Ex k A)\\\n \\ DBMem (trans_tm e t) (trans_tm e u) =\n DBEx (trans_fm_sumC (k # ea, A))\n 9. \\e t u ea ta ua.\n (e, t EQ u) = (ea, ta EQ ua) \\\n DBEq (trans_tm e t) (trans_tm e u) =\n DBEq (trans_tm ea ta) (trans_tm ea ua)\n 10. \\e t u ea A B.\n \\eqvt_at trans_fm_sumC (ea, A);\n eqvt_at trans_fm_sumC (ea, B);\n (case (ea, A) of\n (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (ea, A));\n (case (ea, B) of\n (xs, uu_) \\ fresh_star (atom ` set xs))\n (trans_fm_sumC (ea, B));\n (e, t EQ u) = (ea, A OR B)\\\n \\ DBEq (trans_tm e t) (trans_tm e u) =\n DBDisj (trans_fm_sumC (ea, A))\n (trans_fm_sumC (ea, B))\nA total of 18 subgoals..."} {"_id": "501633", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\0 < k; n < length xs\\\n \\ localState\n (f_Exec_Comp_Stream trans_fun (xs \\ k) c !\n (Suc n * k - Suc 0)) =\n localState\n (f_Exec_Comp trans_fun (xs \\ Suc n \\ k)\n c)"} {"_id": "501634", "text": "proof (prove)\nusing this:\n \\ \\ Cn K\n \\ \\ Cn K\n\ngoal (1 subgoal):\n 1. B \\ \\\\<^sub>T\\<^sub>R K (\\ .\\. \\)"} {"_id": "501635", "text": "proof (state)\nthis:\n random_variable borel (pf asset (Suc n))\n\ngoal (1 subgoal):\n 1. (\\z.\n pf asset (Suc n) z *\n discounted_value r (prices Mkt asset) (Suc n) z)\n \\ borel_measurable N"} {"_id": "501636", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Enc_keys_clean (G \\ H) \\\n analz (implSecureSet_aux Ag G \\ H)\n \\ synth (analz (G \\ H)) \\ - payload"} {"_id": "501637", "text": "proof (prove)\nusing this:\n a \\ allAllocations N G\n finite G\n finite (?r\\) = finite ?r\n Range (?r\\) = Domain ?r\n \\?b \\ ?f ` ?A;\n \\x.\n \\?b = ?f x; x \\ ?A\\\n \\ ?thesis\\\n \\ ?thesis\n (?a \\ possible_allocations_rel ?G ?N) =\n (runiq ?a \\\n runiq (?a\\) \\\n Domain ?a partitions ?G \\ Range ?a \\ ?N)\n runiq ?f \\ finite (Domain ?f) = finite ?f\n \\?a \\ allAllocations ?N ?G; finite ?G\\\n \\ finite (Range ?a)\n\ngoal (1 subgoal):\n 1. finite a"} {"_id": "501638", "text": "proof (prove)\nusing this:\n P \\ Path last Cs to C unique\n P \\ Path last Cs to C via Cs'' \n P \\ Path last Xs to C via Xs'' \n a = a' \\ Cs = Xs\n\ngoal (1 subgoal):\n 1. Cs'' = Xs''"} {"_id": "501639", "text": "proof (prove)\nusing this:\n (restrict_to P A)\\<^sup>+\\<^sup>+ a b\n accessible_on P A b\n\ngoal (1 subgoal):\n 1. accessible_on P A a"} {"_id": "501640", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\paths.\n DisjointPaths G v0 v1 paths \\ card paths = Suc sep_size"} {"_id": "501641", "text": "proof (prove)\nusing this:\n feasible t\n sound P\n\ngoal (1 subgoal):\n 1. wp body (t P) s \\ 1\n \\ wp body (\\s. t P s \\ 1) s"} {"_id": "501642", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in F. norm (f x - c * g x) \\ e * norm (g x)"} {"_id": "501643", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a ratfps, euclidean_ring_gcd_class)"} {"_id": "501644", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (poly_rel ===> poly_rel ===> rel_prod poly_rel poly_rel)\n (bezout_coefficients_i ff_ops) bezout_coefficients"} {"_id": "501645", "text": "proof (prove)\ngoal (1 subgoal):\n 1. AE x in M. norm (s (X (n + N)) x) \\ w x"} {"_id": "501646", "text": "proof (chain)\npicking this:\n bi_unique T"} {"_id": "501647", "text": "proof (prove)\nusing this:\n \\inv2 (Branch B lt z v rta); inv1 (Branch B lt z v rta)\\\n \\ inv2 (rbt_del y (Branch B lt z v rta)) \\\n (color_of (Branch B lt z v rta) = R \\\n bheight (rbt_del y (Branch B lt z v rta)) =\n bheight (Branch B lt z v rta) \\\n inv1 (rbt_del y (Branch B lt z v rta)) \\\n color_of (Branch B lt z v rta) = B \\\n bheight (rbt_del y (Branch B lt z v rta)) =\n bheight (Branch B lt z v rta) - 1 \\\n inv1l (rbt_del y (Branch B lt z v rta)))\n inv2 (Branch B lt z v rta)\n bheight a = bheight (Branch B lt z v rta)\n inv1 (Branch B lt z v rta)\n inv2 a\n inv1 a\n\ngoal (1 subgoal):\n 1. inv2 (rbt_del_from_right y a y' ss (Branch B lt z v rta)) \\\n bheight (rbt_del_from_right y a y' ss (Branch B lt z v rta)) =\n bheight a \\\n (color_of a = B \\\n color_of (Branch B lt z v rta) = B \\\n inv1 (rbt_del_from_right y a y' ss (Branch B lt z v rta)) \\\n (color_of a \\ B \\\n color_of (Branch B lt z v rta) \\ B) \\\n inv1l (rbt_del_from_right y a y' ss (Branch B lt z v rta)))"} {"_id": "501648", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x [^]\\<^bsub>mult_of R\\<^esub> order (mult_of R) =\n \\\\<^bsub>mult_of R\\<^esub>"} {"_id": "501649", "text": "proof (prove)\nusing this:\n Vagree \\ \\' (FVT (args ?i5))\n\ngoal (1 subgoal):\n 1. dterm_sem I ($f f args) \\ = dterm_sem I ($f f args) \\'"} {"_id": "501650", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dist T (midpoint S T) = dist S (midpoint S T)"} {"_id": "501651", "text": "proof (prove)\nusing this:\n arr ?f =\n (?f \\ Null \\\n Dom ?f \\ UNIV \\\n Cod ?f \\ UNIV \\\n Map ?f \\ (if Dom ?f = Cod ?f then {()} else {}))\n\ngoal (1 subgoal):\n 1. arr f = (f = MkIde False \\ f = MkIde True)"} {"_id": "501652", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. \\i\\P. [x = g i] (mod i ^ k i)"} {"_id": "501653", "text": "proof (prove)\ngoal (1 subgoal):\n 1. purge GH (map getGMUserCom (purgeIdle (tl sl))) =\n purge GH (map getGMUserCom (purgeIdle (tl sl')))"} {"_id": "501654", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\b.\n \\(case nth_visible s n of Inl e \\ Inl e\n | Inr x \\\n case nth_visible s (Suc n) of Inl x \\ Inl x\n | Inr xa \\\n Inr (Insert\n (InsertMessage x new_id xa \\))) =\n Inr (Insert (InsertMessage p new_id q \\));\n nth_visible s n = Inr b\\\n \\ extended_to_set q \\ I ` set s"} {"_id": "501655", "text": "proof (prove)\ngoal (1 subgoal):\n 1. binop_v11 A f"} {"_id": "501656", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w \\\\<^sub>n iterate f \\ n =\n w \\\\<^sub>n \\"} {"_id": "501657", "text": "proof (prove)\nusing this:\n s' \\ s \\ r \\ s \\ (s' \\ r) =\n s' \\ s \\ r \\ s' \\ s \\ r\n\ngoal (1 subgoal):\n 1. s' \\ s \\ r \\ s \\ (s' \\ r) =\n \\"} {"_id": "501658", "text": "proof (prove)\nusing this:\n \\ P = I \\\n (\\J1 J2.\n J1 \\ P \\ J2 \\ P \\ J1 \\ J2 \\\n J1 \\ J2 = {})\n \\ Q = I \\\n (\\J1 J2.\n J1 \\ Q \\ J2 \\ Q \\ J1 \\ J2 \\\n J1 \\ J2 = {})\n\ngoal (1 subgoal):\n 1. \\ (P \\ Q) = I"} {"_id": "501659", "text": "proof (prove)\nusing this:\n \\spmf\n (connect_obsf \\\n (obsf_resource (sim |\\<^sub>= 1\\<^sub>C \\ ideal_resource)))\n True -\n spmf (connect_obsf \\ (obsf_resource real_resource)) True\\\n \\ adv\n connect_obsf \\\n (obsf_resource (sim' |\\<^sub>= 1\\<^sub>C \\ ideal_resource)) =\n connect_obsf \\\n (obsf_resource (sim |\\<^sub>= 1\\<^sub>C \\ ideal_resource))\n\ngoal (1 subgoal):\n 1. \\spmf\n (connect_obsf \\\n (obsf_resource\n (sim' |\\<^sub>= 1\\<^sub>C \\ ideal_resource)))\n True -\n spmf (connect_obsf \\ (obsf_resource real_resource)) True\\\n \\ adv"} {"_id": "501660", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\atom g1 \\ (g2, f1, k1, f2, k2, g, x, y, u);\n atom g2 \\ (f1, k1, f2, k2, g, x, y, u);\n atom u \\ (f1, k1, f2, k2, x, y, g, A);\n \\C\\H. atom u \\ C\\\n \\ {HPair x y IN Var g2, ShiftP f2 k2 k1 (Var g2),\n UnionP (Var g1) (Var g2) g,\n RestrictedP f1 k1 (Var g1)} \\\n SyntaxN.Ex u\n (Var u IN k2 AND\n HaddP k1 (Var u) x AND HPair (Var u) y IN f2)"} {"_id": "501661", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. eval_primfuns fs x \\ 0"} {"_id": "501662", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pseudominimal_in (P .\\<^bold>\\ y) w"} {"_id": "501663", "text": "proof (prove)\ngoal (1 subgoal):\n 1. norm (f x ** g x) \\ F * G * K"} {"_id": "501664", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\FDTs'.\n \\P \\ D has_fields FDTs';\n FDTs = map (\\(F, b, T). ((F, C), b, T)) fs @ FDTs'\\\n \\ thesis) \\\n thesis"} {"_id": "501665", "text": "proof (prove)\nusing this:\n (g \\ f \\ A) (f.source.additive.inverse u) =\n g.target.additive.inverse (g (f u))\n\ngoal (1 subgoal):\n 1. f.source.additive.inverse u\n \\ (g \\ f \\ A) \\<^sup>\\ A \\\\<^sub>C"} {"_id": "501666", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lo u < relabel_effect f l u u \\\n ((f, relabel_effect f l u), fo, lo) \\ pr_algo_rel\\<^sup>*"} {"_id": "501667", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite (statesNames s)"} {"_id": "501668", "text": "proof (prove)\ngoal (1 subgoal):\n 1. BVG (the (snd (usubstappp \\ U \\^d)))\n \\ BVG (the (snd (usubstappp \\ U \\)))"} {"_id": "501669", "text": "proof (prove)\nusing this:\n mmonitorable \\\n\ngoal (1 subgoal):\n 1. future_bounded_mfotl \\"} {"_id": "501670", "text": "proof (prove)\nusing this:\n q dvd d \\ q < d \\ \\.induced_modulus q\n [k = 1] (mod q)\n \\.induced_modulus ?d =\n (?d dvd d \\\n (\\a.\n coprime a d \\ [a = 1] (mod ?d) \\ \\ a = 1))\n induced_modulus d\n coprime k n\n induced_modulus ?d =\n (?d dvd n \\\n (\\a.\n coprime a n \\ [a = 1] (mod ?d) \\ \\ a = 1))\n\ngoal (1 subgoal):\n 1. \\ k = 1"} {"_id": "501671", "text": "proof (prove)\nusing this:\n (\\\\'.\n ((?P, ?\\), {#}, \\')\n \\ matchrel\\<^sup>*) \\\n matchers_map ?\\ \\ matchers (set_mset ?P) = {}\n ((P, \\), y) \\ matchrel\\<^sup>*\n (y, {#}, ?\\) \\ matchrel\\<^sup>*\n\ngoal (1 subgoal):\n 1. matchers_map (snd y) \\ matchers (set_mset (fst y)) = {}"} {"_id": "501672", "text": "proof (prove)\nusing this:\n sup a (- a) = (0::'a)\n\ngoal (1 subgoal):\n 1. sup a (- a) + a = a"} {"_id": "501673", "text": "proof (prove)\nusing this:\n trimmed F\n basis_wf basis\n (f expands_to F) basis\n\ngoal (1 subgoal):\n 1. ((\\x. f x ^ n) expands_to power_expansion abort F n) basis"} {"_id": "501674", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length \\ parOf = length \\ parOf \\\n intF = intF \\ intP = intP"} {"_id": "501675", "text": "proof (prove)\ngoal (1 subgoal):\n 1. trg f \\ lunit f =\n (\\[trg f] \\ f) \\ \\' (trg f) (trg f) f"} {"_id": "501676", "text": "proof (prove)\nusing this:\n \\ ?P \\\\ ?Q \\\n supp (distinguishing_weak_formula ?P ?Q) \\ supp ?P\n x = distinguishing_weak_formula P Q\n \\ P \\\\ Q\n\ngoal (1 subgoal):\n 1. supp x \\ supp P"} {"_id": "501677", "text": "proof (prove)\nusing this:\n esk48\\<^sub>0 \\ N \\ (Pow G - {{}})\n\ngoal (1 subgoal):\n 1. summedBidVector (real \\ bids) N G esk48\\<^sub>0 =\n real (summedBidVector bids N G esk48\\<^sub>0)"} {"_id": "501678", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ia\\i.\n coeff (poly_of_vec (vec_first v n)) ia * coeff p (i - ia)) +\n (\\ia\\i.\n coeff (poly_of_vec (vec_last v m)) ia * coeff q (i - ia)) =\n (\\xxG.\n \\F \\ transient A; F ok G;\n F \\ G \\ A co A \\ B\\\n \\ F \\ G \\ transient A"} {"_id": "501680", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\,\\ \\ mk_eq\n (subst_bvs\n(map (\\(x, y). Fv x y) vs) t)\n (subst_bvs\n(map (\\(x, y). Fv x y) vs) u)"} {"_id": "501681", "text": "proof (prove)\nusing this:\n spmf\n (map_spmf (snd \\ snd)\n (exec_gpv (rf_encrypt_bad challenge) (\\2 cipher \\)\n (s, False)))\n True\n \\ 1 / 2 ^ len * real q2\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal (indicat_real {True} x)\n \\measure_spmf\n (map_spmf (snd \\ snd)\n (exec_gpv (rf_encrypt_bad challenge)\n(\\2 cipher \\) (s, False)))\n \\ ennreal (1 / 2 ^ len * real q2)"} {"_id": "501682", "text": "proof (prove)\nusing this:\n evaluate True ?env ?st ?e ?r = (?r = eval ?env ?e ?st)\n evaluate_list True ?env ?st ?es ?r' = (?r' = eval_list ?env ?es ?st)\n evaluate_match True ?env ?st ?v ?pes ?v' ?r =\n (?r = eval_match ?env ?v ?pes ?v' ?st)\n evaluate True ?env ?st ?e ?r = (?r = eval' ?env ?e ?st)\n evaluate_list True ?env ?st ?es ?r' = (?r' = eval_list' ?env ?es ?st)\n evaluate_match True ?env ?st ?v ?pes ?v' ?r =\n (?r = eval_match' ?env ?v ?pes ?v' ?st)\n\ngoal (1 subgoal):\n 1. \\x xa xb. eval_list' x xa xb = eval_list x xa xb"} {"_id": "501683", "text": "proof (state)\nthis:\n nat ` {1..int p - 1} \\ {1..p - 1}\n\ngoal (1 subgoal):\n 1. {1..p - 1} \\ nat ` {1..int p - 1}"} {"_id": "501684", "text": "proof (chain)\npicking this:\n Set_Theory.map \\' M M'\n monoid_homomorphism_axioms \\' M (\\) \\ (\\') \\'"} {"_id": "501685", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>s S =\n wf\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>s' S []"} {"_id": "501686", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\[\\] = \\"} {"_id": "501687", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\a b c.\n \\s = (a, b, c); c l = INIT\\\n \\ (\\s'. blstep l (a, b, c) s') = (l < N)\n 2. \\a b c x2.\n \\s = (a, b, c); c l = WAIT x2\\\n \\ (\\s'. blstep l (a, b, c) s') = (l < N)\n 3. \\a b c x3.\n \\s = (a, b, c); c l = HOLD x3\\\n \\ (\\s'. blstep l (a, b, c) s') = (l < N)\n 4. \\a b c x4.\n \\s = (a, b, c); c l = REL x4\\\n \\ (\\s'. blstep l (a, b, c) s') = (l < N)"} {"_id": "501688", "text": "proof (prove)\nusing this:\n f p \\ dest.carrier\n\ngoal (1 subgoal):\n 1. inv.push_forward ` dest.tangent_space (f p) = src.tangent_space p"} {"_id": "501689", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F \\ Always (- A \\ B) \\\n (B \\w A) \\\n F \\ UNIV \\w (- A \\ B) \\ (A \\ - B)"} {"_id": "501690", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\bs'. as <~~> bs' \\ bs' [\\] cs"} {"_id": "501691", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\allDistinct (pat_bindings p []);\n pmatch (c env) (refs st) p v2 [] = Match x3_\\\n \\ evaluate True\n (update_v\n (\\_.\n nsAppend (alist_to_ns x3_) (sem_env.v env))\n env)\n st e\n (map_prod id (map_result hd id)\n (fun_evaluate st\n (update_v\n (\\_.\n nsAppend (alist_to_ns x3_) (sem_env.v env))\n env)\n [e]))"} {"_id": "501692", "text": "proof (prove)\ngoal (1 subgoal):\n 1. t x \\ makespan T"} {"_id": "501693", "text": "proof (prove)\nusing this:\n \\, \\\n \\\\<^bsub>/F\\<^esub> (P \\ b) c1 Q A, c2\n atom_com Q \\\n \\s t c'.\n \\ \\ (Q, s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` (P \\ b) \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` A \\\n c' = Throw \\ t \\ Normal ` c2\n \\, \\\n \\\\<^bsub>/F\\<^esub> (P \\ - b) P\\<^sub>2_ c\\<^sub>2_\nA, c2\n atom_com c\\<^sub>2_ \\\n \\s t c'.\n \\ \\ (c\\<^sub>2_, s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` (P \\ - b) \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` A \\\n c' = Throw \\ t \\ Normal ` c2\n atom_com (Cond b Q c\\<^sub>2_)\n\ngoal (1 subgoal):\n 1. \\s t c'.\n \\ \\ (Cond b Q c\\<^sub>2_, s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` P \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` A \\\n c' = Throw \\ t \\ Normal ` c2"} {"_id": "501694", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sigma_algebra UNIV UNIV \\\n positive UNIV (measure_of_st_vec' x) \\\n countably_additive UNIV (measure_of_st_vec' x)"} {"_id": "501695", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\fa fb fc.\n \\fa 0 = a; fb 0 = b; fc 0 = c;\n \\n.\n convex hull {fa n, fb n, fc n}\n \\ convex hull {a, b, c};\n \\n. dist (fa n) (fb n) \\ K / 2 ^ n;\n \\n. dist (fb n) (fc n) \\ K / 2 ^ n;\n \\n. dist (fc n) (fa n) \\ K / 2 ^ n;\n \\n.\n e * (K / 2 ^ n)\\<^sup>2\n \\ cmod\n (contour_integral (linepath (fa n) (fb n)) f +\n contour_integral (linepath (fb n) (fc n)) f +\n contour_integral (linepath (fc n) (fa n)) f);\n \\n.\n convex hull {fa (Suc n), fb (Suc n), fc (Suc n)}\n \\ convex hull {fa n, fb n, fc n}\\\n \\ thesis) \\\n thesis"} {"_id": "501696", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ev (holds (\\step. snd step = r)) steps"} {"_id": "501697", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_gpv (=) (eq_onp (\\x. x \\ outs_\\ \\)) gpv' gpv'"} {"_id": "501698", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (u \\ H.isolated_verts) = (u \\ G.isolated_verts)"} {"_id": "501699", "text": "proof (prove)\nusing this:\n RModule FG (\\) U \\ U \\ V\n KerT \\ ker T \\ V\n x \\ U\n y \\ U\n (T \\ U) x = (T \\ U) y\n\ngoal (1 subgoal):\n 1. x - y \\ U \\ KerT"} {"_id": "501700", "text": "proof (prove)\nusing this:\n K \\ carrier R\n\ngoal (1 subgoal):\n 1. a # p = [] \\\n set (a # p) \\ K \\\n lead_coeff (a # p) \\ \\ \\\n a \\ carrier R - {\\}"} {"_id": "501701", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. infdist (g x) A) \\ borel_measurable M"} {"_id": "501702", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a p.\n \\a \\ (0::'a) \\ p \\ 0;\n poly_y_x (poly_x_minus_y p) = poly_y_minus_x p\\\n \\ poly_y_x [:[:a:]:] +\n poly_y_x (poly_x_minus_y p) * poly_y_x x_y =\n poly_y_minus_x (pCons a p)"} {"_id": "501703", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x \\ [n\\,d]; n < x\\\n \\ x - Suc 0 \\ [n\\,d]"} {"_id": "501704", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite (set_pmf (mix_pmf a p q))"} {"_id": "501705", "text": "proof (state)\nthis:\n mbody(CT,m,D) = xs . e\n length xs = length Cs\n\ngoal (1 subgoal):\n 1. \\CT C CDef m D Bs B.\n \\CT C = Some CDef;\n lookup (cMethods CDef) (\\md. mName md = m) = None;\n cSuper CDef = D; mtype(CT,m,D) = Bs \\ B;\n \\xs e.\n mbody(CT,m,D) = xs . e \\ length xs = length Bs\\\n \\ \\xs e.\n mbody(CT,m,C) = xs . e \\\n length xs = length Bs"} {"_id": "501706", "text": "proof (prove)\ngoal (1 subgoal):\n 1. update_b (\\_. 3) (X 1 2) = X 1 3"} {"_id": "501707", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\r.\n evaluate_prog True env s prog r \\\n snd r = Rerr (Rabort Rtimeout_error)"} {"_id": "501708", "text": "proof (prove)\nusing this:\n valid2 tr_\n \\tgtOf ?trn = srcOf (hd tr_); validTrans ?trn\\\n \\ valid2 (?trn # tr_)\n tgtOf (last tr_) = srcOf trn_\n validTrans trn_\n tgtOf trn = srcOf (hd (tr_ ## trn_))\n validTrans trn\n\ngoal (1 subgoal):\n 1. valid2 ((trn # tr_) ## trn_)"} {"_id": "501709", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (dist B A)\\<^sup>2 =\n (dist B D)\\<^sup>2 + (dist A D)\\<^sup>2 -\n 2 * dist A D * dist B D * cos (angle B D A)"} {"_id": "501710", "text": "proof (prove)\nusing this:\n \\(x \\ T) \\ A : \\\n A \\\\ ?M' \\\n \\(x \\ T) \\ ?M' : \\\n Fn x T A \\\\ M'\n\ngoal (1 subgoal):\n 1. (\\A'.\n \\A \\\\ A'; M' = Fn x T A'\\\n \\ thesis) \\\n thesis"} {"_id": "501711", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_sub_rel {}"} {"_id": "501712", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subpath 0 u (flow_to_path x t t') = flow_to_path x t (t + u * (t' - t))"} {"_id": "501713", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i.\n enat i < llength w \\\n (w \\\\ iterates Suc 0) ?! i = (w ?! i, i)"} {"_id": "501714", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ring_ops (finite_field_ops32 pp) mod_ring_rel32"} {"_id": "501715", "text": "proof (prove)\ngoal (1 subgoal):\n 1. extractable K I \\ I"} {"_id": "501716", "text": "proof (prove)\nusing this:\n \\x. g \\z\\ x \\ is_pole g z\n \\\\<^sub>F w in at z. g w = f w\n\ngoal (1 subgoal):\n 1. \\x. f \\z\\ x \\ is_pole f z"} {"_id": "501717", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\element_tag_name h2 new_shadow_root_ptr h3.\n \\h \\ get_tag_name element_ptr\n \\\\<^sub>r element_tag_name;\n element_tag_name \\ safe_shadow_root_element_types;\n h \\ get_shadow_root element_ptr\n \\\\<^sub>r None;\n h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>h h2;\n h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>r new_shadow_root_ptr;\n h2 \\ set_mode new_shadow_root_ptr new_mode\n \\\\<^sub>h h3;\n h3 \\ set_shadow_root element_ptr\n (Some new_shadow_root_ptr)\n \\\\<^sub>h h'\\\n \\ thesis) \\\n thesis"} {"_id": "501718", "text": "proof (state)\nthis:\n N = density M (\\x. ennreal (D x))\n\ngoal (1 subgoal):\n 1. (KL_divergence b M N = 0) = (N = M)"} {"_id": "501719", "text": "proof (prove)\nusing this:\n c \\# add_mset b B\n\ngoal (1 subgoal):\n 1. c \\# B"} {"_id": "501720", "text": "proof (prove)\nusing this:\n \\?X \\ {#}; ?X \\# N; M = N - ?X + ?Y;\n ?k \\# ?Y; \\ ?k < ?a; ?a \\ ?k\\\n \\ ?a < ?k\n \\ (M \\ N \\\n (\\y.\n count N y < count M y \\\n (\\x>y. count M x < count N x)))\n N \\ M\n\ngoal (1 subgoal):\n 1. N \\ M \\\n (\\y.\n count M y < count N y \\\n (\\x>y. count N x < count M x))"} {"_id": "501721", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (R ===> R ===> rel_prod (rel_prod R R) R)\n (euclid_ext_aux_i (arith_ops_record.one ops)\n (arith_ops_record.zero ops) (arith_ops_record.zero ops)\n (arith_ops_record.one ops))\n (euclid_ext_aux (1::'a) (0::'a) (0::'a) (1::'a))"} {"_id": "501722", "text": "proof (prove)\ngoal (1 subgoal):\n 1. composite_functor (\\\\<^sub>A\\<^sub>2\\<^sub>x\\<^sub>A\\<^sub>1)\n (\\\\<^sub>A\\<^sub>1\\<^sub>x\\<^sub>A\\<^sub>2) (\\\\<^sub>B)\n S.map F"} {"_id": "501723", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((c, take i t), The (trace c (take i t)))\n \\ Collect (case_prod (\\(x, y). trace x y))"} {"_id": "501724", "text": "proof (prove)\nusing this:\n j < dim_vec\n (Matrix.vec (dim_row A)\n (\\j.\n tensor_from_lookup (input_sizes m)\n (\\is.\n (A *\\<^sub>v\n map_vec (\\T. lookup T is) (tensors_from_net m)) $\n j)))\n\ngoal (1 subgoal):\n 1. Matrix.vec (dim_row A)\n (\\j.\n tensor_from_lookup (input_sizes m)\n (\\is.\n (A *\\<^sub>v\n map_vec (\\T. lookup T is) (tensors_from_net m)) $\n j)) $\n j =\n tensor_from_lookup (input_sizes m)\n (\\is.\n (A *\\<^sub>v\n map_vec (\\T. lookup T is) (tensors_from_net m)) $\n j)"} {"_id": "501725", "text": "proof (prove)\nusing this:\n ?T \\ VectorSpaceHomSet (\\) ?V ?fsmult' ?N \\\n VectorSpaceHom (\\) ?V ?fsmult' ?T\n ?T \\ VectorSpaceHomSet (\\) ?V ?fsmult' ?W \\\n ?T ` ?V \\ ?W\n VectorSpaceHom (\\) ?V ?smult' ?T \\ GroupHom ?V ?T\n \\GroupHom V ?T; ?T ` V \\ W\\\n \\ ?T \\ GroupHomSet V W\n \\VectorSpaceHom (\\) ?V ?smult' ?T; ?r \\ UNIV;\n ?m \\ ?V\\\n \\ ?T (?r \\ ?m) = ?smult' ?r (?T ?m)\n\ngoal (1 subgoal):\n 1. VectorSpaceHomSet (\\) V (\\) W\n \\ GroupHomSet V W \\\n {T. \\a.\n \\v\\V. T (a \\ v) = a \\ T v}"} {"_id": "501726", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subset_mset.bdd_above A"} {"_id": "501727", "text": "proof (prove)\ngoal (1 subgoal):\n 1. corec_tllist IS_TNIL TNIL THD endORmore TTL_end TTL_more b =\n Lazy_tllist\n (\\_.\n if IS_TNIL b then Inr (TNIL b)\n else Inl (THD b,\n if endORmore b then TTL_end b\n else corec_tllist IS_TNIL TNIL THD endORmore TTL_end\n TTL_more (TTL_more b)))"} {"_id": "501728", "text": "proof (prove)\nusing this:\n wf_plan_action (PAction n args)\n\ngoal (1 subgoal):\n 1. sound_opr\n (instantiate_action_schema (the (resolve_action_schema n)) args)\n (pddl_opr_to_act\n (instantiate_action_schema (the (resolve_action_schema n)) args))"} {"_id": "501729", "text": "proof (prove)\ngoal (1 subgoal):\n 1. periodic_orbit y \\ range (flow0 y) = \\_limit_set x"} {"_id": "501730", "text": "proof (prove)\nusing this:\n isVal z\n\ngoal (1 subgoal):\n 1. (\\y e'. z = Lam [y]. e' \\ atom y \\ c) \\\n (\\b. z = Bool b)"} {"_id": "501731", "text": "proof (prove)\nusing this:\n \\?f : ?a \\\\<^sub>J ?b\\ \\\n \\\\'\n ?f : \\'.A.map\n ?a \\ if ?b = m then a\n else null\\\n \\'.A.map ?f = (if J.arr ?f then a' else null)\n J.arr ?f = (?f \\ {m})\n J.dom ?f = (if ?f \\ {m} then ?f else J.null)\n J.cod ?f = (if ?f \\ {m} then ?f else J.null)\n\ngoal (1 subgoal):\n 1. \\\\' m : a' \\ a\\"} {"_id": "501732", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fv\\<^sub>s\\<^sub>t (S \\\\<^sub>s\\<^sub>t \\)\n \\ fv\\<^sub>s\\<^sub>t S"} {"_id": "501733", "text": "proof (prove)\nusing this:\n x \\ T \\ closure S \\ closure T \\\n (g has_derivative g' x)\n (at x within T \\ closure S \\ closure T)\n\ngoal (1 subgoal):\n 1. bounded_linear (if x \\ T then g' x else (\\x. 0::'b))"} {"_id": "501734", "text": "proof (prove)\ngoal (1 subgoal):\n 1. factorial_monoid mk_monoid"} {"_id": "501735", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (t \\ final_threads s) = final_thread s t"} {"_id": "501736", "text": "proof (prove)\ngoal (1 subgoal):\n 1. degree\n (\\p | p permutes {0..i = 0..\\<^sub>m 1\\<^sub>m (dim_row A) +\n map_mat (\\a. [:- a:]) A) $$\n (i, p i))) =\n n \\\n coeff\n (\\p | p permutes {0..i = 0..\\<^sub>m 1\\<^sub>m (dim_row A) +\n map_mat (\\a. [:- a:]) A) $$\n (i, p i)))\n n =\n (1::'a)"} {"_id": "501737", "text": "proof (prove)\nusing this:\n 0 < k\n h \\ {0<..k}\n a \\ {0<..1}\n s \\ {0, 1}\n\ngoal (1 subgoal):\n 1. complex_of_real (2 * pi) powr - s *\n (\\ powr - s * perzeta (real_of_int (int h) / real k) s +\n \\ powr s * perzeta (real_of_int (- int h) / real k) s) =\n complex_of_real (2 * pi) powr - s * of_nat k powr - s *\n ((\\n = 1..k.\n \\ powr - s * cis (2 * pi * real n * real h / real k) *\n hurwitz_zeta (real n / real k) s) +\n (\\n = 1..k.\n \\ powr s * cis (- 2 * pi * real n * real h / real k) *\n hurwitz_zeta (real n / real k) s))"} {"_id": "501738", "text": "proof (state)\ngoal (1 subgoal):\n 1. (\\x.\n x \\ set [] \\\n x \\ carrier_vec n) \\\n foldr (+) [] (0\\<^sub>v n) $ i = (\\x\\[]. x $ i)"} {"_id": "501739", "text": "proof (prove)\ngoal (2 subgoals):\n 1. ring_id \\ carrier (function_ring (carrier Zp) Zp)\n 2. \\s.\n is_Zp_cauchy s \\ is_Zp_cauchy (ring_id \\ s)"} {"_id": "501740", "text": "proof (state)\nthis:\n \\y\\fv\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s (f # F).\n x \\ fv (\\ y)\n\ngoal (1 subgoal):\n 1. \\x \\ fv\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n (F \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n \\) \\\n \\y\\fv\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s F.\n x \\ fv (\\ y);\n x \\ fv\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n (F \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n \\)\\\n \\ \\y\\fv\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n(f # F).\n x \\ fv (\\ y)"} {"_id": "501741", "text": "proof (state)\nthis:\n \\\\<^sub>F x0 in sequentially.\n \\a\\x0.\n \\b>a.\n cmod\n (\\n\\{a<..b}. \\ n * complex_of_real (f (real n)))\n \\ 2 * real (totient n) * f (real a)\n\ngoal (1 subgoal):\n 1. (\\x. 2 * real (totient n) * f (real x))\n \\ 0"} {"_id": "501742", "text": "proof (prove)\ngoal (1 subgoal):\n 1. trivial_homomorphism (relative_homology_group p X S)\n (homology_group (p - 1) (subtopology X S)) (hom_boundary p X S)"} {"_id": "501743", "text": "proof (prove)\nusing this:\n v \\ gunodes A w\n gurun A w r v\n smap f' (gtrace r v) = sconst k\n shd r \\ gureachable A w v\n\ngoal (1 subgoal):\n 1. (\\shd r \\ gunodes A w; gurun A w (stl r) (shd r);\n smap f' (gtrace (stl r) (shd r)) = sconst k\\\n \\ thesis) \\\n thesis"} {"_id": "501744", "text": "proof (prove)\nusing this:\n solve_lir cs fs g = Some sol\n\ngoal (1 subgoal):\n 1. solve_ratfps fps = Some sol"} {"_id": "501745", "text": "proof (prove)\nusing this:\n \\derivs xs\n \\?r\\\\ =\n \\derivs xs ?r\\\n \\deriv x\n \\derivs xs\n r\\\\ =\n \\deriv x (derivs xs r)\\\n \\deriv x\n \\derivs xs\n \\r\\\\\\ =\n \\deriv x\n (derivs xs\n \\r\\)\\\n\ngoal (1 subgoal):\n 1. \\derivs (xs @ [x])\n \\r\\\\ =\n \\derivs (xs @ [x]) r\\"} {"_id": "501746", "text": "proof (prove)\nusing this:\n \\ \\ \\\n\ngoal (1 subgoal):\n 1. (Var x\n \\ \\ Var y\n \\) \\ (Var x \\,\n Var x \\) =\n (Var y \\, Var x \\)"} {"_id": "501747", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s \\ \\ Tr Up s\\<^sub>0 \\ deadlock_free_v2 (P s)"} {"_id": "501748", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dc_crypto = Some ` {0.. {xs. length xs = n}"} {"_id": "501749", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f \\ borel_measurable M; z < esssup M f\\\n \\ 0 < emeasure M {x \\ space M. z < f x}"} {"_id": "501750", "text": "proof (prove)\nusing this:\n lift n p1 \\1 = Ok (p2, \\2)\n rel_option (\\ar fd. arity fd = ar) (get_arity ?x51) (F ?x51)\n\ngoal (1 subgoal):\n 1. Subx.sp F p2 \\1 = Ok \\2"} {"_id": "501751", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\(b, a, i). \\ m (b, a, i)) = (\\(b, a, i). 0)"} {"_id": "501752", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\ShadowRootClass.known_ptr ptr; ShadowRootClass.type_wf h;\n ptr |\\| object_ptr_kinds h\\\n \\ known_ptr\\<^sub>C\\<^sub>o\\<^sub>r\\<^sub>e\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n ptr \\\n h \\ ok get_child_nodes\\<^sub>C\\<^sub>o\\<^sub>r\\<^sub>e\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n ptr\n 2. \\ShadowRootClass.known_ptr ptr; ShadowRootClass.type_wf h;\n ptr |\\| object_ptr_kinds h\\\n \\ \\ known_ptr\\<^sub>C\\<^sub>o\\<^sub>r\\<^sub>e\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n ptr \\\n h \\ ok invoke local.a_get_child_nodes_tups\n ptr ()"} {"_id": "501753", "text": "proof (prove)\ngoal (1 subgoal):\n 1. UP.eval (map poly_of_const l') q =\n UP.eval (map poly_of_const (\\ # l')) q"} {"_id": "501754", "text": "proof (prove)\ngoal (1 subgoal):\n 1. in_all_resolvents_upon S A P1"} {"_id": "501755", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\A A'.\n \\A \\ A'; \\ x1_ A\\\n \\ \\ x1_ A';\n \\A A'.\n \\A \\ A'; \\ x4_ A\\\n \\ \\ x4_ A';\n \\A A'.\n \\A \\ A'; \\ x5_ A\\\n \\ \\ x5_ A';\n A_ \\ A'_;\n \\ (x1_\\compareAndSwap(x2_\\x3_, x4_, x5_))\n A_\\\n \\ \\\n (x1_\\compareAndSwap(x2_\\x3_, x4_, x5_))\n A'_"} {"_id": "501756", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fixed_length_sublist (concat (map vec As)) (prod_list ds) i =\n vec (As ! i)"} {"_id": "501757", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ra u' v' v u.\n \\Idomain S; t \\ carrier S;\n t \\ \\\\<^bsub>S\\<^esub>;\n maximal_ideal S (S \\\\<^sub>p t);\n PolynRg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y;\n f \\ carrier R; g \\ carrier R; h \\ carrier R;\n deg R S X g\n \\ deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g);\n deg R S X h +\n deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g)\n \\ deg R S X f;\n P_mod R S X (S \\\\<^sub>p (t^\\<^bsup>S m\\<^esup>))\n (t^\\<^bsup>S m\\<^esup> \\\\<^sub>r ra);\n 0 < m; Subring R S; Ring S; aGroup R;\n Corps (S /\\<^sub>r (S \\\\<^sub>p t));\n ideal S (S \\\\<^sub>p t);\n pj S (S \\\\<^sub>p t)\n \\ rHom S (S /\\<^sub>r (S \\\\<^sub>p t));\n t \\ carrier R; Ring (S /\\<^sub>r (S \\\\<^sub>p t));\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g \\\n \\\\<^bsub>R'\\<^esub>;\n g \\ \\;\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) h \\\n \\\\<^bsub>R'\\<^esub>;\n 0 \\ deg R S X h; h \\ \\; f \\ \\;\n ra \\ carrier R;\n f \\ -\\<^sub>a g \\\\<^sub>r h =\n t^\\<^bsup>S m\\<^esup> \\\\<^sub>r ra;\n g \\\\<^sub>r h \\ carrier R; deg R S X ra \\ deg R S X f;\n t^\\<^bsup>S m\\<^esup> \\ carrier S;\n t^\\<^bsup>S m\\<^esup> \\ \\\\<^bsub>S\\<^esub>;\n deg R S X (t^\\<^bsup>S m\\<^esup> \\\\<^sub>r ra) = deg R S X ra;\n u' \\ carrier R'; v' \\ carrier R';\n deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y u'\n \\ amax\n (deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) ra) -\n deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g))\n (deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) h));\n deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y v'\n \\ deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g);\n u' \\\\<^sub>r\\<^bsub>R'\\<^esub>\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t))\n g \\\\<^bsub>R'\\<^esub>\n v' \\\\<^sub>r\\<^bsub>R'\\<^esub>\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) h =\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) ra;\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g\n \\ carrier R';\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) h\n \\ carrier R';\n v \\ carrier R; u \\ carrier R;\n deg R S X v\n \\ deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y v';\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) v =\n v';\n deg R S X u\n \\ deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y u';\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) u =\n u';\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t))\n u \\\\<^sub>r\\<^bsub>R'\\<^esub>\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) g =\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) (u \\\\<^sub>r g);\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t))\n v \\\\<^sub>r\\<^bsub>R'\\<^esub>\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) h =\n erH R S X R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t)) (v \\\\<^sub>r h);\n t^\\<^bsup>S m\\<^esup> \\ carrier R\\\n \\ deg R S X\n (g \\\n t^\\<^bsup>S m\\<^esup> \\\\<^sub>r v)\n \\ deg R'\n (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R'\n (S /\\<^sub>r (S \\\\<^sub>p t))\n Y (pj S (S \\\\<^sub>p t))\n (g \\\n t^\\<^bsup>S m\\<^esup> \\\\<^sub>r\n v)) \\\n P_mod R S X\n (S \\\\<^sub>p (t^\\<^bsup>S m\\<^esup>))\n (g \\\n -\\<^sub>a (g \\\nt^\\<^bsup>S m\\<^esup> \\\\<^sub>r v)) \\\n deg R S X\n (h \\\n t^\\<^bsup>S m\\<^esup> \\\\<^sub>r u) +\n deg R' (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (erH R S X R'\n (S /\\<^sub>r (S \\\\<^sub>p t)) Y\n (pj S (S \\\\<^sub>p t))\n (g \\\n t^\\<^bsup>S m\\<^esup> \\\\<^sub>r v))\n \\ deg R S X f \\\n P_mod R S X\n (S \\\\<^sub>p (t^\\<^bsup>S m\\<^esup>))\n (h \\\n -\\<^sub>a (h \\\nt^\\<^bsup>S m\\<^esup> \\\\<^sub>r u)) \\\n P_mod R S X\n (S \\\\<^sub>p\n (t^\\<^bsup>S Suc m\\<^esup>))\n (f \\\n -\\<^sub>a (g \\\nt^\\<^bsup>S m\\<^esup> \\\\<^sub>r v) \\\\<^sub>r\n (h \\\nt^\\<^bsup>S m\\<^esup> \\\\<^sub>r u))"} {"_id": "501758", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Z\\<^sub>1 Z\\<^sub>2.\n \\mwb_lens Z;\n Z\\<^sub>1 ;\\<^sub>L Z \\ Z\\<^sub>2 ;\\<^sub>L Z;\n vwb_lens Z\\<^sub>1; X = Z\\<^sub>1 ;\\<^sub>L Z; vwb_lens Z\\<^sub>2;\n Y = Z\\<^sub>2 ;\\<^sub>L Z\\\n \\ vwb_lens (Z\\<^sub>1 +\\<^sub>L Z\\<^sub>2) \\\n Z\\<^sub>1 ;\\<^sub>L Z +\\<^sub>L\n Z\\<^sub>2 ;\\<^sub>L Z =\n (Z\\<^sub>1 +\\<^sub>L Z\\<^sub>2) ;\\<^sub>L Z"} {"_id": "501759", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Hv (sum f S) = (\\x\\S. Hv (f x))"} {"_id": "501760", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\.\n finite \\ \\\n \\ \\ \\ \\\n {x \\ topspace X. f x = y} \\ \\ \\"} {"_id": "501761", "text": "proof (prove)\ngoal (1 subgoal):\n 1. up\\x * up\\y = up\\(x * y)"} {"_id": "501762", "text": "proof (prove)\nusing this:\n \\ \\ e : t'\n\ngoal (1 subgoal):\n 1. the (expr_type \\ e) = t'"} {"_id": "501763", "text": "proof (prove)\nusing this:\n ?t'13 \\ t \\\n tbisim (ws ?t'13 = None) ?t'13 (ts1 ?t'13) m1 (ts2 ?t'13) m2\n ts1 t = \\xln\\\n r2.cond_action_oks (ls, (ts2, m2), ws, is) t cts\n\ngoal (1 subgoal):\n 1. (\\mRed1 (ls, (ts1, m1), ws, is) s1' &&&\n (\\t'.\n t' \\ t \\\n tbisim (ws t' = None) t' (thr s1' t') m1 (ts2 t') m2)) &&&\n r1.cond_action_oks s1' t cts &&& thr s1' t = \\xln\\"} {"_id": "501764", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fully_faithful_functor (\\) (\\) H.R"} {"_id": "501765", "text": "proof (prove)\nusing this:\n DIM_precond TYPE('a) n\n\ngoal (1 subgoal):\n 1. (Basis_list_impl n, Basis_list) \\ \\lv_rel\\list_rel"} {"_id": "501766", "text": "proof (prove)\nusing this:\n altnst @ cs = us @ [u, x, z] @ zs\n xs = us @ [u]\n Suc (length xs) \\ n\n altnst = alternating_list n s t\n\ngoal (1 subgoal):\n 1. \\ order.greater n (length xs)"} {"_id": "501767", "text": "proof (prove)\nusing this:\n wf_formula n (FOr \\ \\)\n\ngoal (1 subgoal):\n 1. wf_formula n (FOr \\ \\) \\\n slow.check_eqvRE enum_class.enum n\n (Plus (rexp_of'' n (norm \\)) One)\n (Plus (rexp_of'' n (norm \\)) One)"} {"_id": "501768", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P is \\ \\ \\<^bold>\\ \\ P"} {"_id": "501769", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coeffs (Poly ys) = ys"} {"_id": "501770", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (RETURN\n (find_lasso_tr node_eq_impl node_hash_impl node_def_hash_size G_impl),\n (OP op_find_lasso_spec :::\n igbg_impl_rel_ext Re node_rel \\\n \\fl_rel node_rel\\nres_rel) $\n G)\n \\ \\fl_rel node_rel\\nres_rel"} {"_id": "501771", "text": "proof (prove)\nusing this:\n (\\n\\real -` {0<..x}. a n * f (real n)) =\n sum_upto (\\n. a n * f (real n)) x\n\ngoal (1 subgoal):\n 1. ((\\t. A t * f' t) has_integral\n A x * f x - sum_upto (\\n. a n * f (real n)) x)\n {0..x}"} {"_id": "501772", "text": "proof (prove)\nusing this:\n \\ (order x f = 0 \\ order x f = 1)\n\ngoal (1 subgoal):\n 1. (\\n.\n order x f = Suc (Suc n) \\ thesis) \\\n thesis"} {"_id": "501773", "text": "proof (prove)\nusing this:\n \\ (set (A # As)) \\ carrier\n is_weak_ranking (A # As)\n of_weak_ranking (A # As) = restrict_relation (\\ (set (A # As))) le\n\ngoal (1 subgoal):\n 1. Max_wrt_among (of_weak_ranking (A # As)) (\\ (set (A # As))) = A"} {"_id": "501774", "text": "proof (state)\nthis:\n nc = Suc ncc\n\ngoal (1 subgoal):\n 1. \\a m1 m2 i nc.\n \\\\m2 i nc.\n \\length m1 = nc \\ Ball (set m1) (vec nr);\n length m2 = nc \\ Ball (set m2) (vec nr); i < nc;\n length m1 = nc \\ Ball (set m1) (vec nr);\n length m2 = nc \\ Ball (set m2) (vec nr);\n i < nc\\\n \\ vec_plusI pl (m1 ! i) (m2 ! i) ! j =\n pl (m1 ! i ! j) (m2 ! i ! j);\n length (a # m1) = nc \\ Ball (set (a # m1)) (vec nr);\n length m2 = nc \\ Ball (set m2) (vec nr); i < nc;\n length (a # m1) = nc \\ Ball (set (a # m1)) (vec nr);\n length m2 = nc \\ Ball (set m2) (vec nr); i < nc\\\n \\ vec_plusI pl ((a # m1) ! i) (m2 ! i) ! j =\n pl ((a # m1) ! i ! j) (m2 ! i ! j)"} {"_id": "501775", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_nondetT (BNF_Def.Grp A f) =\n BNF_Def.Grp {x. set_nondetT x \\ A} (map_nondetT f)"} {"_id": "501776", "text": "proof (chain)\npicking this:\n x1 = x2\n y1 = y2"} {"_id": "501777", "text": "proof (prove)\ngoal (1 subgoal):\n 1. at_infinity = sup at_top at_bot"} {"_id": "501778", "text": "proof (prove)\nusing this:\n R \\ S\n\ngoal (1 subgoal):\n 1. (\\k.\n \\strict_mono k;\n \\x.\n x \\ R \\\n \\l.\n (\\n. \\ (k n) x)\n \\ l\\\n \\ thesis) \\\n thesis"} {"_id": "501779", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\e\\challenge_space.\n (\\(h, w)\\R.\n local.\\_protocols_base.R h w e =\n local.\\_protocols_base.S h e) \\\n (\\h\\valid_pub.\n \\(a, z)\\set_spmf (S2 h e). check h a e z)"} {"_id": "501780", "text": "proof (prove)\ngoal (1 subgoal):\n 1. insertion f (Const 0) = 0"} {"_id": "501781", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((h', a) \\ allocate h hT \\\n P,h' \\ v :\\ T) &&&\n (heap_write h a al v' h' \\ P,h' \\ v :\\ T)"} {"_id": "501782", "text": "proof (state)\ngoal (1 subgoal):\n 1. S \\ \\ \\ \\ {}"} {"_id": "501783", "text": "proof (state)\nthis:\n identity stk1.carrier_stalk (stk1.add_stalk X' Y') =\n stk2.add_stalk (identity stk1.carrier_stalk X')\n (identity stk1.carrier_stalk Y')\n\ngoal (1 subgoal):\n 1. identity stk1.carrier_stalk (stk1.mult_stalk X' Y') =\n stk2.mult_stalk (identity stk1.carrier_stalk X')\n (identity stk1.carrier_stalk Y')"} {"_id": "501784", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_formula n \\ \\\n local.wf n (rexp.Inter (rexp_of_alt' n \\) (ENC n (FOV \\)))"} {"_id": "501785", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ Collect sg"} {"_id": "501786", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\xs ys.\n us = xs \\ [u] \\ ys \\\n thesis) \\\n thesis"} {"_id": "501787", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\d' t'.\n \\(d2, t) \\*c (d', t'); s' \\ t';\n discr d'\\\n \\ (\\d' t'.\n (d1 | d2, t) \\*c (d', t') \\\n s' \\ t' \\\n ((\\c1a c2.\n c1 = c1a | c2 \\\n (\\d1 d2.\nd' = d1 | d2 \\ c1a \\w d1 \\ c2 \\w d2)) \\\n c1 \\w d')) \\\n (\\t'.\n (d1 | d2, t) \\*t t' \\\n s' \\ t' \\ discr c1)\n 2. \\t'.\n \\(d2, t) \\*t t'; s' \\ t'\\\n \\ (\\d' t'.\n (d1 | d2, t) \\*c (d', t') \\\n s' \\ t' \\\n (c1, d')\n \\ {(c1 | c2, d1 | d2) |c1 c2 d1 d2.\n c1 \\w d1 \\\n c2 \\w d2} \\\n Wbis) \\\n (\\t'.\n (d1 | d2, t) \\*t t' \\\n s' \\ t' \\ discr c1)"} {"_id": "501788", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\m \\ Suc 0; n \\ [n\\]\\\n \\ [n\\] \\ [ r, mod m ]"} {"_id": "501789", "text": "proof (prove)\ngoal (15 subgoals):\n 1. \\P. P \\ P\n 2. \\P Q.\n \\P \\\\<^sub>s Q; P \\ Q\\\n \\ Q \\ P\n 3. \\P Q R.\n \\P \\\\<^sub>s Q; P \\ Q; Q \\\\<^sub>s R;\n Q \\ R\\\n \\ P \\ R\n 4. \\P Q. P \\ Q \\ Q \\ P\n 5. \\P Q R.\n P \\ Q \\ R \\ P \\ (Q \\ R)\n 6. \\P. P \\ \\ \\ P\n 7. \\P Q. P \\ Q \\ Q \\ P\n 8. \\P Q R. P \\ Q \\ R \\ P \\ (Q \\ R)\n 9. \\P. P \\ \\ \\ P\n 10. \\x. \\\\x\\\\ \\ \\\nA total of 15 subgoals..."} {"_id": "501790", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Mod_Type_Connect.HMA_M ===>\n Mod_Type_Connect.HMA_I ===>\n Mod_Type_Connect.HMA_I ===> (=) ===> Mod_Type_Connect.HMA_M)\n (\\A a b q. addrow q a b A) row_add"} {"_id": "501791", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\in_carrier l; Imon_pol_list l mps\\\n \\ Ipol l (psubstl1 P1 mps n) = Ipol l P1"} {"_id": "501792", "text": "proof (prove)\nusing this:\n ?a2 \\ arcs G \\ edge_rev M ?a2 = ?a2\n ?a2 \\ arcs G \\ edge_rev M ?a2 \\ ?a2\n ?a2 \\ arcs G \\ edge_rev M (edge_rev M ?a2) = ?a2\n ?a2 \\ arcs G \\ tail G (edge_rev M ?a2) = head G ?a2\n edge_succ M permutes arcs G\n \\?v2 \\ verts G; out_arcs G ?v2 \\ {}\\\n \\ cyclic_on (edge_succ M) (out_arcs G ?v2)\n\ngoal (1 subgoal):\n 1. digraph_map G M"} {"_id": "501793", "text": "proof (prove)\nusing this:\n 2 * (bernoulli_fps oo - fps_X) = 2 * (bernoulli_fps + fps_X)\n\ngoal (1 subgoal):\n 1. bernoulli_fps oo - fps_X = bernoulli_fps + fps_X"} {"_id": "501794", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_prefix' xs ys\n \\ ESPEC (\\_. False)\n (\\r. r = (xs = take (length xs) ys))"} {"_id": "501795", "text": "proof (prove)\nusing this:\n bprv (imp (PP \\\\L \\\\) \\)\n\ngoal (1 subgoal):\n 1. bprv (\\L \\)"} {"_id": "501796", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Group.group H \\\n (f \\ Group.iso G (subgroup_generated H (f ` carrier G))) =\n (f \\ mon G H)"} {"_id": "501797", "text": "proof (chain)\npicking this:\n ops \\ set (List_Supplement.embed \\)\n op \\ set \\\n ops = [op]"} {"_id": "501798", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (atoms (tseytin_tran F)) \\ size F + card (atoms F)"} {"_id": "501799", "text": "proof (prove)\nusing this:\n dist\n (sum (confine (\\i. (orthonormal_coeff S w f i)\\<^sup>2) I) {..n})\n (sum (confine (\\i. (orthonormal_coeff S w f i)\\<^sup>2) I) {..m})\n < e\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\(\\i\\I \\ {..n}.\n (orthonormal_coeff S w f i)\\<^sup>2) -\n (\\i\\I \\ {..m}.\n (orthonormal_coeff S w f i)\\<^sup>2)\\\n < e\\<^sup>2"} {"_id": "501800", "text": "proof (prove)\nusing this:\n V.arr ?\\ = arrow_of_spans (\\) ?\\\n V.ide ?\\ = (arrow_of_spans (\\) ?\\ \\ C.ide (Chn ?\\))\n identity_arrow_of_spans_axioms ?C ?\\ \\\n partial_magma.ide ?C (Chn ?\\)\n identity_arrow_of_spans ?C ?\\ \\\n arrow_of_spans ?C ?\\ \\ identity_arrow_of_spans_axioms ?C ?\\\n identity_arrow_of_spans ?C ?\\ \\\n arrow_of_spans ?C ?\\\n identity_arrow_of_spans ?C ?\\ \\\n partial_magma.ide ?C (Chn ?\\)\n\ngoal (1 subgoal):\n 1. V.ide \\ = identity_arrow_of_spans (\\) \\"} {"_id": "501801", "text": "proof (prove)\ngoal (1 subgoal):\n 1. COMP f g = f"} {"_id": "501802", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Select0 (LiftInitData [V0 0, V1 0]) = 0"} {"_id": "501803", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\\\<^sub>\\.\n (subst_domain \\\\<^sub>\\ = \\ \\\n ground (subst_range \\\\<^sub>\\)) \\\n \\\\<^sub>\\ \\ \\\\\\ \\\n wt\\<^sub>s\\<^sub>u\\<^sub>b\\<^sub>s\\<^sub>t \\\\<^sub>\\ \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s (subst_range \\\\<^sub>\\)"} {"_id": "501804", "text": "proof (prove)\nusing this:\n smem_ind s i = Some j\n\ngoal (1 subgoal):\n 1. \\s;vs;vs_to_es ves @\n [$b_e]\\ \\_ i \\s';vs';es'\\"} {"_id": "501805", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {(X, Y). X < Y} = depth_nmset <*mlex*> {(X, Y). |X| = |Y| \\ X < Y}"} {"_id": "501806", "text": "proof (prove)\ngoal (1 subgoal):\n 1. interpret_floatariths (map Var d1s) xs1 =\n interpret_floatariths (map Var d2s) xs2"} {"_id": "501807", "text": "proof (prove)\nusing this:\n invar (a # bq)\n\ngoal (1 subgoal):\n 1. invar bq"} {"_id": "501808", "text": "proof (prove)\nusing this:\n order \\ \\ 0\n \\[?b = ?c] (mod ?a); [?d = ?e] (mod ?a)\\\n \\ [?b + ?d = ?c + ?e] (mod ?a)\n [?b = ?c] (mod ?a) = (?b mod ?a = ?c mod ?a)\n ?a * (?b mod ?c) mod ?c = ?a * ?b mod ?c\n\ngoal (1 subgoal):\n 1. [r *\n ((order \\ * order \\ -\n s * (nat (fst (bezw t (order \\))) mod order \\) +\n x) mod\n order \\) +\n (r * s * nat (fst (bezw t (order \\))) + xa) mod\n order\n \\ = r *\n (order \\ * order \\ -\n s * (nat (fst (bezw t (order \\))) mod order \\) +\n x) +\n (r * s * nat (fst (bezw t (order \\))) +\n xa)] (mod order \\)"} {"_id": "501809", "text": "proof (prove)\nusing this:\n \\ \\ P \\ M\\\\*(xvec @\n yvec)\\\\N\\ \\ P'\n x \\ supp N\n Prop ?Cb2 \\ P\n (M\\\\*(xvec @ yvec)\\\\N\\) P' A\\<^sub>P\n \\\\<^sub>P\n y \\ \\\n y \\ x\n y \\ P\n y \\ xvec\n y \\ yvec\n y \\ \\\n y \\ P'\n y \\ A\\<^sub>P\n y \\ \\\\<^sub>P\n y \\ M\n y \\ N\n y \\ C\n y \\ p\n x \\ \\\n x \\ M\n x \\ xvec\n x \\ yvec\n xvec \\* \\\n xvec \\* P\n xvec \\* M\n yvec \\* \\\n yvec \\* P\n yvec \\* M\n yvec \\* C\n x \\ C\n xvec \\* C\n distinct xvec\n distinct yvec\n extractFrame P = \\A\\<^sub>P, \\\\<^sub>P\\\n distinct A\\<^sub>P\n x \\ A\\<^sub>P\n xvec \\* yvec\n xvec \\* \\\\<^sub>P\n A\\<^sub>P \\* \\\n A\\<^sub>P \\* P\n A\\<^sub>P \\* M\n A\\<^sub>P \\* xvec\n A\\<^sub>P \\* yvec\n A\\<^sub>P \\* N\n A\\<^sub>P \\* P'\n A\\<^sub>P \\* C\n\ngoal (1 subgoal):\n 1. Prop C \\ (\\\\x\\P)\n (M\\\\*(xvec @\n y #\n yvec)\\\\([(x, y)] \\ N)\\)\n ([(x, y)] \\ P') (x # A\\<^sub>P) \\\\<^sub>P"} {"_id": "501810", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_get_dom_component_insert_before\\<^sub>C\\<^sub>o\\<^sub>r\\<^sub>e\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n heap_is_wellformed parent_child_rel DocumentClass.type_wf\n DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order\n get_parent get_parent_locs get_child_nodes get_child_nodes_locs\n get_dom_component is_strongly_dom_component_safe\n is_weakly_dom_component_safe get_root_node get_root_node_locs\n get_ancestors get_ancestors_locs get_disconnected_nodes\n get_disconnected_nodes_locs get_element_by_id get_elements_by_tag_name\n set_child_nodes set_child_nodes_locs set_disconnected_nodes\n set_disconnected_nodes_locs get_owner_document remove_child\n remove_child_locs remove adopt_node adopt_node_locs insert_before\n insert_before_locs append_child get_scdom_component\n is_strongly_scdom_component_safe is_weakly_scdom_component_safe"} {"_id": "501811", "text": "proof (prove)\nusing this:\n rvars (\\ s') = rvars (\\ s) - {x\\<^sub>j} \\ {x\\<^sub>i}\n x \\ rvars (\\ s) \\ x = x\\<^sub>i\n x \\ x\\<^sub>j\n x \\ x\\<^sub>i \\ x \\ lvars (\\ s)\n \\\\ (\\ s); \\ s; x\\<^sub>i \\ lvars (\\ s);\n x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ s) x\\<^sub>i)\\\n \\ Mapping.lookup\n (\\ (pivot_and_update x\\<^sub>i x\\<^sub>j b s))\n x\\<^sub>i =\n Some b\n \\\\ (\\ s); \\ s; x\\<^sub>i \\ lvars (\\ s);\n x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ s) x\\<^sub>i);\n x \\ lvars (\\ s); x \\ x\\<^sub>j\\\n \\ Mapping.lookup\n (\\ (pivot_and_update x\\<^sub>i x\\<^sub>j b s)) x =\n Mapping.lookup (\\ s) x\n x\\<^sub>i \\ lvars (\\ s)\n x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ s) x\\<^sub>i)\n \\ (\\ s)\n \\ s\n s' = pivot_and_update x\\<^sub>i x\\<^sub>j b s\n b = the (\\\\<^sub>l s x\\<^sub>i) \\\n b = the (\\\\<^sub>u s x\\<^sub>i)\n\ngoal (1 subgoal):\n 1. \\\\ s'\\ x = \\\\ s\\ x \\\n \\\\ s'\\ x = the (\\\\<^sub>l s x) \\\n \\\\ s'\\ x = the (\\\\<^sub>u s x)"} {"_id": "501812", "text": "proof (prove)\nusing this:\n xs \\ []\n run (xs @- ys) \\\n steps xs \\ run ys \\ E (last xs) (shd ys)\n run ((x # xs) @- ys)\n\ngoal (1 subgoal):\n 1. steps (x # xs) \\ run ys \\ E (last (x # xs)) (shd ys)"} {"_id": "501813", "text": "proof (prove)\nusing this:\n ins ! pc = Invoke M n\n\ngoal (1 subgoal):\n 1. (ta', xcp', h', (stk' @ stk'', loc', C, M, pc') # frs)\n \\ exec_instr (ins ! pc) P t h (stk @ stk'') loc C M pc frs"} {"_id": "501814", "text": "proof (prove)\nusing this:\n \\

\\<^sub>1[ra, ru.prj\\<^sub>1 \\ \\.chine] =\n (ru.prj\\<^sub>1 \\ \\.chine) \\\n \\

\\<^sub>0[ra, ru.prj\\<^sub>1 \\ \\.chine]\n\ngoal (1 subgoal):\n 1. ru.prj\\<^sub>1 \\ \\.chine =\n \\

\\<^sub>1[ra, ru.prj\\<^sub>1 \\ \\.chine] \\\n C.inv \\

\n (?w +o \\{?s}\\) \\\n dual_order.poset_simplex_map \\

{?w}\n\ngoal (1 subgoal):\n 1. \\w \\ W; s \\ S\\\n \\ dual_order.poset_simplex_map \\

{w, w + s} \\\n dual_order.poset_simplex_map \\

{w}"} {"_id": "502006", "text": "proof (prove)\nusing this:\n merges f qs ss ! q = [] \\\\<^bsub>f\\<^esub> ss ! q\n\ngoal (1 subgoal):\n 1. merges f qs ss ! q = ss ! q"} {"_id": "502007", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fx0.brent f x = ?code"} {"_id": "502008", "text": "proof (prove)\nusing this:\n ((bi_unique poly_rel \\ right_total poly_rel) \\\n poly_rel (arith_ops_record.zero (poly_ops ops)) 0 \\\n poly_rel (arith_ops_record.one (poly_ops ops)) 1 \\\n (poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.plus (poly_ops ops)) (+)) \\\n ((poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.minus (poly_ops ops)) (-) \\\n (poly_rel ===> poly_rel) (arith_ops_record.uminus (poly_ops ops))\n uminus_class.uminus) \\\n (poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.times (poly_ops ops)) (*) \\\n (poly_rel ===> poly_rel ===> (=)) (=) (=) \\\n Domainp poly_rel = arith_ops_record.DP (poly_ops ops)\n\ngoal (1 subgoal):\n 1. ((bi_unique poly_rel \\ right_total poly_rel) \\\n poly_rel (arith_ops_record.zero (poly_ops ops)) 0 \\\n poly_rel (arith_ops_record.one (poly_ops ops)) 1 \\\n (poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.plus (poly_ops ops)) (+)) \\\n ((poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.minus (poly_ops ops)) (-) \\\n (poly_rel ===> poly_rel) (arith_ops_record.uminus (poly_ops ops))\n uminus_class.uminus) \\\n (poly_rel ===> poly_rel ===> poly_rel)\n (arith_ops_record.times (poly_ops ops)) (*) \\\n (poly_rel ===> poly_rel ===> (=)) (=) (=) \\\n Domainp poly_rel = arith_ops_record.DP (poly_ops ops)"} {"_id": "502009", "text": "proof (prove)\ngoal (1 subgoal):\n 1. program_inv (fst (setUp ast))"} {"_id": "502010", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {0..Suc k} = insert (Suc k) {0..k} \\ Suc k \\ n"} {"_id": "502011", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sv \\ atoms\n (encode_operator_precondition\n (\\ prob_with_noop abs_prob ) t op);\n t < h;\n op \\ set ((\\ prob_with_noop abs_prob )\\<^sub>\\)\\\n \\ valid_state_var sv"} {"_id": "502012", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ {n. P n x} = n"} {"_id": "502013", "text": "proof (prove)\ngoal (1 subgoal):\n 1. arctan (sqrt (real_of_float y))\n \\ {sqrt (real_of_float y) *\n real_of_float\n (lb_arctan_horner prec (get_even n) 1\n y)..sqrt (real_of_float y) *\n real_of_float (ub_arctan_horner prec (get_odd n) 1 y)}"} {"_id": "502014", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x \\ carrier L; \\1.\\_protocol;\n (a, b) \\ set_spmf (S1_raw x1 x); aa = x\\\n \\ check1 x1 a x b"} {"_id": "502015", "text": "proof (prove)\nusing this:\n \\\n \\ \\ \\, n \\ \\ \\ \\ \\\\<^sub>c\\<^sub>o\\<^sub>n\\<^sub>f\\<^sub>i\\<^sub>g\n\ngoal (1 subgoal):\n 1. \\\\\\<^sub>k \\\\<^sub>k \\\\<^sub>k k.\n \\, n \\ \\ \\ \\ \\\\<^bsup>k\\<^esup> \\\\<^sub>k, 0 +\n Suc n \\ \\\\<^sub>k \\ \\\\<^sub>k \\\n \\\n \\ \\ \\\\<^sub>k, 0 +\n Suc n \\ \\\\<^sub>k \\ \\\\<^sub>k \\\\<^sub>c\\<^sub>o\\<^sub>n\\<^sub>f\\<^sub>i\\<^sub>g"} {"_id": "502016", "text": "proof (prove)\nusing this:\n ((f', l'), f, l) \\ pr_algo_rel\n\ngoal (1 subgoal):\n 1. Height_Bounded_Labeling c s t f l"} {"_id": "502017", "text": "proof (state)\nthis:\n formulaEntailsClause F0 (getC state)\n\ngoal (1 subgoal):\n 1. let l = getCl state; bClause = getC state; bLiteral = opposite l;\n level = getBackjumpLevel state;\n prefix = prefixToLevel level (getM state);\n state'' = applyBackjump state\n in (formulaEntailsClause F0 bClause \\\n isUnitClause bClause bLiteral (elements prefix) \\\n getM state'' = prefix @ [(bLiteral, False)]) \\\n getF state'' = getF state"} {"_id": "502018", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < y * (z / (x + y))"} {"_id": "502019", "text": "proof (prove)\ngoal (1 subgoal):\n 1. RF (\\ (TER f \\ TER g)) \\ RF (TER (f \\ g))"} {"_id": "502020", "text": "proof (prove)\ngoal (1 subgoal):\n 1. op_eventually_within_sctn X sctn S\n \\ SPEC\n (\\b.\n b \\\n (\\x\\X.\n x \\ S \\\n (\\\\<^sub>F x in at x within plane_of sctn.\n x \\ S)))"} {"_id": "502021", "text": "proof (prove)\ngoal (1 subgoal):\n 1. emeasure M (G i j) = \\ i j"} {"_id": "502022", "text": "proof (prove)\nusing this:\n bounded ?S = (\\b>0. \\x\\?S. norm x \\ b)\n\ngoal (1 subgoal):\n 1. (\\C.\n \\0 < C;\n \\x. x \\ S \\ norm (f x) \\ C\\\n \\ thesis) \\\n thesis"} {"_id": "502023", "text": "proof (prove)\nusing this:\n {card {y \\ {0.. {0.. {len ?m1 ?i1 ?j1 xs |xs.\n set xs \\ {0..?k1} \\\n ?i1 \\ set xs \\\n ?j1 \\ set xs \\ distinct xs} \\\n Min {len ?m1 ?i1 ?j1 xs |xs.\n set xs \\ {0..?k1} \\\n ?i1 \\ set xs \\ ?j1 \\ set xs \\ distinct xs}\n \\ ?x\n\ngoal (1 subgoal):\n 1. \\set xs \\ {0..k}; i \\ set xs;\n j \\ set xs; distinct xs\\\n \\ D m i j k \\ len m i j xs"} {"_id": "502025", "text": "proof (prove)\nusing this:\n u \\ ginitial A\n gupath A w t u\n v = gtarget t u\n gurun A w s v\n\ngoal (1 subgoal):\n 1. gtarget (stake (Suc l) (t @- s)) u \\ gunodes A w"} {"_id": "502026", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ovalidNF (\\s. P (f s) s) (ogets f) P"} {"_id": "502027", "text": "proof (prove)\ngoal (1 subgoal):\n 1. restrict_indets ` ideal F = ideal (restrict_indets ` F)"} {"_id": "502028", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_all (\\f. \\ \\ \\ f) l"} {"_id": "502029", "text": "proof (state)\nthis:\n (?eqa4, ?st'a4) \\ code_status_status_rel \\\n (merge_cstatus cst ?eqa4, merge_status cst' ?st'a4)\n \\ code_status_status_rel\n\ngoal (1 subgoal):\n 1. PAC_checker_l_step spec Ast st\n \\ \\\n (code_status_status_rel \\\\<^sub>r\n \\var_rel\\set_rel \\\\<^sub>r\n \\nat_rel,\n sorted_poly_rel O mset_poly_rel\\fmap_rel)\n (PAC_checker_step spec' Bst st')"} {"_id": "502030", "text": "proof (state)\nthis:\n \\ (h \\ l) = h (\\ (l \\ h))\n\ngoal (1 subgoal):\n 1. \\ (h \\ l) \\ \\ (h \\ g)"} {"_id": "502031", "text": "proof (prove)\nusing this:\n S.ide f\n S.ide f \\\n \\\\[f] : R f \\\\<^sub>S f\\\n S.ide (R f) \\\n \\\\[R f] : R (R f) \\\\<^sub>S R\n f\\\n\ngoal (1 subgoal):\n 1. S.seq \\[f] \\[R f]"} {"_id": "502032", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xs.\n (distinct xs \\\n set (injections_alg xs Y) = injections (set xs) Y) \\\n distinct (x # xs) \\\n set (injections_alg (x # xs) Y) = injections (set (x # xs)) Y"} {"_id": "502033", "text": "proof (prove)\nusing this:\n strong_low_bisim_mm (\\ ?\\')\n \\\\ ?\\ {?c} ?\\';\n mds_consistent ?mds ?\\;\n \\x.\n to_total ?\\ x = Low \\\n ?mem\\<^sub>1 x = ?mem\\<^sub>2 x\\\n \\ \\?c, ?mds, ?mem\\<^sub>1\\ \\\\<^sup>u\\<^bsub>?\\'\\<^esub> \\?c, ?mds, ?mem\\<^sub>2\\\n\ngoal (1 subgoal):\n 1. \\c, mds, mem\\<^sub>1\\ \\\n \\c, mds, mem\\<^sub>2\\"} {"_id": "502034", "text": "proof (chain)\npicking this:\n [(p - 1) ^ (2 ^ x * m) = 1] (mod p)\n [(p - 1) ^ (2 ^ x * m) = p - 1] (mod p)"} {"_id": "502035", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\m f.\n (\\f.\n map_poly hom (graeffe_poly_impl_main c f m) =\n graeffe_poly_impl_main (hom c) (map_poly hom f)\n m) \\\n map_poly hom (graeffe_one_step c (graeffe_poly_impl_main c f m)) =\n graeffe_one_step (hom c)\n (graeffe_poly_impl_main (hom c) (map_poly hom f) m)"} {"_id": "502036", "text": "proof (prove)\nusing this:\n B * cnj a = cnj B * a\n C = cnj B\n cor (Re (- 4 * a / B)) = - 4 * a / B\n B \\ 0\n\ngoal (1 subgoal):\n 1. - 4 * cnj a = cor (Re (- 4 * a / B)) * C"} {"_id": "502037", "text": "proof (prove)\nusing this:\n ?l \\ lists A \\ (?l, ?l) \\ listrel r\n\ngoal (1 subgoal):\n 1. refl_on (lists A) (listrel r)"} {"_id": "502038", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a aa b ab ac ba.\n \\semilat (err a, Err.le aa, lift2 b);\n semilat (err ab, Err.le ac, lift2 ba)\\\n \\ order (Err.le (Product.le aa ac)) \\\n closed (err (a \\ ab))\n (lift2 (Product.sup b ba)) \\\n (\\x\\err (a \\ ab).\n \\y\\err (a \\ ab).\n x \\\\<^bsub>Err.le\n (Product.le aa\n ac)\\<^esub> x \\\\<^bsub>lift2\n (Product.sup b ba)\\<^esub> y) \\\n (\\x\\err (a \\ ab).\n \\y\\err (a \\ ab).\n y \\\\<^bsub>Err.le\n (Product.le aa\n ac)\\<^esub> x \\\\<^bsub>lift2\n (Product.sup b ba)\\<^esub> y) \\\n (\\x\\err (a \\ ab).\n \\y\\err (a \\ ab).\n \\z\\err (a \\ ab).\n x \\\\<^bsub>Err.le\n (Product.le aa ac)\\<^esub> z \\\n y \\\\<^bsub>Err.le\n (Product.le aa ac)\\<^esub> z \\\n x \\\\<^bsub>lift2\n (Product.sup b ba)\\<^esub> y \n \\\\<^bsub>Err.le\n (Product.le aa ac)\\<^esub> z)"} {"_id": "502039", "text": "proof (prove)\nusing this:\n \\heap.\n P heap \\\n Q heap \\ state_dp_consistency.cmem heap \\\n (case execute b heap of None \\ False\n | Some (v', heap') \\\n P heap' \\\n Q heap' \\ R a v' \\ state_dp_consistency.cmem heap')\n P heap\n Q heap\n state_dp_consistency.cmem heap\n\ngoal (1 subgoal):\n 1. (\\x heap'.\n \\execute b heap = Some (x, heap'); P heap'; Q heap';\n state_dp_consistency.cmem heap'; R a x\\\n \\ thesis) \\\n thesis"} {"_id": "502040", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ valid2 []"} {"_id": "502041", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Abs_freeword (map (\\s. (s, b)) (xs @ ys)) =\n Abs_freeword (map (\\s. (s, b)) xs) +\n Abs_freeword (map (\\s. (s, b)) ys)"} {"_id": "502042", "text": "proof (prove)\nusing this:\n pointermap_p_valid p m\n pointermap_getmk a m = (x, u)\n\ngoal (1 subgoal):\n 1. pointermap_p_valid p u"} {"_id": "502043", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_set_shadow_root_get_shadow_root ShadowRootClass.type_wf\n set_shadow_root set_shadow_root_locs get_shadow_root\n get_shadow_root_locs"} {"_id": "502044", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z. - - (x \\ y) \\ (- x \\ z) = bot"} {"_id": "502045", "text": "proof (prove)\ngoal (1 subgoal):\n 1. typ_of (mk_all x ty A) = Some propT"} {"_id": "502046", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_Msg m =\n (case m of NoMsg \\ False | Msg x \\ True)"} {"_id": "502047", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a expr', card_UNIV_class)"} {"_id": "502048", "text": "proof (prove)\ngoal (1 subgoal):\n 1. M\\<^sup>\\\\<^sup>\\ $$ (i, j) = M $$ (i, j)"} {"_id": "502049", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b.\n \\P a; Q b\\\n \\ \\aa ba ab.\n aa ## ab \\\n (\\bb.\n ba ## bb \\\n a = aa + ab \\\n b = ba + bb \\\n P aa \\\n ba = (0::'d) \\\n ab = (0::'c) \\ Q bb)"} {"_id": "502050", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((M, M'), M \\ N, M' \\ N')\n \\ measures\n [\\(M, N). card (vars_of M \\ vars_of N),\n \\(M, N). size M]"} {"_id": "502051", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ b; \\x\\{a..b}. isCont f x;\n f b \\ k \\ y; y \\ f a \\ k\\\n \\ \\x\\{a..b}. f x \\ k = y"} {"_id": "502052", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (closure {ou_witness.a} = {ou_witness.a} &&&\n closure {ou_witness.b} = {ou_witness.a, ou_witness.b} &&&\n closure {ou_witness.c} = {ou_witness.a, ou_witness.c}) &&&\n (closure {ou_witness.a, ou_witness.b} = {ou_witness.a, ou_witness.b} &&&\n closure {ou_witness.a, ou_witness.c} =\n {ou_witness.a, ou_witness.c}) &&&\n closure {ou_witness.a, ou_witness.b, ou_witness.c} =\n {ou_witness.a, ou_witness.b, ou_witness.c} &&&\n closure {ou_witness.b, ou_witness.c} =\n {ou_witness.a, ou_witness.b, ou_witness.c}"} {"_id": "502053", "text": "proof (prove)\nusing this:\n vwb_lens x\n \\vwb_lens ?x; Idempotent id;\n uex ?x \\ id = id \\ uex ?x\\\n \\ retract\n ((uex ?x \\\n id) \\\\uex\n ?x,id\\\\ id)\n\ngoal (1 subgoal):\n 1. retract (uex x \\\\uex x,id\\\\ id)"} {"_id": "502054", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distributed M S (\\x. (X x, Y x, Z x))\n (\\x. ennreal (P x))"} {"_id": "502055", "text": "proof (prove)\nusing this:\n finite Z\n z \\ Z\n \\Z \\ carrier V;\n a \\ Z \\ {\\\\<^bsub>K\\<^esub>}\\\n \\ lincomb a (A \\ Z) = lincomb a A\n insert z Z \\ carrier V\n a \\ insert z Z \\ {\\\\<^bsub>K\\<^esub>}\n\ngoal (1 subgoal):\n 1. lincomb a (A \\ Z) = lincomb a A"} {"_id": "502056", "text": "proof (prove)\nusing this:\n \\x\\free_vars (Var x). val_type (\\ x) = \\ x\n\ngoal (1 subgoal):\n 1. sets (expr_sem \\ (Var x)) = sets (stock_measure (\\ x))"} {"_id": "502057", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\a b aa ba W \\ Wtl Wn \\ W' \\'.\n \\(a, b) \\ dfs_invar \\i R; b \\ [];\n a = \\; b = W; aa = \\'; ba = W'; W = \\ # Wtl;\n distinct Wn; set Wn = R `` {\\} - \\; W' = Wn @ Wtl;\n \\' = R `` {\\} \\ \\\\\n \\ \\ \\ set b\n 2. \\a b aa ba W \\ Wtl Wn \\ W' \\'.\n \\(a, b) \\ dfs_invar \\i R; b \\ [];\n a = \\; b = W; aa = \\'; ba = W'; W = \\ # Wtl;\n distinct Wn; set Wn = R `` {\\} - \\; W' = Wn @ Wtl;\n \\' = R `` {\\} \\ \\\\\n \\ aa = a \\ R `` {\\}\n 3. \\a b aa ba W \\ Wtl Wn \\ W' \\'.\n \\(a, b) \\ dfs_invar \\i R; b \\ [];\n a = \\; b = W; aa = \\'; ba = W'; W = \\ # Wtl;\n distinct Wn; set Wn = R `` {\\} - \\; W' = Wn @ Wtl;\n \\' = R `` {\\} \\ \\\\\n \\ set ba =\n set b - {\\} \\ (R `` {\\} - a)\n 4. \\aa ba.\n (aa, ba) \\ dfs_initial \\i \\\n (aa, set ba) = sse_initial \\i\n 5. \\aa ba.\n (aa, ba) \\ dfs_invar \\i R \\\n (aa, set ba) \\ sse_invar \\i R"} {"_id": "502058", "text": "proof (prove)\nusing this:\n Finite_Set.fold f z B = Finite_Set.fold g z B\n f a = g a\n finite B\n a \\ B\n\ngoal (1 subgoal):\n 1. Finite_Set.fold f z (insert a B) = Finite_Set.fold g z (insert a B)"} {"_id": "502059", "text": "proof (prove)\nusing this:\n a = (l, b)\n\ngoal (1 subgoal):\n 1. trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\n (dual\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t (a # A)) =\n trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t (a # A)"} {"_id": "502060", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {v. atom v \\ supp (Bool b)} = {}"} {"_id": "502061", "text": "proof (prove)\nusing this:\n \\\\<^sub>D\\<^sub>G I \\ i :\n dg_prod I\n \\ \\\\\\<^sub>D\\<^sub>G\\<^bsub>\\\\<^esub> \\ i\n \\ :\n \\ \\\\\\<^sub>D\\<^sub>G\\<^bsub>\\\\<^esub> dg_prod I\n \\\n a \\\\<^sub>\\ \\\\Arr\\\n i \\\\<^sub>\\ I\n\ngoal (1 subgoal):\n 1. dghm_up I \\ \\\n \\\\ArrMap\\\\a\\\\i\\ =\n \\\\ArrMap\\\\a\\\\i\\"} {"_id": "502062", "text": "proof (prove)\nusing this:\n Fstd_get F1 f = Some fd1\n rel_fundef (\\x y. x = norm_instr y) fd1 fd2\n pc = length (body fd2)\n length (body fd1) = length (body fd2)\n\ngoal (1 subgoal):\n 1. Sstd.final (State F1 H [Frame f pc \\])"} {"_id": "502063", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_weakly_respects_barbs SRel SWB"} {"_id": "502064", "text": "proof (prove)\nusing this:\n \\ p dvd p'\n prime p\n p dvd z\n p dvd gauss_cnj z\n\ngoal (1 subgoal):\n 1. p * p' dvd z"} {"_id": "502065", "text": "proof (prove)\ngoal (1 subgoal):\n 1. avl t \\\n AVL_Bal_Set.insert x t \\\n \\l, (a, Rh), \\\\\\ \\\n AVL_Bal_Set.insert x t \\\n \\\\\\, (a, Lh), r\\"} {"_id": "502066", "text": "proof (prove)\ngoal (1 subgoal):\n 1. V \\ attractor p K \\ (V - W1 - attractor p K) \\\n W1"} {"_id": "502067", "text": "proof (prove)\nusing this:\n map_of (zip xs ys)(x \\ y) = map_of (zip xs zs)(x \\ z)\n\ngoal (1 subgoal):\n 1. y = z"} {"_id": "502068", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ideal_generated S = {y. \\f. (\\i\\S. f i * i) = y}"} {"_id": "502069", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\init rest.\n \\pes = init @ (p, e) # rest; distinct (pat_bindings p []);\n list_all\n (\\(p, e).\n cupcake_pmatch cenv p v0 [] = No_match \\\n distinct (pat_bindings p []))\n init;\n cupcake_pmatch cenv p v0 [] = Match env'\\\n \\ thesis) \\\n thesis"} {"_id": "502070", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inres (ASSERT (\\x. P x) \\ (\\_. SPEC P)) x =\n ((\\x. \\ P x) \\ P x)"} {"_id": "502071", "text": "proof (prove)\nusing this:\n xs \\ []\n\ngoal (1 subgoal):\n 1. 0 < natural_of_nat (length xs)"} {"_id": "502072", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rat_of (mediant X) < rat_of (mediant (X \\ UR \\ Y))"} {"_id": "502073", "text": "proof (prove)\nusing this:\n c_row + c_rec \\ (\\ii = i..\n cost (dmu_array_cost dmus i)\n \\ (\\ii = i..0 < ?rs;\n {\\tp. tp = (Bk \\ 0, )} p\n {\\tp. tp = (Bk \\ ?m, Oc \\ ?rs @ Bk \\ ?k)}\\\n \\ {\\tp. tp = ([], )} UTM\n {\\tp.\n \\m n.\n tp = (Bk \\ m, Oc \\ ?rs @ Bk \\ n)}\n\ngoal (1 subgoal):\n 1. {\\tp. tp = ([], )} UTM\n {\\tp. \\m k. tp = (Bk \\ m, @ Bk \\ k)}"} {"_id": "502075", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x \\ y; y \\ z\\ \\ x \\ z"} {"_id": "502076", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a $ k = \\x\\ k"} {"_id": "502077", "text": "proof (prove)\ngoal (1 subgoal):\n 1. oclosed\n (opnet (\\i. optoy i \\\\\\<^bsub>i\\<^esub> qmsg)\n n) \\ (\\_ _ _. True,\n other nos_inc (net_tree_ips n) \\)\n global\n (\\\\.\n \\i\\net_tree_ips n.\n no (\\ i)\n \\ no (\\ (nhid (\\ i))))"} {"_id": "502078", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\s. s \\ productive_on g) \\ productive g"} {"_id": "502079", "text": "proof (prove)\nusing this:\n ms \\ LPDs mr\n\ngoal (1 subgoal):\n 1. (fv_regex (from_mregex ms \\s) = fv_regex r &&&\n safe_regex Futu Strict (from_mregex ms \\s)) &&&\n ok (length \\s) ms &&& LPD ms \\ LPDs mr"} {"_id": "502080", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite X; fun S1 = fun S2 on - X; S2 \\ S1;\n S1 \\ S2 < S1\\\n \\ n\\<^sub>s (S1 \\ S2) X < n\\<^sub>s S1 X"} {"_id": "502081", "text": "proof (prove)\nusing this:\n Gmod3 = Gmod4\n\ngoal (1 subgoal):\n 1. Gmod1 \\ Gmod2"} {"_id": "502082", "text": "proof (prove)\ngoal (3 subgoals):\n 1. 1-lipschitz_on (c ` {A..B}) to_Bonk_Schramm_extension\n 2. geodesic_segment_between (to_Bonk_Schramm_extension ` G)\n ((to_Bonk_Schramm_extension \\ c) A)\n ((to_Bonk_Schramm_extension \\ c) B)\n 3. to_Bonk_Schramm_extension x \\ to_Bonk_Schramm_extension ` G"} {"_id": "502083", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\H \\ carrier G; x \\ carrier G\\\n \\ x <+ H \\ carrier G"} {"_id": "502084", "text": "proof (prove)\ngoal (1 subgoal):\n 1. continuous_on {a..b} f"} {"_id": "502085", "text": "proof (prove)\nusing this:\n x * y < (0::'a)\n xs \\ []\n hd xs \\ (0::'a)\n last xs \\ (0::'a)\n\ngoal (1 subgoal):\n 1. even (sign_changes (y # xs'')) =\n (sgn (hd (y # xs'')) = sgn (last (y # xs'')))"} {"_id": "502086", "text": "proof (prove)\nusing this:\n i \\\\<^sub>\\ \\\\<^sub>\\ x\n J \\\\<^sub>\\ I\n\ngoal (1 subgoal):\n 1. x\\i\\ =\n ((\\i\\\\<^sub>\\I.\n if i \\\\<^sub>\\ J \\ x\\i\\ \n | otherwise \\ SOME x.\n x \\\\<^sub>\\ A i) \\\\<^sup>l\\<^sub>\\\n J)\\i\\"} {"_id": "502087", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(nterm, pre_term_class)"} {"_id": "502088", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ccTTree S G \\\\ ccTTree S' G' \\\n ccTTree (S \\ S') (G \\ G' \\ S G\\ S')"} {"_id": "502089", "text": "proof (prove)\ngoal (1 subgoal):\n 1. consts t |\\|\n ffUnion (consts |`| fmimage (fmdrop x env) (frees t)) =\n consts t |\\| ffUnion (consts |`| fmimage env (frees t |-| {|x|}))"} {"_id": "502090", "text": "proof (prove)\ngoal (1 subgoal):\n 1. derivesp g =\n (\\lsl rsl.\n \\s1 s2 lhs rhs.\n s1 @ [NTS lhs] @ s2 = lsl \\\n s1 @ rhs @ s2 = rsl \\ fst g (rule lhs rhs))"} {"_id": "502091", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\x y.\n \\x \\ carrier P; y \\ carrier P\\\n \\ to_fun (x \\\\<^bsub>P\\<^esub> y) =\n to_fun\n x \\\\<^bsub>function_ring (carrier R) R\\<^esub>\n to_fun y\n 2. \\x y.\n \\x \\ carrier P; y \\ carrier P\\\n \\ to_fun (x \\\\<^bsub>P\\<^esub> y) =\n to_fun\n x \\\\<^bsub>function_ring (carrier R) R\\<^esub>\n to_fun y\n 3. to_fun \\\\<^bsub>P\\<^esub> =\n \\\\<^bsub>function_ring (carrier R) R\\<^esub>"} {"_id": "502092", "text": "proof (prove)\nusing this:\n AE \\ in M. False\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502093", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mono_setup_loc (\\)"} {"_id": "502094", "text": "proof (prove)\ngoal (1 subgoal):\n 1. correctCompositionDiffLevels sA9"} {"_id": "502095", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\b \\ a; d \\ c\\\n \\ (a - b) * (c - d) =\n a * c + b * d - (a * d + b * c) \\\n a * d + b * c \\ a * c + b * d"} {"_id": "502096", "text": "proof (prove)\nusing this:\n 0 \\ y\n y < x\n n \\ {i. 0 < i \\ real i \\ x}\n\ngoal (1 subgoal):\n 1. (f' has_integral f x - f (max (real n) y)) {max (real n) y..x}"} {"_id": "502097", "text": "proof (state)\nthis:\n LINT (x, y, z)\n |(count_space (X ` space M) \\\\<^sub>M\n count_space (Y ` space M) \\\\<^sub>M\n count_space (Z ` space M)).\n (if (x, y, z) \\ (\\x. (X x, Y x, Z x)) ` space M\n then Pxyz (x, y, z) else 0) *\n log b\n ((if (x, y, z) \\ (\\x. (X x, Y x, Z x)) ` space M\n then Pxyz (x, y, z) else 0) /\n ((if (x, z) \\ (\\x. (X x, Z x)) ` space M then Pxz (x, z)\n else 0) *\n ((if (y, z) \\ (\\x. (Y x, Z x)) ` space M then Pyz (y, z)\n else 0) /\n Pz z))) =\n (\\(x, y, z)\\(\\x. (X x, Y x, Z x)) ` space M.\n Pxyz (x, y, z) *\n log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y, z) / Pz z))))\n\ngoal (5 subgoals):\n 1. \\x.\n x \\ space (count_space (X ` space M)) \\\n 0 \\ prob (X -` {x} \\ space M)\n 2. \\z.\n z \\ space (count_space (Z ` space M)) \\\n 0 \\ Pz z\n 3. \\y z.\n \\y \\ space (count_space (Y ` space M));\n z \\ space (count_space (Z ` space M))\\\n \\ 0 \\ (if (y, z)\n \\ (\\x. (Y x, Z x)) `\n space M\n then Pyz (y, z) else 0)\n 4. \\x z.\n \\x \\ space (count_space (X ` space M));\n z \\ space (count_space (Z ` space M))\\\n \\ 0 \\ (if (x, z)\n \\ (\\x. (X x, Z x)) `\n space M\n then Pxz (x, z) else 0)\n 5. \\x y z.\n \\x \\ space (count_space (X ` space M));\n y \\ space (count_space (Y ` space M));\n z \\ space (count_space (Z ` space M))\\\n \\ 0 \\ (if (x, y, z)\n \\ (\\x. (X x, Y x, Z x)) `\n space M\n then Pxyz (x, y, z) else 0)"} {"_id": "502098", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ {Inf_aform' p X..Sup_aform' p X} \\ xs \\ Joints XS"} {"_id": "502099", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isCont f a \\ isCont (\\x. Re (f x)) a"} {"_id": "502100", "text": "proof (prove)\ngoal (1 subgoal):\n 1. matpow (A + mat 1) n ** A = A ** matpow (A + mat 1) n"} {"_id": "502101", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\h \\ r.\n \\\\ = Some (snd (Ref.alloc x (the_state (Some h))));\n r = fst (Ref.alloc x (the_state (Some h)));\n Ref.get (the_state \\) r = x\\\n \\ \\ is_exn \\ \\\n Ref.get (the_state \\) r = x \\\n new_addrs h {} (the_state \\) =\n {addr_of_ref r} \\\n addr_of_ref r < lim (the_state \\) \\\n relH {a. a < lim h} h (the_state \\) \\\n lim h \\ lim (the_state \\)"} {"_id": "502102", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x\\#A. if x = p then 1 else 0) =\n (if p \\ S then f p else 0)"} {"_id": "502103", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hn_refine emp (return None) emp (option_assn P) (RETURN $ None)"} {"_id": "502104", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n (\\xa.\n xa \\ x \\\n (\\va u.\n move ts ts' v=va\\u \\\n True \\\n (\\v ua.\n u=v\\ua \\\n (\\va u.\n v=va--u \\\n True \\\n (\\v ua.\n u=v--ua \\\n ((0 < \\ext v\\ \\\n len v ts' x = ext v \\\n restrict v (clm ts') x = lan v \\\n |lan v| = 1) \\\n (0 < \\ext v\\ \\\n len v ts' xa = ext v \\\n restrict v (res ts') xa = lan v \\\n |lan v| = 1 \\\n 0 < \\ext v\\ \\\n len v ts' xa = ext v \\\n restrict v (clm ts') xa = lan v \\\n |lan v| = 1)) \\\n True)) \\\n True))) \\\n (\\ts'.\n ts' \\<^bold>\\r( x ) \\<^bold>\\ ts' \\\n False)"} {"_id": "502105", "text": "proof (prove)\nusing this:\n (a1 \\\\<^sub>g b1) = Some d1\n \\\\Catch a1 a2 \\ s\n s = Normal s'\n\ngoal (1 subgoal):\n 1. \\\\d1 \\ Normal s'"} {"_id": "502106", "text": "proof (prove)\nusing this:\n toList ?x \\ rel_ext (\\x. sorted x \\ distinct x)\n\ngoal (1 subgoal):\n 1. sorted (toList xs)"} {"_id": "502107", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.countable_complete_distrib_lattice Inf Sup inf (\\) (<) sup bot\n top"} {"_id": "502108", "text": "proof (prove)\nusing this:\n ?r2 \\ g3ip \\ ?r2 \\ upRules\n\ngoal (1 subgoal):\n 1. g3ip \\ upRules"} {"_id": "502109", "text": "proof (prove)\nusing this:\n x * (n - 1) mod n = 1\n \\x \\ carrier G; n - 1 \\ carrier G;\n x \\\\<^bsub>G\\<^esub> (n - 1) = \\\\<^bsub>G\\<^esub>;\n (n - 1) \\\\<^bsub>G\\<^esub> x = \\\\<^bsub>G\\<^esub>\\\n \\ n - 1 = inv\\<^bsub>G\\<^esub> x\n \\x \\ carrier G; y \\ carrier G;\n x \\\\<^bsub>G\\<^esub> y = \\\\<^bsub>G\\<^esub>;\n y \\\\<^bsub>G\\<^esub> x = \\\\<^bsub>G\\<^esub>\\\n \\ y = inv\\<^bsub>G\\<^esub> x\n 2 \\ n \\ n - 1 \\ totatives n\n 1 < n\n x \\ totatives n\n y \\ totatives n\n [x * y = 1] (mod n)\n\ngoal (1 subgoal):\n 1. y = n - 1"} {"_id": "502110", "text": "proof (state)\nthis:\n c' = csp + c\n ?s \\# csp \\\n \\p u v.\n ?s = [entry fg p] \\\n (u, Spawn p, v) \\ edges fg \\ initialproc fg p\n\ngoal (1 subgoal):\n 1. \\csp.\n \\c' = csp + c;\n \\s.\n s \\# csp \\\n \\p u v.\n s = [entry fg p] \\\n (u, Spawn p, v) \\ edges fg \\\n initialproc fg p\\\n \\ P"} {"_id": "502111", "text": "proof (prove)\ngoal (1 subgoal):\n 1. HInv4b s' q"} {"_id": "502112", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\s. s \\ terminates_on g) \\ terminates g"} {"_id": "502113", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (case bezout (A $ from_nat i $ from_nat i)\n (A $ from_nat j $ from_nat j) of\n (p, q, u, v, d) \\ p,\n case bezout (A $ from_nat i $ from_nat i)\n (A $ from_nat j $ from_nat j) of\n (p, q, u, v, d) \\ q,\n case bezout (A $ from_nat i $ from_nat i)\n (A $ from_nat j $ from_nat j) of\n (p, q, u, v, d) \\ u,\n case bezout (A $ from_nat i $ from_nat i)\n (A $ from_nat j $ from_nat j) of\n (p, q, u, v, d) \\ v,\n case bezout (A $ from_nat i $ from_nat i)\n (A $ from_nat j $ from_nat j) of\n (p, q, u, v, d) \\ d) =\n bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j)"} {"_id": "502114", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Bot < x \\ \\z. x = Value z"} {"_id": "502115", "text": "proof (prove)\ngoal (1 subgoal):\n 1. root n (real_of_rat (((l' + r') / 2) ^ n)) =\n root n (real_of_rat ((l' + r') / 2) ^ n)"} {"_id": "502116", "text": "proof (prove)\nusing this:\n Abs_fps (\\n. of_nat n ^ k) = 1 / (1 - fps_X)\n\ngoal (1 subgoal):\n 1. Abs_fps (\\n. of_nat n ^ k) =\n fps_of_poly (fps_monom_poly (1::'a) k) / (1 - fps_X) ^ (k + 1)"} {"_id": "502117", "text": "proof (prove)\nusing this:\n A \\ Q\n Bet a P b\n Bet A Q B\n P a Le Q A\n P b Le Q B\n \\Bet ?a ?P ?b; Bet ?A ?Q ?B; ?P ?a Le ?Q ?A; ?P ?b Le ?Q ?B;\n ?B = ?Q\\\n \\ ?a ?b Le ?A ?B\n \\Bet ?a ?Po ?b; Bet ?A ?PO ?B; ?Po ?a Le ?PO ?A; ?Po ?b Le ?PO ?B;\n ?A \\ ?PO; ?B \\ ?PO\\\n \\ ?a ?b Le ?A ?B\n\ngoal (1 subgoal):\n 1. a b Le A B"} {"_id": "502118", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_poly_comm_monoid_add_hom (\\x. monom x i)"} {"_id": "502119", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a \\ 'b, card_UNIV_class)"} {"_id": "502120", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\equivalence_map f;\n \\g : trg f \\ src f\\; ide g;\n \\\\ : src f \\ g \\\n f\\;\n local.iso \\; ide f;\n \\f : src f \\ trg f\\;\n obj (src f) \\\n \\UP.unit\n (src f) : UP.map\\<^sub>0\n (src f) \\\\<^sub>S S.UP\n (src f)\\;\n obj (src f) \\ S.iso (UP.unit (src f));\n obj (src f) \\\n UP.unit (src f) \\\\<^sub>S S.\\ (UP.map\\<^sub>0 (src f)) =\n (S.UP \\[src f] \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (src f, src f)) \\\\<^sub>S\n (UP.unit (src f) \\\\<^sub>S UP.unit (src f));\n obj (src f) \\\n \\!\\.\n \\\\ : UP.map\\<^sub>0\n (src f) \\\\<^sub>S S.UP\n (src f)\\ \\\n S.iso \\ \\\n \\ \\\\<^sub>S S.\\ (UP.map\\<^sub>0 (src f)) =\n (S.UP \\[src f] \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (src f, src f)) \\\\<^sub>S\n (\\ \\\\<^sub>S \\);\n \\f a b.\n \\S.iso f;\n \\f : a \\\\<^sub>S b\\\\\n \\ \\S.inv\n f : b \\\\<^sub>S a\\\\\n \\ S.src (S.UP f) = S.trg (S.UP (src f))"} {"_id": "502121", "text": "proof (chain)\npicking this:\n homeomorphism (C \\ g -` S) S g f"} {"_id": "502122", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Var x\n \\ \\ Var y\n \\) \\ (Var x \\,\n Var x \\) =\n (Var y \\, Var x \\)"} {"_id": "502123", "text": "proof (prove)\nusing this:\n guessed_runs Rb = \\role = Resp, owner = B, partner = A\\\n guessed_frame Rb xpkE = Some KE\n KE = epubKF (Ra $ kE)\n guessed_runs Ra = \\role = Init, owner = A, partner = B\\\n\ngoal (1 subgoal):\n 1. Aenc (NonceF (Rb $ sk)) KE \\ synth (analz generators)"} {"_id": "502124", "text": "proof (prove)\ngoal (1 subgoal):\n 1. q = p"} {"_id": "502125", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gurun A w r x"} {"_id": "502126", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\k\\d. norm (integral k f)) =\n (\\k\\d. norm (set_lebesgue_integral lebesgue k f))"} {"_id": "502127", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a \\ lists A \\ b \\ lists A"} {"_id": "502128", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a b.\n \\a \\ set (x # xs); b \\ set (x # xs)\\\n \\ (f a \\\\ f) b =\n (f b \\\\ f) a) \\\n fold f (x # xs) = (f x \\\\\\ fold) f xs"} {"_id": "502129", "text": "proof (prove)\nusing this:\n a = a' * gcd a b\n a' dvd c\n\ngoal (1 subgoal):\n 1. a = gcd a b * a' \\ gcd a b dvd b \\ a' dvd c"} {"_id": "502130", "text": "proof (prove)\nusing this:\n (\\x.\n \\k\\n2. a2 k * sin (real k * x) + b2 k * cos (real k * x)) =\n (\\x.\n \\k\\n2. a2 k * sin (real k * x) + b2 k * cos (real k * x))\n\ngoal (1 subgoal):\n 1. trigpoly\n (\\x.\n (\\k\\n1.\n a1 k * sin (real k * x) + b1 k * cos (real k * x)) *\n (\\k\\n2.\n a2 k * sin (real k * x) + b2 k * cos (real k * x)))"} {"_id": "502131", "text": "proof (prove)\nusing this:\n supp \\ - {atom i} \\ atom ` Vs\n\ngoal (1 subgoal):\n 1. subst i' (Var sn')\n (ssubst \\\\\\(insert i Vs) (insert i Vs) Fi) =\n ssubst \\\\(i::=Var sn)\\V' V' F'"} {"_id": "502132", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mpt M Tinv \\ finite_measure M"} {"_id": "502133", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\s.\n \\p q.\n p \\ q = Some s \\ sep_true p \\ sep_true q) =\n sep_true"} {"_id": "502134", "text": "proof (prove)\nusing this:\n finite (A \\ {xs. set xs \\ A \\ length xs = n})\n\ngoal (1 subgoal):\n 1. finite A"} {"_id": "502135", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\as = as' @ [a']; valid_node m'; valid_call_list cs' m';\n \\i get_return_edges (cs' ! i);\n valid_return_list rs' m'; length rs' = length cs';\n ms' = targetnodes rs'; upd_cs cs (as' @ [a']) = cs'\\\n \\ valid_node m' \\\n valid_call_list cs' m' \\\n (\\i get_return_edges (cs' ! i)) \\\n valid_return_list rs' m' \\\n length rs' = length cs' \\\n ms' = targetnodes rs' \\ upd_cs cs as = cs'"} {"_id": "502136", "text": "proof (prove)\nusing this:\n ?x \\# image_mset fst\n (SkewBinomialHeapStruc.queue_to_multiset\n (map bsmapt qy)) \\\n ay \\ eprio ?x\n ?x \\# image_mset fst\n (SkewBinomialHeapStruc.queue_to_multiset\n (map bsmapt qy)) \\\n elem_invar ?x\n ?y \\# image_mset fst\n (SkewBinomialHeapStruc.queue_to_multiset\n (map bsmapt q)) \\\n eprio (findMin q) \\ eprio ?y\n ?y \\# image_mset fst\n (SkewBinomialHeapStruc.queue_to_multiset\n (map bsmapt q)) \\\n elem_invar ?y\n\ngoal (1 subgoal):\n 1. elem_invar (deleteMin' (Element e a q))"} {"_id": "502137", "text": "proof (prove)\nusing this:\n ((S.UP g \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (f, g)) \\\\<^sub>S\n (S.inv (S.cmp\\<^sub>U\\<^sub>P (g, f)) \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.UP g)) \\\\<^sub>S\n S.inv (S.cmp\\<^sub>U\\<^sub>P (trg g, g)) =\n ((S.UP g \\\\<^sub>S UP.unit (src g)) \\\\<^sub>S\n (S.inv (UP.unit (trg g)) \\\\<^sub>S S.UP g)) \\\\<^sub>S\n S.inv (S.cmp\\<^sub>U\\<^sub>P (trg g, g))\n S.epi (S.inv (S.cmp\\<^sub>U\\<^sub>P (trg g, g)))\n S.seq\n ((S.UP g \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (f, g)) \\\\<^sub>S\n (S.inv (S.cmp\\<^sub>U\\<^sub>P (g, f)) \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.UP g))\n (S.inv (S.cmp\\<^sub>U\\<^sub>P (trg g, g)))\n S.seq\n ((S.UP g \\\\<^sub>S UP.unit (src g)) \\\\<^sub>S\n (S.inv (UP.unit (trg g)) \\\\<^sub>S S.UP g))\n (S.inv (S.cmp\\<^sub>U\\<^sub>P (trg g, g)))\n\ngoal (1 subgoal):\n 1. (S.UP g \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.cmp\\<^sub>U\\<^sub>P (f, g)) \\\\<^sub>S\n (S.inv (S.cmp\\<^sub>U\\<^sub>P (g, f)) \\\\<^sub>S\n S.UP \\ \\\\<^sub>S\n S.UP g) =\n (S.UP g \\\\<^sub>S UP.unit (src g)) \\\\<^sub>S\n (S.inv (UP.unit (trg g)) \\\\<^sub>S S.UP g)"} {"_id": "502138", "text": "proof (state)\nthis:\n U <#> V <#> inv\\<^bsub>G Mod N\\<^esub> U =\n N #> g <#> (N #> h) <#> (N #> inv g)\n\ngoal (1 subgoal):\n 1. U <#> V <#> inv\\<^bsub>G Mod N\\<^esub> U =\n N #> g \\ h \\ inv g"} {"_id": "502139", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\n g v v'.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n n \\ set (\\n g); v \\ allDefs g n;\n v' \\ allDefs g n; v \\ v'\\\n \\ var g v' \\ var g v\n 2. \\g.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g.\n Mapping.keys (uses g) \\ set (\\n g)\\\n \\ Mapping.keys (uses g) \\ set (\\n g)"} {"_id": "502140", "text": "proof (prove)\nusing this:\n s \\ directed_towards S2 (SIGMA x:S\\<^sub>r. set_pmf (ct x))\n s \\ S\\<^sub>r\n \\simple ct t \\ valid_cfg; v (simple ct t) \\ 0\\\n \\ state (simple ct t) \\ S\\<^sub>r \\ S2\n \\s \\ S; t \\ set_pmf (ct s)\\\n \\ t \\ S\n cfg = simple ct t\n t \\ set_pmf (ct s)\n s \\ S\n\ngoal (1 subgoal):\n 1. (\\s.\n cfg = simple ct s \\\n s \\ directed_towards S2\n (SIGMA x:S\\<^sub>r. set_pmf (ct x)) \\\n s \\ S\\<^sub>r) \\\n v cfg = 0"} {"_id": "502141", "text": "proof (prove)\ngoal (1 subgoal):\n 1. infinite (range \\)"} {"_id": "502142", "text": "proof (prove)\nusing this:\n t1 = N0\n t \\ U (Suc h)\n split_min t = Some (a, t')\n split_min t1 = None\n\ngoal (1 subgoal):\n 1. t' \\ Um (Suc h)"} {"_id": "502143", "text": "proof (prove)\nusing this:\n invar s_\n\ngoal (1 subgoal):\n 1. Tree2.set_tree (Interval_Tree.insert x_ s_) =\n Tree2.set_tree s_ \\ {x_}"} {"_id": "502144", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. f x ^ n \\ g x ^ n"} {"_id": "502145", "text": "proof (prove)\nusing this:\n II1 \\ II2 \\ {}\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502146", "text": "proof (state)\nthis:\n Polynomial.smult a (\\a\\as. [:- a, 1::'a:]) =\n Polynomial.smult a (\\a\\bs. [:- a, 1::'a:])\n\ngoal (1 subgoal):\n 1. mset as = mset bs"} {"_id": "502147", "text": "proof (prove)\nusing this:\n \\0 < s; s \\ 1\\\n \\ \\c>0.\n \\x\\1.\n \\sum_upto (\\x. real x powr - s) x -\n (x powr (1 - s) / (1 - s) +\n Re (zeta (complex_of_real s)))\\\n \\ c * x powr - s\n 1 < s\n \\ \\ Re (zeta (complex_of_real s))\n\ngoal (1 subgoal):\n 1. (\\c.\n \\0 < c;\n \\x\\1.\n \\sum_upto (\\x. real x powr - s) x -\n (x powr (1 - s) / (1 - s) + \\)\\\n \\ c * x powr - s\\\n \\ thesis) \\\n thesis"} {"_id": "502148", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (lookup t k = Some v) = ((k, v) \\ set (entries t))"} {"_id": "502149", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (RETURN \\\\\\ f,\n \\x y z.\n ASSERT (P x y z) \\ (\\_. RETURN (f x y z)))\n \\ Id \\\n Id \\ Id \\ \\Id\\nres_rel"} {"_id": "502150", "text": "proof (prove)\nusing this:\n trg\\<^sub>C g = src\\<^sub>C f\n src\\<^sub>C g = trg\\<^sub>C f\n \\D.arr ?f; D.dom ?f = ?a\\\n \\ ?f \\\\<^sub>D ?a = ?f\n \\D.arr ?f; D.cod ?f = ?b\\\n \\ ?b \\\\<^sub>D ?f = ?f\n\ngoal (1 subgoal):\n 1. F ((f \\\\<^sub>C g) \\\\<^sub>C f) \\\\<^sub>D\n F \\\\<^sub>C\\<^sup>-\\<^sup>1[f, g, f] \\\\<^sub>D\n F (f \\\\<^sub>C g \\\\<^sub>C f) =\n F \\\\<^sub>C\\<^sup>-\\<^sup>1[f, g, f]"} {"_id": "502151", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ topspace X \\\n \\j.\n j \\ I \\\n x \\ T j \\\n f (SOME i. i \\ I \\ x \\ T i) x = f j x"} {"_id": "502152", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\atom\n l \\ ((s, k, x, y, x', y', v, i), sl, sl', m, n, sm,\n sm', sn, sn');\n atom\n sl \\ ((s, k, x, y, x', y', v, i), sl', m, n, sm, sm', sn, sn');\n atom sl' \\ ((s, k, x, y, x', y', v, i), m, n, sm, sm', sn, sn');\n atom m \\ ((s, k, x, y, x', y', v, i), n, sm, sm', sn, sn');\n atom n \\ ((s, k, x, y, x', y', v, i), sm, sm', sn, sn');\n atom sm \\ ((s, k, x, y, x', y', v, i), sm', sn, sn');\n atom sm' \\ ((s, k, x, y, x', y', v, i), sn, sn');\n atom sn \\ ((s, k, x, y, x', y', v, i), sn');\n atom sn' \\ (s, k, x, y, x', y', v, i);\n atom s \\ (k, x, y, x', y', v, i);\n atom k \\ (x, y, x', y', v, i)\\\n \\ {Var l EQ Zero, SubstTermP v i x x',\n SubstTermP v i y y'} \\\n SubstAtomicP v i (Q_Eq x y) (Q_Eq x' y') OR\n SyntaxN.Ex m\n (SyntaxN.Ex n\n (SyntaxN.Ex sm\n (SyntaxN.Ex sm'\n (SyntaxN.Ex sn\n (SyntaxN.Ex sn'\n (Var m IN Var l AND\n Var n IN Var l AND\n HPair (Var m)\n (HPair (Var sm) (Var sm')) IN\n Eats Zero\n (HPair Zero\n (HPair (Q_Eq x y) (Q_Eq x' y'))) AND\n HPair (Var n)\n (HPair (Var sn) (Var sn')) IN\n Eats Zero\n (HPair Zero\n (HPair (Q_Eq x y) (Q_Eq x' y'))) AND\n (Q_Eq x y EQ Q_Disj (Var sm) (Var sn) AND\n Q_Eq x' y' EQ\n Q_Disj (Var sm') (Var sn') OR\n Q_Eq x y EQ Q_Neg (Var sm) AND\n Q_Eq x' y' EQ Q_Neg (Var sm') OR\n Q_Eq x y EQ Q_Ex (Var sm) AND\n Q_Eq x' y' EQ Q_Ex (Var sm'))))))))"} {"_id": "502153", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mset_neg (abs_zmultiset (Mp, Mn)) = Mn - Mp"} {"_id": "502154", "text": "proof (prove)\nusing this:\n \\measure_pmf.random_variable M (count_space UNIV) g;\n f \\ borel_measurable (count_space UNIV)\\\n \\ integral\\<^sup>L\n (distr (measure_pmf M) (count_space UNIV) g) f =\n measure_pmf.expectation M (\\x. f (g x))\n\ngoal (1 subgoal):\n 1. integral\\<^sup>L (distr (measure_pmf M) (count_space UNIV) g) f =\n measure_pmf.expectation M (\\x. f (g x))"} {"_id": "502155", "text": "proof (prove)\ngoal (6 subgoals):\n 1. \\\\ exp c.\n \\atom ` domA \\ \\* c;\n \\c. \\(x, e)\\set \\. P c e;\n \\c. P c exp\\\n \\ P c (Let \\ exp)\n 2. \\var exp c.\n \\{atom var} \\* c; \\c. P c exp\\\n \\ P c (Lam [var]. exp)\n 3. \\b c. P c (Bool b)\n 4. \\scrut e1 e2 c.\n \\\\c. P c scrut; \\c. P c e1;\n \\c. P c e2\\\n \\ P c (scrut ? e1 : e2)\n 5. \\c. \\(x, e)\\set []. P c e\n 6. \\var exp \\ c.\n \\\\c. P c exp;\n \\c. \\(x, e)\\set \\. P c e\\\n \\ \\(x, e)\\set ((var, exp) # \\).\n P c e"} {"_id": "502156", "text": "proof (prove)\nusing this:\n g = 0\n\ngoal (1 subgoal):\n 1. f = 0 &&& c = 0 &&& fs = []"} {"_id": "502157", "text": "proof (prove)\nusing this:\n x \\ y\n card x = 1\n card y = 1\n\ngoal (1 subgoal):\n 1. x \\ y = {}"} {"_id": "502158", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distributed M S (\\x. (X x, Y x)) (\\x. ennreal (P x))"} {"_id": "502159", "text": "proof (prove)\nusing this:\n 0 < p * q\n 2 ^ (length (nat_to_bv (p * q)) - Suc 0) \\ p * q\n\ngoal (1 subgoal):\n 1. 2 ^ (length (nat_to_bv (p * q)) - Suc 0) < p * q"} {"_id": "502160", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\p.\n \\p \\ cball a r;\n \\y.\n y \\ cball a r \\\n cmod (g y) \\ cmod (g p)\\\n \\ thesis) \\\n thesis"} {"_id": "502161", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f x \\ (\\s. emeasure (M s) {x \\ space N. F top x})"} {"_id": "502162", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lower (arctan_float_interval p x) = lb_arctan p (lower x)"} {"_id": "502163", "text": "proof (prove)\nusing this:\n split_vals_e xs = (as, bs)\n\ngoal (1 subgoal):\n 1. xs = ($$* as) @ bs"} {"_id": "502164", "text": "proof (prove)\nusing this:\n U.Problem TE_wtFsym wtPsym (length \\ TE_arOf) (length \\ parOf)\n TE_\\ \\\n U.Struct TE_wtFsym wtPsym (length \\ TE_arOf) (length \\ parOf)\n D eintF eintP \\\n U.Model_axioms TE_\\ D eintF eintP\n\ngoal (1 subgoal):\n 1. M_Problem TE_wtFsym wtPsym TE_arOf TE_resOf parOf TE_\\ \\\n (M_Signature TYPE('tp) TE_wtFsym wtPsym \\\n U.Struct TE_wtFsym wtPsym (length \\ TE_arOf)\n (length \\ parOf) D eintF eintP) \\\n U.Problem TE_wtFsym wtPsym (length \\ TE_arOf)\n (length \\ parOf) TE_\\ \\\n U.Struct TE_wtFsym wtPsym (length \\ TE_arOf)\n (length \\ parOf) D eintF eintP \\\n U.Model_axioms TE_\\ D eintF eintP"} {"_id": "502165", "text": "proof (prove)\nusing this:\n k < n\n ?x < ?y \\\n max_stutter_sampler ?\\1 ?x < max_stutter_sampler ?\\1 ?y\n\ngoal (1 subgoal):\n 1. max_stutter_sampler \\ k < max_stutter_sampler \\ n"} {"_id": "502166", "text": "proof (prove)\ngoal (2 subgoals):\n 1. reflp (\\h h'. object_ptr_kinds h = object_ptr_kinds h')\n 2. transp (\\h h'. object_ptr_kinds h = object_ptr_kinds h')"} {"_id": "502167", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map f xs \\ g = xs \\ g \\ f"} {"_id": "502168", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P\\<^sup>\\ \\ II \\ (P ;; P\\<^sup>\\)"} {"_id": "502169", "text": "proof (prove)\ngoal (1 subgoal):\n 1. concat\n (map (\\n. if P (l ! n) then [n] else [])\n [0..\n []"} {"_id": "502170", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\cs.\n (\\c\\set cs.\n c \\ carrier G \\ irreducible G c) \\\n fmset G cs = Cs"} {"_id": "502171", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map f (take n xs) \\ map f xs"} {"_id": "502172", "text": "proof (prove)\nusing this:\n p = q\n prime ?p \\ 1 < ?p\n fst (p, m) ^ snd (p, m) = fst (q, n) ^ snd (q, n)\n 0 < m\n prime p\n 0 < n\n prime q\n\ngoal (1 subgoal):\n 1. m = n"} {"_id": "502173", "text": "proof (prove)\nusing this:\n Obtuse A' B C\n\ngoal (1 subgoal):\n 1. A B C LtA A' B C"} {"_id": "502174", "text": "proof (prove)\nusing this:\n sets x = sets y\n\ngoal (1 subgoal):\n 1. y \\ sup_measure' x y"} {"_id": "502175", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ range f \\ (\\n. f (Suc n))"} {"_id": "502176", "text": "proof (prove)\ngoal (1 subgoal):\n 1. locally_trans R (A - B) (C - D) (E - F)"} {"_id": "502177", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bij (rsuml \\ map_sum swap_sum id \\ lsumr)"} {"_id": "502178", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\z za.\n \\\\v.\n (v \\<^bold>\\ RPb) =\n (\\u\\b. v = u \\ a);\n \\u. (u \\<^bold>\\ Pb) = (u \\ b);\n \\v.\n (v \\<^bold>\\ RPb) =\n (\\u. u \\<^bold>\\ Pb \\ v = u \\ a);\n \\u. (u \\<^bold>\\ Pb) = (u \\ b);\n \\u.\n (u \\<^bold>\\ z) =\n (u \\ b \\ (\\ua\\b. u = ua \\ a));\n \\u.\n (u \\<^bold>\\ za) =\n (u \\<^bold>\\ Pb \\ u \\<^bold>\\ RPb)\\\n \\ \\u.\n (u \\<^bold>\\ z) =\n (u \\ b \\\n (\\ua.\n ua \\ a = u \\\n a \\<^bold>\\ ua \\ ua \\ b))"} {"_id": "502179", "text": "proof (chain)\npicking this:\n inv\\<^bsub>K\\<^esub> a \\\\<^bsub>K\\<^esub> a =\n \\\\<^bsub>K\\<^esub>"} {"_id": "502180", "text": "proof (prove)\ngoal (1 subgoal):\n 1. len m k j bs < len m k j ys"} {"_id": "502181", "text": "proof (prove)\nusing this:\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 1}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) =\n rat_of_int\n (int (card\n {x. poly p x = 0 \\\n (\\p\\set (retrieve_polys qs (fst I)).\n poly p x = 0) \\\n 0 < poly (prod_list (retrieve_polys qs (snd I))) x}))\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = - 1}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) =\n - rat_of_int\n (int (card\n {x. poly p x = 0 \\\n (\\p\\set (retrieve_polys qs (fst I)).\n poly p x = 0) \\\n poly (prod_list (retrieve_polys qs (snd I))) x < 0}))\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 0}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) =\n 0\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 1} \\\n set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = - 1}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) +\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 0}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) =\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 1}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) +\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = - 1}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s}))) +\n (\\s\\set (map (consistent_sign_vec qs)\n (characterize_root_list_p p)) \\\n {s. z_R I s = 0}.\n z_R I s *\n rat_of_int\n (int (card {x. poly p x = 0 \\ consistent_sign_vec qs x = s})))\n\ngoal (1 subgoal):\n 1. vec_of_list (mtx_row_R signs I) \\\n construct_lhs_vector_R p qs signs =\n rat_of_int\n (int (card\n {x. poly p x = 0 \\\n (\\p\\set (retrieve_polys qs (fst I)).\n poly p x = 0) \\\n 0 < poly (prod_list (retrieve_polys qs (snd I))) x})) -\n rat_of_int\n (int (card\n {x. poly p x = 0 \\\n (\\p\\set (retrieve_polys qs (fst I)).\n poly p x = 0) \\\n poly (prod_list (retrieve_polys qs (snd I))) x < 0}))"} {"_id": "502182", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\t s.\n L = Pos (Eq t s) \\ L = Neg (Eq t s) \\\n thesis) \\\n thesis"} {"_id": "502183", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (m, n) \\ Le_enat \\ m \\ n"} {"_id": "502184", "text": "proof (prove)\ngoal (1 subgoal):\n 1. B C D' CongA C B E"} {"_id": "502185", "text": "proof (prove)\nusing this:\n atom x \\ u'\n atom x \\ e\\<^sub>1\n atom x \\ e\\<^sub>1'\n \\[], Syntax.Let e\\<^sub>1 x\n e\\<^sub>2\\ I\\ \\[], u'\\\n value e\\<^sub>1 \\ False\n\ngoal (1 subgoal):\n 1. Syntax.Let e\\<^sub>1' x e\\<^sub>2 = u'"} {"_id": "502186", "text": "proof (prove)\nusing this:\n map_pmf (snd \\ snd)\n (config'_rand BIT\n (fst BIT ?s0.0 \\ (\\is. return_pmf (?s0.0, is))) ?qs) =\n return_pmf ?s0.0 \\\n map_pmf (fst \\ snd)\n (config'_rand BIT\n (fst BIT ?s0.0 \\ (\\is. return_pmf (?s0.0, is))) ?qs) =\n bv (length ?s0.0)\n\ngoal (1 subgoal):\n 1. map_pmf (fst \\ snd)\n (config'_rand BIT\n (fst BIT s0 \\ (\\is. return_pmf (s0, is))) qs) =\n bv (length s0)"} {"_id": "502187", "text": "proof (prove)\nusing this:\n \\?G1.0 \\ \\; ?G2.0 \\ \\; ?G1.0 \\ ?G2.0;\n ?x \\ ?G1.0; ?y \\ ?G2.0\\\n \\ int (\\ index {?x, ?y}) = \\\n\ngoal (1 subgoal):\n 1. \\ps \\ \\; card ps = 2;\n \\G.\n G \\ \\ \\ \\ ps \\ G\\\n \\ int (\\ index ps) = \\"} {"_id": "502188", "text": "proof (prove)\nusing this:\n At i xs \\# \\ + \\' \\\n At i xs \\# \\ + \\'\n\ngoal (1 subgoal):\n 1. ( \\ + \\' \\* \\ + \\', 0)\n \\ derivable R*"} {"_id": "502189", "text": "proof (prove)\nusing this:\n \\ \\ M\\\\*p \\\n xvec p \\\n N\\.p \\\n P \\ K\\(p \\\nN)[(p \\\n xvec)::=Tvec]\\ \\ (p \\ P)[(p \\ xvec)::=Tvec]\n\ngoal (1 subgoal):\n 1. \\ \\ M\\\\*xvec N\\.P \\ K\\N[xvec::=Tvec]\\ \\ P[xvec::=Tvec]"} {"_id": "502190", "text": "proof (prove)\nusing this:\n yun_rel ?F ?c ?f \\ yun_rel (?F ^ ?n) (?c ^ ?n) (?f ^ ?n)\n\ngoal (1 subgoal):\n 1. \\c. yun_rel F c f \\\n \\c. yun_rel (F ^ n) c (f ^ n)"} {"_id": "502191", "text": "proof (prove)\nusing this:\n ?e \\ E \\\n fst (trace_core' core1 p tr) ?e = fst (trace_core' core2 q tr) ?e\n ?ia \\ IA \\\n fst (snd (trace_core' core1 p tr)) ?ia =\n fst (snd (trace_core' core2 q tr)) ?ia\n ?iu \\ IU \\\n snd (snd (trace_core' core1 p tr)) ?iu =\n snd (snd (trace_core' core2 q tr)) ?iu\n p' = trace_core_aux core1 p tr\n q' = trace_core_aux core2 q tr\n\ngoal (1 subgoal):\n 1. (e \\ E \\\n weight_spmf (p' \\ (\\s. cpoke core1 s e)) =\n weight_spmf (q' \\ (\\s. cpoke core2 s e))) &&&\n (ia \\ IA \\\n p' \\ (\\s1. map_spmf fst (cfunc_adv core1 s1 ia)) =\n q' \\ (\\s2. map_spmf fst (cfunc_adv core2 s2 ia))) &&&\n (iu \\ IU \\\n p' \\ (\\s1. map_spmf fst (cfunc_usr core1 s1 iu)) =\n q' \\ (\\s2. map_spmf fst (cfunc_usr core2 s2 iu)))"} {"_id": "502192", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P Q OS A C"} {"_id": "502193", "text": "proof (prove)\nusing this:\n steps0 (Suc 0, l, r) (tp @ shift (mopup n) off) stp =\n (Suc off, Bk # Bk # ires, @ Bk \\ k)\n steps0 (Suc 0 + off, Bk # Bk # ires, @ Bk \\ k)\n (tp @ shift (mopup n) off) stpb =\n (0, Bk \\ i @ Bk # Bk # ires, Oc # Oc \\ rs @ Bk \\ j)\n\ngoal (1 subgoal):\n 1. \\stp i j.\n steps0 (Suc 0, l, r) (tp @ shift (mopup n) off) stp =\n (0, Bk \\ i @ Bk # Bk # ires, Oc \\ Suc rs @ Bk \\ j)"} {"_id": "502194", "text": "proof (state)\nthis:\n length x = DIM('h)\n\ngoal (1 subgoal):\n 1. \\aa a'a x y z.\n \\(aa, a'a) \\ \\lv_rel\\sctn_rel;\n (x, y) \\ appr_rell;\n (y, z) \\ \\lv_rel\\set_rel\\\n \\ ((nres_of o2 inter_appr_plane ops optns) x aa,\n inter_sctn_spec z a'a)\n \\ \\appr_rell O\n \\lv_rel\\set_rel\\nres_rel"} {"_id": "502195", "text": "proof (prove)\nusing this:\n u < Suc u\n sgn1 (sf (u + 1) - u) \\ 0\n\ngoal (1 subgoal):\n 1. b_least2 (\\u z. sgn1 (sf (z + 1) - u)) u (Suc u) =\n (LEAST z. sgn1 (sf (z + 1) - u) \\ 0)"} {"_id": "502196", "text": "proof (prove)\nusing this:\n eventually ((<) 0) sequentially\n \\0 < ?n; ?z \\ \\\\<^sub>\\\\<^sub>0\\\n \\ exp (ln_Gamma_series ?z ?n) = Gamma_series ?z ?n\n z \\ \\\\<^sub>\\\\<^sub>0\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially.\n exp (- ln_Gamma_series z n) = rGamma_series z n"} {"_id": "502197", "text": "proof (prove)\nusing this:\n y \\ \\ I\n\ngoal (1 subgoal):\n 1. \\S\\I. y \\ S"} {"_id": "502198", "text": "proof (prove)\ngoal (1 subgoal):\n 1. row A i \\ col C j \\ row B i \\ col C j"} {"_id": "502199", "text": "proof (prove)\nusing this:\n M (rinit rest1) (rinit rest2)\n (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S))\n (rfunc_adv rest1) (rfunc_adv rest2)\n (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S))\n (rfunc_usr rest1) (rfunc_usr rest2)\n\ngoal (1 subgoal):\n 1. rel_rest' S E IA IU OA OU M rest1 rest2"} {"_id": "502200", "text": "proof (prove)\nusing this:\n finite A\n ?i \\ A \\ convex (B ?i)\n\ngoal (1 subgoal):\n 1. convex (sum B A)"} {"_id": "502201", "text": "proof (prove)\ngoal (7 subgoals):\n 1. \\\\.\n is_nnf_mset \\ \\ is_disj (disj_of_clause R')\n 2. \\\\.\n is_nnf_mset \\ \\\n is_nnf \\<^bold>\\map disj_of_clause S'\n 3. \\\\ x xa.\n \\is_nnf_mset \\; x \\ cnf (disj_of_clause R');\n x \\ {}; xa \\ x\\\n \\ xa \\ R\n 4. \\\\ x xa.\n \\is_nnf_mset \\; x \\ cnf (disj_of_clause R');\n x \\ {}; xa \\ R\\\n \\ xa \\ x\n 5. \\\\ x.\n is_nnf_mset \\ \\\n R \\ cnf (disj_of_clause R')\n 6. \\\\ x.\n \\is_nnf_mset \\;\n x \\ cnf \\<^bold>\\map disj_of_clause S'\\\n \\ x \\ S\n 7. \\\\.\n \\is_nnf_mset \\;\n disj_of_clause\n R', \\<^bold>\\map disj_of_clause\n S', \\ \\\\<^sub>n\\\n \\ \\<^bold>\\map disj_of_clause\n S', \\ \\\\<^sub>n"} {"_id": "502202", "text": "proof (prove)\nusing this:\n Tree2.set_tree (inter_rbt (list_to_rbt ?xs2) (list_to_rbt ?xs1)) =\n Tree2.set_tree (list_to_rbt ?xs2) \\\n Tree2.set_tree (list_to_rbt ?xs1)\n\ngoal (1 subgoal):\n 1. dimacs_model ls cs =\n (let tls = list_to_rbt ls\n in (\\c\\set cs.\n size (inter_rbt tls (list_to_rbt c)) \\ 0) \\\n distinct (map dimacs_lit_to_var ls))"} {"_id": "502203", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa xb xc xd.\n \\heap_is_wellformed h; type_wf h; known_ptrs h;\n h \\ get_owner_document ptr \\\\<^sub>r x;\n h \\ get_disconnected_nodes x \\\\<^sub>r xa;\n h \\ to_tree_order\n (cast\\<^sub>d\\<^sub>o\\<^sub>c\\<^sub>u\\<^sub>m\\<^sub>e\\<^sub>n\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n x)\n \\\\<^sub>r xb;\n h \\ map_M\n (to_tree_order \\\n cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r)\n xa\n \\\\<^sub>r xc;\n sc = xb @ concat xc; xd \\ set xc; ptr' \\ set xd\\\n \\ ptr' |\\| object_ptr_kinds h"} {"_id": "502204", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x.\n \\finite A; card A = Suc 0; A \\ {}; x \\ A\\\n \\ \\f.\n f \\ {0} \\ A \\\n surj_to f {0} A\n 2. \\finite A; card A = Suc 0; A \\ {}\\\n \\ A \\ {}"} {"_id": "502205", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (((f \\ g \\ h) \\\n arrow_of_spans_of_maps.the_\\ (\\) (\\)\n (tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0)\n ((tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0)\n (arrow_of_tabulations_in_maps.chine (\\) (\\) \\ \\ src\n trg\n (\\\\<^sup>-\\<^sup>1[f \\\n g, h, tab\\<^sub>0 h \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>0] \\\n ((f \\ g) \\\n \\[h, tab\\<^sub>0 h, fg\\<^sub>0h\\<^sub>1.p\\<^sub>0]) \\\n ((f \\ g) \\\n h.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n ((f \\ g) \\ fg\\<^sub>0h\\<^sub>1.\\) \\\n \\[f \\\n g, tab\\<^sub>0 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n (Tfg.\\\\.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>1))\n (tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0)\n (f \\ g \\ h) TfTgh.tab\n ((tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0)\n (tab\\<^sub>1 f \\ f\\<^sub>0gh\\<^sub>1.p\\<^sub>1)\n \\[f, g, h])) \\\n \\[f \\\n g \\\n h, (tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0, arrow_of_tabulations_in_maps.chine\n (\\) (\\) \\ \\ src trg\n (\\\\<^sup>-\\<^sup>1[f \\\n g, h, tab\\<^sub>0 h \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>0] \\\n ((f \\ g) \\\n \\[h, tab\\<^sub>0 h, fg\\<^sub>0h\\<^sub>1.p\\<^sub>0]) \\\n ((f \\ g) \\\n h.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n ((f \\ g) \\ fg\\<^sub>0h\\<^sub>1.\\) \\\n \\[f \\\n g, tab\\<^sub>0 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n (Tfg.\\\\.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>1))\n (tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0)\n (f \\ g \\ h) TfTgh.tab\n ((tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0)\n (tab\\<^sub>1 f \\ f\\<^sub>0gh\\<^sub>1.p\\<^sub>1)\n \\[f, g, h]] \\\n (TfTgh.tab \\\n arrow_of_tabulations_in_maps.chine (\\) (\\) \\ \\ src\n trg\n (\\\\<^sup>-\\<^sup>1[f \\\n g, h, tab\\<^sub>0 h \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>0] \\\n ((f \\ g) \\\n \\[h, tab\\<^sub>0 h, fg\\<^sub>0h\\<^sub>1.p\\<^sub>0]) \\\n ((f \\ g) \\\n h.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n ((f \\ g) \\ fg\\<^sub>0h\\<^sub>1.\\) \\\n \\[f \\\n g, tab\\<^sub>0 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n (Tfg.\\\\.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>1))\n (tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0)\n (f \\ g \\ h) TfTgh.tab\n ((tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0)\n (tab\\<^sub>1 f \\ f\\<^sub>0gh\\<^sub>1.p\\<^sub>1)\n \\[f, g, h])) \\\n arrow_of_spans_of_maps.the_\\ (\\) (\\)\n ((tab\\<^sub>1 f \\ f\\<^sub>0g\\<^sub>1.p\\<^sub>1) \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1)\n (tab\\<^sub>1 f \\ f\\<^sub>0gh\\<^sub>1.p\\<^sub>1)\n (arrow_of_tabulations_in_maps.chine (\\) (\\) \\ \\ src\n trg\n (\\\\<^sup>-\\<^sup>1[f \\\n g, h, tab\\<^sub>0 h \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>0] \\\n ((f \\ g) \\\n \\[h, tab\\<^sub>0 h, fg\\<^sub>0h\\<^sub>1.p\\<^sub>0]) \\\n ((f \\ g) \\\n h.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n ((f \\ g) \\ fg\\<^sub>0h\\<^sub>1.\\) \\\n \\[f \\\n g, tab\\<^sub>0 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n (Tfg.\\\\.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>1))\n (tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0)\n (f \\ g \\ h) TfTgh.tab\n ((tab\\<^sub>0 h \\ g\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n f\\<^sub>0gh\\<^sub>1.p\\<^sub>0)\n (tab\\<^sub>1 f \\ f\\<^sub>0gh\\<^sub>1.p\\<^sub>1)\n \\[f, g, h]) =\n (\\[f, g, h] \\\n tab\\<^sub>0 h \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n \\\\<^sup>-\\<^sup>1[f \\\n g, h, tab\\<^sub>0 h \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>0] \\\n ((f \\ g) \\\n \\[h, tab\\<^sub>0 h, fg\\<^sub>0h\\<^sub>1.p\\<^sub>0]) \\\n ((f \\ g) \\\n h.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>0) \\\n ((f \\ g) \\ fg\\<^sub>0h\\<^sub>1.\\) \\\n \\[f \\\n g, tab\\<^sub>0 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n (Tfg.\\\\.tab \\ fg\\<^sub>0h\\<^sub>1.p\\<^sub>1)"} {"_id": "502206", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i : opaodv\n i \\\\\\<^bsub>i\\<^esub> qmsg : R\\\\<^sub>o \\\\<^sub>A (\\\\\n _. oarrivemsg\n (\\\\ m.\n msg_fresh \\ m \\ msg_zhops m)\n \\,\n other quality_increases {i} \\)\n globala\n (\\(\\, a, \\').\n a \\ \\ \\\n (\\d.\n a \\\n i:deliver(d)) \\\n \\ i = \\' i)"} {"_id": "502207", "text": "proof (prove)\nusing this:\n \\ \\ App t1 t2 : T\n\ngoal (1 subgoal):\n 1. \\T'.\n \\ \\ t1 : T' \\ T \\\n \\ \\ t2 : T'"} {"_id": "502208", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coreflexive\n ((f \\ - q) * (f \\ - q)\\<^sup>T \\\n (f \\ - q) * (f \\ q) \\\n (f \\ - q) * e\\<^sup>T \\\n (f \\ q)\\<^sup>T * (f \\ - q)\\<^sup>T \\\n (f \\ q)\\<^sup>T * (f \\ q) \\\n (f \\ q)\\<^sup>T * e\\<^sup>T \\\n e * (f \\ - q)\\<^sup>T \\\n e * (f \\ q) \\\n e * e\\<^sup>T)"} {"_id": "502209", "text": "proof (state)\nthis:\n poss AA \\# map_clause f ?C \\\n \\AA0. poss AA0 \\# ?C \\ AA = image_mset f AA0\n poss (add_mset A AA) \\# map_clause f C\n\ngoal (2 subgoals):\n 1. \\C.\n poss {#} \\# map_clause f C \\\n \\AA0. poss AA0 \\# C \\ {#} = image_mset f AA0\n 2. \\x AA C.\n \\\\C.\n poss AA \\# map_clause f C \\\n \\AA0.\n poss AA0 \\# C \\ AA = image_mset f AA0;\n poss (add_mset x AA) \\# map_clause f C\\\n \\ \\AA0.\n poss AA0 \\# C \\\n add_mset x AA = image_mset f AA0"} {"_id": "502210", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x y.\n x - y = Abs_bit1 ((Rep_bit1 x - Rep_bit1 y) mod int CARD('a bit1))\n 2. \\x. - x = Abs_bit1 (- Rep_bit1 x mod int CARD('a bit1))"} {"_id": "502211", "text": "proof (prove)\ngoal (1 subgoal):\n 1. partial_state.encode1 a.dims0 a.vars1'\n (partial_state.encode1 b.dims0 b.vars1' i) =\n partial_state.encode1 d.dims0 d.vars1' i"} {"_id": "502212", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (SUP F\\{F. finite F \\ F \\ A}. ereal (sum f F)) =\n ereal (Sup (sum f ` {F. finite F \\ F \\ A}))"} {"_id": "502213", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\r = get h a ! i; h' = update a i x h;\n i < Array.length h a\\\n \\ thesis) \\\n thesis"} {"_id": "502214", "text": "proof (state)\nthis:\n ?u \\ N2 \\ (y, ?u) \\ (edges T2)\\<^sup>*\n\ngoal (1 subgoal):\n 1. (\\T1 T2.\n \\tree T1; tree T2; nodes T1 \\ nodes T2 = {};\n nodes T = nodes T1 \\ nodes T2;\n edges T1 \\ edges T2 = E';\n nodes T1 = {u. (x, u) \\ E'\\<^sup>*};\n nodes T2 = {u. (y, u) \\ E'\\<^sup>*}; x \\ nodes T1;\n y \\ nodes T2\\\n \\ thesis) \\\n thesis"} {"_id": "502215", "text": "proof (prove)\ngoal (1 subgoal):\n 1. maintained\n (transl_rule (A_Cmp (A_Lbl idt) (A_Lbl idt) \\ A_Lbl idt))\n G &&&\n maintained (transl_rule (A_Cnv (A_Lbl idt) \\ A_Lbl idt))\n G &&&\n maintained\n (transl_rule\n (A_Cmp (A_Cmp (A_Lbl idt) (A_Lbl l)) (A_Lbl idt) \\\n A_Lbl l))\n G"} {"_id": "502216", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n.\n is_final\n (steps0 (Suc 0, [], ) (t_wcode_prepare |+| t_wcode_main)\n n) \\\n (\\(l, r).\n l = Bk # Oc # Oc \\ m \\\n (\\ln rn.\n r =\n Bk #\n Oc #\n Bk \\ ln @\n Bk #\n Bk #\n Oc \\ bl_bin () @\n Bk \\ rn)) holds_for steps0 (Suc 0, [], )\n (t_wcode_prepare |+| t_wcode_main) n"} {"_id": "502217", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\l.\n \\\\n. (\\i<2 * n. (- 1) ^ i * a i) \\ l;\n (\\n. \\i<2 * n. (- 1) ^ i * a i)\n \\ l;\n \\n. l \\ (\\i<2 * n + 1. (- 1) ^ i * a i);\n (\\n. \\i<2 * n + 1. (- 1) ^ i * a i)\n \\ l\\\n \\ thesis) \\\n thesis"} {"_id": "502218", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rho &[y1 // y]_ys &[z1 // z]_zs =\n rho &[z1 // z]_zs &[y1 @ys[z1 / z]_zs // y]_ys"} {"_id": "502219", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cmod (to_complex (moebius_pt (moebius_rotation \\) u)) =\n cmod (to_complex u)"} {"_id": "502220", "text": "proof (prove)\nusing this:\n Unit \\\\ M'\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502221", "text": "proof (prove)\ngoal (1 subgoal):\n 1. normalization_euclidean_semiring_class.Gcd = Gcd"} {"_id": "502222", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ generat.\n (case generat of Pure x \\ f x\n | IO out c \\\n \\r\\responses_\\ \\ out.\n expectation_gpv' (c r))\n \\measure_spmf (the_gpv gpv) +\n fail * ennreal (pmf (the_gpv gpv) None)\n \\ expectation_gpv fail \\' f gpv"} {"_id": "502223", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i\\degree (map_poly hom p).\n monom (monom (coeff (map_poly hom p) i) i) i) =\n map_poly (map_poly hom)\n (\\i\\degree p. monom (monom (coeff p i) i) i)"} {"_id": "502224", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n. inf (F n x) (G n x))\n \\ (if x \\ S then inf (f x) (g x)\n else (0::'b))"} {"_id": "502225", "text": "proof (prove)\nusing this:\n u \\ timpl_closure s TI\n u \\ timpl_closure t TI\n set TI' = {(a, b). (a, b) \\ TI\\<^sup>+ \\ a \\ b}\n\ngoal (1 subgoal):\n 1. equal_mod_timpls TI' s t"} {"_id": "502226", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prv (imp (eql t1 t2) (eql (suc t1) (suc t2)))"} {"_id": "502227", "text": "proof (prove)\nusing this:\n arr f\n X \\<^bold>\\ ide_to_hf (local.dom f)\n \\X, Y\\ \\<^bold>\\ arr_to_hfun f\n arr_to_hfun ?f =\n \\XY \\<^bold>\\ ide_to_hf (local.dom ?f) * ide_to_hf (cod ?f).\n hsnd XY = DOWN (Fun ?f (UP (hfst XY)))\\\n\ngoal (1 subgoal):\n 1. Y = DOWN (Fun f (UP X))"} {"_id": "502228", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0..i = 0.. * adjoint M1))"} {"_id": "502231", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\D. (f has_derivative D) (at x within s);\n \\D. (g has_derivative D) (at x within s)\\\n \\ \\D.\n ((\\x. cinner (f x) (g x)) has_derivative D)\n (at x within s)"} {"_id": "502232", "text": "proof (prove)\nusing this:\n (0::'c) =\n (\\v\\f ` {v. r v \\ (0::'a)}. r (the_inv_into B f v) *b v)\n\ngoal (1 subgoal):\n 1. r v \\ (0::'a) \\\n r (the_inv_into B f (f v)) = (0::'a)"} {"_id": "502233", "text": "proof (prove)\nusing this:\n (\\(x, y). x \\ y)\n \\ hom (subgroup_generated G A \\\\ subgroup_generated G B)\n G\n\ngoal (1 subgoal):\n 1. \\x\\A. \\y\\B. x \\ y = y \\ x"} {"_id": "502234", "text": "proof (prove)\nusing this:\n \\ \\ e1 : t1\n case_nat t1 \\ \\ e2 : t2\n \\randomfree e1; free_vars e1 \\ ?V\\\n \\ (\\\\. expr_sem_rf \\ e1)\n \\ state_measure ?V \\ \\\\<^sub>M\n stock_measure t1\n \\randomfree e2; free_vars e2 \\ ?V\\\n \\ (\\\\. expr_sem_rf \\ e2)\n \\ state_measure ?V\n (\\a.\n case a of 0 \\ t1\n | Suc x \\\n \\ x) \\\\<^sub>M\n stock_measure t2\n randomfree (LET e1 IN e2)\n free_vars (LET e1 IN e2) \\ V\n Suc -` ?A \\ ?V \\ ?A \\ shift_var_set ?V\n\ngoal (1 subgoal):\n 1. (\\\\. expr_sem_rf \\ e1)\n \\ state_measure V \\ \\\\<^sub>M\n stock_measure t1 &&&\n (\\\\. expr_sem_rf \\ e2)\n \\ state_measure (shift_var_set V)\n (case_nat t1 \\) \\\\<^sub>M\n stock_measure t2"} {"_id": "502235", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite V; n = 0 \\ (\\v\\V. v < n)\\\n \\ local.wf n\n (INTERSECT (sorted_list_of_set (valid_ENC n ` V)))"} {"_id": "502236", "text": "proof (prove)\nusing this:\n (c::'a::refinement_lattice)\\<^sup>\\ \\\n c ; c\\<^sup>\\ ; c\\<^sup>\\ =\n (nil \\ c ; c\\<^sup>\\) ; c\\<^sup>\\\n nil \\\n (c::'a::refinement_lattice) ; c\\<^sup>\\ ;\n c\\<^sup>\\ \\\n nil \\ c ; c\\<^sup>\\ \\\n c ; c\\<^sup>\\ ; c\\<^sup>\\\n nil \\\n (?c::'a::refinement_lattice) ; (?x::'a::refinement_lattice) \\\n ?x \\\n ?c\\<^sup>\\ \\ ?x\n (?c::'a::refinement_lattice)\\<^sup>\\ =\n nil \\ ?c ; ?c\\<^sup>\\\n (?a::'a::refinement_lattice) ; (?b::'a::refinement_lattice) ;\n (?c::'a::refinement_lattice) =\n ?a ; (?b ; ?c)\n\ngoal (1 subgoal):\n 1. c\\<^sup>\\ \\ c\\<^sup>\\ ; c\\<^sup>\\\nvariables:\n c, nil :: 'a\n (;) :: 'a \\ 'a \\ 'a\ntype variables:\n 'a :: refinement_lattice"} {"_id": "502237", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\F.\n \\bij h; F \\ increasing (func \\ inv h);\n \\G. F = rename h G\\\n \\ F \\ rename h ` increasing func\n 2. \\F.\n \\bij h; F \\ increasing (func \\ inv h)\\\n \\ \\G. F = rename h G\n 3. \\xa.\n \\bij h; xa \\ increasing func\\\n \\ rename h xa \\ increasing (func \\ inv h)"} {"_id": "502238", "text": "proof (prove)\nusing this:\n 0 < x\n\ngoal (1 subgoal):\n 1. deriv (stirling_sum' (Suc j) m) x = stirling_sum' (Suc (Suc j)) m x"} {"_id": "502239", "text": "proof (prove)\nusing this:\n nat \\x\\ < m\n 0 < x\n\ngoal (1 subgoal):\n 1. cmod (\\k\\real -` {x<..real m}. \\ k * of_nat k powr - s)\n \\ real (totient n) * (2 + cmod s / \\) / x powr \\"} {"_id": "502240", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(set xs, X)\n \\ \\\\A\\default_rel\n d\\set_rel;\n those xs = Some z_\\\n \\ (concat z_, op_Union_default $ X) \\ A"} {"_id": "502241", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z.\n (x \\ - - (y \\ z) \\ y) \\ z =\n (x \\ y) \\ z"} {"_id": "502242", "text": "proof (state)\nthis:\n f p \\ p\n\ngoal (2 subgoals):\n 1. p \\ f p\n 2. f p = p"} {"_id": "502243", "text": "proof (prove)\nusing this:\n p \\# prime_factorization j\n\ngoal (1 subgoal):\n 1. multiplicity p j \\ 0"} {"_id": "502244", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xa_\n \\ snd `\n (SIGMA xa:{\\ \\ A. \\ \\ x \\ 0}.\n {i \\ N. a i = xa \\ x \\ W i}) \\\n (case (a xa_, xa_) of (\\, i) \\ g i x) = g xa_ x"} {"_id": "502245", "text": "proof (prove)\ngoal (1 subgoal):\n 1. blinfun_apply (comp12 f g) b =\n blinfun_apply f (fst b) + blinfun_apply g (snd b)"} {"_id": "502246", "text": "proof (state)\nthis:\n {cf e |e. e \\ set p} \\ {}\n\ngoal (1 subgoal):\n 1. (\\e.\n \\e \\ set p; cf e = resCap p\\\n \\ thesis) \\\n thesis"} {"_id": "502247", "text": "proof (prove)\nusing this:\n rlex_fun s t \\ rlex_fun (s + u) (t + u)\n finite (supp_fun s)\n finite (supp_fun t)\n finite (supp_fun u)\n dord_fun rlex_fun s t\n\ngoal (1 subgoal):\n 1. dord_fun rlex_fun (s + u) (t + u)"} {"_id": "502248", "text": "proof (prove)\nusing this:\n LB M2 M1 vs' xs' (TS M2 M1 \\ V m j \\ V) S1 \\ V''\n \\ m\n m < LB M2 M1 vs' xs' (TS M2 M1 \\ V m j \\ V) S1 \\ V''\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502249", "text": "proof (prove)\nusing this:\n map (fst \\ meval n (\\ \\ j) db) \\s = xss\n list_all2 (\\i t. t = \\ \\ i)\n [progress_regex \\ P r j.. \\ j])\n list_all2 (\\i t. t = \\ \\ i)\n [progress_regex \\ P r j.. \\ j])\n Monitor.progress \\ P (formula.MatchF I r) j + length aux =\n progress_regex \\ P r j\n wf_matchF_aux \\ V n R I r aux\n (Monitor.progress \\ P (formula.MatchF I r) j) 0\n list_all2 (wf_mformula \\ j P V n R) \\s \\s\n\ngoal (1 subgoal):\n 1. wf_matchF_aux \\ V n R I r aux'\n (Monitor.progress \\ P (formula.MatchF I r) j) 0 \\\n Monitor.progress \\ P (formula.MatchF I r) j + length aux' =\n progress_regex \\ P' r (Suc j)"} {"_id": "502250", "text": "proof (prove)\nusing this:\n \\approx_form prec f ?vs ss; bounded_by xs ?vs\\\n \\ interpret_form f xs\n approx_form prec (Assign x a f) vs ss\n bounded_by xs vs\n\ngoal (1 subgoal):\n 1. (\\n. x = Var n \\ thesis) \\ thesis"} {"_id": "502251", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_gpv'' A C R gpv\n (map_gpv (relcompp_witness A A') (relcompp_witness C C')\n (rel_witness_gpv (A OO A') (C OO C') R R' (gpv, gpv'')))"} {"_id": "502252", "text": "proof (prove)\nusing this:\n wf_until_aux \\ n R pos \\'' I \\'' ((t, a1, a2) # aux') i\n\ngoal (1 subgoal):\n 1. qtable n (fv \\'') (mem_restr R)\n (\\v.\n \\j\\i.\n j < Suc (i + length aux') \\\n mem (\\ \\ j - \\ \\ i) I \\\n MFOTL.sat \\ (map the v) j \\'' \\\n (\\k\\{i.. (map the v) k \\''\n else \\ MFOTL.sat \\ (map the v) k \\''))\n a2"} {"_id": "502253", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (msum_appr ops, (+))\n \\ appr_rel \\ appr_rel \\ appr_rel"} {"_id": "502254", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Generalizations2.matrix inner_rows inner_cols"} {"_id": "502255", "text": "proof (prove)\nusing this:\n 3 * C / \\ \\ R\n 1 \\ R\n 0 < \\\n \\ \\ 1 / 2\n \\ < l\n\ngoal (1 subgoal):\n 1. (((path A &&& path B1) &&& path B3 &&& path B2 &&& valid_path A) &&&\n (valid_path B1 &&& valid_path B3) &&&\n valid_path B2 &&& arc A &&& arc B1) &&&\n ((arc B3 &&& arc B2) &&&\n pathstart A = - \\ * complex_of_real R &&&\n pathfinish A = \\ * complex_of_real R &&&\n pathstart B1 = \\ * complex_of_real R) &&&\n (pathfinish B1 = complex_of_real R * \\ - complex_of_real \\ &&&\n pathstart B3 =\n complex_of_real (- R) * \\ - complex_of_real \\) &&&\n pathfinish B3 = - \\ * complex_of_real R &&&\n pathstart B2 = complex_of_real R * \\ - complex_of_real \\ &&&\n pathfinish B2 = complex_of_real (- R) * \\ - complex_of_real \\"} {"_id": "502256", "text": "proof (prove)\nusing this:\n normal \\_ \\\n I (lift_nnf_qe qe \\_) ?xs = I \\_ ?xs\n normal (ExQ \\_)\n\ngoal (1 subgoal):\n 1. I (lift_nnf_qe qe (ExQ \\_)) xs = I (ExQ \\_) xs"} {"_id": "502257", "text": "proof (prove)\ngoal (10 subgoals):\n 1. \\n g.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n n \\ set (\\n g)\\\n \\ defs g n \\ phiDefs g n = {}\n 2. \\n g m.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n n \\ set (\\n g); m \\ set (\\n g);\n n \\ m\\\n \\ allDefs g n \\ allDefs g m = {}\n 3. \\v g n.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n v \\ CFG_SSA_base.allUses \\n inEdges (usesOf \\ uses)\n (\\g. Mapping.lookup (phis g)) g n;\n n \\ set (\\n g)\\\n \\ defAss g n v\n 4. \\g v.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g.\n Mapping.keys (uses g) \\ set (\\n g)\\\n \\ Mapping.lookup (phis g) (Entry g, v) = None\n 5. \\g n.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g.\n Mapping.keys (uses g) \\ set (\\n g)\\\n \\ oldDefs g n = var g ` defs g n\n 6. \\n g.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n n \\ set (\\n g)\\\n \\ oldUses g n = var g ` (usesOf \\ uses) g n\n 7. \\g n ns m v x v'.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n old.path2 g n ns m; n \\ set (tl ns); v \\ allDefs g n;\n v \\ CFG_SSA_base.allUses \\n inEdges (usesOf \\ uses)\n (\\g. Mapping.lookup (phis g)) g m;\n x \\ set (tl ns); v' \\ allDefs g x\\\n \\ var g v' \\ var g v\n 8. \\g n v vs v'.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n Mapping.lookup (phis g) (n, v) = Some vs; v' \\ set vs\\\n \\ var g v' = var g v\n 9. \\n g v v'.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g. Mapping.keys (uses g) \\ set (\\n g);\n n \\ set (\\n g); v \\ allDefs g n;\n v' \\ allDefs g n; v \\ v'\\\n \\ var g v' \\ var g v\n 10. \\g.\n \\CFG_SSA_Transformed \\e \\n invar inEdges Entry\n oldDefs oldUses defs\n (\\g. lookup_multimap (uses g))\n (\\g. Mapping.lookup (phis g)) var;\n \\g.\n Mapping.keys (uses g) \\ set (\\n g)\\\n \\ Mapping.keys (uses g)\n \\ set (\\n g)"} {"_id": "502258", "text": "proof (prove)\nusing this:\n igWlsDisj MOD \\ igWlsAbsDisj MOD\n igWlsDisj MOD\n igWlsAbsIsInBar MOD\n igVarIPresIGWls MOD \\ igAbsIPresIGWls MOD \\ igOpIPresIGWls MOD\n igSwapIPresIGWls MOD \\ igSwapAbsIPresIGWlsAbs MOD\n igSwapIGVar MOD \\ igSwapIGAbs MOD \\ igSwapIGOp MOD\n \\igVarIPresIGWls MOD; igSwapIGVar MOD\\\n \\ igSwapIGVar (errMOD MOD)\n \\igAbsIPresIGWls MOD; igWlsDisj MOD; igSwapIPresIGWls MOD;\n igSwapIGAbs MOD\\\n \\ igSwapIGAbsSTR (errMOD MOD)\n \\igWlsAbsIsInBar MOD; igOpIPresIGWls MOD; igSwapIPresIGWls MOD;\n igSwapAbsIPresIGWlsAbs MOD; igWlsDisj MOD; igWlsAbsDisj MOD;\n igSwapIGOp MOD\\\n \\ igSwapIGOpSTR (errMOD MOD)\n\ngoal (1 subgoal):\n 1. igSwapIGVar (errMOD MOD) \\\n igSwapIGAbsSTR (errMOD MOD) \\ igSwapIGOpSTR (errMOD MOD)"} {"_id": "502259", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\string_binop_type op \\ \\ \\\\<^sub>1;\n string_binop_type op \\ \\ \\\\<^sub>2\\\n \\ \\\\<^sub>1 = \\\\<^sub>2"} {"_id": "502260", "text": "proof (prove)\nusing this:\n a \\ snd ` set (Atoms (preprocess' t start))\n\ngoal (1 subgoal):\n 1. atom_var a < FirstFreshVariable (preprocess' (hh # t) start)"} {"_id": "502261", "text": "proof (state)\nthis:\n f \\ g C1_differentiable_on {0..1} - S\n\ngoal (1 subgoal):\n 1. valid_path (f \\ g)"} {"_id": "502262", "text": "proof (prove)\nusing this:\n \\X\\\\\\: F \\\\: G\\\n \\ set (a # A)\n \\X\\\\\\: F \\\\: G\\\n \\ set A \\\n \\X\\\\\\: (F \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n rm_vars (set X)\n\\) \\\\: G \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n rm_vars (set X) \\\\\n \\ set (A \\\\<^sub>s\\<^sub>s\\<^sub>t \\)\n a # A \\\\<^sub>s\\<^sub>s\\<^sub>t \\ =\n (a \\\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p \\) #\n (A \\\\<^sub>s\\<^sub>s\\<^sub>t \\)\n\ngoal (1 subgoal):\n 1. \\X\\\\\\: (F \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n rm_vars (set X)\n \\) \\\\: G \\\\<^sub>p\\<^sub>a\\<^sub>i\\<^sub>r\\<^sub>s\n rm_vars (set X) \\\\\n \\ set (a # A \\\\<^sub>s\\<^sub>s\\<^sub>t \\)"} {"_id": "502263", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finfun_curry (K$ c) = (K$ K$ c)"} {"_id": "502264", "text": "proof (prove)\ngoal (1 subgoal):\n 1. path (x\\<^sup>T\\<^sup>\\ ; p \\ x)"} {"_id": "502265", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map norm_instr ys = xs"} {"_id": "502266", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\s x y es s'.\n \\((y, es), s') \\ set_spmf (rfunc_adv rest s x);\n x \\ outs_\\ \\_adv; I s\\\n \\ y \\ responses_\\ \\_adv x \\ I s') &&&\n (\\s x y es s'.\n \\((y, es), s') \\ set_spmf (rfunc_usr rest s x);\n x \\ outs_\\ \\_usr; I s\\\n \\ y \\ responses_\\ \\_usr x \\ I s') &&&\n I (rinit rest)"} {"_id": "502267", "text": "proof (prove)\nusing this:\n (X', ?Y90) \\ rel \\ phi ?Y90\n\ngoal (1 subgoal):\n 1. phiAbs (Abs xs x X)"} {"_id": "502268", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\w.\n \\cmod w = r;\n \\z.\n cmod z < r \\\n cmod (h z) \\ cmod (h w)\\\n \\ thesis) \\\n thesis"} {"_id": "502269", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_bij ((\\ w x y z)\\<^sup>\\)"} {"_id": "502270", "text": "proof (state)\nthis:\n k' \\ subterms\\<^sub>s\\<^sub>e\\<^sub>t M \\\n \\ ((set \\ fst \\ Ana) ` M) -\n M\n\ngoal (1 subgoal):\n 1. k' \\ M \\\n (\\\\.\n wt\\<^sub>s\\<^sub>u\\<^sub>b\\<^sub>s\\<^sub>t \\ \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s (subst_range \\) \\\n k \\ M \\\\<^sub>s\\<^sub>e\\<^sub>t \\) \\\n k \\ SMP (subterms\\<^sub>s\\<^sub>e\\<^sub>t M \\\n \\ ((set \\ fst \\ Ana) ` M) -\n M)"} {"_id": "502271", "text": "proof (prove)\nusing this:\n \\\\<^sub># (mset Cs) \\ negs (mset As)\n \\B\\atms_of (\\\\<^sub># (mset Cs)). B < A_max\n Neg A_max \\# \\\\<^sub># (mset Cs)\n Neg A_max \\# negs (mset As)\n\ngoal (1 subgoal):\n 1. \\\\<^sub># (mset Cs) \\ negs (mset As) \\\n (\\y.\n count (negs (mset As)) y\n < count (\\\\<^sub># (mset Cs)) y \\\n (\\x>y.\n count (\\\\<^sub># (mset Cs)) x < count (negs (mset As)) x))"} {"_id": "502272", "text": "proof (prove)\nusing this:\n x \\ carrier R\n y \\ carrier R\n\ngoal (1 subgoal):\n 1. (\\a\\I +> x. \\b\\I +> y. I +> a \\ b) =\n I +> x \\ y"} {"_id": "502273", "text": "proof (prove)\nusing this:\n store_typing s \\\n i < length (s_inst \\)\n store_typing ?s ?\\ \\\n length (inst ?s) = length (s_inst ?\\)\n \\store_typing ?s ?\\; ?i < length (inst ?s)\\\n \\ inst_typing ?\\ (inst ?s ! ?i) (s_inst ?\\ ! ?i)\n\ngoal (1 subgoal):\n 1. list_all2 (globi_agree (s_globs \\)) (inst.globs (inst s ! i))\n (global (s_inst \\ ! i))"} {"_id": "502274", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\s.\n \\alw (holds\n (\\c. cri.InvMsgInGlob (exchange_config c)))\n s;\n alw Next s; InvGlobPointstampsEq (shd s);\n inv_init_imp_prop_safe (shd s);\n \\p. prop_invs (prop_config (shd s) p);\n \\ alw (holds all_invs) (stl s)\\\n \\ inv_init_imp_prop_safe (shd (stl s))\n 2. \\s p.\n \\alw (holds\n (\\c. cri.InvMsgInGlob (exchange_config c)))\n s;\n alw Next s; InvGlobPointstampsEq (shd s);\n inv_init_imp_prop_safe (shd s);\n \\p. prop_invs (prop_config (shd s) p);\n \\ alw (holds all_invs) (stl s)\\\n \\ inv_implications_nonneg\n (prop_config (shd (stl s)) p)\n 3. \\s p.\n \\alw (holds\n (\\c. cri.InvMsgInGlob (exchange_config c)))\n s;\n alw Next s; InvGlobPointstampsEq (shd s);\n inv_init_imp_prop_safe (shd s);\n \\p. prop_invs (prop_config (shd s) p);\n \\ alw (holds all_invs) (stl s)\\\n \\ inv_imps_work_sum (prop_config (shd (stl s)) p)\n 4. \\s.\n \\alw (holds\n (\\c. cri.InvMsgInGlob (exchange_config c)))\n s;\n alw Next s; InvGlobPointstampsEq (shd s);\n inv_init_imp_prop_safe (shd s);\n \\p. prop_invs (prop_config (shd s) p);\n \\ alw (holds all_invs) (stl s)\\\n \\ alw (holds\n (\\c.\n cri.InvMsgInGlob (exchange_config c)))\n (stl s)\n 5. \\s.\n \\alw (holds\n (\\c. cri.InvMsgInGlob (exchange_config c)))\n s;\n alw Next s; InvGlobPointstampsEq (shd s);\n inv_init_imp_prop_safe (shd s);\n \\p. prop_invs (prop_config (shd s) p);\n \\ alw (holds all_invs) (stl s)\\\n \\ alw Next (stl s)"} {"_id": "502275", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue ((+) a ` S) =\n Sigma_Algebra.measure lebesgue S"} {"_id": "502276", "text": "proof (prove)\nusing this:\n A \\ sets Invariants\n\ngoal (1 subgoal):\n 1. A \\ local.P.events"} {"_id": "502277", "text": "proof (prove)\nusing this:\n \\\\<^sub>F n in sequentially. \\.C n x < \\\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F n in sequentially. x < \\.I n \\"} {"_id": "502278", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lim (h\\lim := l\\) = l"} {"_id": "502279", "text": "proof (state)\nthis:\n (cmod\n (complex_of_real r1 * exp (\\ * complex_of_real \\1) -\n complex_of_real r2 * exp (\\ * complex_of_real \\2)))\\<^sup>2 =\n (cmod\n (complex_of_real r1 * exp (\\ * complex_of_real \\1) +\n complex_of_real r2 *\n exp (\\ * complex_of_real (\\2 + pi))))\\<^sup>2\n\ngoal (1 subgoal):\n 1. (cmod\n (complex_of_real r1 * exp (\\ * complex_of_real \\1) -\n complex_of_real r2 *\n exp (\\ * complex_of_real \\2)))\\<^sup>2 =\n r1\\<^sup>2 + r2\\<^sup>2 - 2 * r1 * r2 * cos (\\1 - \\2)"} {"_id": "502280", "text": "proof (prove)\ngoal (1 subgoal):\n 1. spec s \\\n alw (holds (\\c. \\t. 0 \\ zcount (c_records c) t)) s"} {"_id": "502281", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B \\ f \\ folding.\\;\n C \\ folding.\\ - f \\ folding.\\;\n D \\ f \\ folding.\\;\n folding.gallery (B # C # Cs @ [D])\\\n \\ \\ min_gallery (B # C # Cs @ [D])"} {"_id": "502282", "text": "proof (state)\nthis:\n L1 = ta_lang TA1\n ranked_tree_automaton TA1 A\n L2 = ta_lang TA2\n ranked_tree_automaton TA2 A\n\ngoal (1 subgoal):\n 1. \\TA TAa.\n \\L1 = ta_lang TA; ranked_tree_automaton TA A;\n L2 = ta_lang TAa; ranked_tree_automaton TAa A\\\n \\ L1 \\ L2 \\ regular_languages A"} {"_id": "502283", "text": "proof (prove)\ngoal (1 subgoal):\n 1. t \\\\<^sub>t \\\\<^sub>1 = t \\\\<^sub>t \\\\<^sub>2"} {"_id": "502284", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coprime a' b'"} {"_id": "502285", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bottom r (OK None)"} {"_id": "502286", "text": "proof (prove)\ngoal (1 subgoal):\n 1. numeral n \\ \\\\<^sub>z"} {"_id": "502287", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ set (stake (LEAST n. xs !! n = x) xs)"} {"_id": "502288", "text": "proof (prove)\nusing this:\n \\ separating \\ (TER h)\n\ngoal (1 subgoal):\n 1. (\\x y p.\n \\x \\ A \\; y \\ B \\;\n path \\ x p y; x \\ TER h;\n \\z. z \\ set p \\ z \\ TER h\\\n \\ thesis) \\\n thesis"} {"_id": "502289", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\V.\n V \\ set (build_ip_partition c rs) \\\n s \\ wordinterval_to_set V"} {"_id": "502290", "text": "proof (state)\nthis:\n Domain a \\ X \\ {}\n\ngoal (1 subgoal):\n 1. sum b (Outside' X a) \\ Max (sum b ` allAllocations (N - X) G)"} {"_id": "502291", "text": "proof (prove)\ngoal (1 subgoal):\n 1. while\\<^sub>T\n (\\(f, PRED, C, N, d). f = False \\ C \\ {})\n (\\(f, PRED, C, N, d).\n assert (C \\ {}) \\\n (\\_.\n IICF_Set.op_set_pick C \\\n (\\v.\n let C = C - {v}\n in assert (v \\ V) \\\n (\\_.\n succ v \\\n (\\sl.\n inner_loop2 dst sl v PRED N \\\n (\\(f, PRED, N).\n if f then return (f, PRED, C, N, d + 1)\n else assert\n(assn1 src dst (f, PRED, C, N, d)) \\\n (\\_.\n if C = {} then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d)\n else return (f, PRED, C, N, d))))))))\n (False, [src \\ src], {src}, {}, 0) \\\n (\\(f, PRED, uu_, uu_, d).\n if f then return (Some (d, PRED)) else return None)\n \\ \\ Id\n (while\\<^sub>T\\<^bsup>outer_loop_invar src dst\\<^esup>\n (\\(f, PRED, C, N, d). f = False \\ C \\ {})\n (\\(f, PRED, C, N, d).\n (spec v. v \\ C) \\\n (\\v.\n let C = C - {v}\n in assert (v \\ V) \\\n (\\_.\n let succ = E `` {v}\n in assert (finite succ) \\\n (\\_.\n add_succ_spec dst succ v PRED N \\\n (\\(f, PRED, N).\n if f then return (f, PRED, C, N, d + 1)\n else assert (assn1 src dst (f, PRED, C, N, d)) \\\n (\\_.\n if C = {}\n then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d)\n else return (f, PRED, C, N, d)))))))\n (False, [src \\ src], {src}, {}, 0) \\\n (\\(f, PRED, uu_, uu_, d).\n if f then return (Some (d, PRED)) else return None))"} {"_id": "502292", "text": "proof (prove)\nusing this:\n (?i::nat) < (j::nat) \\\n (b::'a::refinement_lattice) \\<^sup>;^ j =\n (b \\<^sup>;^ ?i) ; (b \\<^sup>;^ (j - ?i))\n (i::nat) < Suc (j::nat)\n ((?m::nat) < Suc (?n::nat)) = (?m < ?n \\ ?m = ?n)\n (?a::'a::refinement_lattice) ; (?b::'a::refinement_lattice) ;\n (?c::'a::refinement_lattice) =\n ?a ; (?b ; ?c)\n\ngoal (1 subgoal):\n 1. b \\<^sup>;^ Suc j = b ; (b \\<^sup>;^ i) ; (b \\<^sup>;^ (j - i))\nvariables:\n i, j :: nat\n b, nil :: 'a\n (;) :: 'a \\ 'a \\ 'a\ntype variables:\n 'a :: refinement_lattice"} {"_id": "502293", "text": "proof (prove)\nusing this:\n \\v. closed (\\ v) \\ closed (\\' v)\n \\v.\n evalDdb (\\ v)\\env_empty_db =\n evalDdb (\\' v)\\env_empty_db\n\ngoal (1 subgoal):\n 1. evalDdb (closing_subst (DBVar x_) \\ k)\\\\ =\n evalDdb (closing_subst (DBVar x_) \\' k)\\\\"} {"_id": "502294", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ set xs \\ propos x \\ propos (Or xs)"} {"_id": "502295", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (((R ===> M) ===> X) ===> local.rel_envT R M ===> X) case_envT case_envT"} {"_id": "502296", "text": "proof (prove)\nusing this:\n ((\\, k), \\', k') \\ cop\n\ngoal (1 subgoal):\n 1. (path \\, k) \\ scop \\ (path \\', k') \\ scp"} {"_id": "502297", "text": "proof (prove)\nusing this:\n s \\ correct_jvm_state \\\n P \\ s -t'\\ta'\\\\<^bsub>jvmd\\<^esub> s'\n thr s t = \\((xcp, frs), no_wait_locks)\\\n execd_mthr.can_sync P t (xcp, frs) (shr s') L\n\ngoal (1 subgoal):\n 1. \\L'\\L. execd_mthr.can_sync P t (xcp, frs) (shr s) L'"} {"_id": "502298", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_get_host_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.known_ptr\n ShadowRootClass.known_ptrs ShadowRootClass.type_wf get_host"} {"_id": "502299", "text": "proof (prove)\nusing this:\n module R M1\n module R M2\n module R M1\n module R M2\n\ngoal (1 subgoal):\n 1. module_axioms R (direct_sum M1 M2)"} {"_id": "502300", "text": "proof (prove)\nusing this:\n \\\\x. x \\ set xs \\ g (f (fst x)) = fst x;\n f (g x) = x\\\n \\ map_of\n (map (\\a.\n case a of (k, a) \\ (f k, a))\n xs)\n x =\n map_of xs (g x)\n ?x \\ set (a # xs) \\ g (f (fst ?x)) = fst ?x\n f (g x) = x\n\ngoal (1 subgoal):\n 1. map_of\n (map (\\a. case a of (k, a) \\ (f k, a)) (a # xs))\n x =\n map_of (a # xs) (g x)"} {"_id": "502301", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ \\ fml_sem I (FsubstFO (Exists x \\) \\)) =\n (\\r. repv \\ x r \\ fml_sem I (FsubstFO \\ \\))"} {"_id": "502302", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sorted_wrt (\\\\<^sub>t)\n (fold (\\p. merge_wrt (\\\\<^sub>t) (keys_to_list p)) ps [])"} {"_id": "502303", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_mbufn k js Ps buf' &&& list_all2 R [k..n.\n if n = 0 then a else if n = 1 then b else if n = 2 then c else d)\n ord {a, b, c, d}"} {"_id": "502305", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f `# P = {#y#};\n \\x.\n \\P = {#x#}; f x = y\\ \\ Q\\\n \\ Q"} {"_id": "502306", "text": "proof (prove)\ngoal (1 subgoal):\n 1. iexp x =\n (\\k\\0. (\\ * complex_of_real x) ^ k / fact k) +\n \\ ^ Suc 0 / fact 0 * (CLBINT s=0..ereal x. f s 0 * iexp s)"} {"_id": "502307", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\\\v\\fset V. wf_ass (\\ v);\n \\e\\set E. wf_com (snd3 e);\n acyclic (\\e\\set E. fset (fst3 e) \\ fset (thd3 e));\n distinct E \\\n (\\e\\set E.\n \\f\\set E.\n e \\ f \\\n fset (fst3 e |\\| fst3 f) = fset {||} \\\n fset (thd3 e |\\| thd3 f) = fset {||});\n \\e\\set E.\n fset (fst3 e |\\| thd3 e) \\ fset V;\n e \\ set E; fset (fst3 e) \\ fset S;\n fset S\n \\ fset\n (ffilter\n (\\v.\n \\e\\set (Graph V \\ E)^E.\n v \\ fset (thd3 e))\n (Graph V \\ E)^V)\\\n \\ acyclic ?s14\n 2. \\\\v\\fset V. wf_ass (\\ v);\n \\e\\set E. wf_com (snd3 e);\n acyclic (\\e\\set E. fset (fst3 e) \\ fset (thd3 e));\n distinct E \\\n (\\e\\set E.\n \\f\\set E.\n e \\ f \\\n fset (fst3 e |\\| fst3 f) = fset {||} \\\n fset (thd3 e |\\| thd3 f) = fset {||});\n \\e\\set E.\n fset (fst3 e |\\| thd3 e) \\ fset V;\n e \\ set E; fset (fst3 e) \\ fset S;\n fset S\n \\ fset\n (ffilter\n (\\v.\n \\e\\set (Graph V \\ E)^E.\n v \\ fset (thd3 e))\n (Graph V \\ E)^V)\\\n \\ (\\e\\set (removeAll e E).\n fset (fst3 e) \\ fset (thd3 e))\n \\ ?s14\n 3. \\\\v\\fset V. wf_ass (\\ v);\n \\e\\set E. wf_com (snd3 e);\n acyclic (\\e\\set E. fset (fst3 e) \\ fset (thd3 e));\n distinct E \\\n (\\e\\set E.\n \\f\\set E.\n e \\ f \\\n fset (fst3 e |\\| fst3 f) = fset {||} \\\n fset (thd3 e |\\| thd3 f) = fset {||});\n \\e\\set E.\n fset (fst3 e |\\| thd3 e) \\ fset V;\n e \\ set E; fset (fst3 e) \\ fset S;\n fset S\n \\ fset\n (ffilter\n (\\v.\n \\e\\set (Graph V \\ E)^E.\n v \\ fset (thd3 e))\n (Graph V \\ E)^V)\\\n \\ distinct (removeAll e E) \\\n (\\ea\\set (removeAll e E).\n \\f\\set (removeAll e E).\n ea \\ f \\\n fset (fst3 ea |\\| fst3 f) =\n fset {||} \\\n fset (thd3 ea |\\| thd3 f) = fset {||})\n 4. \\\\v\\fset V. wf_ass (\\ v);\n \\e\\set E. wf_com (snd3 e);\n acyclic (\\e\\set E. fset (fst3 e) \\ fset (thd3 e));\n distinct E \\\n (\\e\\set E.\n \\f\\set E.\n e \\ f \\\n fset (fst3 e |\\| fst3 f) = fset {||} \\\n fset (thd3 e |\\| thd3 f) = fset {||});\n \\e\\set E.\n fset (fst3 e |\\| thd3 e) \\ fset V;\n e \\ set E; fset (fst3 e) \\ fset S;\n fset S\n \\ fset\n (ffilter\n (\\v.\n \\e\\set (Graph V \\ E)^E.\n v \\ fset (thd3 e))\n (Graph V \\ E)^V)\\\n \\ \\ea\\set (removeAll e E).\n fset (fst3 ea |\\| thd3 ea)\n \\ fset (V |-| fst3 e)"} {"_id": "502308", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f \\ o[F](g) \\ L F (\\x. f x - g x) = L F g"} {"_id": "502309", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < k \\\n f \\ k \\\\<^sub> [ 0, mod k, n ] = f \\\\<^sub> [\\n]"} {"_id": "502310", "text": "proof (prove)\nusing this:\n z = of_nat p\n prime p\n [p = 3] (mod 4)\n prime z\n gauss_int_norm z = p\\<^sup>2\n\ngoal (1 subgoal):\n 1. prime p \\ z = of_nat p"} {"_id": "502311", "text": "proof (prove)\nusing this:\n \\spmf (DDH0 ?\\) True - spmf (DDH1 ?\\) True\\\n \\ advantage ?\\ +\n \\spmf (ddh_1 ?\\) True - spmf (DDH1 ?\\) True\\\n advantage (DDH_\\' ?\\) =\n \\spmf (ddh_1 ?\\) True - spmf (DDH1 ?\\) True\\\n DDH_advantage ?\\ =\n \\spmf (DDH0 ?\\) True - spmf (DDH1 ?\\) True\\\n\ngoal (1 subgoal):\n 1. DDH_advantage \\ \\ advantage \\ + advantage (DDH_\\' \\)"} {"_id": "502312", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (ffUnion \\ fset_of_list \\ map frees) (x # xs) =\n frees x |\\| (ffUnion \\ fset_of_list \\ map frees) xs"} {"_id": "502313", "text": "proof (prove)\ngoal (1 subgoal):\n 1. n < length cms''"} {"_id": "502314", "text": "proof (prove)\ngoal (1 subgoal):\n 1. conv_mirror \\ \\ conv ars \\\n set (labels_conv (conv_mirror \\)) =\n set (labels_conv \\) \\\n fst \\ = lst_conv (conv_mirror \\) \\\n lst_conv \\ = fst (conv_mirror \\)"} {"_id": "502315", "text": "proof (prove)\ngoal (1 subgoal):\n 1. min_set\n (\\\\<^sub>m\n {min_dnf (subst \\ m) |\\. \\ \\ fset \\}) =\n \\\\<^sub>m\n {min_dnf (subst \\ m) |\\. \\ \\ fset \\}"} {"_id": "502316", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Rel_match B (f x) (g y)"} {"_id": "502317", "text": "proof (prove)\nusing this:\n Ord \\\n ?y \\ elts \\ \\ P ?y\n\ngoal (1 subgoal):\n 1. P \\"} {"_id": "502318", "text": "proof (prove)\nusing this:\n AX10 \\ ?y1 \\\\<^sub>H\n (?F3 \\<^bold>\\ ?H3) \\<^bold>\\\n (?G3 \\<^bold>\\ ?H3) \\<^bold>\\\n (?F3 \\<^bold>\\ ?G3) \\<^bold>\\ ?H3\n \\?\\ \\\\<^sub>H ?F;\n ?\\ \\\\<^sub>H ?F \\<^bold>\\ ?G\\\n \\ ?\\ \\\\<^sub>H ?G\n AX10 \\ \\ \\\\<^sub>H F \\<^bold>\\ H\n AX10 \\ \\ \\\\<^sub>H G \\<^bold>\\ H\n\ngoal (1 subgoal):\n 1. AX10 \\ \\ \\\\<^sub>H\n (F \\<^bold>\\ G) \\<^bold>\\ H"} {"_id": "502319", "text": "proof (state)\nthis:\n c * a < c * b\n\ngoal (2 subgoals):\n 1. \\a. \\a\\ = (if a < 0 then - a else a)\n 2. \\x. sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"} {"_id": "502320", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f g h.\n \\B.ide f; B.ide g; B.ide h; src\\<^sub>B f = trg\\<^sub>B g;\n src\\<^sub>B g = trg\\<^sub>B h\\\n \\ local.map \\\\<^sub>B[f, g, h] \\\\<^sub>B\n cmp (f \\\\<^sub>B g, h) \\\\<^sub>B\n (cmp (f, g) \\\\<^sub>B local.map h) =\n cmp (f, g \\\\<^sub>B h) \\\\<^sub>B\n (local.map f \\\\<^sub>B\n cmp (g, h)) \\\\<^sub>B\n \\\\<^sub>B[local.map f, local.map g, local.map h]"} {"_id": "502321", "text": "proof (prove)\nusing this:\n A \\ list.set as\n\ngoal (1 subgoal):\n 1. sorted A"} {"_id": "502322", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ = \\"} {"_id": "502323", "text": "proof (prove)\ngoal (1 subgoal):\n 1. transformation_by_components A B F G \\"} {"_id": "502324", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom_mult (1::'b) (0::'a) p = p"} {"_id": "502325", "text": "proof (prove)\nusing this:\n local_ring stk1.carrier_stalk stk1.add_stalk stk1.mult_stalk\n (stk1.zero_stalk (f \\<^sup>\\ X U))\n (stk1.one_stalk (f \\<^sup>\\ X U))\n ring_isomorphism (identity stk1.carrier_stalk) stk1.carrier_stalk\n stk1.add_stalk stk1.mult_stalk\n (stk1.zero_stalk (f \\<^sup>\\ X U))\n (stk1.one_stalk (f \\<^sup>\\ X U)) stk2.carrier_stalk\n stk2.add_stalk stk2.mult_stalk\n (stk2.zero_stalk (f \\<^sup>\\ X U))\n (stk2.one_stalk (f \\<^sup>\\ X U))\n\ngoal (1 subgoal):\n 1. local_ring_morphism (identity stk1.carrier_stalk) stk1.carrier_stalk\n stk1.add_stalk stk1.mult_stalk\n (stk1.zero_stalk (f \\<^sup>\\ X U))\n (stk1.one_stalk (f \\<^sup>\\ X U)) stk2.carrier_stalk\n stk2.add_stalk stk2.mult_stalk\n (stk2.zero_stalk (f \\<^sup>\\ X U))\n (stk2.one_stalk (f \\<^sup>\\ X U))"} {"_id": "502326", "text": "proof (prove)\nusing this:\n bounded_hashcode_nat (length l) x \\\n bounded_hashcode_nat (length l) k\n\ngoal (1 subgoal):\n 1. map_of (abs_update k v l ! bounded_hashcode_nat (length l) x) x =\n map_of (l ! bounded_hashcode_nat (length l) x) x"} {"_id": "502327", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\ t {t}. Q t) \\\n (\\ ta {t}.\n P ta \\ (\\ t' {t} \\\\ ta. Q t'))) =\n Q t"} {"_id": "502328", "text": "proof (prove)\nusing this:\n has_channel\\<^sup>+\\<^sup>+ p q\n\ngoal (1 subgoal):\n 1. (\\r s. has_channel r s \\ thesis) \\\n thesis"} {"_id": "502329", "text": "proof (prove)\nusing this:\n V < length xs\n\ngoal (1 subgoal):\n 1. True,P,t \\1 \\sync\\<^bsub>V\\<^esub> (null) e2,\n (h, xs)\\ -\\\\\\\n \\Throw (addr_of_sys_xcpt NullPointer),\n (h, xs[V := Null])\\"} {"_id": "502330", "text": "proof (prove)\nusing this:\n fold_graph f z (insert x B) v\n x \\ B\n insert x B \\ A\n z \\ A\n\ngoal (1 subgoal):\n 1. (\\y.\n \\v = f x y; fold_graph f z B y\\\n \\ thesis) \\\n thesis"} {"_id": "502331", "text": "proof (prove)\nusing this:\n 2 \\ card X\n\ngoal (1 subgoal):\n 1. (\\Y.\n \\Y \\ X; card Y = 2\\\n \\ thesis) \\\n thesis"} {"_id": "502332", "text": "proof (state)\nthis:\n (g ?x = Some ?y) = (P ?x \\ ?y = fst ?x)\n\ngoal (1 subgoal):\n 1. set_iterator_genord (map_iterator_dom_filter P it)\n {k. \\v. m k = Some v \\ P (k, v)} R'"} {"_id": "502333", "text": "proof (prove)\nusing this:\n (D::'a::refinement_lattice set) =\n {d \\ (i::'a::refinement_lattice) |d::'a::refinement_lattice.\n (c::'a::refinement_lattice) \\ d \\ i}\n (C::'a::refinement_lattice set) = {c::'a::refinement_lattice}\n (?a::?'a::type)\n \\ {x::?'a::type.\n (?P::?'a::type \\ bool) x} \\\n ?P ?a\n (?a::?'a::type) \\ {?a}\n\ngoal (1 subgoal):\n 1. \\d::'a\\D. \\c::'a\\C. c \\ d\nvariables:\n C, D :: 'a set\ntype variables:\n 'a :: refinement_lattice"} {"_id": "502334", "text": "proof (prove)\nusing this:\n approx_tm prec ord I a (Max l r) env = Some t\n\ngoal (1 subgoal):\n 1. (\\t1 t2.\n \\t = tm_max prec I a t1 t2;\n Some t1 = approx_tm prec ord I a l env;\n approx_tm prec ord I a r env = Some t2\\\n \\ thesis) \\\n thesis"} {"_id": "502335", "text": "proof (prove)\ngoal (1 subgoal):\n 1. src.saturated_upto\n (Liminf_llist\n (lmap\n (\\St.\n grounding_of_clss\n (N_of_wstate St \\ P_of_wstate St \\\n Q_of_wstate St))\n Sts))"} {"_id": "502336", "text": "proof (prove)\nusing this:\n d_IN j ?x1 \\ \\\n\ngoal (1 subgoal):\n 1. weight \\ a + d_IN j \\a\\ - \\\n \\ weight \\ a"} {"_id": "502337", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ s' \\\n (set (the (\\\\<^sub>c s')) \\ indices_state s' \\\n (\\v.\n v \\\\<^sub>t \\ s' \\\n (\\x c.\n \\\\<^sub>l s' x = Some c \\\n \\\\<^sub>l s' x\n \\ set (the (\\\\<^sub>c s')) \\\n c \\ v x) \\\n (\\x c.\n \\\\<^sub>u s' x = Some c \\\n \\\\<^sub>u s' x\n \\ set (the (\\\\<^sub>c s')) \\\n v x \\ c))) \\\n (distinct_indices_state s' \\\n (\\I\\set (the (\\\\<^sub>c s')).\n \\v.\n v \\\\<^sub>t \\ s' \\\n (\\x c.\n \\\\<^sub>l s' x = Some c \\\n \\\\<^sub>l s' x \\ I \\ v x = c) \\\n (\\x c.\n \\\\<^sub>u s' x = Some c \\\n \\\\<^sub>u s' x \\ I \\ v x = c)))) =\n (\\ s' \\\n (set (the (\\\\<^sub>c s')) \\ indices_state s' \\\n (\\v.\n v \\\\<^sub>t \\ s' \\\n (\\x c i.\n \\\\<^sub>l s' x = Some c \\\n \\\\<^sub>l s' x = i \\\n i \\ set (the (\\\\<^sub>c s')) \\\n c \\ v x) \\\n (\\x c i.\n \\\\<^sub>u s' x = Some c \\\n \\\\<^sub>u s' x = i \\\n i \\ set (the (\\\\<^sub>c s')) \\\n v x \\ c))) \\\n (distinct_indices_state s' \\\n (\\I\\set (the (\\\\<^sub>c s')).\n \\v.\n v \\\\<^sub>t \\ s' \\\n (\\x c i.\n \\\\<^sub>l s' x = Some c \\\n \\\\<^sub>l s' x = i \\\n i \\ I \\ v x = c) \\\n (\\x c i.\n \\\\<^sub>u s' x = Some c \\\n \\\\<^sub>u s' x = i \\\n i \\ I \\ v x = c))))"} {"_id": "502338", "text": "proof (prove)\nusing this:\n incseq (\\n x. F n x * indicator (\\n. (T ^^ n) --` A) x)\n\ngoal (1 subgoal):\n 1. (\\n. set_nn_integral M (\\n. (T ^^ n) --` A) (F n)) =\n \\\\<^sup>+ x. (\\n.\n F n x *\n indicator (\\n. (T ^^ n) --` A) x)\n \\M"} {"_id": "502339", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\c\\{[Pos (Eq (lOfWax \\) (rOfWax \\))] |\n \\. \\ isRes \\ \\ protFw \\}.\n list_all TE.wtL c"} {"_id": "502340", "text": "proof (prove)\nusing this:\n ?xc \\ set lc \\ invar ?xc\n \\invar ?xc; \\ ?xc \\ set (map \\ lc)\\\n \\ fc ?xc = fa (\\ ?xc)\n\ngoal (1 subgoal):\n 1. foldli (map \\ lc) c fa \\0 = foldli lc c fc \\0"} {"_id": "502341", "text": "proof (prove)\nusing this:\n prefs_from_table_wf agents alts xs\n prefs_from_table_wf agents alts ys\n h permutes {..\n set V \\ vec.span (set (Gram_Schmidt2 V)) \\\n card (set (Gram_Schmidt2 V)) = vec.dim (set V) \\\n pairwise orthogonal (set (Gram_Schmidt2 V))"} {"_id": "502343", "text": "proof (prove)\nusing this:\n ide f\n src ?\\ \\\n if arr ?\\ then mkObj (C.cod (Leg0 (Dom ?\\))) else null\n\ngoal (1 subgoal):\n 1. f.dsrc = src_f.dom.leg1"} {"_id": "502344", "text": "proof (prove)\nusing this:\n vangle u v = 0\n \\u \\ v / (norm u * norm v)\\ \\ 1 \\\n \\1\\ \\ 1 \\\n (arccos (u \\ v / (norm u * norm v)) = arccos 1) =\n (u \\ v / (norm u * norm v) = 1)\n \\u \\ v\\ \\ norm u * norm v\n\ngoal (1 subgoal):\n 1. u \\ v = norm u * norm v"} {"_id": "502345", "text": "proof (prove)\ngoal (1 subgoal):\n 1. L \\ K"} {"_id": "502346", "text": "proof (prove)\nusing this:\n T_slow_median ys \\ 5\\<^sup>2 + 3 * 5 + 4\n\ngoal (1 subgoal):\n 1. T_slow_median ys \\ 44"} {"_id": "502347", "text": "proof (prove)\nusing this:\n finite C\n\ngoal (1 subgoal):\n 1. sum c V \\ (\\i\\C. sum c (S i))"} {"_id": "502348", "text": "proof (prove)\nusing this:\n a dvdm prod_list xs\n\ngoal (1 subgoal):\n 1. of_int_poly a dvd of_int_poly (prod_list xs)"} {"_id": "502349", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ltl (llist_of_tllist xs) = llist_of_tllist (ttl xs)"} {"_id": "502350", "text": "proof (prove)\nusing this:\n Nml (t \\<^bold>\\ u)\n\ngoal (1 subgoal):\n 1. t = \\<^bold>\\un_Prim t\\<^bold>\\ \\\n arr (un_Prim t) \\\n Nml t \\\n Nml u \\\n \\ is_Prim\\<^sub>0 u \\\n \\<^bold>\\src (un_Prim t)\\<^bold>\\\\<^sub>0 = Trg u"} {"_id": "502351", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inorder (n2 l a r) = inorder l @ a # inorder r"} {"_id": "502352", "text": "proof (prove)\ngoal (1 subgoal):\n 1. valid_mmsaux args cur\n (join_mmsaux_abs args X\n (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since))\n (map (\\(t, rel). (t, join rel (args_pos args) X)) auxlist)"} {"_id": "502353", "text": "proof (prove)\nusing this:\n F (\\A. if g integrable_on A then Some (integral A g) else None)\n p =\n Some i\n\ngoal (1 subgoal):\n 1. (g has_integral i) (cbox a b)"} {"_id": "502354", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < Im ((b - a) * cnj (b - z)) \\\n 0 < Im ((c - b) * cnj (c - z)) \\\n 0 < Im ((a - c) * cnj (a - z)) \\\n Im ((b - a) * cnj (b - z)) < 0 \\\n 0 < Im ((b - c) * cnj (b - z)) \\\n 0 < Im ((a - b) * cnj (a - z)) \\ 0 < Im ((c - a) * cnj (c - z))"} {"_id": "502355", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\ja.\n \\distinct_pds K n P; ideal (O\\<^bsub>K P n\\<^esub>) I;\n I \\ {\\\\<^bsub>O\\<^bsub>K P n\\<^esub>\\<^esub>};\n I \\ carrier (O\\<^bsub>K P n\\<^esub>); j \\ n; Ring K;\n aGroup K; j \\ n; valuation K (\\\\<^bsub>K P j\\<^esub>);\n Zl_mI K P I j \\ I;\n (\\\\<^bsub>K P j\\<^esub>) (Zl_mI K P I j) =\n LI K (\\\\<^bsub>K P j\\<^esub>) I;\n ja \\ n - Suc 0; (\\\\<^bsub>j n\\<^esub>) ja \\ n\\\n \\ ant (m_zmax_pdsI K n P I)\n \\ (\\\\<^bsub>K P j\\<^esub>)\n (Zl_mI K P I\n ((\\\\<^bsub>j n\\<^esub>) ja)) +\n (\\\\<^bsub>K P j\\<^esub>)\n (mprod_exp K\n (K_gamma ((\\\\<^bsub>j n\\<^esub>) ja))\n (Kb\\<^bsub>K n P\\<^esub>)\n n\\<^bsub>K\\<^esub>\\<^bsup>m_zmax_pdsI K n\n P I\\<^esup>)\n 2. \\distinct_pds K n P; ideal (O\\<^bsub>K P n\\<^esub>) I;\n I \\ {\\\\<^bsub>O\\<^bsub>K P n\\<^esub>\\<^esub>};\n I \\ carrier (O\\<^bsub>K P n\\<^esub>); j \\ n; Ring K;\n aGroup K; j \\ n; valuation K (\\\\<^bsub>K P j\\<^esub>);\n Zl_mI K P I j \\ I;\n (\\\\<^bsub>K P j\\<^esub>) (Zl_mI K P I j) =\n LI K (\\\\<^bsub>K P j\\<^esub>) I;\n ant (m_zmax_pdsI K n P I)\n \\ (\\\\<^bsub>K P j\\<^esub>)\n (\\\\<^sub>e K (\\x.\n Zl_mI K P I\n ((\\\\<^bsub>j n\\<^esub>)\n x) \\\\<^sub>r\n mprod_exp K\n (K_gamma\n ((\\\\<^bsub>j n\\<^esub>) x))\n (Kb\\<^bsub>K n P\\<^esub>)\n n\\<^bsub>K\\<^esub>\\<^bsup>m_zmax_pdsI K\n n P I\\<^esup>) (n - Suc 0))\\\n \\ LI K (\\\\<^bsub>K P j\\<^esub>) I\n < (\\\\<^bsub>K P j\\<^esub>)\n (\\\\<^sub>e K (\\x.\n Zl_mI K P I ((\\\\<^bsub>j n\\<^esub>) x) \\\\<^sub>r\n mprod_exp K (K_gamma ((\\\\<^bsub>j n\\<^esub>) x))\n (Kb\\<^bsub>K n P\\<^esub>)\n n\\<^bsub>K\\<^esub>\\<^bsup>m_zmax_pdsI K n P\n I\\<^esup>) (n - Suc 0))\n 3. \\distinct_pds K n P; ideal (O\\<^bsub>K P n\\<^esub>) I;\n I \\ {\\\\<^bsub>O\\<^bsub>K P n\\<^esub>\\<^esub>};\n I \\ carrier (O\\<^bsub>K P n\\<^esub>); j \\ n; Ring K;\n aGroup K; j \\ n;\n valuation K (\\\\<^bsub>K P j\\<^esub>)\\\n \\ (\\\\<^bsub>K P j\\<^esub>)\n (Zl_mI K P I j \\\\<^sub>r\n mprod_exp K (K_gamma j) (Kb\\<^bsub>K n P\\<^esub>)\n n\\<^bsub>K\\<^esub>\\<^bsup>m_zmax_pdsI K n P\n I\\<^esup>) =\n LI K (\\\\<^bsub>K P j\\<^esub>) I"} {"_id": "502356", "text": "proof (prove)\nusing this:\n mset_factors ?F ?p \\\n mset_factors (image_mset poly_lift ?F) (poly_lift ?p)\n mset_factors G q\n\ngoal (1 subgoal):\n 1. mset_factors (image_mset poly_lift G) (poly_lift q)"} {"_id": "502357", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (if sWlsVal SEM val\n then if sWlsVal SEM (val (y := Y val)_ys)\n then sOp SEM delta\n (lift (\\X. X (val (y := Y val)_ys)) inp)\n (lift (\\A. A (val (y := Y val)_ys)) binp)\n else undefined\n else undefined) =\n (if sWlsVal SEM val\n then sOp SEM delta\n (lift (\\X. X val) (igSubstInp (asIMOD SEM) ys Y y inp))\n (lift (\\A. A val) (igSubstBinp (asIMOD SEM) ys Y y binp))\n else undefined)"} {"_id": "502358", "text": "proof (prove)\nusing this:\n poss \\ []\n list_all safe_formula (map remove_neg negs)\n \\ (fv ` set negs) \\ \\ (fv ` set poss)\n (poss, negs) = partition safe_formula l\n\ngoal (1 subgoal):\n 1. safe_formula (formula.Ands l)"} {"_id": "502359", "text": "proof (prove)\nusing this:\n [m] \\g\\ [l] \\ [l] \\f\\ [l] \\\n [m] \\g\\ [l] \\\n ms \\g\\ [l] \\ [l] \\f\\ [l] \\\n ms \\g\\ [l]\n \\distinct ms;\n ms \\g\\ [l] \\ [l] \\f\\ [l] \\\n ms \\g\\ [l]\\\n \\ ms \\g\\ [l] \\\n [l] \\star \\ f\\ [l] \\\n ms \\g\\ [l]\n distinct (m # ms)\n\ngoal (1 subgoal):\n 1. ms \\g\\ [l] \\\n [l] \\star \\ f\\ [l] \\\n ms \\g\\ [l]"} {"_id": "502360", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A $ a $ b = (P ** A ** Q) $ a $ b"} {"_id": "502361", "text": "proof (prove)\ngoal (1 subgoal):\n 1. newpkt p \\\n newpkt (p\\p_iiface := iface_sel iface\\)"} {"_id": "502362", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (rec_option Map.empty (\\x f. f x),\n rec_option Map.empty (\\x f. f x))\n \\ \\S\\option_rel \\\n (S \\ \\R\\option_rel) \\\n \\R\\option_rel"} {"_id": "502363", "text": "proof (prove)\ngoal (2 subgoals):\n 1. length (fst z) = 2\n 2. x \\ y"} {"_id": "502364", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Y \\ {} \\\n \\ (c ` X) ; \\ (d ` Y) =\n (\\x::'c\\X. \\y::'b\\Y. c x ; d y)\nvariables:\n d :: 'b \\ 'a\n X :: 'c set\n c :: 'c \\ 'a\n (;) :: 'a \\ 'a \\ 'a\n Y :: 'b set\ntype variables:\n 'b, 'c :: type\n 'a :: refinement_lattice"} {"_id": "502365", "text": "proof (prove)\ngoal (1 subgoal):\n 1. App s u \\\\<^sub>\\\\<^sup>* App t v"} {"_id": "502366", "text": "proof (prove)\nusing this:\n B i \\ (T ^^ (n - i - 1)) --` (T ^^ (i + 1)) --` A\n (T ^^ (n - i - 1)) --` (T ^^ (i + 1)) --` A =\n (T ^^ (n - i - 1 + (i + 1))) --` A\n i < n\n\ngoal (1 subgoal):\n 1. B i \\ (T ^^ n) --` A"} {"_id": "502367", "text": "proof (prove)\ngoal (9 subgoals):\n 1. Tree2.inorder PST_RBT.empty = []\n 2. \\t a.\n rbt t \\ sorted1 (Tree2.inorder t) \\\n lookup t a = AList_Upd_Del.map_of (Tree2.inorder t) a\n 3. \\t a b.\n rbt t \\ sorted1 (Tree2.inorder t) \\\n Tree2.inorder (update a b t) = upd_list a b (Tree2.inorder t)\n 4. \\t a.\n rbt t \\ sorted1 (Tree2.inorder t) \\\n Tree2.inorder (delete a t) =\n AList_Upd_Del.del_list a (Tree2.inorder t)\n 5. rbt PST_RBT.empty\n 6. \\t a b.\n rbt t \\ sorted1 (Tree2.inorder t) \\\n rbt (update a b t)\n 7. \\t a.\n rbt t \\ sorted1 (Tree2.inorder t) \\\n rbt (delete a t)\n 8. \\t.\n \\rbt t; sorted1 (Tree2.inorder t)\\\n \\ rbt_is_empty t = (Tree2.inorder t = [])\n 9. \\t a b.\n \\rbt t; sorted1 (Tree2.inorder t);\n Tree2.inorder t \\ []; pst_getmin t = (a, b)\\\n \\ is_min2 (a, b) (set (Tree2.inorder t))"} {"_id": "502368", "text": "proof (prove)\nusing this:\n \\isOK (mapM (concurrent ?w ?l ?u) ?t);\n ?x \\ set (concat (projr (mapM (concurrent ?w ?l ?u) ?t)))\\\n \\ ?x \\ (InString \\ I) ` set ?t\n \\?x \\ (InString \\ I) ` set (substr ?w ?l ?u);\n isOK (idx ?w ?x)\\\n \\ ?l < projr (idx ?w ?x) \\ projr (idx ?w ?x) < ?u\n mapM (concurrent w l u) (substr w l u) = Inr d\n\ngoal (1 subgoal):\n 1. \\x y.\n \\x \\ set (concat d); idx w x = Inr y\\\n \\ l < y \\ y < u"} {"_id": "502369", "text": "proof (prove)\nusing this:\n Px \\ borel_measurable S\n Py \\ borel_measurable T\n\ngoal (1 subgoal):\n 1. density S Px \\\\<^sub>M density T Py =\n density (S \\\\<^sub>M T) (\\(x, y). Px x * Py y)"} {"_id": "502370", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a'.\n \\rtrancl3p r a bs a'; rtrancl3p r a' bs' a''\\\n \\ thesis) \\\n thesis"} {"_id": "502371", "text": "proof (prove)\ngoal (1 subgoal):\n 1. well_complete B (\\)"} {"_id": "502372", "text": "proof (prove)\ngoal (1 subgoal):\n 1. G \\ dgrad_sig_set' (length fs) d"} {"_id": "502373", "text": "proof (prove)\nusing this:\n \\ n < dfg\n df \\ fls_subdegree f\n dg \\ fls_subdegree g\n\ngoal (1 subgoal):\n 1. (f * g) $$ n =\n (\\i = 0..nat (n - dfg).\n f $$ (df + int i) * g $$ (dg + int (nat (n - dfg) - i)))"} {"_id": "502374", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf {(x, y). strictly_generalizes_atm x y}"} {"_id": "502375", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (toFull \\ a = toFull \\ b) = (a = b)"} {"_id": "502376", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>p {wlp S (wlp (While M S) P)} S {wlp (While M S) P}"} {"_id": "502377", "text": "proof (prove)\nusing this:\n ?y \\# mset_set (\\ - {x}) \\ ?y \\ x\n\ngoal (1 subgoal):\n 1. \\y.\n y \\# mset_set (\\ - {x}) \\\n size\n (filter_mset ((\\) {x, y})\n (filter_mset ((\\) x) \\)) =\n nat \\"} {"_id": "502378", "text": "proof (prove)\ngoal (1 subgoal):\n 1. |Field (cardSuc |UNIV|)| =o |{} <+> Field (cardSuc |UNIV|)|"} {"_id": "502379", "text": "proof (prove)\ngoal (1 subgoal):\n 1. IArray\n (map (\\x. vec_to_iarray ((A + B) $ mod_type_class.from_nat x))\n [0..x. vec_to_iarray (A $ mod_type_class.from_nat x))\n [0..x. vec_to_iarray (B $ mod_type_class.from_nat x))\n [0..H.\n \\hermitean H \\ H \\ mat_zero;\n circline_type_cmat H < 0; is_diag_circline_cmat H\\\n \\ \\A\\{v. v \\ vec_zero}.\n \\B\\{v. v \\ vec_zero}.\n \\C\\{v. v \\ vec_zero}.\n \\ A \\\\<^sub>v B \\\n \\ A \\\\<^sub>v C \\\n \\ B \\\\<^sub>v C \\\n on_circline_cmat_cvec H A \\\n on_circline_cmat_cvec H B \\\n on_circline_cmat_cvec H C"} {"_id": "502381", "text": "proof (prove)\nusing this:\n length xvec = length Tvec\n distinct xvec\n xvec \\* Tvec\n\ngoal (1 subgoal):\n 1. xvec \\* subs P xvec Tvec &&&\n xvec \\* subs' I xvec Tvec &&& xvec \\* subs'' C xvec Tvec"} {"_id": "502382", "text": "proof (state)\nthis:\n \\ zcount (c_work c loc1) t' < 0\n\ngoal (1 subgoal):\n 1. \\zcount (c_work c loc1) t' < 0 \\\n t' \\#\\<^sub>z zmset_of\n (mset_set\n (set_antichain\n (frontier (c_pts c loc1)))) +\n (\\loc'\\UNIV.\n after_summary\n (zmset_of\n (mset_set\n (set_antichain (frontier (c_imp c loc')))))\n (summary loc' loc1));\n \\ zcount (c_work c loc1) t' < 0\\\n \\ \\t' loc1 xs.\n path loc1 loc2 xs \\\n t =\n results_in t'\n (sum_weights\n (map (\\(s, l, t). l) xs)) \\\n (t' \\\\<^sub>A frontier (c_pts c loc1) \\\n zcount (c_work c loc1) t' < 0)"} {"_id": "502383", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card\n (inversions xs \\\n map_prod ((+) m) ((+) m) ` inversions ys \\\n map_prod id ((+) m) ` inversions_between xs ys) =\n card (inversions xs \\ map_prod ((+) m) ((+) m) ` inversions ys) +\n card (map_prod id ((+) m) ` inversions_between xs ys)"} {"_id": "502384", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\xa x.\n \\extr (bad xa) (ik xa) (chan xa)\n \\ synth (analz (generators xa));\n chan xa \\ faked_chan_msgs xa \\ chan_generators;\n x \\ extr (bad xa) (ik xa) (chan xa)\\\n \\ x \\ synth\n (analz\n (generators\n(xa\\ik := insert M (ik xa)\\)))\n 2. \\xa xb.\n \\M \\ dy_fake_msg (bad xa) (ik xa) (chan xa);\n extr (bad xa) (ik xa) (chan xa)\n \\ synth (analz (generators xa));\n chan xa \\ faked_chan_msgs xa \\ chan_generators;\n xb \\ extr (bad xa) (ik xa) (chan xa)\\\n \\ xb \\ synth\n (analz\n (generators\n (xa\\ik := insert M (ik xa)\\)))\n 3. \\xa xb.\n \\M \\ dy_fake_msg (bad xa) (ik xa) (chan xa);\n extr (bad xa) (ik xa) (chan xa)\n \\ synth (analz (generators xa));\n chan xa \\ faked_chan_msgs xa \\ chan_generators;\n xb \\ chan xa; xb \\ chan_generators\\\n \\ xb \\ faked_chan_msgs\n (xa\\ik :=\n insert M (ik xa)\\)"} {"_id": "502385", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum (X <| Y) (X \\ Y) = sum (chi X) (X \\ Y)"} {"_id": "502386", "text": "proof (state)\nthis:\n to_fract y' / to_fract z' = to_fract r\n\ngoal (1 subgoal):\n 1. z' \\ (0::'a) \\ x = to_fract x'"} {"_id": "502387", "text": "proof (prove)\nusing this:\n wf_fmla_atom tys a\n list_all2 (is_of_type tys) ?xs16 ?Ts16 \\\n list_all2 (is_of_type tys') ?xs16 ?Ts16\n\ngoal (1 subgoal):\n 1. wf_fmla_atom tys' a"} {"_id": "502388", "text": "proof (prove)\nusing this:\n is_final (steps c p n2)\n\ngoal (1 subgoal):\n 1. is_final (steps c p n1)"} {"_id": "502389", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\f x. return_spmf f \\ return_spmf x = return_spmf (f x)\n 2. \\g f x.\n return_spmf (\\g f x. g (f x)) \\ g \\\n f \\\n x =\n g \\ (f \\ x)\n 3. \\f x y.\n return_spmf (\\f x y. f y x) \\ f \\\n x \\\n y =\n f \\ y \\ x\n 4. \\x. return_spmf (\\x. x) \\ x = x"} {"_id": "502390", "text": "proof (prove)\nusing this:\n pmf (bv (Suc n)) xs = pmf (bv n) as * pmf (bernoulli_pmf (5 / 10)) a\n pmf (bv n) as = (1 / 2) ^ n\n\ngoal (1 subgoal):\n 1. pmf (bv (Suc n)) xs = (1 / 2) ^ n * 1 / 2"} {"_id": "502391", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\ l < u \\\n (if c (l, s0) then f (l, s0) \\ WHILE\\<^sub>T c f\n else RETURN (l, s0))\n \\ SPEC\n (\\(uu_, r).\n r = F {list ! i |i. l \\ i \\ i < u} s0)\n 2. \\ \\ l < u \\\n WHILE\\<^sub>T c f (l, s0)\n \\ SPEC\n (\\(uu_, r).\n r = F {list ! i |i. l \\ i \\ i < u} s0)"} {"_id": "502392", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\f.\n continuous_on U f \\ f ` U \\ S \\\n (\\c.\n homotopic_with_canon (\\x. True) U S f\n (\\x. c))) =\n (\\f.\n continuous_on U f \\ f ` U \\ T \\\n (\\c.\n homotopic_with_canon (\\x. True) U T f (\\x. c)))"} {"_id": "502393", "text": "proof (prove)\nusing this:\n f integrable_on s\n g integrable_on s\n ?x \\ s \\ f ?x \\ g ?x\n\ngoal (1 subgoal):\n 1. integral s f \\ integral s g"} {"_id": "502394", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.hanoi (Suc n) =\n repeat (2 ^ n - 1 + 1 + (2 ^ n - 1)) (tick (return ()))"} {"_id": "502395", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sep < p"} {"_id": "502396", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (learn_lim \\ U s \\\n (\\f\\U.\n \\n k.\n k \\ n \\\n \\ (the (s (f \\ n))) k = f k))"} {"_id": "502397", "text": "proof (prove)\nusing this:\n C.ide a\n \\o ?e ?b =\n S.mkArr (Hom.set (?b, a)) (Ya'.SET ?b)\n (\\x. Ya'.FUN (\\ (?b, a) x) ?e)\n Cop.ide ?b \\ S.arr (\\ e ?b)\n Cop.ide ?b \\\n \\ e ?b =\n S.mkArr (Hom.set (?b, a)) (Ya'.SET ?b)\n (\\x. Ya'.FUN (\\ (?b, a) x) e)\n\ngoal (1 subgoal):\n 1. \\ e a =\n S.mkArr (Hom.set (a, a)) (Ya'.SET a)\n (\\x. Ya'.FUN (\\ (a, a) x) e)"} {"_id": "502398", "text": "proof (prove)\nusing this:\n r = \\\\\n invar \\l, (xy, n), r\\\n\ngoal (1 subgoal):\n 1. l = \\\\"} {"_id": "502399", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eventually ((\\) M) sequentially"} {"_id": "502400", "text": "proof (prove)\nusing this:\n set (concat\n (map (possible_assignments_for \\)\n (filter (\\v. s v \\ None)\n (\\\\<^sub>\\\\<^sub>+)))) =\n (\\v\\{v |v.\n v \\ set (\\\\<^sub>\\\\<^sub>+) \\\n s v \\ None}.\n set (possible_assignments_for \\ v))\n\ngoal (1 subgoal):\n 1. set (concat\n (map (possible_assignments_for \\)\n (filter (\\v. s v \\ None)\n (\\\\<^sub>\\\\<^sub>+)))) =\n (\\v\\{v |v.\n v \\ set (\\\\<^sub>\\\\<^sub>+) \\\n s v \\ None}.\n {(v, a) |a. a \\ \\\\<^sub>+ \\ v})"} {"_id": "502401", "text": "proof (prove)\nusing this:\n simulation as\\<^sub>0 alstep bs\\<^sub>0 blstep R\\<^sub>1\n simulation bs\\<^sub>0 blstep cs\\<^sub>0 clstep R\\<^sub>2\n\ngoal (1 subgoal):\n 1. simulation as\\<^sub>0 alstep cs\\<^sub>0 clstep (R\\<^sub>1 OO R\\<^sub>2)"} {"_id": "502402", "text": "proof (prove)\nusing this:\n f = prod_list us\n u \\ set us\n\ngoal (1 subgoal):\n 1. u dvd f"} {"_id": "502403", "text": "proof (prove)\nusing this:\n \\\\ (\\\\ p \\ !r) = !r \\ \\\\ p \\ !r\n H (\\\\ p \\ \\\\ r) x p\n H ?p ?x ?q =\n \\\\\\ ?p\\?x\\?x \\ \\\\ ?q\\\n H (\\\\ ?p \\ \\\\ ?r) ?x ?q \\\n H ?p (\\\\ ?r \\ ?x) ?q\n \\?y\\?x\\?x \\ ?y\\ \\\n \\?y\\?x\\<^sup>\\\\?x\\<^sup>\\ \\\n ?y\\\n \\\\?z\\?w\\?w \\ ?z \\ ?w\\;\n \\?x\\?y\\?y \\ ?z\\\\\n \\ \\?x\\?y \\\n?w\\?y \\ ?w \\ (?z \\ ?w)\\\n while ?b do ?x = (\\\\ ?b \\ ?x)\\<^sup>\\ \\ !?b\n\ngoal (1 subgoal):\n 1. H p (while r do x) (\\\\ p \\ !r)"} {"_id": "502404", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\set M.\n case x of\n (m_impl, m_spec) \\\n SecurityInvariant_complies_formal_def m_impl m_spec;\n wf_list_graph G; valid_reqs (get_spec M);\n TopoS_Composition_Theory.all_security_requirements_fulfilled\n (get_spec M) (list_graph_to_graph G);\n \\ =\n \\hostsL = nodesL G, flows_fixL = edgesL G,\n flows_stateL =\n inefficient_list_intersect\n (TopoS_Stateful_Policy_impl.filter_IFS_no_violations G\n (get_impl M))\n (TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS G\n (get_impl M))\\\\\n \\ set (nodesL G) = nodes (list_graph_to_graph G) \\\n set (edgesL G) = edges (list_graph_to_graph G) \\\n set (inefficient_list_intersect\n (TopoS_Stateful_Policy_impl.filter_IFS_no_violations\n G (get_impl M))\n (TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS\n G (get_impl M))) =\n set (TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations\n (list_graph_to_graph G) (get_spec M)\n (edgesL G)) \\\n set (TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS\n (list_graph_to_graph G) (get_spec M) (edgesL G))"} {"_id": "502405", "text": "proof (prove)\nusing this:\n xs ! j = x'\n j < length xs - 1\n j < length xs\n ys ! k = y'\n k < length ys - 1\n k < length ys\n \\j \\ {0.. {0..\n \\ xs ! j = ys ! k \\\n j = length xs - 1 \\ k = length ys - 1\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502406", "text": "proof (prove)\nusing this:\n split ts x = (ls, rrs)\n a = (sub, sep)\n rrs = a # rs\n abs_split.ins k x sub = abs_split.Up\\<^sub>i l w r\n x \\ sep\n\ngoal (2 subgoals):\n 1. \\abs_split.ins k x sub = abs_split.Up\\<^sub>i l w r;\n sep \\ x; tia__ = Some p; split (ls @ a # rs) x = (ls, a # rs);\n rrs = a # rs; ts = ls @ a # rs;\n (ab_, bb_) \\\n is_pfa (2 * k) (zs1 @ (suba, sepi) # zs2) (tsil, tsin) *\n blist_assn k ls zs1 *\n (btree_assn k \\\\<^sub>a id_assn) a (suba, sepi) *\n blist_assn k rs zs2 *\n p \\\\<^sub>r Btnode (tsil, tsin) ti *\n btree_assn k t ti *\n true;\n n = Suc (length zs1 + length zs2);\n length ls < Suc (length zs1 + length zs2);\n (zs1 @ (suba, sepi) # zs2) ! length ls = (suba, sepa);\n z = (suba, sepa); sepa \\ x; subi = suba\\\n \\ \\<^sub>a id_assn) a\n (suba, sepi) *\n blist_assn k rs zs2 *\n p \\\\<^sub>r Btnode (tsil, tsin) ti *\n btree_assn k t ti *\n true> ins k x suba \\\n case_btupi\n (\\lp.\n pfa_set (tsil, tsin) (length ls)\n (lp, sepa) \\\n (\\_. return (T\\<^sub>i (Some p))))\n (\\lp x' rp.\n pfa_set (tsil, tsin) (length ls)\n (rp, sepa) \\\n (\\_.\nif Suc (length zs1 + length zs2) < 2 * k\nthen pfa_insert (tsil, tsin) (length ls) (lp, x') \\\n (\\kvs'.\n p := Btnode kvs' ti \\\n (\\_. return (T\\<^sub>i (Some p))))\nelse pfa_insert_grow (tsil, tsin) (length ls) (lp, x') \\\n (\\kvs'.\n node\\<^sub>i k kvs'\n ti))) i k (ls @ (l, w) # (r, sep) # rs)\n t)>\\<^sub>t\n 2. \\abs_split.ins k x sub = abs_split.Up\\<^sub>i l w r;\n sep \\ x; tia__ = Some p; split (ls @ a # rs) x = (ls, a # rs);\n rrs = a # rs; ts = ls @ a # rs;\n (ab_, bb_) \\\n is_pfa (2 * k) (zs1 @ (subi, sepi) # zs2) (tsil, tsin) *\n blist_assn k ls zs1 *\n (btree_assn k \\\\<^sub>a id_assn) a (subi, sepi) *\n blist_assn k rs zs2 *\n p \\\\<^sub>r Btnode (tsil, tsin) ti *\n btree_assn k t ti *\n true;\n n = length (zs1 @ (subi, sepi) # zs2);\n length ls < length (zs1 @ (subi, sepi) # zs2);\n (zs1 @ (subi, sepi) # zs2) ! length ls = (suba, sepa);\n z = (suba, sepa); sepa \\ x\\\n \\ subi = suba"} {"_id": "502407", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {fin_cut_same (any, replicate (length I) False) xa @\n replicate m (any, replicate (length I) False) |\n xa m. xa \\ map \\ ` enc (w, x # I)} =\n enc (w, I)"} {"_id": "502408", "text": "proof (prove)\nusing this:\n \\x.\n (\\y.\n (y, x) \\ Tree_wf \\ P y) \\\n P x\n\ngoal (1 subgoal):\n 1. P (tPred x1a_ x2a_)"} {"_id": "502409", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {k. k \\ n \\\n \\ (\\i xs.\n i \\ n \\\n set xs \\ {0..k} \\\n (0::'a) \\ len m i i xs)} \\\n {}"} {"_id": "502410", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_graph\n \\nodes = hosts \\,\n edges =\n flows_fix \\ \\ flows_state \\ \\\n backflows (flows_state \\)\\ \\\n wf_graph\n \\nodes = hosts \\,\n edges =\n flows_fix \\ \\\n (flows_state \\ - backflows (flows_fix \\)) \\\n (backflows (flows_state \\) - flows_fix \\)\\"} {"_id": "502411", "text": "proof (prove)\ngoal (1 subgoal):\n 1. block_design \\\\<^sup>+ \\\\<^sup>+ \\"} {"_id": "502412", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f x.\n measure_pmf.expectation (K' (Some x)) f =\n measure_pmf.expectation (K x) (\\x. f (Some x))"} {"_id": "502413", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nc\\<^sup>\\ = U"} {"_id": "502414", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A = UNIV \\ bij_lens typedef\\<^sub>L"} {"_id": "502415", "text": "proof (state)\nthis:\n r x a\n rtrancl_path r a (as @ y # ys) z\n\ngoal (1 subgoal):\n 1. \\a xs x.\n \\\\x.\n \\\\rtrancl_path r x (xs @ [y]) y;\n rtrancl_path r y ys z\\\n \\ thesis;\n rtrancl_path r x (xs @ y # ys) z\\\n \\ thesis;\n \\rtrancl_path r x ((a # xs) @ [y]) y;\n rtrancl_path r y ys z\\\n \\ thesis;\n rtrancl_path r x ((a # xs) @ y # ys) z\\\n \\ thesis"} {"_id": "502416", "text": "proof (prove)\ngoal (1 subgoal):\n 1. top_pres ((`) map_IInf)"} {"_id": "502417", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (range T) = (\\n. T (f n))"} {"_id": "502418", "text": "proof (state)\nthis:\n target (w || r2) p2 = last r2\n\ngoal (1 subgoal):\n 1. \\nat.\n \\length w = 0 \\\n target (w || r1 || r2) (p1, p2) = (p1, p2);\n 0 < length w \\\n target (w || r1 || r2) (p1, p2) = last (r1 || r2);\n length w = Suc nat\\\n \\ target (w || r1) p1 =\n fst (target (w || r1 || r2) (p1, p2)) \\\n target (w || r2) p2 =\n snd (target (w || r1 || r2) (p1, p2))"} {"_id": "502419", "text": "proof (prove)\nusing this:\n j < n\n j = i + Suc k\n\ngoal (1 subgoal):\n 1. (replicate i ze @ on # replicate (n - 1 - i) ze) ! j =\n (if i = j then on else ze)"} {"_id": "502420", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\q @ p \\ []; x \\ dom (C a)\\\n \\ C (list2FWpolicy (q @ p @ [a])) x =\n C (list2FWpolicy (q @ p)) x"} {"_id": "502421", "text": "proof (prove)\ngoal (1 subgoal):\n 1. abs_Gromov_completion u \\ Gromov_boundary"} {"_id": "502422", "text": "proof (prove)\nusing this:\n distinct l0\n x \\ set l0\n x \\ S0\n set l0 \\ S0\n distinct (map \\ l0)\n sorted_wrt Ra (map \\ l0)\n ?x' \\ set l0 \\ Rc x ?x'\n\ngoal (1 subgoal):\n 1. distinct (map \\ (x # l0)) \\\n sorted_wrt Ra (map \\ (x # l0))"} {"_id": "502423", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\w a wa aa wb ab.\n \\f\\a = Writer\\wa\\aa;\n g\\aa = Writer\\wb\\ab\\\n \\ bindU\\\n (bindU\\(Writer\\w\\a)\\\n f)\\\n g =\n bindU\\(Writer\\w\\a)\\\n (\\ x. bindU\\(f\\x)\\g)"} {"_id": "502424", "text": "proof (prove)\ngoal (1 subgoal):\n 1. homeomorphism (UNIV - {bot_sphere}) UNIV st_proj2 st_proj2_inv"} {"_id": "502425", "text": "proof (prove)\nusing this:\n A Out P B \\ Cong A B X Y\n Cong ?A ?B ?C ?D \\\n Cong ?A ?B ?C ?D \\\n Cong ?A ?B ?D ?C \\\n Cong ?B ?A ?C ?D \\\n Cong ?B ?A ?D ?C \\\n Cong ?C ?D ?A ?B \\\n Cong ?C ?D ?B ?A \\ Cong ?D ?C ?A ?B \\ Cong ?D ?C ?B ?A\n ?A Out ?B ?C \\ ?A Out ?C ?B \\ ?A Out ?B ?C\n \\X0 Y0. Cong X Y X0 Y0 = l X0 Y0\n\ngoal (1 subgoal):\n 1. \\B. l A B \\ A Out B P"} {"_id": "502426", "text": "proof (prove)\nusing this:\n 0 < degree_m f \\\n (\\g h.\n degree_m g < degree_m f \\\n degree_m h < degree_m f \\ Mp f \\ Mp (g * h))\n f =m g\n\ngoal (1 subgoal):\n 1. 0 < degree_m g \\\n (\\ga h.\n degree_m ga < degree_m g \\\n degree_m h < degree_m g \\ Mp g \\ Mp (ga * h))"} {"_id": "502427", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a admS, order_class)"} {"_id": "502428", "text": "proof (prove)\nusing this:\n A B Perp C D\n \\ ?A ?B Perp ?A ?B\n \\?A ?B Perp ?X ?Y; ?C \\ ?D; Col ?A ?B ?C;\n Col ?A ?B ?D\\\n \\ ?C ?D Perp ?X ?Y\n ?A ?B Perp ?C ?D \\ ?A \\ ?B \\ ?C \\ ?D\n\ngoal (1 subgoal):\n 1. \\ Col A B C \\ \\ Col A B D"} {"_id": "502429", "text": "proof (state)\ngoal (1 subgoal):\n 1. 2 < M \\\n icount_impl M <\\r. \\ (r = lcount M)>\\<^sub>t"} {"_id": "502430", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\RETURN\n (msort f_\n (take (length (v_ # vb_ # vc_) div 2) (v_ # vb_ # vc_))) =\n REC\\<^sub>T\n (\\g xs.\n case xs of [] \\ RETURN []\n | [x] \\ RETURN [x]\n | x # aa # lista \\\n g (take (length xs div 2) xs) \\\n (\\a.\n g (drop (length xs div 2) xs) \\\n (\\b.\n RETURN (PAC_Checker_Init.merge f_ a b))))\n (take (length (v_ # vb_ # vc_) div 2) (v_ # vb_ # vc_));\n RETURN\n (msort f_ (drop (length (v_ # vb_ # vc_) div 2) (v_ # vb_ # vc_))) =\n REC\\<^sub>T\n (\\g xs.\n case xs of [] \\ RETURN []\n | [x] \\ RETURN [x]\n | x # aa # lista \\\n g (take (length xs div 2) xs) \\\n (\\a.\n g (drop (length xs div 2) xs) \\\n (\\b. RETURN (PAC_Checker_Init.merge f_ a b))))\n (drop (length (v_ # vb_ # vc_) div 2) (v_ # vb_ # vc_))\\\n \\ RETURN (msort f_ (v_ # vb_ # vc_)) =\n REC\\<^sub>T\n (\\g xs.\n case xs of [] \\ RETURN []\n | [x] \\ RETURN [x]\n | x # aa # lista \\\n g (take (length xs div 2) xs) \\\n (\\a.\n g (drop (length xs div 2) xs) \\\n (\\b.\n RETURN (PAC_Checker_Init.merge f_ a b))))\n (v_ # vb_ # vc_)"} {"_id": "502431", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a.\n \\(c, a) \\ R; \\ a\\\n \\ thesis) \\\n thesis"} {"_id": "502432", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\<^bold>g [^] y \\ g' [^] ya \\\n ((\\<^bold>g [^] fst w') [^] e \\ (g' [^] snd w') [^] e) =\n \\<^bold>g [^] y \\ g' [^] ya \\\n (\\<^bold>g [^] fst w' \\ g' [^] snd w') [^] e"} {"_id": "502433", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n x \\ y \\ prod a {..x} \\ prod a {..y}"} {"_id": "502434", "text": "proof (state)\nthis:\n satPB \\ Wax\n\ngoal (1 subgoal):\n 1. eintP (Guard \\) [eintF (Wit \\) []]"} {"_id": "502435", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ a'; b \\ b'\\\n \\ appendS\\a\\b \\\n appendS\\a'\\b'"} {"_id": "502436", "text": "proof (prove)\ngoal (1 subgoal):\n 1. plain_floatarith 0\n (Num (of_nat n) * Num (float_of xe) *\n Power (Abs (Num (float_of x0))) (n - 1) +\n Sum\\<^sub>e\n (\\k.\n Num (of_nat (n choose k)) *\n Power (Abs (Num (float_of x0))) (n - k) *\n Power (Num (float_of xe) + Num (float_of t)) k)\n [2..\\ : src\\<^sub>C\n f \\\\<^sub>C g \\\\<^sub>C\n f\\\n \\\\ : f \\\\<^sub>C\n g \\\\<^sub>C src\\<^sub>C\n g\\\n adjunction_in_bicategory (\\\\<^sub>D) (\\\\<^sub>D) \\\\<^sub>D\n \\\\<^sub>D src\\<^sub>D trg\\<^sub>D (F f) (F g)\n (D.inv (\\ (g, f)) \\\\<^sub>D\n F \\ \\\\<^sub>D unit (src\\<^sub>C f))\n (D.inv (unit (trg\\<^sub>C f)) \\\\<^sub>D\n F \\ \\\\<^sub>D \\ (f, g))\n trg\\<^sub>C g = src\\<^sub>C f\n src\\<^sub>C g = trg\\<^sub>C f\n\ngoal (1 subgoal):\n 1. C.par\n ((g \\\\<^sub>C \\) \\\\<^sub>C\n \\\\<^sub>C[g, f, g] \\\\<^sub>C (\\ \\\\<^sub>C g))\n (\\\\<^sub>C\\<^sup>-\\<^sup>1[g] \\\\<^sub>C \\\\<^sub>C[g])"} {"_id": "502438", "text": "proof (prove)\nusing this:\n \\ \\ s \\ t : T\n\ngoal (1 subgoal):\n 1. s \\| &&& t \\|"} {"_id": "502439", "text": "proof (prove)\nusing this:\n - (- - x \\ - - y) = - x \\ - y\n\ngoal (1 subgoal):\n 1. - - x \\ - y \\ - x \\ - z =\n (- x \\ - y) \\ (- - x \\ - z)"} {"_id": "502440", "text": "proof (prove)\nusing this:\n new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t\n h =\n (new_shadow_root_ptr, h')\n\ngoal (1 subgoal):\n 1. object_ptr_kinds h' =\n object_ptr_kinds h |\\| {|cast (cast new_shadow_root_ptr)|}"} {"_id": "502441", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum (card \\\n (\\i.\n (\\l. replicate i R @ B # l) `\n {l. length l = n - i - 1 \\ valid l}))\n {3..i = 3.. Qi) (ls.\\ \\))\n 2. wa_invar (det_wa_wa (brec_det_algo rqrm Qi \\)) =\n ?addi \\\n {s. brec_\\ s\n \\ wa_invar (brw_algo (hs.\\ Qi) (ls.\\ \\))}\n 3. \\s s'.\n \\s \\ ?addi;\n s \\ wa_cond (det_wa_wa (brec_det_algo rqrm Qi \\));\n brec_\\ s\n \\ wa_invar (brw_algo (hs.\\ Qi) (ls.\\ \\));\n (s, s')\n \\ wa_step (det_wa_wa (brec_det_algo rqrm Qi \\))\\\n \\ s' \\ ?addi\n 4. wa_initial (det_wa_wa (brec_det_algo rqrm Qi \\))\n \\ ?addi"} {"_id": "502444", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\<^sub>\\ r\n a \\\\<^sub>\\ A\n\ngoal (1 subgoal):\n 1. vinsert (r\\a\\) (r `\\<^sub>\\ A) = r `\\<^sub>\\ A"} {"_id": "502445", "text": "proof (prove)\nusing this:\n 0 < x\n (x * 2 - inverse x * 2) / (2 * x + 2 * inverse x) =\n (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)\n\ngoal (1 subgoal):\n 1. tanh (ln x) = (x\\<^sup>2 - 1) / (x\\<^sup>2 + 1)"} {"_id": "502446", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i\\B. (f x \\ i) *\\<^sub>R i) = f x"} {"_id": "502447", "text": "proof (prove)\nusing this:\n \\' \\ is_rreq \\\n\ngoal (1 subgoal):\n 1. \\hops' rreqid' dip' dsn' dsk' oip' osn' sip'.\n msg \\ = Rreq hops' rreqid' dip' dsn' dsk' oip' osn' sip' \\\n \\' = \\\n \\hops := hops', rreqid := rreqid', dip := dip', dsn := dsn',\n dsk := dsk', oip := oip', osn := osn', sip := sip'\\"} {"_id": "502448", "text": "proof (prove)\nusing this:\n r'\\<^sub>1 \\ r1 \\ (w1 \\\\<^sub>E r'\\<^sub>1)\n r'\\<^sub>2 \\ r2 \\ (w2 \\\\<^sub>E r'\\<^sub>2)\n \\?a \\ ?b; ?c \\ ?d\\\n \\ ?a + ?c \\ ?b + ?d\n r'\\<^sub>1 + r'\\<^sub>2 =\n real_of_int\n (sint\n (SCAST(64 \\ 32)\n (SCAST(32 \\ 64) w1 + SCAST(32 \\ 64) w2)))\n sint\n (SCAST(64 \\ 32)\n (SCAST(32 \\ 64) w1 + SCAST(32 \\ 64) w2)) =\n sint w1 + sint w2\n sint w1 + sint w2 < 2147483647\n sint w1 + sint w2 < 2147483647\n - 2147483647 < sint w1 + sint w2\n - 2147483647 < sint w1 + sint w2\n r'\\<^sub>1 + r'\\<^sub>2 =\n real_of_int\n (sint\n (SCAST(64 \\ 32)\n (SCAST(32 \\ 64) w1 + SCAST(32 \\ 64) w2)))\n\ngoal (1 subgoal):\n 1. r'\\<^sub>1 + r'\\<^sub>2 =\n real_of_int\n (sint\n (SCAST(64 \\ 32)\n (SCAST(32 \\ 64) w1 +\n SCAST(32 \\ 64) w2))) \\\n r'\\<^sub>1 + r'\\<^sub>2 < 2147483647 \\\n - 2147483647 < r'\\<^sub>1 + r'\\<^sub>2"} {"_id": "502449", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ring (S\\carrier := h ` carrier R, zero := h \\\\)"} {"_id": "502450", "text": "proof (prove)\nusing this:\n DN.map\\<^sub>0 (UP.map\\<^sub>0 a') = a'\n B.obj a'\n B.obj ?a \\ B.equivalent_objects ?a ?a\n\ngoal (1 subgoal):\n 1. B.equivalent_objects (DN.map\\<^sub>0 (UP.map\\<^sub>0 a')) a'"} {"_id": "502451", "text": "proof (state)\nthis:\n \\\\

\\<^sub>1[g.leg0, f.leg1] \\g.cod.leg0, f.cod.leg1\\ \\

\\<^sub>0[g.leg0, f.leg1]\\ =\n \\\\

\\<^sub>1[g.leg0, f.leg1] \\\n (g.leg0 \\\\\n f.leg1) \\g.cod.leg0, f.cod.leg1\\ \\

\\<^sub>0[g.leg0, f.leg1] \\\n (g.leg0 \\\\ f.leg1)\\\n\ngoal (1 subgoal):\n 1. chine_hcomp g f = g.leg0 \\\\ f.leg1"} {"_id": "502452", "text": "proof (prove)\ngoal (3 subgoals):\n 1. finite\n (ta_rules\n \\ta_initial = ta.Qi \\ ta'.Qi,\n ta_rules = ta.\\ \\ ta'.\\\\)\n 2. finite\n (ta_initial\n \\ta_initial = ta.Qi \\ ta'.Qi,\n ta_rules = ta.\\ \\ ta'.\\\\)\n 3. \\q f qs.\n q \\ f qs\n \\ ta_rules\n \\ta_initial = ta.Qi \\ ta'.Qi,\n ta_rules =\n ta.\\ \\\n ta'.\\\\ \\\n A f = Some (length qs)"} {"_id": "502453", "text": "proof (prove)\nusing this:\n evaluate True ?env ?s ?e\n ((SOME x. \\env e s. evaluate True env s e (x env e s)) ?env ?e\n ?s)\n\ngoal (1 subgoal):\n 1. evaluate True env_ s_ e_\n ((SOME f. \\env e s. evaluate True env s e (f env e s)) env_ e_\n s_)"} {"_id": "502454", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\t. exp (t *\\<^sub>R - s) * f' t) has_integral s * L - f0)\n {0..}"} {"_id": "502455", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\\\\\ \\ [$C v] : ts _> ts'"} {"_id": "502456", "text": "proof (state)\nthis:\n \\twoC\\{diamond_cube}.\n analytically_valid (cubeImage twoC) (\\x. F x \\ i) j\n\ngoal (1 subgoal):\n 1. integral (cubeImage diamond_cube)\n (\\x.\n partial_vector_derivative (\\x. F x \\ j) i x -\n partial_vector_derivative (\\x. F x \\ i) j x) =\n one_chain_line_integral F {i, j} (boundary diamond_cube)"} {"_id": "502457", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\f 0 = (c # cs, s); \\i. f i \\ f (Suc i);\n \\i. fst (f i) = cs\\\n \\ \\i\\LEAST i. fst (f i) = cs.\n \\p.\n (p \\ []) =\n (i < (LEAST i. fst (f i) = cs)) \\\n fst (f i) = p @ cs"} {"_id": "502458", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Sorting_Algorithms.sort cmp ys = xs"} {"_id": "502459", "text": "proof (prove)\ngoal (1 subgoal):\n 1. uniformly_continuous_on s (\\x. c)"} {"_id": "502460", "text": "proof (prove)\nusing this:\n (P, B, Q) = diagonal_to_Smith_PQ A bezout\n\ngoal (1 subgoal):\n 1. (P, B, Q) =\n diagonal_to_Smith_aux_PQ [0.. C (=) ===> rel_gpv A C ===> (=)) (gen_lossless_gpv b)\n (gen_lossless_gpv b)"} {"_id": "502462", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Tree2.set_tree (list_to_rbt xs) = set xs"} {"_id": "502463", "text": "proof (state)\nthis:\n bad1 s1'' \\ X_bad s1'' s2'\n\ngoal (2 subgoals):\n 1. \\a b aa ba.\n \\(a, b) \\ set_spmf (exec_gpv oracle1 (c r1) s1');\n (aa, ba) \\ set_spmf (exec_gpv oracle2 (c r2) s2')\\\n \\ bad1 b = bad2 ba\n 2. \\a b aa ba.\n \\(a, b) \\ set_spmf (exec_gpv oracle1 (c r1) s1');\n (aa, ba) \\ set_spmf (exec_gpv oracle2 (c r2) s2')\\\n \\ if bad2 ba then X_bad b ba\n else a = aa \\ X b ba"} {"_id": "502464", "text": "proof (prove)\nusing this:\n S'.arr f'\n \\ ?f =\n (if S.arr ?f\n then S'.mkArr (S'.set (\\o (S.dom ?f))) (S'.set (\\o (S.cod ?f)))\n (\\a ?f)\n else S'.null)\n S'.arr ?f \\\n \\ ?f \\ S.hom (\\o (S'.dom ?f)) (\\o (S'.cod ?f))\n S'.ide ?a' \\ \\o (\\o ?a') = ?a'\n\ngoal (1 subgoal):\n 1. \\ (\\ f') =\n S'.mkArr (S'.set (S'.dom f')) (S'.set (S'.cod f')) (\\a (\\ f'))"} {"_id": "502465", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(of_match_field, linorder_class)"} {"_id": "502466", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite S"} {"_id": "502467", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a \\ b \\ rank a < rank b"} {"_id": "502468", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum f {..n} = Suc n"} {"_id": "502469", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\dom f = dom lz; dom f' = dom f; finite L1;\n \\l\\dom f.\n \\s p.\n s \\ L1 \\\n p \\ L1 \\ s \\ p \\\n (\\t.\n (the (f l)\\<^bsup>[Fvar s,Fvar p]\\<^esup> => t \\\n (\\z.\n the (f l)\\<^bsup>[Fvar s,Fvar p]\\<^esup> =>\n z \\\n (\\u. t => u \\ z => u))) \\\n the (f' l) = \\[s,p] t);\n finite L2;\n \\l\\dom lz.\n \\s p.\n s \\ L2 \\\n p \\ L2 \\ s \\ p \\\n (\\t.\n the (f l)\\<^bsup>[Fvar s,Fvar p]\\<^esup> => t \\\n the (lz l) = \\[s,p] t)\\\n \\ \\L'.\n finite L' \\\n (\\lu.\n dom lu = dom f \\\n (\\l\\dom f.\n \\s p.\n s \\ L' \\\n p \\ L' \\\n s \\ p \\\n (\\t.\n the (f' l)\\<^bsup>[Fvar s,Fvar p]\\<^esup> => t \\\n the (lu l) = \\[s,p] t)) \\\n (\\l\\dom f.\n body (the (f' l))) \\\n (\\l\\dom f.\n \\s p.\n s \\ L' \\\n p \\ L' \\\n s \\ p \\\n (\\t.\n the (lz l)\\<^bsup>[Fvar s,Fvar p]\\<^esup> => t \\\n the (lu l) = \\[s,p] t)) \\\n (\\l\\dom f. body (the (lz l))))"} {"_id": "502470", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a\\(b, i) #\n bs. case a of (a, i) \\ a * a ^ i) =\n smult\n (\\x\\(b, i) #\n bs. fst (case x of\n (b, i) \\\n case f b of\n (d, fs) \\\n (d ^ Suc i, map (\\f. (f, i)) fs)))\n (\\a\\concat\n (map (\\x.\n snd (case x of\n (b, i) \\\n case f b of\n (d, fs) \\ (d ^ Suc i, map (\\f. (f, i)) fs)))\n ((b, i) #\n bs)). case a of\n (a, i) \\ a * a ^ i)"} {"_id": "502471", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f \\ lim f"} {"_id": "502472", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fun.\n inf (pred_fun top bounded_linear) bounded_bilinear\n fun \\\n inf (pred_fun top bounded_linear) bounded_bilinear\n (\\x y. fun y x)"} {"_id": "502473", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x\\range left \\ range right \\ \n \\)\n \\ COPY"} {"_id": "502474", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vars_term_ms (SCF (Fun f (ss1 @ t # ss2))) \\#\n vars_term_ms (SCF (Fun f (ss1 @ s # ss2)))"} {"_id": "502475", "text": "proof (prove)\nusing this:\n \\g. g \\ f = id \\ f \\ g = id\n\ngoal (1 subgoal):\n 1. inj f"} {"_id": "502476", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ssubst x v k = k"} {"_id": "502477", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pmf_prob p (set xs) = sum (pmf p) (set xs) &&&\n pmf_prob p (List.coset xs) = 1 - sum (pmf p) (set xs)"} {"_id": "502478", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\local.inv t\\<^sub>0u\\<^sub>1.\\ \\\n chine : (u\\<^sub>1 \\\n t\\<^sub>0u\\<^sub>1.p\\<^sub>0) \\\n chine \\ (t\\<^sub>0 \\\n t\\<^sub>0u\\<^sub>1.p\\<^sub>1) \\\n chine\\"} {"_id": "502479", "text": "proof (prove)\nusing this:\n valid_edge a''\n kind a'' = Q''\\\\<^bsub>p\\<^esub>f''\n\ngoal (1 subgoal):\n 1. method_exit (sourcenode a'')"} {"_id": "502480", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {y. \\v\\0.\n \\u.\n (\\x\\{b, c}. 0 \\ u x) \\\n sum u {b, c} = 1 - v \\\n (\\x\\{b, c}. u x *\\<^sub>R x) = y - v *\\<^sub>R a} =\n {u *\\<^sub>R a + v *\\<^sub>R b + w *\\<^sub>R c |u v w.\n 0 \\ u \\ 0 \\ v \\ 0 \\ w \\ u = 1 - v - w}"} {"_id": "502481", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mono f \\ gfp f = \\ Fix f"} {"_id": "502482", "text": "proof (prove)\ngoal (1 subgoal):\n 1. freshEnv xs x rho = qFreshEnv xs x (pickE rho)"} {"_id": "502483", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ifex_ordered (IF v t e); v' < v\\\n \\ restrict_top (IF v t e) v' val = IF v t e"} {"_id": "502484", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fv s \\ fv t \\\n fv (s \\ \\) \\ fv (t \\ \\)"} {"_id": "502485", "text": "proof (prove)\nusing this:\n Range\n (a \\ {seller} \\ ({\\ - \\ (Range a)} - {{}})) =\n Range a \\ ({\\ - \\ (Range a)} - {{}})\n ?A \\ (?B - ?A) = ?A \\ ?B\n ?B - ?A \\ ?A = ?B \\ ?A\n \\ (?A \\ ?B) = \\ ?A \\ \\ ?B\n \\ {} = {}\n \\ (insert ?a ?B) = ?a \\ \\ ?B\n\ngoal (1 subgoal):\n 1. \\\n (Range\n (a \\\n {seller} \\ ({\\ - \\ (Range a)} - {{}}))) =\n \\"} {"_id": "502486", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x\\<^sup>T = x"} {"_id": "502487", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_finite_Float TYPE((11, 52) IEEE.float) f =\n (let e = Float.exponent f; bm = bitlen \\mantissa f\\\n in bm \\ 53 \\ e + bm < 1025 \\ - 1075 < e)"} {"_id": "502488", "text": "proof (prove)\nusing this:\n (\\\\.\n ?w \\\\<^sub>n (rewrite_modal \\ rewrite_X) \\ =\n ?w \\\\<^sub>n \\) \\\n ?w \\\\<^sub>n iterate (rewrite_modal \\ rewrite_X) ?\\\n ?n =\n ?w \\\\<^sub>n ?\\\n\ngoal (1 subgoal):\n 1. w \\\\<^sub>n rewrite_iter_fast \\ =\n w \\\\<^sub>n \\"} {"_id": "502489", "text": "proof (prove)\nusing this:\n 2 * n + c_main \\ 2 * n + (2 * n + 1 + 2 * i) * i\n\ngoal (1 subgoal):\n 1. result (dmu_array_row_cost dmus i) = dmu_array_row fs dmus i \\\n cost (dmu_array_row_cost dmus i) \\ 2 * n + (2 * n + 1 + 2 * i) * i"} {"_id": "502490", "text": "proof (prove)\nusing this:\n iwf fc t ent\n\ngoal (1 subgoal):\n 1. length (iAnts t) \\ length (inPorts' (iNodeOf (tree_at t [])))"} {"_id": "502491", "text": "proof (state)\nthis:\n ?w \\ ball \\ (min r0 r1) \\\n f ?w = (?w - \\) ^ n * g ?w\n\ngoal (4 subgoals):\n 1. 0 < n\n 2. 0 < min r0 r1\n 3. ball \\ (min r0 r1) \\ S\n 4. \\w. w \\ ball \\ (min r0 r1) \\ g w \\ 0"} {"_id": "502492", "text": "proof (prove)\nusing this:\n Limit \\\n 0 \\\\<^sub>\\ \\\n\ngoal (1 subgoal):\n 1. \\ * \\ + 1\n \\\\<^sub>\\ \\ * \\ + \\"} {"_id": "502493", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x [^]\\<^bsub>R\\<^esub> (CARD('a) ^ m' - 1) = \\\\<^bsub>R\\<^esub>"} {"_id": "502494", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (positive (b - a) \\ a = b) \\ positive (a - b) \\ b = a"} {"_id": "502495", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dim_row\n (mat_of_rows_list (degree f')\n (map (\\cs.\n cs @\n replicate (degree f' - length cs)\n (arith_ops_record.zero ff_ops))\n (power_polys_i ff_ops\n (power_poly_f_mod_i ff_ops\n (\\v. mod_field_poly_i ff_ops v f)\n [arith_ops_record.zero ff_ops, arith_ops_record.one ff_ops]\n (nat p))\n f [arith_ops_record.one ff_ops] (degree f')))) =\n dim_row\n (mat_of_rows_list (degree f')\n (map (\\cs.\n coeffs cs @ replicate (degree f' - length (coeffs cs)) 0)\n (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1\n (degree f')))) \\\n dim_col\n (mat_of_rows_list (degree f')\n (map (\\cs.\n cs @\n replicate (degree f' - length cs)\n (arith_ops_record.zero ff_ops))\n (power_polys_i ff_ops\n (power_poly_f_mod_i ff_ops\n (\\v. mod_field_poly_i ff_ops v f)\n [arith_ops_record.zero ff_ops, arith_ops_record.one ff_ops]\n (nat p))\n f [arith_ops_record.one ff_ops] (degree f')))) =\n dim_col\n (mat_of_rows_list (degree f')\n (map (\\cs.\n coeffs cs @ replicate (degree f' - length (coeffs cs)) 0)\n (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1\n (degree f')))) \\\n (\\i j.\n i < dim_row\n (mat_of_rows_list (degree f')\n (map (\\cs.\n coeffs cs @\n replicate (degree f' - length (coeffs cs)) 0)\n (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1\n (degree f')))) \\\n j < dim_col\n (mat_of_rows_list (degree f')\n (map (\\cs.\n coeffs cs @\n replicate (degree f' - length (coeffs cs)) 0)\n (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1\n (degree f')))) \\\n R (mat_of_rows_list (degree f')\n (map (\\cs.\n cs @\n replicate (degree f' - length cs)\n (arith_ops_record.zero ff_ops))\n (power_polys_i ff_ops\n (power_poly_f_mod_i ff_ops\n (\\v. mod_field_poly_i ff_ops v f)\n [arith_ops_record.zero ff_ops,\n arith_ops_record.one ff_ops]\n (nat p))\n f [arith_ops_record.one ff_ops] (degree f'))) $$\n (i, j))\n (mat_of_rows_list (degree f')\n (map (\\cs.\n coeffs cs @\n replicate (degree f' - length (coeffs cs)) 0)\n (power_polys (power_poly_f_mod f' [:0, 1:] (nat p)) f' 1\n (degree f'))) $$\n (i, j)))"} {"_id": "502496", "text": "proof (prove)\nusing this:\n vertex_subtree w \\ T.left_tree s t\n vertex_subtree w \\ T.left_tree t s \\ {}\n ?s \\\\<^bsub>T\\<^esub> ?t \\\n T.left_tree ?s ?t \\ T.left_tree ?t ?s = {}\n s \\\\<^bsub>T\\<^esub> t\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502497", "text": "proof (prove)\nusing this:\n Domain r \\ Range r = {r.zero}\n r.zero \\ Domain r \\ Range r \\\n (r.zero, r.zero) \\ r\n\ngoal (1 subgoal):\n 1. r = {(r.zero, r.zero)}"} {"_id": "502498", "text": "proof (prove)\nusing this:\n x \\ X\n \\xa. real xa \\ u x\n \\ real (k x) < u x\n \\x\\X. 0 \\ u x\n (\\?x\\ < ?z) = (?x < of_int ?z)\n 0 \\ ?w \\ (nat ?w < ?m) = (?w < int ?m)\n\ngoal (1 subgoal):\n 1. valid_intv (k x) (Intv (nat \\u x\\))"} {"_id": "502499", "text": "proof (prove)\nusing this:\n p \\ phull (monomial (1::'a) ` deg_sect X z)\n\ngoal (1 subgoal):\n 1. (\\M u.\n \\M \\ monomial (1::'a) ` deg_sect X z; finite M;\n p = (\\m\\M. u m \\ m)\\\n \\ thesis) \\\n thesis"} {"_id": "502500", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum f {.. range g)"} {"_id": "502501", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_language letter_eq (Language1 (init1 r)) (Language2 (init2 s)) =\n (\\(r', s')\n \\{((r, s), delta1 a r, delta2 b s) |a b r s.\n a \\ set alphabet1 \\\n b \\ set alphabet2 \\ letter_eq a b}\\<^sup>* ``\n {(init1 r, init2 s)}.\n accept1 r' = accept2 s')"} {"_id": "502502", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\y\\A \\ range (X i). X i -` {y}) =\n (\\y\\A \\ range (X i).\n proj_stoch_proc X n -`\n \\\n (stream_space_single (proj_stoch_proc X n) ` comp_proj_i X n i y))"} {"_id": "502503", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ p m a rs.\n \\\\\\ matches \\ m a p;\n Undecided = Undecided\\\n \\ approximating_bigstep_fun \\ p\n (cut_off_after_match_any rs) Undecided =\n approximating_bigstep_fun \\ p rs\n Undecided;\n \\\\ \\ matches \\ m a p; a = Log;\n Undecided = Undecided\\\n \\ approximating_bigstep_fun \\ p\n (cut_off_after_match_any rs) Undecided =\n approximating_bigstep_fun \\ p rs Undecided;\n \\\\ \\ matches \\ m a p; a = Empty;\n Undecided = Undecided\\\n \\ approximating_bigstep_fun \\ p\n (cut_off_after_match_any rs) Undecided =\n approximating_bigstep_fun \\ p rs Undecided;\n Undecided = Undecided\\\n \\ approximating_bigstep_fun \\ p\n (cut_off_after_match_any (Rule m a # rs))\n Undecided =\n approximating_bigstep_fun \\ p\n (Rule m a # rs) Undecided"} {"_id": "502504", "text": "proof (prove)\nusing this:\n (h, as) \\ P\n (h, as') \\ Q\n as \\ as' = {}\n\ngoal (1 subgoal):\n 1. (h, as \\ as') \\\n Abs_assn (times_assn_raw (Rep_assn P) (Rep_assn Q))"} {"_id": "502505", "text": "proof (prove)\nusing this:\n \\ \\ {}\n finite (af_abs.abs_reach (Abs ?\\))\n\ngoal (1 subgoal):\n 1. finite (reach \\ \\af (Abs \\))"} {"_id": "502506", "text": "proof (prove)\nusing this:\n \\set (labels_conv\n (fst (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))) @\n snd \\3))\n \\ ?S;\n set (labels_conv (conv_mirror \\3)) \\ ?T\\\n \\ set (labels_conv\n (fst (fst (conv_mirror\n(fst \\1''', map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n(fst \\1''', map (Pair True) (snd \\1'''))) @\n snd \\3),\n snd (fst (conv_mirror\n(fst \\1''', map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n(fst \\1''', map (Pair True) (snd \\1'''))) @\n snd \\3) @\n snd (conv_mirror \\3)))\n \\ ?S \\ ?T\n set (labels_conv\n (fst (conv_mirror\n (fst \\1''', map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n (fst \\1''', map (Pair True) (snd \\1'''))) @\n snd \\3))\n \\ r \\s {fst \\_step, fst \\_step}\n conv_mirror \\3 \\ conv ars\n set (labels_conv (conv_mirror \\3))\n \\ r \\s {fst \\_step, fst \\_step}\n fst (conv_mirror \\3) = lst_conv \\3\n lst_conv (conv_mirror \\3) = fst \\1'''\n\ngoal (1 subgoal):\n 1. set (labels_conv\n (fst (fst (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))) @\n snd \\3),\n snd (fst (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))),\n snd (conv_mirror\n (fst \\1''',\n map (Pair True) (snd \\1'''))) @\n snd \\3) @\n snd (conv_mirror \\3)))\n \\ r \\s {fst \\_step, fst \\_step}"} {"_id": "502507", "text": "proof (prove)\nusing this:\n i.lpp g \\ keys g\n\ngoal (1 subgoal):\n 1. except (i.lpp g) {x} \\ (\\t. except t {x}) ` keys g"} {"_id": "502508", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\S\\S.\n wf\\<^sub>s\\<^sub>t (wfrestrictedvars\\<^sub>e\\<^sub>s\\<^sub>t A)\n (dual\\<^sub>s\\<^sub>t S)"} {"_id": "502509", "text": "proof (prove)\ngoal (1 subgoal):\n 1. birkhoff_sum (indicator A) n \\ borel_measurable M"} {"_id": "502510", "text": "proof (state)\nthis:\n \\a\\set (atoms\\<^sub>0 \\).\n divisor a dvd zlcms (map divisor (atoms\\<^sub>0 \\))\n\ngoal (1 subgoal):\n 1. Z.I (qe_cooper\\<^sub>1 \\) xs = (\\x. Z.I \\ (x # xs))"} {"_id": "502511", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\length key = len; length plain = len\\\n \\ encrypt key plain \\\n (\\cipher. return_spmf (decrypt key cipher)) =\n return_spmf (Some plain)"} {"_id": "502512", "text": "proof (prove)\nusing this:\n S.arr m \\\n S.set (S.dom m) \\ S.set (S.cod m) \\\n m = S.mkArr (S.set (S.dom m)) (S.set (S.cod m)) (\\x. x)\n S.ide ?a \\ \\ ?a = \\o ?a\n\ngoal (1 subgoal):\n 1. S'.mkArr (S'.set (\\o (S.dom m))) (S'.set (\\o (S.cod m)))\n (\\a m) =\n S'.mkArr (S'.set (S'.dom (\\ m))) (S'.set (S'.cod (\\ m)))\n (\\a m)"} {"_id": "502513", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s \\ \\\\<^sub>s S \\\n aval a s \\ \\ (aval' a S)"} {"_id": "502514", "text": "proof (prove)\nusing this:\n HNext s s'\n HInv1 s \\ HInv2 s \\ HInv2 s' \\ HInv4 s\n\ngoal (1 subgoal):\n 1. HInv4a s' p"} {"_id": "502515", "text": "proof (prove)\nusing this:\n finite S\n\ngoal (1 subgoal):\n 1. extract_var (p + p') v = extract_var p v + extract_var p' v"} {"_id": "502516", "text": "proof (prove)\ngoal (1 subgoal):\n 1. total1 (amalgamation (the (f 0)) (the (f 1)))"} {"_id": "502517", "text": "proof (prove)\ngoal (1 subgoal):\n 1. safe_formula \\ \\\n mstate_n\n \\mstate_i = 0, mstate_m = minit0 (Formula.nfv \\) \\,\n mstate_n = Formula.nfv \\\\ =\n Formula.nfv \\ \\\n (\\\\.\n prefix_of pnil \\ \\\n mstate_i\n \\mstate_i = 0,\n mstate_m = minit0 (Formula.nfv \\) \\,\n mstate_n = Formula.nfv \\\\ =\n Monitor.progress \\ Map.empty \\ (plen pnil) \\\n wf_mformula \\ (plen pnil) Map.empty Map.empty\n (mstate_n\n \\mstate_i = 0,\n mstate_m = minit0 (Formula.nfv \\) \\,\n mstate_n = Formula.nfv \\\\)\n R (mstate_m\n \\mstate_i = 0,\n mstate_m = minit0 (Formula.nfv \\) \\,\n mstate_n = Formula.nfv \\\\)\n \\)"} {"_id": "502518", "text": "proof (prove)\nusing this:\n wset_final_ok ws ts\n ws t = \\w\\\n \\ts t = \\(?x5, ?ln5)\\; \\ final ?x5\\\n \\ thesis\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "502519", "text": "proof (prove)\nusing this:\n t0 \\ T'\n is_interval T'\n T' \\ existence_ivl t0 x0\n (z solves_ode f) T' X\n z t0 = flow t0 x0 t0\n t \\ T'\n t0 \\ T'\n is_interval T'\n T' \\ existence_ivl t0 x0\n (z solves_ode f) T' X\n z t0 = flow t0 x0 t0\n t \\ T'\n existence_ivl t0 ?x0.0 \\ T\n ?t \\ existence_ivl t0 ?x0.0 \\ t0 \\ T\n ?t \\ existence_ivl t0 ?x0.0 \\ ?x0.0 \\ X\n\ngoal (1 subgoal):\n 1. t0 \\ T \\ x0 \\ X"} {"_id": "502520", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vars p = vars\\<^sub>0 \\\n (\\s\\<^sub>H.\n body_split p s\\<^sub>L\\<^sub>0 s\\<^sub>H \\\n wfs \\\\<^sub>0 s\\<^sub>H \\\n high_proc\\<^sub>2 s\\<^sub>H \\\n high_proc_no_low_output_change s\\<^sub>H) \\\n vars p = vars\\<^sub>0 \\\n (\\s\\<^sub>L s\\<^sub>H.\n body_split p s\\<^sub>L s\\<^sub>H \\\n wfs \\\\<^sub>0 s\\<^sub>L \\\n wfs \\\\<^sub>0 s\\<^sub>H \\\n low_proc_non0\\<^sub>2 s\\<^sub>L \\\n low_proc_0\\<^sub>2 s\\<^sub>L \\\n low_proc_no_input_change s\\<^sub>L \\\n high_proc\\<^sub>2 s\\<^sub>H \\\n high_proc_no_low_output_change s\\<^sub>H)"} {"_id": "502521", "text": "proof (prove)\ngoal (1 subgoal):\n 1. blinfun_apply (blinfun_scaleR a b) c = blinfun_apply a c *\\<^sub>R b"} {"_id": "502522", "text": "proof (prove)\nusing this:\n a \\ carrier Zp\n b \\ carrier Zp\n c \\ carrier Zp\n val_Zp a = val_Zp b\n val_Zp b < val_Zp (c \\ a)\n\ngoal (1 subgoal):\n 1. val_Zp c = val_Zp (c \\ a \\ a)"} {"_id": "502523", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.triangular_column l A'"} {"_id": "502524", "text": "proof (state)\nthis:\n (\\tst d.\n compatTst tst \\ ?ca3 = while tst do d \\ siso0 d) \\\n (\\tst d1 d.\n compatTst tst \\\n ?ca3 = d1 ;; while tst do d \\\n siso0 d1 \\ siso0 d) \\\n siso0 ?ca3\n\ngoal (1 subgoal):\n 1. siso0 (while tst do c)"} {"_id": "502525", "text": "proof (prove)\ngoal (1 subgoal):\n 1. r -`\\<^sub>\\ A -\\<^sub>\\ r -`\\<^sub>\\ B\n \\\\<^sub>\\ r -`\\<^sub>\\ (A -\\<^sub>\\ B)"} {"_id": "502526", "text": "proof (prove)\nusing this:\n CC \\ N\n\ngoal (1 subgoal):\n 1. CC \\ N -\n {C. \\DD\\N.\n DD \\ {C} \\\n (\\D\\DD. D < C)}"} {"_id": "502527", "text": "proof (prove)\nusing this:\n X \\ {..n + 1}\n\ngoal (1 subgoal):\n 1. (if R `` X' \\ {n + 1..} = {} then 0 else g' n (n + 1)) +\n sum (g' n) (Y' \\ {..n}) =\n sum (g' n) (R' n `` X)"} {"_id": "502528", "text": "proof (prove)\nusing this:\n path ?v ?v []\n terminal_vertex (c, [])\n\ngoal (1 subgoal):\n 1. \\pth v'. path (c, []) v' pth \\ terminal_vertex v'"} {"_id": "502529", "text": "proof (prove)\nusing this:\n invariant composition P18\n reachable composition (s1, s2)\n\ngoal (1 subgoal):\n 1. P18 (s1, s2)"} {"_id": "502530", "text": "proof (prove)\ngoal (1 subgoal):\n 1. advantage3 \\ = local.dis_log.advantage (adversary3 \\)"} {"_id": "502531", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fgmodule_condition R f i s A z; ideal R (carrier R);\n R module fgmodule R A z i f s\\\n \\ linear_span R (fgmodule R A z i f s) (carrier R) A =\n carrier (fgmodule R A z i f s)"} {"_id": "502532", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rsquarefree ([:- a, 1::'a:] * (p * q))"} {"_id": "502533", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x. a \\ x \\ x \\ P \\ x = A\n 2. \\x.\n a \\ x \\ x \\ P \\\n a \\ S \\ {b \\ S. (b, a) \\ Equivalence} = x"} {"_id": "502534", "text": "proof (prove)\nusing this:\n \\?u \\ ?v = \\; ?v \\ ?u = \\; ?u \\ M;\n ?v \\ M\\\n \\ invertible ?u\n\ngoal (1 subgoal):\n 1. \\sub.invertible u; u \\ N\\\n \\ invertible u"} {"_id": "502535", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cyc_post blues reds onstack u0 (init_wit_blue_early u0 t)"} {"_id": "502536", "text": "proof (state)\nthis:\n 0 < k\n prime q\n p ^ n = q ^ k\n\ngoal (2 subgoals):\n 1. primepow (p ^ n) \\ primepow p\n 2. \\primepow p; 0 < n\\ \\ primepow (p ^ n)"} {"_id": "502537", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (weak_ranking R1 = weak_ranking R2) = (R1 = R2)"} {"_id": "502538", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\b l aa ba.\n \\\\s' r.\n recognises_execution e s' r t \\\n \\c1 s. obtains s c1 e s' r t;\n recognises_execution e s' r ((l, b) # t); a = (l, b);\n (aa, ba) |\\| possible_steps e s' r l b;\n recognises_execution e aa (evaluate_updates ba b r) t\\\n \\ \\c1 s.\n \\(s'', T)\n |\\|possible_steps e s' r l b.\n obtains s c1 e s'' (evaluate_updates T b r) t"} {"_id": "502539", "text": "proof (prove)\nusing this:\n a \\ b'\n b \\ b'\n D - c * g b' \\ c * g x\n b' \\ x\n f \\ O(g')\n filterlim g at_top at_top\n \\a \\ ?a'; ?a' \\ ?x\\\n \\ f integrable_on {?a'..?x}\n a \\ ?x \\\n (g has_real_derivative g' ?x) (at ?x within {a..})\n continuous_on {a..} g'\n a \\ ?x \\ 0 \\ g' ?x\n 0 < c\n (\\x. \\g' x\\) integrable_on {b'..x}\n\ngoal (1 subgoal):\n 1. norm (integral {b'..x} f)\n \\ integral {b'..x} (\\x. c * norm (g' x))"} {"_id": "502540", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ ((\\x\\state_set (state_list s as).\n last x \\ valid_states PROB) \\\n (\\x\\state_set (state_list s as).\n \\y\\state_set (state_list s as).\n last x = last y \\ x = y))) =\n (\\ (\\x\\state_set (state_list s as).\n \\y\\state_set (state_list s as).\n last x = last y \\ x = y))"} {"_id": "502541", "text": "proof (prove)\nusing this:\n d = degree u\n d \\ n\n u \\ 0\n k \\ (0::'a)\n i < n\n\ngoal (1 subgoal):\n 1. i < n - degree u \\\n degree (u * monom (1::'a) (n - Suc (degree u + i))) = n - Suc i"} {"_id": "502542", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nearly_healthy\n (wlp (\\x.\n body ;;\n x \\<^bsub>\\ G \\\\<^esub>\\ Skip))"} {"_id": "502543", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\e\\challenge_space.\n (\\(h, w)\\R_DL.\n Schnorr_\\.R h w e = Schnorr_\\.S h e) \\\n (\\h\\valid_pub.\n \\(a, z)\\set_spmf (S2 h e). check h a e z)"} {"_id": "502544", "text": "proof (prove)\nusing this:\n e_take\n (Suc (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take (Suc j) (f \\ n\\<^sub>1)])))\n (f \\ n\\<^sub>1) =\n e_take\n (Suc (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take (Suc j) (f \\ n\\<^sub>1)])))\n (f \\ n\\<^sub>2)\n (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take (Suc j) (f \\ n\\<^sub>1)]))\n < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take\n (Suc (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1),\n i, e_take (Suc j)\n(f \\ n\\<^sub>1)])))\n (f \\ n\\<^sub>1)])\n (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take (Suc j) (f \\ n\\<^sub>1)]))\n < e_length (f \\ n\\<^sub>2)\n\ngoal (1 subgoal):\n 1. 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1), i,\n e_take\n (Suc (GREATEST j.\n j < e_length (f \\ n\\<^sub>1) \\\n 0 < the (eval r_result1\n [e_length (f \\ n\\<^sub>1),\n i, e_take (Suc j)\n (f \\ n\\<^sub>1)])))\n (f \\ n\\<^sub>2)])"} {"_id": "502545", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pairwise (\\i j. disjnt (ball (a i) (r i)) (ball (a j) (r j))) C"} {"_id": "502546", "text": "proof (prove)\nusing this:\n \\ \\ des.\\ - 1\n ?bl \\# \\\\<^sup>C \\ int (card ?bl) = des.\\ - \\\n\ngoal (1 subgoal):\n 1. block_design \\ \\\\<^sup>C (des.\\ - \\)"} {"_id": "502547", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_pmf con\n (map_pmf noc\n (Sum_pmf (8 / 10) (Partial_Cost_Model.config'_rand BIT Da qs)\n (Partial_Cost_Model.config'_rand\n (Partial_Cost_Model.embed (rTS [])) Db qs))) =\n Sum_pmf (8 / 10) (Partial_Cost_Model.config'_rand BIT Da qs)\n (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS [])) Db\n qs)"} {"_id": "502548", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a\\a. a \\ Alts e1 e2) =\n ((\\a\\a. a \\ e1) \\\n (\\a\\a. a \\ e2))"} {"_id": "502549", "text": "proof (prove)\nusing this:\n l_set_shadow_root ShadowRootClass.type_wf set_shadow_root\n set_shadow_root_locs\n l_get_tag_name ShadowRootClass.type_wf get_tag_name get_tag_name_locs\n\ngoal (1 subgoal):\n 1. l_set_shadow_root_get_tag_name set_shadow_root_locs get_tag_name_locs"} {"_id": "502550", "text": "proof (prove)\nusing this:\n \\?P \\ S; ?P \\\\ ?Q\\\n \\ ?Q \\ S\n P \\ Disj (Abs_bset (characteristic_weak_formula ` S)) =\n (\\x\\characteristic_weak_formula ` S. P \\ x)\n ?Q \\ characteristic_weak_formula ?P = ?P \\\\ ?Q\n ?Q \\ characteristic_weak_formula ?P = (?P \\\\ ?Q)\n ?P \\\\ ?Q \\\n \\x.\n weak_formula x \\ ?P \\ x = ?Q \\ x\n\ngoal (1 subgoal):\n 1. P \\ Disj (Abs_bset (characteristic_weak_formula ` S)) =\n (P \\ S)"} {"_id": "502551", "text": "proof (prove)\ngoal (1 subgoal):\n 1. r *\\<^sub>R b = One \\ b = (1 / r) *\\<^sub>R One"} {"_id": "502552", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\connectedin X C; separatedin X S T;\n C \\ S \\ T\\\n \\ C \\ S \\ C \\ T"} {"_id": "502553", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. arcosh (f x)) \\ arcosh a) F"} {"_id": "502554", "text": "proof (prove)\nusing this:\n (rev_H uw \\\\<^sub>F rev_H vw)\n (edge_succ HM (rev_G a)) =\n edge_succ HM (rev_G a)\n\ngoal (1 subgoal):\n 1. perm_rem (rev_H vw) (perm_rem (rev_H uw) (edge_succ HM)) (rev_G a) =\n edge_succ HM (rev_G a)"} {"_id": "502555", "text": "proof (prove)\ngoal (1 subgoal):\n 1. of_nat (card X) * Bnd \\ of_nat k * Bnd"} {"_id": "502556", "text": "proof (prove)\ngoal (1 subgoal):\n 1. real_of_rat c *\n (\\ai\\set fs.\n case ai of (a, i) \\ rpoly (a ^ Suc i) x) =\n real_of_rat c *\n (\\ai\\set fs.\n case ai of (a, i) \\ rpoly a x ^ Suc i)"} {"_id": "502557", "text": "proof (prove)\ngoal (1 subgoal):\n 1. integrable (lebesgue_on {a..b})\n (\\x. if x = a then d\\<^sup>2 else 0)"} {"_id": "502558", "text": "proof (prove)\ngoal (1 subgoal):\n 1. non_inf {(a, b). b < a}"} {"_id": "502559", "text": "proof (state)\nthis:\n a \\\\<^sub>B \\[a] =\n \\[a] \\\\<^sub>B cmp (a, a) \\\\<^sub>B a \\\\<^sub>B a\n\ngoal (1 subgoal):\n 1. a \\\\<^sub>B \\[map\\<^sub>0 a] =\n (local.map \\[a] \\\\<^sub>B cmp (a, a)) \\\\<^sub>B\n a \\\\<^sub>B a"} {"_id": "502560", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sigma. Implementation sigma \\ EEx Hist sigma"} {"_id": "502561", "text": "proof (prove)\nusing this:\n Discr (Inr (call.Let l binds c', \\, ve, b)) \\ arg_poss p\n\ngoal (1 subgoal):\n 1. (l \\ labels p &&&\n c' \\ calls p &&& fst ` set binds \\ vars p) &&&\n snd ` set binds \\ lambdas p &&&\n \\ \\ maps_over (labels p) UNIV &&&\n ve \\ smaps_over (vars p \\ UNIV) (proc_poss p)"} {"_id": "502562", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ X \\ X) = false"} {"_id": "502563", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (roots_of_real_main p) = {x. poly p x = 0}"} {"_id": "502564", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i < length ts; \\sop\\write_sops sb. valid_sop sop;\n \\sop\\store_sops is. valid_sop sop\\\n \\ valid_sops\n (ts[i := (p, is, xs, sb, \\, \\, \\)])"} {"_id": "502565", "text": "proof (prove)\nusing this:\n b \\ ideal (ideal B \\

x)\n\ngoal (1 subgoal):\n 1. b \\ ideal B \\
x"} {"_id": "502566", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. the_elem (f ` x)) \\ (+>) I\n \\ ring_hom R (image_ring f R)"} {"_id": "502567", "text": "proof (prove)\ngoal (1 subgoal):\n 1. integrable M (\\x. indicat_real {a..} x *\\<^sub>R norm (f x))"} {"_id": "502568", "text": "proof (prove)\nusing this:\n E F D CongA G F D\n\ngoal (1 subgoal):\n 1. F D E CongA F D G"} {"_id": "502569", "text": "proof (prove)\ngoal (1 subgoal):\n 1. strong_reduction_simulation (TRel\\<^sup>+) Target \\\n (\\P Q Q'.\n (P, Q) \\ TRel\\<^sup>+ \\\n Q \\Target Q' \\\n (\\P'.\n P \\Target P' \\ (P', Q') \\ TRel\\<^sup>+))"} {"_id": "502570", "text": "proof (chain)\npicking this:\n p\\<^sub>0 \\ ps\\<^sub>L \\ p\\<^sub>1 \\ ps\\<^sub>L\n p\\<^sub>0 \\ ps\\<^sub>R \\ p\\<^sub>1 \\ ps\\<^sub>R"} {"_id": "502571", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lincomb\n (\\v.\n (if v \\ b1 then a1 v else \\) \\\n (if v \\ b2 then a2 v else \\))\n (b1 \\ b2) =\n x"} {"_id": "502572", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xa xaa.\n \\xa \\ {.. {..\n \\ cnj (x $ xa) *\\<^sub>C\n y $ xaa *\\<^sub>C\n (canonical_basis ! xa \\\\<^sub>C\n canonical_basis ! xaa) =\n cnj (x $ xa) *\\<^sub>C\n y $ xaa *\\<^sub>C (if xa = xaa then 1 else 0)"} {"_id": "502573", "text": "proof (prove)\nusing this:\n finite A\n A \\ {}\n x \\ A\n\ngoal (1 subgoal):\n 1. card (insert x ` Set.filter (odd \\ card) (Pow A)) =\n card (Set.filter (odd \\ card) (Pow A))"} {"_id": "502574", "text": "proof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` vec_of_basis_enum ` cspan X =\n basis_enum_of_vec ` local.span (vec_of_basis_enum ` X)"} {"_id": "502575", "text": "proof (prove)\nusing this:\n \\v F. [\\F,x\\ in v] = [\\F,y\\ in v]\n\ngoal (1 subgoal):\n 1. \\v.\n (\\\\ o\\<^sub>1 \\ ex1 r v) =\n (\\\\ o\\<^sub>2 \\ ex1 r v)"} {"_id": "502576", "text": "proof (prove)\nusing this:\n fixed_length_sublist (concat (map vec As)) (prod_list ds) i = vec (As ! i)\n (\\A.\n A \\ set ?As \\ dims A = ?ds) \\\n dims (subtensor_combine ?ds ?As) = length ?As # ?ds\n (\\A.\n A \\ set ?As \\ dims A = ?ds) \\\n vec (subtensor_combine ?ds ?As) = concat (map vec ?As)\n\ngoal (1 subgoal):\n 1. tensor_from_vec (tl (dims (subtensor_combine ds As)))\n (fixed_length_sublist (vec (subtensor_combine ds As))\n (prod_list (tl (dims (subtensor_combine ds As)))) i) =\n As ! i"} {"_id": "502577", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((A ===> A ===> A) ===> A ===> (A ===> A ===> A) ===> (=))\n (comm_semiring_0_ow (Collect (Domainp A)))\n (\\a b c.\n class.comm_semiring_0 a b c \\\n (\\a b.\n a \\ UNIV \\\n b \\ UNIV \\ c a b \\ UNIV) \\\n b \\ UNIV)"} {"_id": "502578", "text": "proof (prove)\nusing this:\n Digraph_Component.connected G\n u \\ verts G\n v \\ verts G\n \\?w2 \\ verts G; u \\ ?w2; v \\ ?w2\\\n \\ in_degree G ?w2 = out_degree G ?w2\n in_degree G u + 1 = out_degree G u\n out_degree G v + 1 = in_degree G v\n\ngoal (1 subgoal):\n 1. \\u p v. euler_trail u p v \\ u \\ v"} {"_id": "502579", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fpxs_const (1::'a) = 1"} {"_id": "502580", "text": "proof (prove)\ngoal (2 subgoals):\n 1. t \\ set (args\n (worst_chain (\\t s. ground t \\ t >\\<^sub>t s)\n (\\t s. hsize s < hsize t) i)) \\\n {u. worst_chain (\\t s. ground t \\ t >\\<^sub>t s)\n (\\t s. hsize s < hsize t) i \\\\<^sub>e\\<^sub>m\\<^sub>b\n u}\n \\ range\n (\\i.\n {u. worst_chain (\\t s. ground t \\ t >\\<^sub>t s)\n (\\t s. hsize s < hsize t)\n i \\\\<^sub>e\\<^sub>m\\<^sub>b\n u})\n 2. t \\ set (args\n (worst_chain (\\t s. ground t \\ t >\\<^sub>t s)\n (\\t s. hsize s < hsize t) i)) \\\n t \\ {u. worst_chain (\\t s. ground t \\ t >\\<^sub>t s)\n (\\t s. hsize s < hsize t)\n i \\\\<^sub>e\\<^sub>m\\<^sub>b\n u}"} {"_id": "502581", "text": "proof (prove)\nusing this:\n {0..t} = {0--t}\n\ngoal (1 subgoal):\n 1. {s--0} \\ {0--t}"} {"_id": "502582", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B \\ A2; F \\ A1 - B co A1 \\ B;\n F \\ A2 - C co A2 \\ C\\\n \\ F \\ A1 \\ A2 - C co\n A1 \\ A2 \\ C"} {"_id": "502583", "text": "proof (state)\nthis:\n \\X x.\n derivation X \\\n x \\ X \\ f x \\ f x\n\ngoal (1 subgoal):\n 1. \\p.\n extreme {q \\ A. f q = q} (\\x y. y \\ x) p"} {"_id": "502584", "text": "proof (prove)\nusing this:\n A \\ \\\n C \\ A\n B \\ \\\n chamber C\n chamber C'\n C' \\ B\n ChamberComplexIsomorphism A (CoxeterComplex.TheComplex S) \\\n B' \\ \\\n C \\ B'\n C' \\ B'\n\ngoal (1 subgoal):\n 1. \\\\.\n ChamberComplexIsomorphism B (CoxeterComplex.TheComplex S) \\"} {"_id": "502585", "text": "proof (prove)\nusing this:\n finite L\n finite (L \\ FV t) \\\n \\s p.\n s \\ L \\ FV t \\\n p \\ L \\ FV t \\ s \\ p\n\ngoal (1 subgoal):\n 1. (\\s p.\n \\s \\ L; p \\ L; s \\ p; s \\ FV t;\n p \\ FV t\\\n \\ thesis) \\\n thesis"} {"_id": "502586", "text": "proof (prove)\nusing this:\n lookup_pp (rep_nat_pp (PP_oalist xs)) (rep_nat y) =\n lookup_pp (rep_nat_pp (PP_oalist ys)) (rep_nat y)\n\ngoal (1 subgoal):\n 1. OAlist_tc_lookup xs y = OAlist_tc_lookup ys y"} {"_id": "502587", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sorted (map fst kvs); distinct (map fst kvs)\\\n \\ is_rbt (rbtreeify kvs)"} {"_id": "502588", "text": "proof (state)\nthis:\n r1 < r2 \\ r1 = r2 \\ r2 < r1\n\ngoal (1 subgoal):\n 1. v = w"} {"_id": "502589", "text": "proof (prove)\nusing this:\n finite (pderivs_lang ?A ?r)\n\ngoal (1 subgoal):\n 1. card (Timess (pderivs_lang A r) s) \\ card (pderivs_lang A r)"} {"_id": "502590", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((M ===> (S ===> M) ===> M) ===>\n (S ===> (A ===> M) ===> M) ===>\n rel_nondetT M ===> (A ===> rel_nondetT M) ===> rel_nondetT M)\n nondetM_base.bind_nondet nondetM_base.bind_nondet"} {"_id": "502591", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map of_int_hom.vec_hom fs ! i - gs2.gso i\n \\ gs.span (gs1.gso ` {0.. \\ border ?y ?x)\n \\border ?a ?b; border ?b ?c\\\n \\ border ?a ?c\n border (take (\\ ?s ?i - 1) ?s) (take ?i ?s)\n\ngoal (1 subgoal):\n 1. \\y = \\ s (i - 1);\n strict_border (take (i - 1) s) (take (j - 1) s)\\\n \\ strict_border (take (y - 1) s) (take (j - 1) s)"} {"_id": "502593", "text": "proof (prove)\nusing this:\n a \\ (\\a. if a \\ A then the_inv_into A f' (f a) else a) ` A\n\ngoal (1 subgoal):\n 1. (\\a'.\n \\a' \\ A; a = the_inv_into A f' (f a')\\\n \\ thesis) \\\n thesis"} {"_id": "502594", "text": "proof (prove)\nusing this:\n A \\ X\n\ngoal (1 subgoal):\n 1. A B Perp P A \\ Col A B T \\ Bet C T P"} {"_id": "502595", "text": "proof (prove)\ngoal (1 subgoal):\n 1. degree p = 4 \\\n \\a b c d e. p = [:e, d, c, b, a:] \\ a \\ (0::'a)"} {"_id": "502596", "text": "proof (prove)\nusing this:\n Postdomination.standard_control_dependence src trg\n (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit)\n NewExit n n'\n\ngoal (1 subgoal):\n 1. \\as.\n CFG.path src trg\n (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) n\n as n' \\\n as \\ []"} {"_id": "502597", "text": "proof (prove)\nusing this:\n types_agree t1 a\n\ngoal (1 subgoal):\n 1. \\s;vs;vs_to_es ves @\n [$b_e]\\ \\_ i \\s';vs';es'\\"} {"_id": "502598", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Fun f U \\ \\ = Fun f U &&&\n Fun f T \\ \\ = Fun f T"} {"_id": "502599", "text": "proof (prove)\nusing this:\n \\Ana ?t = (?K, ?R);\n \\g S. Fun g S \\ ?t \\ length S = arity g;\n ?k \\ set ?K; Fun ?f ?T \\ ?k\\\n \\ length ?T = arity ?f\n\ngoal (1 subgoal):\n 1. \\Ana t = (K, T);\n \\f T.\n Fun f T \\ t \\ length T = arity f;\n k \\ set K\\\n \\ \\f T.\n Fun f T \\ k \\\n length T = arity f"} {"_id": "502600", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\D \\ C; project h C F \\ stable A\\\n \\ project h D F \\ stable A"} {"_id": "502601", "text": "proof (prove)\nusing this:\n ide g\n trg ?F \\ if arr ?F then MkIde (E.Trg (Dom ?F)) else null\n\ngoal (1 subgoal):\n 1. E.Trg (Dom g) = Dom (trg g)"} {"_id": "502602", "text": "proof (prove)\nusing this:\n SP \\S P\n\ngoal (1 subgoal):\n 1. P \\\\\\\\LT P"} {"_id": "502603", "text": "proof (state)\ngoal (1 subgoal):\n 1. set_kdt r \\ cbox p\\<^sub>0 p\\<^sub>1 = {}"} {"_id": "502604", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\node_ptr.\n \\\\node_ptr\\fset (node_ptr_kinds h').\n (\\document_ptr.\n (document_ptr =\n cast\\<^sub>s\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>_\\<^sub>r\\<^sub>o\\<^sub>o\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>d\\<^sub>o\\<^sub>c\\<^sub>u\\<^sub>m\\<^sub>e\\<^sub>n\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n shadow_root_ptr \\\n document_ptr |\\| document_ptr_kinds h') \\\n node_ptr\n \\ set |h2 \\ get_disconnected_nodes\n document_ptr|\\<^sub>r) \\\n (\\parent_ptr.\n parent_ptr |\\| object_ptr_kinds h2 \\\n node_ptr\n \\ set |h2 \\ get_child_nodes\n parent_ptr|\\<^sub>r);\n node_ptr |\\| node_ptr_kinds h';\n \\parent_ptr.\n parent_ptr |\\| object_ptr_kinds h' \\\n node_ptr\n \\ set |h' \\ get_child_nodes\n parent_ptr|\\<^sub>r\\\n \\ \\document_ptr.\n document_ptr |\\|\n document_ptr_kinds h' \\\n node_ptr\n \\ set |h'\n \\ get_disconnected_nodes document_ptr|\\<^sub>r"} {"_id": "502605", "text": "proof (prove)\nusing this:\n d \\ top * e\\<^sup>T * H \\\n (H * d\\<^sup>T)\\<^sup>\\ * H * a\\<^sup>T * top \\\n bot\n\ngoal (1 subgoal):\n 1. times_top_class.total\n (top *\n (d \\ top * e\\<^sup>T * H \\\n (H * d\\<^sup>T)\\<^sup>\\ * H * a\\<^sup>T * top))"} {"_id": "502606", "text": "proof (prove)\ngoal (1 subgoal):\n 1. move_matrix (A * B) j i = move_matrix A j 0 * move_matrix B 0 i"} {"_id": "502607", "text": "proof (prove)\nusing this:\n LLL_invariant upw i fs\n set (map of_int_hom.vec_hom fs) \\ Rn\n ?i < length (map of_int_hom.vec_hom fs) \\\n map of_int_hom.vec_hom fs ! ?i \\ Rn\n ?i < length (map of_int_hom.vec_hom fs) \\\n gs'.gso ?i \\ Rn\n\ngoal (1 subgoal):\n 1. ((gs.lin_indpt_list (map of_int_hom.vec_hom fs) \\\n length (map of_int_hom.vec_hom fs) = m) \\\n set fs \\ carrier_vec n \\\n (\\i carrier_vec n) \\\n (\\i Rn)) \\\n (length fs = m \\\n lattice_of fs = L \\ gs'.weakly_reduced \\ i) \\\n i \\ m \\\n gs'.reduced \\ i \\ (upw \\ \\_small fs i)"} {"_id": "502608", "text": "proof (prove)\nusing this:\n c' = While e c \\ mds' \\ mdsa\n \\While ?e ?c, ?mds, ?mem\\ \\\n \\Stmt.If ?e (?c ;; While ?e ?c) Stop, ?mds, ?mem\\\n \\While ?e ?c, ?mds, ?mem\\ \\\n \\?c', ?mds', ?mem'\\ \\\n ?c' = Stmt.If ?e (?c ;; While ?e ?c) Stop \\\n ?mds' = ?mds \\ ?mem' = ?mem\n\ngoal (1 subgoal):\n 1. (x \\ mds' GuarNoReadOrWrite \\\n (\\mds mem c'a mds' mem'.\n \\c', mds, mem\\ \\\n \\c'a, mds', mem'\\ \\\n (\\x\\{x} \\ \\_vars x.\n \\v.\n \\c', mds, mem(x := v)\\ \\\n \\c'a, mds', mem'(x := v)\\))) \\\n (x \\ mds' GuarNoWrite \\\n (\\mds mem c'a mds' mem'.\n \\c', mds, mem\\ \\\n \\c'a, mds', mem'\\ \\\n mem x = mem' x \\ dma mem x = dma mem' x))"} {"_id": "502609", "text": "proof (prove)\nusing this:\n select_return_top cts c1 c2 = ctm\n ctm \\ Bot\n\ngoal (1 subgoal):\n 1. \\xs. ctm = TopType xs"} {"_id": "502610", "text": "proof (prove)\nusing this:\n finite \\' \\\n \\' \\ \\ \\ \\' \\ F\n \\' = set \\''\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "502611", "text": "proof (prove)\nusing this:\n PROP ?psi \\ PROP ?psi\n integral {0..1}\n (\\s.\n (1 - s) *\\<^sub>R\n (h\\<^sup>2 * p\\<^sup>2) *\\<^sub>R\n f'' (t + s * (h * p), x t + (s * (h * p)) *\\<^sub>R f (t, x t)) $\n (1, f (t, x t)) $\n (1, f (t, x t))) =\n (discrete_evolution (rk2_increment p (\\t x. f (t, x))) (t + h) t\n (x t) -\n x t -\n h *\\<^sub>R f (t, x t) -\n (h\\<^sup>2 / 2) *\\<^sub>R f' (t, x t) $ (1, f (t, x t))) /\\<^sub>R\n (h / (p * 2))\n ((\\s.\n (1 - s) *\\<^sub>R\n (h\\<^sup>2 * p\\<^sup>2) *\\<^sub>R\n f'' (t + s * (h * p), x t + (s * (h * p)) *\\<^sub>R f (t, x t)) $\n (1, f (t, x t)) $\n (1, f (t, x t))) has_integral\n f ((t, x t) + (h * p) *\\<^sub>R (1, f (t, x t))) - f (t, x t) -\n f' (t, x t) $ (h * p, (h * p) *\\<^sub>R f (t, x t)))\n {0..1}\n\ngoal (1 subgoal):\n 1. ((\\s.\n (1 - s) *\\<^sub>R\n (h\\<^sup>2 * p\\<^sup>2) *\\<^sub>R\n f'' (t + s * (h * p), x t + (s * (h * p)) *\\<^sub>R f (t, x t)) $\n (1, f (t, x t)) $\n (1, f (t, x t))) has_integral\n (discrete_evolution (rk2_increment p (\\t x. f (t, x))) (t + h)\n t (x t) -\n x t -\n h *\\<^sub>R f (t, x t) -\n (h\\<^sup>2 / 2) *\\<^sub>R f' (t, x t) $ (1, f (t, x t))) /\\<^sub>R\n (h / (p * 2)))\n {0..1}"} {"_id": "502612", "text": "proof (prove)\nusing this:\n \\c\\{wax \\ |\\.\n \\ unprot \\ \\\n (\\ isRes \\ \\ protCl \\)}.\n satC \\ c\n \\ unprot \\\n \\ isRes \\ \\ protCl \\\n\ngoal (1 subgoal):\n 1. satC \\ (wax \\)"} {"_id": "502613", "text": "proof (prove)\nusing this:\n Calculus SWB = Source\n Calculus TWB = Target\n\ngoal (1 subgoal):\n 1. Calculus\n \\Calculus = STCal (Calculus SWB) (Calculus TWB),\n HasBarb =\n \\P a.\n (\\SP. SP \\S P \\ SP\\a) \\\n (\\TP. TP \\T P \\ TP\\a)\\ =\n STCal Source Target"} {"_id": "502614", "text": "proof (prove)\nusing this:\n Class a = Class g\n\ngoal (1 subgoal):\n 1. a \\ Class g"} {"_id": "502615", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\dist_perm xs ys; Suc n < length xs\\\n \\ x < y in swap n xs =\n (x < y in xs \\\n \\ (x = xs ! n \\ y = xs ! Suc n) \\\n x = xs ! Suc n \\ y = xs ! n)"} {"_id": "502616", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\h.\n \\\\p. h p 0 = 0;\n \\p c1 c2. h p (c1 - c2) = h p c1 - h p c2;\n \\p X c.\n singular_chain p X c \\\n singular_chain (Suc p) X (h p c);\n \\p X c.\n singular_chain p X c \\\n chain_boundary (Suc p) (h p c) +\n h (p - Suc 0) (chain_boundary p c) =\n (singular_subdivision p ^^ m) c - c\\\n \\ thesis) \\\n thesis"} {"_id": "502617", "text": "proof (prove)\ngoal (1 subgoal):\n 1. opnet (\\i. optoy i \\\\\\<^bsub>i\\<^esub> qmsg)\n n \\ (otherwith nos_inc (net_tree_ips n) (oarrivemsg msg_ok),\n other nos_inc (net_tree_ips n) \\)\n global\n (\\\\.\n \\i\\net_tree_ips n.\n no (\\ i)\n \\ no (\\ (nhid (\\ i))))"} {"_id": "502618", "text": "proof (prove)\ngoal (1 subgoal):\n 1. countable (SOME B. countable B \\ topological_basis B) &&&\n topological_basis (SOME B. countable B \\ topological_basis B)"} {"_id": "502619", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ simple_matches simple_match_none p"} {"_id": "502620", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n.\n \\x < a; x * ordinal_of_nat n < a\\\n \\ x * ordinal_of_nat (Suc n) < a"} {"_id": "502621", "text": "proof (prove)\nusing this:\n euler_liar 1 p\n int x * int y mod int p = 1\n 1 < p\n\ngoal (1 subgoal):\n 1. euler_liar (int x * int y) p"} {"_id": "502622", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\equiv V R; (x, y) \\ R\\\n \\ (y, x) \\ R"} {"_id": "502623", "text": "proof (prove)\nusing this:\n a = (l, b)\n\ngoal (1 subgoal):\n 1. t \\ trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\n (S \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t \\) \\\n t \\ trms\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p\n (b \\\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p \\)"} {"_id": "502624", "text": "proof (prove)\ngoal (1 subgoal):\n 1. start_heap P (addr_of_sys_xcpt C) = \\blank P C\\"} {"_id": "502625", "text": "proof (prove)\nusing this:\n \\basis_wf bs; (l expands_to L) bs; trimmed_pos L\\\n \\ \\\\<^sub>F x in at_top. 0 < l x\n (l expands_to L) bs\n \\\\<^sub>F x in at_top. l x \\ f x\n trimmed_pos L\n basis_wf bs\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. 0 < l x"} {"_id": "502626", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y A.\n \\bi_unique A; rel_set (rel_set A) x y\\\n \\ rel_set A\n (if finite (\\ x) then \\ x else {})\n (if finite (\\ y) then \\ y else {})"} {"_id": "502627", "text": "proof (prove)\nusing this:\n enat n < llength P\n\ngoal (1 subgoal):\n 1. vmc_path G (ldropn n P) (P $ n) p \\"} {"_id": "502628", "text": "proof (prove)\ngoal (1 subgoal):\n 1. invariant3p (mredT P)\n ({s. sync_es_ok (thr s) (shr s) \\\n lock_ok (locks s) (thr s)} \\\n {s. \\Es. sconf_type_ts_ok Es (thr s) (shr s)} \\\n {s. def_ass_ts_ok (thr s) (shr s)})"} {"_id": "502629", "text": "proof (prove)\nusing this:\n \\?n \\ num; ?\\ \\ fmla; Fvars ?\\ = {};\n isTrue (B.neg (PPf ?n \\?\\\\))\\\n \\ bprv (B.neg (PPf ?n \\?\\\\))\n\ngoal (1 subgoal):\n 1. \\n \\.\n \\n \\ num; \\ \\ fmla; Fvars \\ = {}\\\n \\ isTrue (B.neg (PPf n \\\\\\)) =\n bprv (B.neg (PPf n \\\\\\))"} {"_id": "502630", "text": "proof (prove)\ngoal (1 subgoal):\n 1. concrete_category {A. |A| }\n (\\A B. extensional A \\ (A \\ B))\n (restrict (\\x. x)) (\\C B. compose)"} {"_id": "502631", "text": "proof (prove)\nusing this:\n i < dim_row U\\<^sub>f\n j < dim_col U\\<^sub>f\n\ngoal (1 subgoal):\n 1. U\\<^sub>f\\<^sup>t $$ (i, j) = U\\<^sub>f $$ (i, j)"} {"_id": "502632", "text": "proof (prove)\nusing this:\n derivable D P S \\ FirstOrder C'\n \\P C' D \\.\n derivable D P S \\ FirstOrder C' \\ D \\ S\n\ngoal (1 subgoal):\n 1. D \\ S"} {"_id": "502633", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ \\<\\> = x \\ \\"} {"_id": "502634", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ F.Inf_from M"} {"_id": "502635", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\par body l0 G C m T MI k la codea a b ins.\n \\\\\\(a, b) = Some n;\n MST\\(a, b) = Some (mkSPEC (Cachera n) Anno2);\n mbody_is a b (par, body, l0); MST\\(C, m) = Some (T, MI, Anno);\n derivAssum G C m (la + 1) (Cachera k); f = (a, b);\n Anno\\la = None; ins_is C m la ins;\n codea[la\\invokeS a b]\\la = Some ins\\\n \\ deriv G C m la (Cachera (n + k))\n 2. \\par body l0 l code code1 l1 G C m T MI k la codea x fa args OUT.\n \\\\\\f = Some n;\n MST\\f = Some (mkSPEC (Cachera n) Anno2);\n mbody_is (fst f) (snd f) (par, body, l0);\n MST\\(C, m) = Some (T, MI, Anno); Segment C m l l1 code1;\n derivAssum G C m l1 (Cachera k); l = la; code = codea;\n CallPrim f [] = CallPrim fa (x # args); (code1, l1) = OUT;\n (la + 1, codea[la\\load x], CallPrim fa args, OUT)\n \\ compilePrim\\\n \\ deriv G C m l (Cachera (n + k))\n 3. \\a args.\n \\\\f = Some n \\\n MST\\(fst f, snd f) =\n Some (mkSPEC (Cachera n) Anno2) \\\n (\\par body l0.\n mbody_is (fst f) (snd f) (par, body, l0)) \\\n (\\l code code1 l1 G C m T MI k.\n (l, code, CallPrim f args, code1, l1)\n \\ compilePrim \\\n MST\\(C, m) = Some (T, MI, Anno) \\\n Segment C m l l1 code1 \\\n derivAssum G C m l1 (Cachera k) \\\n deriv G C m l (Cachera (n + k))) \\\n \\\\f = Some n \\\n MST\\(fst f, snd f) =\n Some (mkSPEC (Cachera n) Anno2) \\\n (\\par body l0.\n mbody_is (fst f) (snd f) (par, body, l0)) \\\n (\\l code code1 l1 G C m T MI k.\n (l, code, CallPrim f (a # args), code1, l1)\n \\ compilePrim \\\n MST\\(C, m) = Some (T, MI, Anno) \\\n Segment C m l l1 code1 \\\n derivAssum G C m l1 (Cachera k) \\\n deriv G C m l (Cachera (n + k)))"} {"_id": "502636", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. f (g x)) has_vderiv_on\n (\\x. g' x *\\<^sub>R f' (g x)))\n T"} {"_id": "502637", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Formula.formula.Abs_formula\n (L_transform_formula (FL_Formula.Pred f \\)) =\n Formula.Act (Eff f) (Formula.Pred \\)"} {"_id": "502638", "text": "proof (prove)\nusing this:\n ?A \\ carrier_mat ?n ?n \\\n invertible_mat ?A = (det ?A \\ (0::?'a))\n M_mat signs subsets \\ carrier_mat (length signs) (length signs)\n\ngoal (1 subgoal):\n 1. invertible_mat\n (take_rows_from_matrix\n (reduce_mat_cols (M_mat signs subsets)\n (solve_for_lhs p qs subsets (M_mat signs subsets)))\n (rows_to_keep\n (reduce_mat_cols (M_mat signs subsets)\n (solve_for_lhs p qs subsets (M_mat signs subsets)))))"} {"_id": "502639", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\U.\n openin (top_of_set T) U \\\n (\\v.\n Q v \\\n y \\ U \\ U \\ v \\ v \\ W)"} {"_id": "502640", "text": "proof (state)\nthis:\n rational_preference (\\

outcomes) \\ \\\n independent_vnm (\\

outcomes) \\ \\\n continuous_vnm (\\

outcomes) \\\n\ngoal (2 subgoals):\n 1. rational_preference (\\

outcomes) \\ \\\n independent_vnm (\\

outcomes) \\ \\\n continuous_vnm (\\

outcomes) \\ \\\n \\u. vNM_utility outcomes \\ u\n 2. \\u. vNM_utility outcomes \\ u \\\n rational_preference (\\

outcomes) \\ \\\n independent_vnm (\\

outcomes) \\ \\\n continuous_vnm (\\

outcomes) \\"} {"_id": "502641", "text": "proof (prove)\ngoal (1 subgoal):\n 1. plus_p p (to_int_mod_ring x) (to_int_mod_ring y) =\n to_int_mod_ring (x + y)"} {"_id": "502642", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (v \\ fmdom' (fst a) \\\n v \\ fmdom' (fst (action_proj a (prob_dom PROB - vs)))) \\\n (v \\ fmdom' (snd a) \\\n v \\ fmdom' (snd (action_proj a (prob_dom PROB - vs))))"} {"_id": "502643", "text": "proof (prove)\nusing this:\n FW M n (v c) 0 = len M (v c) 0 xs\n set xs \\ {0..n}\n 0 \\ set xs\n v c \\ set xs\n distinct xs\n \\k\\n. 0 < k \\ (\\c. v c = k)\n v c \\ n\n\ngoal (1 subgoal):\n 1. dbm_entry_val u (Some c) None (len M (v c) 0 xs)"} {"_id": "502644", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (rel_id A ===> (A ===> rel_id A) ===> rel_id A) bind_id bind_id"} {"_id": "502645", "text": "proof (prove)\nusing this:\n val_Zp (f \\ a) < val_Zp (f \\ ns (Suc k))\n\ngoal (1 subgoal):\n 1. val_Zp (a \\ ns (Suc k))\n < val_Zp (ns (Suc k) \\ ns (Suc (Suc k)))"} {"_id": "502646", "text": "proof (prove)\ngoal (1 subgoal):\n 1. on_init param \\ (\\e. RETURN (empty_state e)) \\\\<^sub>n\n SPEC I"} {"_id": "502647", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\a. to (from a) = a; \\b. from (to b) = b\\\n \\ local.iso from to"} {"_id": "502648", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P * addrow a i' i B = P * addrow_mat nr a i' i * B"} {"_id": "502649", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card E =\n card\n ((\\x. x * 2 mod p) ` {0<..(p - 1) div 2} \\\n {(p - 1) div 2<..})"} {"_id": "502650", "text": "proof (prove)\nusing this:\n u \\ V - {s, t}\n QD_invar u f Qr\n\ngoal (1 subgoal):\n 1. fifo_discharge f l Q\n \\ SPEC\n (\\(f', l', Q').\n Height_Bounded_Labeling c s t f' l' \\\n Q_invar f' Q' \\\n ((f', l'), f, l) \\ gap_algo_rel\\<^sup>+)"} {"_id": "502651", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Height_Bounded_Labeling c s t f\n (let l' = relabel_effect f l u\n in if gap_precond l' (l u) then gap_effect l' (l u) else l')"} {"_id": "502652", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ [0] \\\\<^bsub>r\\<^esub> rev [0.. L \\ \\ K \\\n (\\x. Plus_Times_pre_cong R (\\

L x) (\\
K x))"} {"_id": "502654", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Chernoff (CD_on ds)"} {"_id": "502655", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\mRed1 (ls, (ts1, m1), ws, is) s1' &&&\n (\\t'.\n t' \\ t \\\n tbisim (ws t' = None) t' (thr s1' t') m1 (ts2 t') m2)) &&&\n r1.cond_action_oks s1' t cts &&& thr s1' t = \\xln\\"} {"_id": "502656", "text": "proof (prove)\nusing this:\n fmpred P (foldl (++\\<^sub>f) (init ++\\<^sub>f x) xs)\n\ngoal (1 subgoal):\n 1. fmpred P (foldl (++\\<^sub>f) init (x # xs))"} {"_id": "502657", "text": "proof (prove)\ngoal (1 subgoal):\n 1. evaluate True env s (exp0.If e1 e2 e3) (?s', ?r1)"} {"_id": "502658", "text": "proof (prove)\nusing this:\n xs ! V = Addr A\n V < length xs\n\ngoal (1 subgoal):\n 1. True,P,t \\1 \\insync\\<^bsub>V\\<^esub> (a) Throw a',\n (h, xs)\\ -\\(Unlock, A),\n SyncUnlock A\\\\\n \\Throw a',(h, xs)\\ &&&\n True,P,t \\1 \\insync\\<^bsub>V\\<^esub> (a) Throw a',\n (h, xs)\\ -\\(UnlockFail,\n A)\\\\\n \\Throw\n (addr_of_sys_xcpt IllegalMonitorState),\n (h, xs)\\"} {"_id": "502659", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\<^sub>\\\n ((\\ \\\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\n dghm_id \\)\\ArrMap\\)\n\ngoal (1 subgoal):\n 1. (\\ \\\\<^sub>D\\<^sub>G\\<^sub>H\\<^sub>M\n dghm_id \\)\\ArrMap\\\\a\\ =\n \\\\ArrMap\\\\a\\"} {"_id": "502660", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p *\\<^sub>p (1::'a)\\<^sub>p = p &&& (1::'a)\\<^sub>p *\\<^sub>p p = p"} {"_id": "502661", "text": "proof (prove)\ngoal (1 subgoal):\n 1. j < length xs \\ local.times a xs ! j = a * xs ! j"} {"_id": "502662", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (uncurry ?f2, uncurry (RETURN \\\\ lexord_eq))\n \\ monom_assn\\<^sup>k *\\<^sub>a\n monom_assn\\<^sup>k \\\\<^sub>a bool_assn"} {"_id": "502663", "text": "proof (prove)\ngoal (1 subgoal):\n 1. UNIV // eq_nextl \\ left_lang ` states M"} {"_id": "502664", "text": "proof (prove)\nusing this:\n \\encF N t \\ trm; FvarsT (encF N t) = {}\\\n \\ prv (imp (PP (encF N t))\n (PP \\PP (encF N t)\\))\n PP (encF N t) = inst \\ t\n \\J ?\\ \\ subst ?\\ (tJ ?\\) xx\n \\ \\ \\J \\\n \\ \\ PP (encF N (Var xx))\n inst ?\\ ?t = subst ?\\ ?t xx\n t \\ tJ \\\n\ngoal (1 subgoal):\n 1. prv (imp \\ (PP \\\\\\))"} {"_id": "502665", "text": "proof (prove)\nusing this:\n \\f. inj_on f (elts A) \\ f ` elts A \\ elts B\n\ngoal (1 subgoal):\n 1. (\\f.\n \\inj_on f (elts A); f ` elts A \\ elts B\\\n \\ thesis) \\\n thesis"} {"_id": "502666", "text": "proof (prove)\nusing this:\n span f g\n tuple ?f ?g =\n mkArr (Dom ?f) (local.set (local.prod (cod ?f) (cod ?g)))\n (\\x. UP \\DOWN (Fun ?f x), DOWN (Fun ?g x)\\)\n HF'.tuple ?f ?g \\\n if span ?f ?g\n then THE l.\n local.comp (pr1 (cod ?f) (cod ?g)) l = ?f \\\n local.comp (pr0 (cod ?f) (cod ?g)) l = ?g\n else null\n\ngoal (1 subgoal):\n 1. tuple f g = HF'.tuple f g"} {"_id": "502667", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x\\ps\\<^sub>R.\n (case x of (x, y) \\ (real_of_int x, real_of_int y))\n \\ LSQ \\\n (case x of (x, y) \\ (real_of_int x, real_of_int y))\n \\ RSQ"} {"_id": "502668", "text": "proof (state)\ngoal (1 subgoal):\n 1. mon_ww fg w2 \\ mon_ww fg (map le_rem_s (env w))"} {"_id": "502669", "text": "proof (prove)\nusing this:\n {r \\ \\. qw \\ set (rhsq r) \\ the (rcm r) \\ Suc 0} =\n {r \\ \\.\n qw \\ set (rhsq r) \\ set (rhsq r) \\ dom Q - (W - {qw})}\n\ngoal (1 subgoal):\n 1. dsqr =\n {r \\ \\.\n qw \\ set (rhsq r) \\\n set (rhsq r) \\ dom Q - (W - {qw})}"} {"_id": "502670", "text": "proof (prove)\ngoal (1 subgoal):\n 1. valof topfloat = largest TYPE(('e, 'f) IEEE.float)"} {"_id": "502671", "text": "proof (prove)\nusing this:\n {} \\ {a. irreducible a}\n \\c\\{}. [a = b] (mod c)\n \\p1 p2.\n p1 \\ {} \\ p2 \\ {} \\ p1 \\ p2 \\\n comm_monoid_mult_class.coprime p1 p2\n\ngoal (1 subgoal):\n 1. [a = b] (mod \\{})"} {"_id": "502672", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\T. connectedin X T \\ x \\ T \\ y \\ T;\n \\T. connectedin X T \\ y \\ T \\ z \\ T\\\n \\ \\T.\n connectedin X T \\ x \\ T \\ z \\ T"} {"_id": "502673", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (trd (xs @ ys) a # ys)\n \\ set (comp_red_basis_aux xs (trd (xs @ ys) a # ys))"} {"_id": "502674", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a Q'.\n \\x \\ P;\n \\y R.\n y \\ R \\\n (\\\\y\\R, R) \\ Rel;\n Id \\ Rel;\n P \\\n \\\\x\\Q \\a \\ Q'\\\n \\ \\P'.\n \\\\x\\(P \\\n Q) \\a \\ P' \\\n (P', Q') \\ Rel"} {"_id": "502675", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\x. x \\ cbox t (t + h) \\ 0 \\ t + h - x\n 2. bounded F'\n 3. closed F'\n 4. convex F'\n 5. cbox t (t + h) \\ {}"} {"_id": "502676", "text": "proof (prove)\ngoal (1 subgoal):\n 1. key_sets (Hash X) (crypts H) \\ key_sets X (crypts H)"} {"_id": "502677", "text": "proof (prove)\nusing this:\n x \\ s\n y \\ s\n convex s\n ?x \\ s \\\n (f has_derivative (\\h. 0::'b)) (at ?x within s)\n\ngoal (1 subgoal):\n 1. norm (f x - f y) \\ 0 * norm (x - y)"} {"_id": "502678", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ y. j (y, \\a\\)\n \\embed_measure (count_space UNIV) Some\n \\ (\\\\<^sup>+ x. j (x, \\a\\))"} {"_id": "502679", "text": "proof (prove)\ngoal (1 subgoal):\n 1. has_sort oss (Tv idn S) S' \\\n sort_leq (subclass oss) S S'"} {"_id": "502680", "text": "proof (state)\nthis:\n (SUP \\\\elts \\. \\)\n \\ \\ ((+) \\ ` elts \\)\n\ngoal (1 subgoal):\n 1. \\\\'.\n \\Limit \\';\n \\\\.\n \\ \\ elts \\' \\\n \\ \\ \\ + \\\\\n \\ (SUP \\\\elts \\'. \\)\n \\ \\ +\n (SUP \\\\elts \\'. \\)"} {"_id": "502681", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_option (\\v'. v' + m\\<^sub>1(k $$:= v) $$! k')\n (if k = k' then Some v' else {$$} $$ k') =\n {k $$:= v' + v} $$ k'"} {"_id": "502682", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\w.\n \\infdist w {x--y} \\ 4 * deltaG TYPE('a);\n infdist w {x--z} \\ 4 * deltaG TYPE('a);\n infdist w {y--z} \\ 4 * deltaG TYPE('a);\n dist w x = Gromov_product_at x y z\\\n \\ thesis) \\\n thesis"} {"_id": "502683", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (map (inv p)\n (filter (\\x. x \\ range p)\n (xs @\n ipurge_tr (con_comp_pol I) (con_comp_map D E p q)\n (con_comp_map D E p q y) ys)),\n inv p `\n ipurge_ref (con_comp_pol I) (con_comp_map D E p q)\n (con_comp_map D E p q y) ys R)\n \\ failures P"} {"_id": "502684", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s + t + punit.lt (rep_list f) \\ keys (rep_list (monom_mult c s p))"} {"_id": "502685", "text": "proof (state)\ngoal (4 subgoals):\n 1. \\x y. (x < y) = (x \\ y \\ \\ y \\ x)\n 2. \\x. x \\ x\n 3. \\x y z.\n \\x \\ y; y \\ z\\ \\ x \\ z\n 4. \\x y.\n \\x \\ y; y \\ x\\ \\ x = y"} {"_id": "502686", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\S.\n S \\ fix f (pset cl) \\\n \\L.\n isLub S\n \\pset = fix f (pset cl),\n order = induced (fix f (pset cl)) (order cl)\\\n L"} {"_id": "502687", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>F y in at a within C. y \\ C"} {"_id": "502688", "text": "proof (prove)\nusing this:\n match_list ?d [(p, t)] = Some ?\\ \\\n ?\\ \\ matchers (set [(p, t)])\n match_list ?d [(p, t)] = None \\\n matchers (set [(p, t)]) = {}\n\ngoal (1 subgoal):\n 1. matches t p = (\\\\. p \\ \\ = t)"} {"_id": "502689", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Event Source Trigger Guard Action Update Target SA.\n \\(Source, (Trigger, Guard, Action, Update), Target) \\ TS;\n Event \\ Action; SA \\ SAs (HA ST);\n (Source, (Trigger, Guard, Action, Update), Target) \\ Delta SA;\n Source \\ Conf ST;\n (Conf ST, Events ST, Value ST) |= (Trigger, Guard, Action, Update);\n (Trigger, Guard, Action, Update) \\ Label (Delta SA)\\\n \\ Event \\ HAEvents (HA ST)"} {"_id": "502690", "text": "proof (prove)\nusing this:\n ((\\, k), \\', k') \\ cp\n\ngoal (1 subgoal):\n 1. ((\\, k), \\', k') \\ cop"} {"_id": "502691", "text": "proof (prove)\nusing this:\n Card_order (natLeq_on n) \\\n Card_order (cardSuc (natLeq_on n)) \\ Card_order (natLeq_on (Suc n))\n (natLeq_on ?m Card_order ?r; Card_order ?r'\\\n \\ (?r o ?r')\n\ngoal (1 subgoal):\n 1. cardSuc (natLeq_on n) \\o natLeq_on (Suc n)"} {"_id": "502692", "text": "proof (prove)\nusing this:\n s \\ carrier R\\<^bsup>\\\\<^esup>\n f \\ carrier (function_ring (carrier R) R)\n ?s \\ carrier ?R\\<^bsup>\\\\<^esup> \\\n ?s ?k \\ carrier ?R\n\ngoal (1 subgoal):\n 1. \\k. (f \\ s) k \\ carrier R"} {"_id": "502693", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Set.filter P (set xs) = set (filter P xs)"} {"_id": "502694", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 \\ valof x"} {"_id": "502695", "text": "proof (prove)\ngoal (3 subgoals):\n 1. P +\\<^sub>\\ bexp_neg e \\ P\n 2. \\mds.\n tyenv_wellformed mds \\ \\ P \\\n tyenv_wellformed mds \\ \\ P\n 3. \\mds.\n tyenv_wellformed mds \\ \\\n P +\\<^sub>\\ bexp_neg e \\\n tyenv_wellformed mds \\ \\ P"} {"_id": "502696", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\h some_ptr ancestors child_node parent.\n \\Shadow_DOM.heap_is_wellformed h;\n h \\ Shadow_DOM.get_ancestors some_ptr\n \\\\<^sub>r ancestors;\n cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n child_node\n \\ set ancestors;\n h \\ Shadow_DOM.get_parent child_node\n \\\\<^sub>r Some parent;\n ShadowRootClass.type_wf h; ShadowRootClass.known_ptrs h\\\n \\ parent \\ set ancestors\n 2. \\h ancestor ptr ancestors thesis.\n \\Shadow_DOM.heap_is_wellformed h; ancestor \\ ptr;\n ancestor \\ set ancestors;\n h \\ Shadow_DOM.get_ancestors ptr\n \\\\<^sub>r ancestors;\n ShadowRootClass.type_wf h; ShadowRootClass.known_ptrs h;\n \\children.\n h \\ Shadow_DOM.get_child_nodes ancestor\n \\\\<^sub>r children \\\n (\\ancestor_child.\n ancestor_child \\ set children \\\n cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n ancestor_child\n \\ set ancestors \\\n thesis)\\\n \\ thesis\n 3. \\h child ancestors ptr.\n \\Shadow_DOM.heap_is_wellformed h;\n h \\ Shadow_DOM.get_ancestors child\n \\\\<^sub>r ancestors;\n ShadowRootClass.known_ptrs h; ShadowRootClass.type_wf h;\n (ptr, child) \\ (Shadow_DOM.parent_child_rel h)\\<^sup>*\\\n \\ ptr \\ set ancestors\n 4. \\h child ancestors ptr.\n \\Shadow_DOM.heap_is_wellformed h;\n h \\ Shadow_DOM.get_ancestors child\n \\\\<^sub>r ancestors;\n ShadowRootClass.known_ptrs h; ShadowRootClass.type_wf h;\n ptr \\ set ancestors\\\n \\ (ptr, child)\n \\ (Shadow_DOM.parent_child_rel h)\\<^sup>*"} {"_id": "502697", "text": "proof (prove)\nusing this:\n y \\ unit_disc\n y \\ circline_set H \\ circline_set y_axis\n moebius_circline (moebius_rotation (pi / 2)) y_axis = x_axis\n\ngoal (1 subgoal):\n 1. moebius_pt (moebius_rotation (pi / 2)) y \\ unit_disc \\\n moebius_pt (moebius_rotation (pi / 2)) y\n \\ circline_set\n (moebius_circline (moebius_rotation (pi / 2)) H) \\\n circline_set x_axis"} {"_id": "502698", "text": "proof (state)\nthis:\n u ?n \\ space\\<^sub>N N\n\ngoal (1 subgoal):\n 1. \\u.\n \\cauchy_in\\<^sub>N N u;\n \\n. u n \\ space\\<^sub>N N\\\n \\ \\x\\space\\<^sub>N N.\n tendsto_in\\<^sub>N N u x"} {"_id": "502699", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(\\ X) \\ \\ \\;\n \\ = true \\\\\n \\ False"} {"_id": "502700", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sets\n (gen_subalgebra M\n (sigma_sets (space M)\n (\\i\\{m. m \\ n}.\n {X i -` A \\ space M |A. A \\ sets P}))) =\n sets\n (gen_subalgebra N\n (sigma_sets (space N)\n (\\i\\{m. m \\ n}.\n {X i -` A \\ space N |A. A \\ sets P})))"} {"_id": "502701", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P xs = P (remdups xs)"} {"_id": "502702", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Worder D; x = ordinal_number D; y \\ ODnums;\n y \\ x; ODNmap D y \\ carrier D;\n \\Y\\y. ord_equiv Y (Iod D (segment D (ODNmap D y)));\n \\c.\n c \\ carrier D \\\n y = ordinal_number (Iod D (segment D c))\\\n \\ y = ordinal_number (Iod D (segment D (ODNmap D y)))"} {"_id": "502703", "text": "proof (prove)\nusing this:\n ?a \\ A \\ g (Min A) \\ g ?a\n \\xs = sorted_list_of_set ?A; finite ?A; mono_on g ?A;\n inj_on g ?A\\\n \\ sorted_list_of_set (g ` ?A) =\n map g (sorted_list_of_set ?A)\n x # xs = sorted_list_of_set A\n finite A\n mono_on g A\n inj_on g A\n A \\ {}\n\ngoal (1 subgoal):\n 1. Min (g ` A) = g (Min A)"} {"_id": "502704", "text": "proof (prove)\ngoal (1 subgoal):\n 1. aff_dim {x. r \\ a \\ x} =\n (if a = (0::'a) \\ 0 < r then - 1 else int DIM('a))"} {"_id": "502705", "text": "proof (prove)\ngoal (1 subgoal):\n 1. - mtx [[a, b], [c, d]] = mtx [[- a, - b], [- c, - d]]"} {"_id": "502706", "text": "proof (prove)\nusing this:\n v \\ {v. v \\ EQ x c}\n v \\ EQ x c\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502707", "text": "proof (prove)\nusing this:\n ?i \\ I \\ Group.group (G ?i)\n\ngoal (1 subgoal):\n 1. (x \\\\<^bsub>DirProds G I\\<^esub>\n y \\\\<^bsub>DirProds G I\\<^esub>\n z)\n i =\n (x \\\\<^bsub>DirProds G I\\<^esub> y)\n i \\\\<^bsub>G i\\<^esub>\n z i"} {"_id": "502708", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum card {S' \\ p |p. p \\ P}\n \\ card {S' \\ p |p. p \\ P} * \\"} {"_id": "502709", "text": "proof (state)\nthis:\n (f (real_of_float l) + f (real_of_float u) -\n real_of_float (a * (l + u))) /\n 2\n \\\\<^sub>r bivl\n Y = affine_unop p (real_of_float a) b (real_of_float (upper divl)) X\n (f (real_of_float u) - f (real_of_float l) -\n real_of_float (a * (u - l))) /\n 2 +\n be\n \\\\<^sub>r divl\n real_of_float a \\ f' (real_of_float l)\n real_of_float a \\ f' (real_of_float u)\n b = real_of_float (lower bivl + upper bivl) / 2\n be = real_of_float (upper bivl - lower bivl) / 2\n\ngoal (1 subgoal):\n 1. f x \\ aform_err e Y"} {"_id": "502710", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {x. x ^ n = 1} = {x. 1 * x ^ n - poly 1 x = 0}"} {"_id": "502711", "text": "proof (prove)\ngoal (1 subgoal):\n 1. if tst then c1 else c2 \\01 if tst then d1 else d2"} {"_id": "502712", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ta_seq_consist P vs (lappend obs obs') =\n (ta_seq_consist P vs obs \\\n ta_seq_consist P (mrw_values P vs (list_of obs)) obs')"} {"_id": "502713", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\w f.\n \\\\w r.\n (\\v r) \\\n hoare (p w \\ b) x r;\n \\u. p u \\ q;\n \\v.\n v < w \\ hoare (p v) f (q \\ - b)\\\n \\ hoare (p w) ([\\ b ] * x)\n (wp f (q \\ - b))\n 2. \\w f.\n \\\\w r.\n (\\v r) \\\n hoare (p w \\ b) x r;\n \\u. p u \\ q;\n \\v.\n v < w \\ hoare (p v) f (q \\ - b)\\\n \\ hoare (p w) [\\ - b ] (q \\ - b)"} {"_id": "502714", "text": "proof (prove)\ngoal (1 subgoal):\n 1. all_normalized_and_ex_tcsigs (set cs) (translate_ars arss)"} {"_id": "502715", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i.\n \\x\\S.\n (f i \\ f i x)\n (at x within S)) \\\n \\x\\S.\n ((\\x. \\i. f i x) \\ (\\i. f i x))\n (at x within S)"} {"_id": "502716", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nfa (Reverse_nfa M)"} {"_id": "502717", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\well_formed procs;\n (Main, ins, outs, c) \\ set procs\\\n \\ False"} {"_id": "502718", "text": "proof (prove)\nusing this:\n shd (stl s) = shd s\n \\c_msg := (c_msg (shd s))\n (p := \\q.\n c_msg (shd s) p q @\n [{#t \\#\\<^sub>z c_temp (shd s) p. t \\ tt#}]),\n c_temp := (c_temp (shd s))\n (p := c_temp (shd s) p -\n {#t \\#\\<^sub>z c_temp (shd s) p. t \\ tt#})\\\n p \\ p'\n\ngoal (1 subgoal):\n 1. InfoAt (shd (stl s)) k' p' q' = InfoAt (shd s) k' p' q'"} {"_id": "502719", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\, \\' \\\\<^sub>C fail_converter \\"} {"_id": "502720", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Gauss_Jordan_in_ij A i j =\n (\\s.\n if s = i then A' $ s\n else row_add A' s i (- interchange_A $ s $ j) $ s)"} {"_id": "502721", "text": "proof (prove)\nusing this:\n \\x\\s. \\f'. (f has_field_derivative f') (at x within s)\n w \\ 0\n\ngoal (1 subgoal):\n 1. \\x\\s.\n \\f'.\n ((\\z. w powr f z) has_field_derivative f') (at x within s)"} {"_id": "502722", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < r1 &&& 0 < r2"} {"_id": "502723", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (A ^\\<^sub>m k *\\<^sub>v v) $v j =\n (map_mat cmod (of_real_hom.mat_hom A ^\\<^sub>m k) *\\<^sub>v v) $v j"} {"_id": "502724", "text": "proof (prove)\ngoal (1 subgoal):\n 1. functor (\\\\<^sub>A\\<^sub>1\\<^sub>x\\<^sub>A\\<^sub>2)\n (\\\\<^sub>B) (uncurry G)"} {"_id": "502725", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\\\c. c = 0 \\ b \\ c *\\<^sub>R a;\n a \\ (0::'a); b \\ (0::'a);\n \\c.\n c \\ 0 \\ b \\ c *\\<^sub>R a\\\n \\ \\A.\n (\\x\\A. x \\ (0::'a)) \\\n open A \\\n (\\B.\n (\\x\\B. x \\ (0::'a)) \\\n open B \\\n a \\ A \\\n b \\ B \\\n A \\ B\n \\ - {x.\n (\\c. c \\ 0 \\ snd x = c *\\<^sub>R fst x) \\\n (\\a b.\n x = (a, b) \\\n a \\ (0::'a) \\ b \\ (0::'a))} \\\n A \\ B\n \\ {a. a \\ (0::'a)} \\\n {b. b \\ (0::'a)})\n 2. \\c.\n \\\\c. c = 0 \\ b \\ c *\\<^sub>R a;\n a \\ (0::'a); b \\ (0::'a); c \\ 0\\\n \\ b \\ c *\\<^sub>R a"} {"_id": "502726", "text": "proof (prove)\nusing this:\n \\\\<^sub>F x in at_top. 0 < l x\n \\\\<^sub>F x in at_top. l x \\ f x\n \\\\<^sub>F x in at_top. f x \\ g x\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F x in at_top. g x powr p \\ f x powr p"} {"_id": "502727", "text": "proof (prove)\nusing this:\n x \\ A \\ \\ x \\ RF (TER f)\n\ngoal (1 subgoal):\n 1. x \\ SAT (quotient_web \\ f) (g \\ \\ / f)"} {"_id": "502728", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pnderiv a r = pnorm (deriv a r)"} {"_id": "502729", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {card B |B. basis_in \\ B} = {card B}"} {"_id": "502730", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pmf (Pi_pmf A dflt p) f =\n (if \\x. x \\ A \\ f x = dflt\n then \\x\\A. pmf (p x) (f x) else 0)"} {"_id": "502731", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(real, euclidean_ring_gcd_class)"} {"_id": "502732", "text": "proof (prove)\nusing this:\n \\Cop_S.MkArr (\\'.Y a) (\\'.Y a')\n \\\\'.map : local.map\n a \\\\<^sub>[\\<^sub>C\\<^sub>o\\<^sub>p\\<^sub>,\\<^sub>S\\<^sub>] local.map\n a'\\\n\ngoal (1 subgoal):\n 1. \\f.\n \\f : a \\ a'\\ \\\n local.map f =\n Cop_S.MkArr (\\'.Y a) (\\'.Y a') \\\\'.map"} {"_id": "502733", "text": "proof (prove)\ngoal (1 subgoal):\n 1. X \\ {} \\\n op_set_npick_remove X =\n ASSERT (X \\ {}) \\\n (\\_.\n op_set_pick X \\\n (\\x.\n op_set_ndelete x X \\ (\\X'. RETURN (x, X'))))"} {"_id": "502734", "text": "proof (chain)\npicking this:\n \\\\<^sub>p {wlp S (wlp (While M S) P)} S\n {adjoint (M 0) * P * M 0 +\n adjoint (M 1) * wlp S (wlp (While M S) P) * M 1}"} {"_id": "502735", "text": "proof (state)\ngoal (1 subgoal):\n 1. (\\ns.\n \\DirProds (\\n. Z (ns ! n)) {.. G;\n \\n\\set ns.\n n = 0 \\\n (\\p k.\n normalization_semidom_class.prime p \\\n 0 < k \\ n = p ^ k)\\\n \\ thesis) \\\n thesis"} {"_id": "502736", "text": "proof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (2 * pi) * \\ * of_int (unwinding z) = z - Ln (exp z)"} {"_id": "502737", "text": "proof (prove)\nusing this:\n \\\\r r3.\n (\\r r2 r3.\n Done\\cfcomp \\ c1\\r \\\n r2 \\\n r3 =\n c1\\r \\\n (r2 \\ r3)) \\\n Done\\cfcomp \\\n More\\(fmap\\c1\\p1) \\\n c2\\r \\\n r3 =\n More\\(fmap\\c1\\p1) \\\n (c2\\r \\ r3);\n \\r r2 r3.\n Done\\cfcomp \\ c1\\r \\ r2 \\\n r3 =\n c1\\r \\ (r2 \\ r3)\\\n \\ Done\\cfcomp \\\n More\\(fmap\\c1\\p1) \\\n More\\(fmap\\c2\\p2) \\\n c3\\?r =\n More\\(fmap\\c1\\p1) \\\n (More\\(fmap\\c2\\p2) \\\n c3\\?r)\n (\\r r2 r3.\n Done\\cfcomp \\ c1\\r \\ r2 \\\n r3 =\n c1\\r \\ (r2 \\ r3)) \\\n Done\\cfcomp \\\n More\\(fmap\\c1\\p1) \\\n c2\\?r \\\n ?r3.0 =\n More\\(fmap\\c1\\p1) \\\n (c2\\?r \\ ?r3.0)\n Done\\cfcomp \\ c1\\?r \\ ?r2.0 \\\n ?r3.0 =\n c1\\?r \\ (?r2.0 \\ ?r3.0)\n\ngoal (1 subgoal):\n 1. Done\\cfcomp \\\n More\\(fmap\\c1\\p1) \\\n More\\(fmap\\c2\\p2) \\\n More\\(fmap\\c3\\p3) =\n More\\(fmap\\c1\\p1) \\\n (More\\(fmap\\c2\\p2) \\\n More\\(fmap\\c3\\p3))"} {"_id": "502738", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\q x.\n \\0 < k; 0 < q;\n [ r, mod q * k ] = [ r div k * k, mod q * k ] \\ r mod k;\n x mod (q * k) = r div k * k mod (q * k);\n r div k * k \\ x\\\n \\ x mod k + r mod k mod k < k\n 2. \\q.\n \\0 < k; 0 < q;\n [ r, mod q * k ] = [ r div k, mod q ] \\ k \\ r mod k;\n [ r div k, mod q ] \\ k \\ r mod k \\ k =\n [ r div k, mod q ] \\ k \\ k \\\n r mod k div k\\\n \\ [ r div k, mod q ] \\ k \\\n r mod k \\\n k =\n [ r div k, mod q ]"} {"_id": "502739", "text": "proof (prove)\ngoal (1 subgoal):\n 1. generalize_instr (generalize_instr x) = generalize_instr x"} {"_id": "502740", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fgraph ` TameEnum \\\\<^sub>\\ Archive"} {"_id": "502741", "text": "proof (prove)\nusing this:\n sds (lift (\\x. R (\\ x))) = sds (lift R)\n\ngoal (1 subgoal):\n 1. sds (lift (\\x. R (\\ x))) = sds (lift R)"} {"_id": "502742", "text": "proof (state)\ngoal (1 subgoal):\n 1. quorum_of p Q"} {"_id": "502743", "text": "proof (prove)\ngoal (2 subgoals):\n 1. is_cupcake_ns (alist_to_ns env')\n 2. is_cupcake_ns (sem_env.v env)"} {"_id": "502744", "text": "proof (prove)\nusing this:\n finite A\n A \\ {a..b}\n a \\ A\n b \\ A\n a < b\n\ngoal (1 subgoal):\n 1. A \\ {Max (A \\ {.. \\\n \\\\<^sub>Q \\ P \\ M\\\\*xvec\\\\N\\ \\ P'\n extractFrame P = \\A\\<^sub>P, \\\\<^sub>P\\\n distinct A\\<^sub>P\n (\\ \\ \\\\<^sub>Q) \\\n \\\\<^sub>P \\ M \\ K'"} {"_id": "502746", "text": "proof (prove)\ngoal (1 subgoal):\n 1. d p \\ d r + wp y q"} {"_id": "502747", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (length l = x1 \\ eval F (l @ init @ xs)) =\n (length l = x1 \\\n eval (map_fm_binders (\\a x. liftAtom (i + x) amount a) F x1)\n (l @ init @ I @ xs))"} {"_id": "502748", "text": "proof (prove)\ngoal (1 subgoal):\n 1. module_on U\\<^sub>M s"} {"_id": "502749", "text": "proof (prove)\ngoal (1 subgoal):\n 1. color_of (rbt_map_entry k f t) = color_of t"} {"_id": "502750", "text": "proof (prove)\ngoal (1 subgoal):\n 1. relation_subsumption_impl r subsumes set\n (\\x. case x of h # t \\ (h, t)) (@) length"} {"_id": "502751", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fw_impl n\n Mi <\\r.\n \\\\<^sub>AM'.\n mtx_curry_assn n M' r *\n \\\n (if \\i\\n. M' i i < (0::'a)\n then \\ cycle_free M n\n else \\i\\n.\n \\j\\n.\n M' i j = D M i j n \\\n cycle_free M n)>\\<^sub>t"} {"_id": "502752", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s[x::=v][y::=u] = s[y::=u][x::=v[y::=u]]"} {"_id": "502753", "text": "proof (prove)\nusing this:\n invariant ?ds cop_F_closed_inv\n cop_F_closed_inv ds fp\n x \\ cop_F ds fp\n Xd x \\ Xd ` CH (cop_F ds fp)\n\ngoal (1 subgoal):\n 1. \\fp'\\cop_F ds fp.\n x \\ fp' \\\n above (Pd (Xd x)) x \\ fp' \\ Xd x \\ Xd ` CH fp'"} {"_id": "502754", "text": "proof (prove)\nusing this:\n resumption_ord (h (the (Resumption.resumption.result f')))\n (k (the (Resumption.resumption.result f')))\n resumption_ord f' g'\n \\x. resumption_ord (h x) (k x)\n \\ Resumption.resumption.is_Done (g' \\ k)\n Resumption.resumption.is_Done (f' \\ h)\n\ngoal (1 subgoal):\n 1. Resumption.resumption.result (f' \\ h) = None"} {"_id": "502755", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sa.is_run r =\n (\\r'. r = pidgc_\\ \\ r' \\ pid.is_run r')"} {"_id": "502756", "text": "proof (prove)\nusing this:\n Some cl \\ set (concat (s.tab s))\n list_all (tab_agree \\) (concat (s.tab s))\n\ngoal (1 subgoal):\n 1. \\tf. cl_typing \\ cl tf"} {"_id": "502757", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>\\ (vprojection I A i) = A i"} {"_id": "502758", "text": "proof (prove)\nusing this:\n Xh ` {d2h1, d3h1} = {Xh d1h1}\n Xh ` {d2h2, d3h2} = {Xh d1h2}\n Xh ` {d2h3, d3h3} = {Xh d1h3}\n distinct [Xh d1h1, Xh d1h2, Xh d1h3]\n distinct [d1h1, d1h2, d1h3, d2h1, d2h2, d2h3, d3h1, d3h2, d3h3]\n {Xd d1h1, Xd d2h1, Xd d3h1} \\ ds\n \\ BPd' BCh ds = {d1h1, d3h3}\n \\ BPd' BCh' ds = {d3h1, d1h3}\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502759", "text": "proof (prove)\nusing this:\n \\op\\set ops.\n v \\ set (add_effects_of op) \\\n v \\ set (delete_effects_of op)\n\ngoal (1 subgoal):\n 1. foldl (++) (s ++ (map_of \\ effect_to_assignments) a)\n (map (map_of \\ effect_to_assignments) ops) v =\n (s ++ (map_of \\ effect_to_assignments) a) v"} {"_id": "502760", "text": "proof (prove)\ngoal (1 subgoal):\n 1. c\\<^sub>2 \\ c'';; c\\<^sub>2"} {"_id": "502761", "text": "proof (prove)\nusing this:\n P \\ wp a (wp b R)\n\ngoal (1 subgoal):\n 1. P \\ wp (a ;; b) R"} {"_id": "502762", "text": "proof (prove)\nusing this:\n \\.A.ide a\n\ngoal (1 subgoal):\n 1. D (\\ (H a)) (\\ (C (\\ a) (\\ a))) =\n D (\\ (\\ a)) (\\ (\\ a))"} {"_id": "502763", "text": "proof (prove)\ngoal (1 subgoal):\n 1. norm x \\ CF"} {"_id": "502764", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\LeftDerivation a D b \\ D = []; i < length a;\n is_nonterminal (a ! i); LeftDerivation x D b; Suc i \\ fst d;\n \\d\\set D. Suc i \\ fst d; leftmost (fst d) a;\n Derives1 a (fst d) (snd d) x\\\n \\ False"} {"_id": "502765", "text": "proof (prove)\nusing this:\n NR\\<^sub>N n u l =\n ennreal p * (of_nat l + NR\\<^sub>N (n - 1) u l) +\n ennreal q *\n (of_nat u +\n (\\kN (k + 1) u l * ennreal (B (n - 1) k))) +\n ennreal (q ^ n) * NR\\<^sub>N n u l\n NR\\<^sub>N n u l \\ \\\n 0 < n\n p \\ {0<..<1}\n q ^ n \\ {0<..<1}\n\ngoal (1 subgoal):\n 1. NR\\<^sub>N n u l =\n (ennreal p * (of_nat l + NR\\<^sub>N (n - 1) u l) +\n ennreal q *\n (of_nat u +\n (\\kN (k + 1) u l * ennreal (B (n - 1) k)))) /\n ennreal (1 - q ^ n)"} {"_id": "502766", "text": "proof (prove)\ngoal (1 subgoal):\n 1. col P Q R"} {"_id": "502767", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Tree2.inorder (combine l r) = Tree2.inorder l @ Tree2.inorder r"} {"_id": "502768", "text": "proof (prove)\nusing this:\n \\m. j \\# m \\ P m = \\\n i \\ j\n\ngoal (1 subgoal):\n 1. \\m.\n j \\# m \\\n (if i \\# m then P (m - {#i#}) else \\) = \\"} {"_id": "502769", "text": "proof (prove)\ngoal (1 subgoal):\n 1. topspace X \\ topspace Y\n \\ \\\n ((\\) (topspace X) ` Collect (openin Y) \\\n (\\U. U \\ topspace Y) ` Collect (openin X))"} {"_id": "502770", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a \\ 'b, cancellative_pam_class)"} {"_id": "502771", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\p | p permutes {0..i = 0..p | p permutes {0..i = 0.. = {}"} {"_id": "502773", "text": "proof (prove)\nusing this:\n kind ax = (\\s. True)\\<^sub>\\\n kind a = (\\s. True)\\<^sub>\\\n kind a\\<^sub>1 = \\id\n kind a\\<^sub>2 = \\id\n transfers (kinds as\\<^sub>1) [(cf\\<^sub>1, undefined)] = cfs\\<^sub>1\n map fst cfs\\<^sub>1 = s\\<^sub>1\n transfers (kinds as\\<^sub>2) [(cf\\<^sub>2, undefined)] = cfs\\<^sub>2\n map fst cfs\\<^sub>2 = s\\<^sub>2\n map fst\n (transfers (kinds (ax # (a # as\\<^sub>1) @ [a\\<^sub>1]))\n [(cf\\<^sub>1, undefined)]) \\\\<^sub>L\n map fst\n (transfers (kinds (ax # (a # as\\<^sub>2) @ [a\\<^sub>2]))\n [(cf\\<^sub>2, undefined)])\n\ngoal (1 subgoal):\n 1. s\\<^sub>1 \\\\<^sub>L s\\<^sub>2"} {"_id": "502774", "text": "proof (prove)\nusing this:\n a < b\n \\a \\ ?z; ?z \\ b\\ \\ isCont f ?z\n \\a \\ ?z; ?z \\ b\\ \\ isCont g ?z\n \\a < ?z; ?z < b\\\n \\ (g has_real_derivative g' ?z) (at ?z)\n \\a < ?z; ?z < b\\\n \\ (f has_real_derivative f' ?z) (at ?z)\n\ngoal (1 subgoal):\n 1. \\g'c f'c c.\n (g has_real_derivative g'c) (at c) \\\n (f has_real_derivative f'c) (at c) \\\n a < c \\ c < b \\ (f b - f a) * g'c = (g b - g a) * f'c"} {"_id": "502775", "text": "proof (prove)\nusing this:\n Complete_Partial_Order.chain orda Y\n Y \\ {}\n Complete_Partial_Order.chain ordb (t ` Y)\n\ngoal (1 subgoal):\n 1. f (luba Y) (t (luba Y)) = (\\x\\Y. f x (t x))"} {"_id": "502776", "text": "proof (prove)\nusing this:\n well_base\\<^sub>h (i + 2) (encode (i + 2) 0 (local.goodstein i))\n\ngoal (1 subgoal):\n 1. well_base\\<^sub>h (i + 3) (encode (i + 2) 0 (local.goodstein i))"} {"_id": "502777", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ordinate B - ordinate C \\ radical_sqrt"} {"_id": "502778", "text": "proof (prove)\nusing this:\n p \\ succs (hd w)\n\ngoal (1 subgoal):\n 1. ss ! p \\\\<^bsub>r\\<^esub> ssa ! p"} {"_id": "502779", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n x =\n Matrix.mat CARD('nr) CARD('nc)\n (\\(i, j).\n y $h mod_type_class.from_nat i $h\n mod_type_class.from_nat j) \\\n (\\xa ya.\n xa =\n Matrix.mat CARD('nr) CARD('nc)\n (\\(i, j).\n ya $h mod_type_class.from_nat i $h\n mod_type_class.from_nat j) \\\n x + map_mat uminus xa =\n Matrix.mat CARD('nr) CARD('nc)\n (\\(i, j).\n (y - ya) $h mod_type_class.from_nat i $h\n mod_type_class.from_nat j))"} {"_id": "502780", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OUTfromChCorrect data6"} {"_id": "502781", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(H2, enum_class)"} {"_id": "502782", "text": "proof (prove)\nusing this:\n x \\ G\n y \\ G\n \\z\\G. binop (binop x (\\ y)) z = z\n\ngoal (1 subgoal):\n 1. binop x (binop (\\ y) y) = y"} {"_id": "502783", "text": "proof (prove)\nusing this:\n P \\\\<^sup>s Q\n\ngoal (1 subgoal):\n 1. [a\\b]P \\\\<^sup>s [a\\b]Q"} {"_id": "502784", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\continuous f S T;\n \\f \\ carrier \\ carrier\\<^bsub>T\\<^esub>;\n \\m.\n m open\\<^bsub>T\\<^esub> \\\n carrier \\ f -` m open\\\n \\ P\\\n \\ P"} {"_id": "502785", "text": "proof (prove)\ngoal (1 subgoal):\n 1. surj pp.mapping_of"} {"_id": "502786", "text": "proof (prove)\ngoal (1 subgoal):\n 1. |SIGMA i:I. B (f i)| \\o |Sigma J B|"} {"_id": "502787", "text": "proof (prove)\nusing this:\n (0::'a) \\ \\b\\\n (0::'a) \\ \\b\\\n (0::'a) \\ a / \\b\\\n (\\?a\\ = (0::'a)) = (?a = (0::'a))\n \\(0::'a) \\ ?a; (0::'a) \\ ?b\\\n \\ (0::'a) \\ ?a * ?b\n\ngoal (1 subgoal):\n 1. (0::'a) \\ a \\ b = (0::'a)"} {"_id": "502788", "text": "proof (state)\nthis:\n {olv_ref_rel \\\n (OV_inv2 \\ OV_inv3 \\ OV_inv4) \\\n UNIV} TS.trans obsv_TS, TS.trans olv_TS {> olv_ref_rel}\n\ngoal (4 subgoals):\n 1. {OV_inv2 \\ OV_inv3 \\ OV_inv4 \\\n Domain\n (olv_ref_rel \\\n UNIV \\\n UNIV)} TS.trans obsv_TS {> OV_inv2 \\ OV_inv3 \\ OV_inv4}\n 2. {UNIV \\\n Range\n (olv_ref_rel \\\n (OV_inv2 \\ OV_inv3 \\ OV_inv4) \\\n UNIV)} TS.trans olv_TS {> UNIV}\n 3. init obsv_TS \\ OV_inv2 \\ OV_inv3 \\ OV_inv4\n 4. init olv_TS \\ UNIV"} {"_id": "502789", "text": "proof (prove)\nusing this:\n oalist_inv_raw ko xs'\n\ngoal (1 subgoal):\n 1. oalist_inv_raw ko (filter P xs')"} {"_id": "502790", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_option f (while_option b c s) = while_option b' c' (f s)"} {"_id": "502791", "text": "proof (prove)\ngoal (1 subgoal):\n 1. from_bool (to_bool (x AND 1)) = x AND 1"} {"_id": "502792", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (HMA_M ===> (=)) Spectral_Radius.spectral_radius\n HMA_Connect.spectral_radius"} {"_id": "502793", "text": "proof (state)\nthis:\n safe_free_flowing c\\<^sub>0\n\ngoal (2 subgoals):\n 1. safe_reach_upto 0 safe_free_flowing c\\<^sub>0 \\\n safe_reach_upto 0 safe_delayed c\\<^sub>0\n 2. \\n.\n \\safe_reach_upto n safe_free_flowing\n c\\<^sub>0 \\\n safe_reach_upto n safe_delayed c\\<^sub>0;\n safe_reach_upto (Suc n) safe_free_flowing c\\<^sub>0\\\n \\ safe_reach_upto (Suc n) safe_delayed c\\<^sub>0"} {"_id": "502794", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\non_terminating P; lifelock_free_v2 Q\\\n \\ non_terminating (P \\C\\ Q)"} {"_id": "502795", "text": "proof (prove)\nusing this:\n \\\\<^sub>F n in F. x \\ u\n\ngoal (1 subgoal):\n 1. x \\ u"} {"_id": "502796", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (bisim ===> (=)) dom fst"} {"_id": "502797", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xc = x, yc = y, zc = z\\ = p\n \\xc := x, yc := y, zc := z\\"} {"_id": "502798", "text": "proof (prove)\nusing this:\n j < length fs\n p = Inr j\n Inr j \\ set ps\n\ngoal (1 subgoal):\n 1. is_RB_in dgrad rword (set bs) (term_of_pair (0::'a, j)) \\\n rep_list (monomial (1::'b) (term_of_pair (0::'a, j)))\n \\ ideal (rep_list ` set bs)"} {"_id": "502799", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LIM x F. f x * inverse (g x) :> at_top"} {"_id": "502800", "text": "proof (prove)\nusing this:\n card B < card (C\\<^sub>1 \\ C\\<^sub>2 - {x})\n \\indep_in (C\\<^sub>1 \\ C\\<^sub>2)\n (C\\<^sub>1 \\ C\\<^sub>2 - {x});\n basis_in (C\\<^sub>1 \\ C\\<^sub>2) B\\\n \\ card (C\\<^sub>1 \\ C\\<^sub>2 - {x}) \\ card B\n basis_in (C\\<^sub>1 \\ C\\<^sub>2) B\n C\\<^sub>2 - {y} \\ B\n indep_in (C\\<^sub>1 \\ C\\<^sub>2)\n (C\\<^sub>1 \\ C\\<^sub>2 - {x})\n\ngoal (1 subgoal):\n 1. False"} {"_id": "502801", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\m. obs n (PDG_BS S) = {m}) \\ obs n (PDG_BS S) = {}"} {"_id": "502802", "text": "proof (chain)\npicking this:\n PROP ?psi \\ PROP ?psi\n dickson_less_v d m (seq (Suc ?i27)) (seq ?i27)\n i < j"} {"_id": "502803", "text": "proof (chain)\npicking this:\n fv e \\ fv (RI (D,throw a) ; Cs \\ unit)\n \\ fv (RI (C,e) ; D # Cs \\ unit)"} {"_id": "502804", "text": "proof (prove)\nusing this:\n x \\ ball (0::'a) 2\n\ngoal (1 subgoal):\n 1. 0 < H x"} {"_id": "502805", "text": "proof (prove)\nusing this:\n Vagree ?v ?w (- {z}) \\ term_sem I t ?v = term_sem I t ?w\n\ngoal (1 subgoal):\n 1. term_sem I t (stateinterpol \\ \\' S) =\n term_sem I t (stateinterpol \\ \\' (insert z S))"} {"_id": "502806", "text": "proof (chain)\npicking this:\n x \\ {a--b}"} {"_id": "502807", "text": "proof (state)\nthis:\n order.greater_eq (chamber_distance C D)\n (ChamberComplex.chamber_distance A C D)\n\ngoal (1 subgoal):\n 1. C \\ D \\\n ChamberComplex.chamber_distance A C D = chamber_distance C D"} {"_id": "502808", "text": "proof (prove)\ngoal (1 subgoal):\n 1. connected (\\ (range X))"} {"_id": "502809", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (a * x ^ 3 + b * x\\<^sup>2 + c * x + d = (0::'a)) =\n (y ^ 3 + e * y + f = (0::'a))"} {"_id": "502810", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z.\n x \\ y \\ x \\ z \\\n x \\ (y \\ z) =\n x \\ (y \\ z)"} {"_id": "502811", "text": "proof (prove)\nusing this:\n f < 0 \\ f = 0 \\ 0 < f\n\ngoal (1 subgoal):\n 1. v \\\\<^sub>n\\<^sub>s (f *R ns)"} {"_id": "502812", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\compact S; closedin (top_of_set S) T\\\n \\ compact T"} {"_id": "502813", "text": "proof (prove)\nusing this:\n T =\n (if Ground = FirstOrder\n then filter_trms C\n (dom_trms (subst_cl D \\) (subst_set E \\))\n else dom_trms (subst_cl D \\) (subst_set E \\))\n\ngoal (1 subgoal):\n 1. T = dom_trms (subst_cl D \\) (subst_set E \\)"} {"_id": "502814", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\d = 1..n.\n mangoldt d *\n real_of_int\n \\real (2 * (n div 2) + n mod 2) / real d\\)\n \\ (\\d = 1..n.\n mangoldt d *\n real_of_int\n (2 * \\real (n div 2) / real d\\ + 1))"} {"_id": "502815", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\# mset_pos M) = (0 < zcount M x)"} {"_id": "502816", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (a, b) \\ E \\\n a \\ (\\a\\S. {b \\ S. (b, a) \\ E}) b"} {"_id": "502817", "text": "proof (prove)\nusing this:\n LIM x at_bot. poly q x / poly p x :> at_top\n\ngoal (1 subgoal):\n 1. (if LIM x at_bot. poly q x / poly p x :> at_top then 1 / 2\n else if LIM x at_bot. poly q x / poly p x :> at_bot then - 1 / 2\n else 0) =\n 1 / 2"} {"_id": "502818", "text": "proof (prove)\nusing this:\n x \\ (x \\ - y\\<^sup>\\ \\ y)\\<^sup>\\\n\ngoal (1 subgoal):\n 1. x \\ - y\\<^sup>\\ \\ y = x"} {"_id": "502819", "text": "proof (prove)\nusing this:\n Seed\\<^bsub>p\\<^esub> [next_plane\\<^bsub>p\\<^esub>]\\* g\n tame g\n final g\n p \\ 3\n fgraph ` TameEnum \\\\<^sub>\\ Archive\n\ngoal (1 subgoal):\n 1. fgraph g \\\\<^sub>\\ Archive"} {"_id": "502820", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a + b + c = a + (b + c)"} {"_id": "502821", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bprv (PPf p \\neg \\R\\)"} {"_id": "502822", "text": "proof (state)\nthis:\n dupe_monic (of_int_poly (Mp D)) (of_int_poly (Mp H)) (of_int_poly (Mp S))\n (of_int_poly (Mp T)) (of_int_poly (Mp U)) =\n (A', B')\n\ngoal (4 subgoals):\n 1. A * D + B * H =m U\n 2. B = 0 \\ degree B < degree D\n 3. Mp_rel_i a A\n 4. Mp_rel_i b B"} {"_id": "502823", "text": "proof (prove)\nusing this:\n xo = \\x\\\n x \\ B \\\n x \\ \\<^bold>V\n A \\ \\ B \\ = {}\n\ngoal (1 subgoal):\n 1. (\\\\<^sup>+ xa. f n (xa, \\x\\)) =\n (\\\\<^sup>+ y.\n f n (y, \\x\\) * indicator (range Inner) y +\n f n (y, \\x\\) * indicator {SOURCE} y)"} {"_id": "502824", "text": "proof (prove)\nusing this:\n c_rec + c_sig + 2 * n + 2 \\ (\\jj = j..\n cost (dmu_array_row_main_cost fi i dmus j)\n \\ (\\jj = j.. r)\n (\\n x. \\i) ^ i)"} {"_id": "502827", "text": "proof (prove)\ngoal (1 subgoal):\n 1. upI d a <\\r. gridI a (up dm lm d v)>"} {"_id": "502828", "text": "proof (chain)\npicking this:\n i < length (l # ts)\n (l # ts) ! i =\n (p, is, \\, Ghost\\<^sub>s\\<^sub>b A L R W # sb, \\, \\, \\)"} {"_id": "502829", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rbl_mul (x # xs) ys =\n rbl_plus False (map ((\\) x) ys) (False # rbl_mul xs ys) &&&\n rbl_mul [] ys = []"} {"_id": "502830", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\c l x.\n \\c \\ gPB; l \\ set c; x \\ nv2L l;\n \\ infTp (GE.tpOfV x)\\\n \\ grdOf c l x \\ set c\n 2. \\\\ c. c \\ gPB \\ mcalc2 \\ c\n 3. \\\\1 P \\2. pol \\1 P = pol \\2 P"} {"_id": "502831", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b. (a \\ b) = (a \\ b = b)"} {"_id": "502832", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\d. R\\<^sup>*\\<^sup>* b d \\ R\\<^sup>*\\<^sup>* c d"} {"_id": "502833", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pnorm_alt ` pset r = pset r"} {"_id": "502834", "text": "proof (prove)\nusing this:\n has_subprob_density M lborel f\n X = {a..b}\n\ngoal (1 subgoal):\n 1. emeasure (distr M lborel h) X = emeasure (density lborel g) X"} {"_id": "502835", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a \\ G \\ bij_betw (a)\\<^sub>L G G"} {"_id": "502836", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dist x y \\ infdist x A"} {"_id": "502837", "text": "proof (state)\nthis:\n x \\ u = z\n\ngoal (1 subgoal):\n 1. x \\ y = y \\ x"} {"_id": "502838", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {0..n} \\ {Suc n..m} = {}"} {"_id": "502839", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ssat (fst (tree.root t))"} {"_id": "502840", "text": "proof (prove)\nusing this:\n linears (snd (strip_comb t\\<^sub>1) @ [t\\<^sub>2])\n\ngoal (1 subgoal):\n 1. linears (snd (strip_comb (app t\\<^sub>1 t\\<^sub>2)))"} {"_id": "502841", "text": "proof (state)\nthis:\n (\\t.\n \\,\\,p\\ \\rs_called, Undecided\\ \\ t) \\\n (\\rs_called1 rs_called2 m'.\n \\ chain_name_neu =\n Some (rs_called1 @ [Rule m' Return] @ rs_called2) \\\n matches \\ m' p \\\n \\,\\,p\\ \\rs_called1, Undecided\\ \\ Undecided)\n\ngoal (1 subgoal):\n 1. \\x m.\n \\\\y m.\n \\(y, x) \\ called_by_chain \\;\n matches \\ m p;\n wf_chain \\ [Rule m (Call y)];\n Ball (ran \\) (wf_chain \\);\n \\rsg\\ran \\.\n \\r\\set rsg.\n (\\chain.\n get_action r \\ Goto chain) \\\n get_action r \\ Unknown\\\n \\ Ex\n(iptables_bigstep \\ \\ p [Rule m (Call y)] Undecided);\n matches \\ m p; wf_chain \\ [Rule m (Call x)];\n Ball (ran \\) (wf_chain \\);\n \\rsg\\ran \\.\n \\r\\set rsg.\n (\\chain. get_action r \\ Goto chain) \\\n get_action r \\ Unknown\\\n \\ Ex (iptables_bigstep \\ \\ p\n [Rule m (Call x)] Undecided)"} {"_id": "502842", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < n \\\n [- a, 1::'a] %^ n divides p \\\n \\ [- a, 1::'a] %^ Suc n divides p \\\n poly p a = (0::'a)"} {"_id": "502843", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inverse x \\ aform_err e Y"} {"_id": "502844", "text": "proof (prove)\ngoal (1 subgoal):\n 1. path (E \\ UNIV \\ - R) u (x # xs) v =\n (\\w.\n w \\ R \\\n x = u \\ (u, w) \\ E \\ path (rel_restrict E R) w xs v)"} {"_id": "502845", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\u'\\{u \\ d |u d.\n u \\ Z_ \\\n {u. u \\ inv_of A_ l_} \\\n (0::'c) \\ d} \\\n {u. u \\ inv_of A_ l_}.\n \\u\\Z_.\n A_ \\ \\l_, u\\ \\ \\l_,u'\\"} {"_id": "502846", "text": "proof (prove)\ngoal (1 subgoal):\n 1. igSubstIGOpSTR MOD \\ igSubstIGOp MOD"} {"_id": "502847", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a' \\ I"} {"_id": "502848", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C.arr f0"} {"_id": "502849", "text": "proof (prove)\nusing this:\n finite A\n\ngoal (1 subgoal):\n 1. Mapping.lookup_default 0 (fold_combine_plus.combine.F f A) x =\n (\\y\\A. Mapping.lookup_default 0 (f y) x)"} {"_id": "502850", "text": "proof (state)\nthis:\n 0 \\ a ?k\n\ngoal (1 subgoal):\n 1. \\e.\n 0 < e \\\n \\N. \\n\\N. dist (\\ n) (\\ N) < e"} {"_id": "502851", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f |`| fset_from_list l = fset_from_list (map f l)"} {"_id": "502852", "text": "proof (prove)\nusing this:\n ins n d x \\\\ = Bal t'\n bal_i n (height \\\\ + d)\n\ngoal (1 subgoal):\n 1. bal_i (n + 1) (height t' + d)"} {"_id": "502853", "text": "proof (state)\nthis:\n \\\\\\ \\ vs : [] _> tls\n\ngoal (1 subgoal):\n 1. \\\\\\ \\ vs @ es0 : ts _> ts'"} {"_id": "502854", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (map (inv q)\n (filter (\\x. x \\ range q)\n (xs @\n ipurge_tr (con_comp_pol I) (con_comp_map D E p q)\n (con_comp_map D E p q y) ys)),\n inv q `\n ipurge_ref (con_comp_pol I) (con_comp_map D E p q)\n (con_comp_map D E p q y) ys S)\n \\ failures Q"} {"_id": "502855", "text": "proof (prove)\nusing this:\n dist a (fst x) < r\n dist b (snd x) < s\n\ngoal (1 subgoal):\n 1. sqrt\n ((dist (fst (a, b)) (fst x))\\<^sup>2 +\n (dist (snd (a, b)) (snd x))\\<^sup>2)\n < sqrt (r\\<^sup>2 + s\\<^sup>2)"} {"_id": "502856", "text": "proof (prove)\nusing this:\n y \\ \\_vars x\n\ngoal (1 subgoal):\n 1. y \\ vars_\\ X"} {"_id": "502857", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\ topspace X \\\n (\\x\\topspace X. f x \\ topspace Y) \\\n (\\V.\n openin Y V \\ f x \\ V \\\n (\\U.\n openin X U \\\n x \\ U \\ (\\y\\U. f y \\ V)))) =\n (x \\ topspace X \\\n (\\x\\topspace X. f x \\ topspace Y) \\\n limitin Y f (f x) (atin X x))"} {"_id": "502858", "text": "proof (prove)\nusing this:\n a ! k * maxne0 ys b < a ! k * xs ! k\n\ngoal (1 subgoal):\n 1. a ! k < sum_list ys"} {"_id": "502859", "text": "proof (prove)\nusing this:\n p * x \\ (0::'a)\n prime p\n\ngoal (1 subgoal):\n 1. normalize (\\\\<^sub># (prime_factorization (p * x))) =\n normalize (p * normalize (\\\\<^sub># (prime_factorization x)))"} {"_id": "502860", "text": "proof (prove)\ngoal (1 subgoal):\n 1. t \\ set ts \\\n size t < Suc (Suc (size s + size_list size ts))"} {"_id": "502861", "text": "proof (prove)\ngoal (1 subgoal):\n 1. matchers (set P) = {}"} {"_id": "502862", "text": "proof (prove)\nusing this:\n \\prime i; i \\ i\\\n \\ multiplicity i factor_pr = 0\n prime i\n prime j\n 0 < factor_pr\n j = i\n\ngoal (1 subgoal):\n 1. multiplicity j (factor_pr * i) \\ 1"} {"_id": "502863", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS(finite_2, field_class) &&&\n OFCLASS(finite_2, idom_abs_sgn_class) &&&\n OFCLASS(finite_2, idom_modulo_class)"} {"_id": "502864", "text": "proof (prove)\nusing this:\n \\x. valid_selection Rs (n, E) (fst x) (snd x)\n (case ?prod of (x, xa) \\ ?f x xa) = ?f (fst ?prod) (snd ?prod)\n \\P x. P x \\ P (Eps P)\n\ngoal (1 subgoal):\n 1. \\ valid_selection Rs (n, E)\n (fst (SOME (R, f). valid_selection Rs (n, E) R f))\n (snd (SOME (R, f).\n valid_selection Rs (n, E) R f)) \\\n False"} {"_id": "502865", "text": "proof (prove)\ngoal (1 subgoal):\n 1. i \\ r \\\n arr_mset arr (i - 1) r = arr_mset arr i r + {#arr (i - 1)#}"} {"_id": "502866", "text": "proof (prove)\ngoal (1 subgoal):\n 1. weakSimAct P (\\ \\ Q') P Rel"} {"_id": "502867", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map (Poly_Mapping.map_key PP) (punit.Macaulay_list fs) =\n pp_pm.punit.Macaulay_list (map (Poly_Mapping.map_key PP) fs)"} {"_id": "502868", "text": "proof (prove)\ngoal (1 subgoal):\n 1. add_col_sub_row a i j A $$ (i', j') = (0::'a)"} {"_id": "502869", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Id_on B = (\\x. (x, x)) ` B &&&\n Id_on (Complement A) =\n (case ID CEQ('a) of\n None \\\n Code.abort STR ''Id_on Complement: ceq = None''\n (\\_. Id_on (Complement A))\n | Some eq \\\n Collect_set (\\(x, y). eq x y \\ x \\ A)) &&&\n Id_on (Collect_set P) =\n (case ID CEQ('a) of\n None \\\n Code.abort STR ''Id_on Collect_set: ceq = None''\n (\\_. Id_on (Collect_set P))\n | Some eq \\\n Collect_set (\\(x, y). eq x y \\ P x)) &&&\n Id_on (DList_set dxs) =\n (case ID CEQ('a) of\n None \\\n Code.abort STR ''Id_on DList_set: ceq = None''\n (\\_. Id_on (DList_set dxs))\n | Some x \\ DList_set (DList_Set.Id_on dxs)) &&&\n Id_on (RBT_set rbt) =\n (case ID ccompare of\n None \\\n Code.abort STR ''Id_on RBT_set: ccompare = None''\n (\\_. Id_on (RBT_set rbt))\n | Some x \\ RBT_set (RBT_Set2.Id_on rbt))"} {"_id": "502870", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Min {extended_Gromov_product_at e x y, extended_Gromov_product_at e y z,\n extended_Gromov_product_at e z t} -\n ereal (2 * deltaG TYPE('a))\n \\ extended_Gromov_product_at e x t"} {"_id": "502871", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x_ \\ subterms f_; x_ \\ Mapping.keys M2;\n f_ \\ Mapping.keys M2\\\n \\ x_ = Binop fa1 fa2"} {"_id": "502872", "text": "proof (prove)\nusing this:\n P,E(V \\\n T) \\ \\e,(h, l(V \\ v'))\\ \\*\n \\e',(h', l')\\\n P \\ T casts v to v' \n wf_prog wf_md P\n\ngoal (1 subgoal):\n 1. (\\v'' w.\n \\P,E \\ \\{V:T; V:=Val v;; e},\n (h, l)\\ \\*\n \\{V:T; V:=Val v'';; e'},\n (h', l'(V := l V))\\;\n P \\ T casts v'' to w \\\n \\ thesis) \\\n thesis"} {"_id": "502873", "text": "proof (prove)\nusing this:\n k' \\ k\n\ngoal (1 subgoal):\n 1. {\\ (M k)..} \\ {\\ (M k')..}"} {"_id": "502874", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (older_seniors x n)\n < card (reach \\ \\ q\\<^sub>0 - Collect sink)"} {"_id": "502875", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p \\[sequentially] (\\n.\n 16 ^ n * fact n ^ 4 /\n (fact (2 * n))\\<^sup>2 /\n (2 * real n + 1))"} {"_id": "502876", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f =\n monom P (T\\<^bsub>a\\<^esub> f 0) 0 \\\\<^bsub>P\\<^esub>\n X_poly_minus R a \\\\<^bsub>P\\<^esub>\n Cring_Poly.compose R (poly_shift (T\\<^bsub>a\\<^esub> f))\n (X_poly_minus R a)"} {"_id": "502877", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\A a b.\n \\wf_prog G;\n G,insert\n ({Normal \\} In1l (Methd a b)\\ {G\\})\n A\\{Normal\n \\} In1l\n (body G a\n b)\\ {G\\}\\\n \\ G,A\\{Normal\n \\} In1l (Methd a b)\\ {G\\}\n 2. \\A t.\n \\wf_prog G;\n \\pn\\?U2.\n G,A\\{Normal\n \\} (case pn of\n (C, sig) \\ In1l (Methd C sig))\\ {G\\}\\\n \\ G,A\\{Normal\n \\} t\\ {G\\}\n 3. wf_prog G \\ finite ?U2"} {"_id": "502878", "text": "proof (prove)\nusing this:\n (INF n. f (A n - \\ (range A))) = 0\n (INF n. f (\\ (range A))) = f (\\ (range A))\n\ngoal (1 subgoal):\n 1. (INF n. f (A n - \\ (range A)) + f (\\ (range A))) =\n 0 + f (\\ (range A))"} {"_id": "502879", "text": "proof (prove)\nusing this:\n p \\ \\ \\ p \\ M \\ N\n\ngoal (1 subgoal):\n 1. p \\\n \\ \\ p \\ M \\ p \\ N"} {"_id": "502880", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A \\ List.coset xs = fold Set.remove xs A"} {"_id": "502881", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prv (imp \\\n (cnj (PP \\\\\\)\n (PP (encF N \\\\\\))))"} {"_id": "502882", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prime_factorization\n (\\\\<^sub>#\n (prime_factorization a \\# prime_factorization b)) =\n prime_factorization a \\# prime_factorization b"} {"_id": "502883", "text": "proof (prove)\nusing this:\n B.obj a\n\ngoal (1 subgoal):\n 1. \\a : a \\\\<^sub>B a\\"} {"_id": "502884", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\Phi y.\n \\Phi (monom P \\ (Suc 0)) \\ carrier S;\n \\s.\n s \\ carrier S \\\n eval R S h s (monom P \\ (Suc 0)) = s;\n Phi \\ ring_hom P S; y \\ ring_hom P S;\n Phi \\ extensional (carrier P); y \\ extensional (carrier P);\n \\r\\carrier R. Phi (monom P r 0) = h r;\n s = Phi (monom P \\ (Suc 0));\n y (monom P \\ (Suc 0)) = Phi (monom P \\ (Suc 0));\n \\r\\carrier R. y (monom P r 0) = h r\\\n \\ Phi \\ extensional (?A39 Phi y)\n 2. \\Phi y.\n \\Phi (monom P \\ (Suc 0)) \\ carrier S;\n \\s.\n s \\ carrier S \\\n eval R S h s (monom P \\ (Suc 0)) = s;\n Phi \\ ring_hom P S; y \\ ring_hom P S;\n Phi \\ extensional (carrier P); y \\ extensional (carrier P);\n \\r\\carrier R. Phi (monom P r 0) = h r;\n s = Phi (monom P \\ (Suc 0));\n y (monom P \\ (Suc 0)) = Phi (monom P \\ (Suc 0));\n \\r\\carrier R. y (monom P r 0) = h r\\\n \\ y \\ extensional (?A39 Phi y)\n 3. \\Phi y x.\n \\Phi (monom P \\ (Suc 0)) \\ carrier S;\n \\s.\n s \\ carrier S \\\n eval R S h s (monom P \\ (Suc 0)) = s;\n Phi \\ ring_hom P S; y \\ ring_hom P S;\n Phi \\ extensional (carrier P); y \\ extensional (carrier P);\n \\r\\carrier R. Phi (monom P r 0) = h r;\n s = Phi (monom P \\ (Suc 0));\n y (monom P \\ (Suc 0)) = Phi (monom P \\ (Suc 0));\n \\r\\carrier R. y (monom P r 0) = h r;\n x \\ ?A39 Phi y\\\n \\ Phi x = y x"} {"_id": "502885", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x.\n (LINT xa|lebesgue_on {- pi..pi}.\n sin (real x * xa) * (cos (1 / 2 * xa) * f xa)) +\n (LINT xa|lebesgue_on {- pi..pi}.\n cos (real x * xa) * (sin (1 / 2 * xa) * f xa)))\n \\ 0 + 0"} {"_id": "502886", "text": "proof (prove)\nusing this:\n fn \\ fn'\n l = ?x # ?xs \\ fc ?x ?xs \\ fc' ?x ?xs\n\ngoal (1 subgoal):\n 1. (case l of [] \\ fn | x # xa \\ fc x xa)\n \\ (case l of [] \\ fn' | x # xa \\ fc' x xa)"} {"_id": "502887", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ ({#}, {#}, Q, n) \\\\<^sub>w St"} {"_id": "502888", "text": "proof (prove)\nusing this:\n ?z2 \\ a \\\n ?z2\n \\ {(x, {(x, Y) |Y x.\n Y \\ finestpart (a ,, x) \\ x \\ Domain a} ,,,\n x) |\n x. x \\ Domain\n {(x, Y) |Y x.\n Y \\ finestpart (a ,, x) \\ x \\ Domain a}}\n\ngoal (1 subgoal):\n 1. a =\n {(x, {(x, Y) |Y x.\n Y \\ finestpart (a ,, x) \\ x \\ Domain a} ,,,\n x) |\n x. x \\ Domain\n {(x, Y) |Y x.\n Y \\ finestpart (a ,, x) \\ x \\ Domain a}}"} {"_id": "502889", "text": "proof (prove)\nusing this:\n xs = xs' @ xs''\n xs'' \\ []\n\ngoal (1 subgoal):\n 1. prefix (xs' @ [hd xs'']) xs"} {"_id": "502890", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Red_F_\\_empty_L_q q = local.ord_q_lifting.Red_F_\\"} {"_id": "502891", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s. guard' s \\ lossless_spmf (body1' s)"} {"_id": "502892", "text": "proof (prove)\nusing this:\n numsub ?s ?t = (if ?s = ?t then C 0 else numadd ?s (numneg ?t))\n\ngoal (1 subgoal):\n 1. Inum bs (numsub a b) = Inum bs (Sub a b)"} {"_id": "502893", "text": "proof (state)\ngoal (1 subgoal):\n 1. Opair x (f \\ x) |\\| f"} {"_id": "502894", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_nn_integral M A f \\ set_nn_integral M A g"} {"_id": "502895", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card {f. CK_nf_pos f} \\ {1, 3, 4, 5, 7}"} {"_id": "502896", "text": "proof (prove)\nusing this:\n \\C.arr ?f; C.dom ?f = ?a\\\n \\ ?f \\ ?a = ?f\n\ngoal (1 subgoal):\n 1. arrow_of_spans (\\)\n \\Chn = \\.dom.apex, Dom = Dom \\, Cod = Dom \\\\"} {"_id": "502897", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Us. Xs = Abs_ExpList Us \\ args (FnCall F Xs) = Xs"} {"_id": "502898", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vec (dim_vec v) (\\i. - (1::'b) * v $ i) =\n vec (dim_vec v) (\\i. - v $ i)"} {"_id": "502899", "text": "proof (prove)\nusing this:\n ins (size t) 0 x t = Bal x2_\n bal_i (size t) (height t + 0) \\\n bal_i (size t + 1) (height x2_ + 0)\n bal_i (size t) (height t)\n ins ?n ?d ?x ?t = Bal ?t' \\ size ?t' = size ?t + 1\n ins ?n ?d ?x ?t = Unbal ?t' \\ size ?t' = size ?t + 1\n\ngoal (1 subgoal):\n 1. bal_i (size (local.insert x t)) (height (local.insert x t))"} {"_id": "502900", "text": "proof (prove)\nusing this:\n ?x \\ S \\ f differentiable at ?x\n ?x \\ S \\ g differentiable at ?x\n open S\n \\higher_differentiable_on S ?f n; higher_differentiable_on S ?g n;\n open S\\\n \\ higher_differentiable_on S (\\x. ?f x ** ?g x) n\n higher_differentiable_on S f (Suc n)\n higher_differentiable_on S g (Suc n)\n open S\n\ngoal (1 subgoal):\n 1. higher_differentiable_on S (\\x. f x ** g x) (Suc n)"} {"_id": "502901", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom_mult c t (atomize_poly (idx_pm_of_pm bs s')) =\n atomize_poly (idx_pm_of_pm bs (monomial c t \\ s'))"} {"_id": "502902", "text": "proof (prove)\nusing this:\n ii < brnL cl (length cl)\n\ngoal (1 subgoal):\n 1. locateT cl ii < length cl"} {"_id": "502903", "text": "proof (prove)\nusing this:\n \\class.nontriv TYPE('b);\n class.bezout_ring (*) (1::?'aa) (+) (0::?'aa) (-) uminus\\\n \\ Ball {A. A \\ carrier_mat p p} admits_SNF_JNF\n\ngoal (1 subgoal):\n 1. Ball {A. A \\ carrier_mat p p} admits_SNF_JNF"} {"_id": "502904", "text": "proof (prove)\nusing this:\n deg_pp t \\ 0\n keys_pp t \\ set xs\n\ngoal (1 subgoal):\n 1. t \\ keys (deg_le_sect_pp_aux xs 0)"} {"_id": "502905", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\H \\ carrier M;\n f \\ {k. k \\ Suc n} \\ H;\n s \\ {k. k \\ Suc n} \\ carrier R; l \\ Suc n;\n f l \\ f ` ({k. k \\ Suc n} - {l}); l \\ Suc n\\\n \\ \\g m t.\n g \\ {k. k \\ m} \\ H \\\n inj_on g {k. k \\ m} \\\n t \\ {k. k \\ m} \\\n carrier R \\\n l_comb R M (Suc n) s f = l_comb R M m t g \\\n t m = s l \\ g m = f l"} {"_id": "502906", "text": "proof (prove)\nusing this:\n \\x \\ y; y \\ x; y \\ z\\\n \\ x \\ z\n x \\ y\n y \\ z\n\ngoal (1 subgoal):\n 1. x \\ z"} {"_id": "502907", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subalgebra (nat_filtration n) (G n)"} {"_id": "502908", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n (\\ x) \\ = true \\ \\\n (\\ (\\\\.\n if (\\ x) \\ = true \\\n then \\\\x \\\n \\ \\\\Rep_Set\\<^sub>b\\<^sub>a\\<^sub>s\\<^sub>e\n (H .allInstances()\n \\)\\\\\\\\\n else \\))\n \\ =\n true \\"} {"_id": "502909", "text": "proof (prove)\nusing this:\n c \\ atlas\n x \\ domain c\n k-smooth_on (codomain c) (a \\ inv_chart c)\n\ngoal (1 subgoal):\n 1. \\c1\\atlas.\n x \\ domain c1 \\\n k-smooth_on (codomain c1) (a + b \\ inv_chart c1)"} {"_id": "502910", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LINT x|measure_spmf (map_spmf fst p).\n Sigma_Algebra.measure (measure_spmf p)\n ({x} \\ UNIV \\ (Pair x \\ snd) -` {i}) /\n Sigma_Algebra.measure (measure_spmf p) ({x} \\ UNIV) =\n spmf (map_spmf fst p) (fst i) * spmf p i /\n Sigma_Algebra.measure (measure_spmf p) ({fst i} \\ UNIV)"} {"_id": "502911", "text": "proof (prove)\nusing this:\n \\x\\fv u'.\n \\a. \\\\<^sub>v x = Var (prot_atom.Atom a)\n u = u' \\ \\\n \\ ?t = \\ (?t \\ \\)\n \\x\\fv u'.\n \\a. \\ (Var x) = Var a \\\n \\x\\fv (u' \\ \\).\n \\y\\fv u'. \\ (Var x) = \\ (Var y)\n\ngoal (1 subgoal):\n 1. \\x\\fv u.\n \\a. \\\\<^sub>v x = Var (prot_atom.Atom a)"} {"_id": "502912", "text": "proof (prove)\ngoal (1 subgoal):\n 1. det_int = det"} {"_id": "502913", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_root' h \\\n hom (del_min' h) = Pairing_Heap_Tree.del_min (hom h)"} {"_id": "502914", "text": "proof (prove)\ngoal (1 subgoal):\n 1. exp (2 * ln (ln (real n)) * (4 * (f n / ln (f n)))) =\n 2 powr (c * f n * ln (ln (real n)) / ln (f n))"} {"_id": "502915", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\wtL (Neg (Pr p Tl)); I.wtE \\;\n \\x.\n \\\\ infTp (tpOfV x);\n x \\ nv2L (Neg (Pr p Tl))\\\n \\ \\l'.\n wtL l' \\\n \\ I.satL \\ l' \\ isGuard x l';\n Ik.satL (transE \\) (Neg (Pr p Tl)); polC p = Fext\\\n \\ I.satL \\ (Neg (Pr p Tl))"} {"_id": "502916", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\ \\ \\ af \\ xs \\ af \\ xs"} {"_id": "502917", "text": "proof (prove)\nusing this:\n max_depth \\ = Suc m\n \\belowNext Next.\n (extendLevel ^^ m) (atoms, depth1) = (belowNext, Next) \\\n i.R belowNext = {\\. max_depth \\ < 1 + m} \\\n i.R Next = {\\. max_depth \\ = 1 + m}\n\ngoal (1 subgoal):\n 1. \\ \\ i.R fmlas"} {"_id": "502918", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\m.\n \\0 < N; j \\ Suc n; vals_nonequiv K (Suc n) vv; Ring K;\n aGroup K; (\\\\<^bsub>K vv Suc n\\<^esub>) j \\ carrier K;\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n m \\ Suc n; j \\ m;\n 0 < vv m ((\\\\<^bsub>K vv Suc n\\<^esub>) j);\n int N *\\<^sub>a vv m ((\\\\<^bsub>K vv Suc n\\<^esub>) j) = 0;\n valuation K (vv m);\n vv m ((\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>) =\n 0\\\n \\ False\n 2. \\0 < N; j \\ Suc n; vals_nonequiv K (Suc n) vv; Ring K;\n aGroup K; (\\\\<^bsub>K vv Suc n\\<^esub>) j \\ carrier K;\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n 1\\<^sub>r = (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>;\n \\m\\Suc n.\n j \\ m \\\n 0 < vv m ((\\\\<^bsub>K vv Suc n\\<^esub>) j)\\\n \\ j \\ {h. h \\ Suc n}\n 3. \\0 < N; j \\ Suc n; vals_nonequiv K (Suc n) vv; Ring K;\n aGroup K; (\\\\<^bsub>K vv Suc n\\<^esub>) j \\ carrier K;\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup> \\\n -\\<^sub>a (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\n \\ carrier K;\n 1\\<^sub>r =\n (\\\\<^bsub>K vv Suc n\\<^esub>) j^\\<^bsup>K N\\<^esup>\\\n \\ \\m\\Suc n.\n j \\ m \\\n 0 < vv m ((\\\\<^bsub>K vv Suc n\\<^esub>) j)"} {"_id": "502919", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i\\degree (map_poly (pCons (0::'a)) p).\n \\j\\degree (coeff (map_poly (pCons (0::'a)) p) i).\n monom (monom (coeff (coeff (map_poly (pCons (0::'a)) p) i) j) i)\n j) =\n pCons 0\n (\\i\\degree p.\n \\j\\degree (coeff p i).\n monom (monom (coeff (coeff p i) j) i) j)"} {"_id": "502920", "text": "proof (prove)\nusing this:\n 0 < k\n (\\d | d dvd ?k. \\ d) \\ \\\n\ngoal (1 subgoal):\n 1. Re (\\d | d dvd k. \\ d) =\n (\\p\\prime_factors k.\n Re (\\d | d dvd p ^ multiplicity p k. \\ d))"} {"_id": "502921", "text": "proof (prove)\nusing this:\n 0 \\ pi / 2\n \\0 \\ ?\\; ?\\ \\ pi; cos ?\\ = 0\\\n \\ ?\\ = \\\n\ngoal (1 subgoal):\n 1. pi / 2 = \\"} {"_id": "502922", "text": "proof (prove)\ngoal (1 subgoal):\n 1. uint32_of_uint x = Abs_uint32' (ucast (Rep_uint x))"} {"_id": "502923", "text": "proof (prove)\nusing this:\n c \\ (0::'a) \\ integrable (count_space A) f\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x *\\<^sub>R c =\n integral\\<^sup>L (count_space A) f *\\<^sub>R c"} {"_id": "502924", "text": "proof (prove)\ngoal (1 subgoal):\n 1. twoNetsDistinct a b c d"} {"_id": "502925", "text": "proof (prove)\ngoal (1 subgoal):\n 1. COERCE('c \\ 'd, 'a \\ 'b)\\(sinr\\x) =\n sinr\\(COERCE('d, 'b)\\x)"} {"_id": "502926", "text": "proof (prove)\nusing this:\n local.set a \\ local.set b\n\ngoal (1 subgoal):\n 1. (\\x. x) \\ local.set a \\ local.set b"} {"_id": "502927", "text": "proof (prove)\ngoal (1 subgoal):\n 1. continuous_on UNIV (\\x. Blinfun (\\y'. y' * x))"} {"_id": "502928", "text": "proof (prove)\nusing this:\n finite A\n A \\ {}\n\ngoal (1 subgoal):\n 1. bo a u"} {"_id": "502929", "text": "proof (prove)\nusing this:\n R2 (m0, m1) \\ = S2 \\ (if \\ then m1 else m0)\n\ngoal (1 subgoal):\n 1. sim_def.perfect_sec_P2 (m0, m1) \\"} {"_id": "502930", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\C D.\n prefix C B \\\n prefix D A \\\n C \\ set (tr\\<^sub>p\\<^sub>c D []) \\\n ik\\<^sub>s\\<^sub>t (proj_unl n C) \\\\<^sub>s\\<^sub>e\\<^sub>t\n \\ \\\n t \\ \\"} {"_id": "502931", "text": "proof (prove)\nusing this:\n CT Object = None\n\ngoal (1 subgoal):\n 1. [] = Cf'"} {"_id": "502932", "text": "proof (prove)\nusing this:\n \\1 \\ (\\2 - {F}) \\\n (\\3 - {G}) \\\n H\n\ngoal (1 subgoal):\n 1. \\\\''\\\\.\n finite \\'' \\ (\\'' \\ H)"} {"_id": "502933", "text": "proof (state)\ngoal (3 subgoals):\n 1. ring_isomorphism (identity stk1.carrier_stalk) stk1.carrier_stalk\n stk1.add_stalk stk1.mult_stalk (stk1.zero_stalk U) (stk1.one_stalk U)\n stk2.carrier_stalk stk2.add_stalk stk2.mult_stalk (stk2.zero_stalk U)\n (stk2.one_stalk U)\n 2. stk1.carrier_stalk = stk2.carrier_stalk\n 3. stk1.class_of = stk2.class_of"} {"_id": "502934", "text": "proof (prove)\nusing this:\n \\closedin (top_of_set ?U) S';\n \\V.\n \\openin (top_of_set ?U) V; S' retract_of V\\\n \\ ?thesis\\\n \\ ?thesis\n\ngoal (1 subgoal):\n 1. (\\V.\n \\open V; S' retract_of V\\\n \\ thesis) \\\n thesis"} {"_id": "502935", "text": "proof (prove)\ngoal (18 subgoals):\n 1. \\G.\n G \\ Skip : Sec (\\(s, t, \\).\n s \\\\<^sub>\\ t)\n 2. \\e G x.\n Expr_low e \\\n G \\ Assign x\n e : Sec (\\(s, t, \\).\n \\r.\n s =\n (update (fst r) x (evalE e (fst r)),\n snd r) \\\n r \\\\<^sub>\\ t)\n 3. \\G c1 \\ c2 \\.\n \\(G, c1, Sec \\) \\ Deriv; G \\ c1 : Sec \\;\n (G, c2, Sec \\) \\ Deriv; G \\ c2 : Sec \\\\\n \\ G \\ Comp c1\n c2 : Sec\n (\\(s, t, \\).\n \\r.\n \\ (r, t, \\) \\\n (\\w \\.\n r \\\\<^sub>\\ w \\\n \\ (s, w, \\)))\n 4. \\b G c1 \\ c2 \\.\n \\BExpr_low b; (G, c1, Sec \\) \\ Deriv;\n G \\ c1 : Sec \\; (G, c2, Sec \\) \\ Deriv;\n G \\ c2 : Sec \\\\\n \\ G \\ Iff b c1\n c2 : Sec\n (\\(s, t, \\).\n (evalB b (fst t) \\ \\ (s, t, \\)) \\\n (\\ evalB b (fst t) \\ \\ (s, t, \\)))\n 5. \\x G C.\n CONTEXT x = low \\\n G \\ New x\n C : Sec (\\(s, t, \\).\n \\l r.\n l \\ Dom (snd r) \\\n r \\\\<^sub>\\ t \\\n s =\n (update (fst r) x (RVal (Loc l)),\n (l, C, []) # snd r))\n 6. \\y f G x.\n \\CONTEXT y = low; GAMMA f = low\\\n \\ G \\ Get x y\n f : Sec\n (\\(s, t, \\).\n \\r l C Flds v.\n fst r y = RVal (Loc l) \\\n lookup (snd r) l = Some (C, Flds) \\\n lookup Flds f = Some v \\\n r \\\\<^sub>\\ t \\ s = (update (fst r) x v, snd r))\n 7. \\x f e G.\n \\CONTEXT x = low; GAMMA f = low; Expr_low e\\\n \\ G \\ Put x f\n e : Sec\n (\\(s, t, \\).\n \\r l C F h.\n fst r x = RVal (Loc l) \\\n r \\\\<^sub>\\ t \\\n lookup (snd r) l = Some (C, F) \\\n h = (l, C, (f, evalE e (fst r)) # F) # snd r \\ s = (fst r, h))\n 8. \\b G c \\.\n \\BExpr_low b; (G, c, Sec \\) \\ Deriv;\n G \\ c : Sec \\\\\n \\ G \\ While b c : Sec (PhiWhile b \\)\n 9. \\\\ G.\n \\({Sec (FIX \\)} \\ G, body,\n Sec (\\ (FIX \\)))\n \\ Deriv;\n ({Sec (FIX \\)} \\\n G) \\ body : Sec (\\ (FIX \\));\n Monotone \\\\\n \\ G \\ Call : Sec (FIX \\)\n 10. \\G. G \\ Skip : HighSec\nA total of 18 subgoals..."} {"_id": "502936", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xs ys v.\n \\xs \\ paths; ys \\ paths; xs \\ ys;\n v \\ set xs; v \\ set ys\\\n \\ v = v0 \\ v = v1"} {"_id": "502937", "text": "proof (prove)\nusing this:\n \\b \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c\\ \\\n (\\[\\b \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c\\] \\\n (\\ \\ \\b \\<^bold>\\ c\\)) \\\n \\[\\, \\b\\, \\c\\] =\n \\b \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c\\ \\\n (\\b \\<^bold>\\ c\\ \\\n \\[\\b\\ \\ \\c\\]) \\\n \\[\\, \\b\\, \\c\\]\n\ngoal (1 subgoal):\n 1. \\b \\<^bold>\\ c\\ \\\n (\\b\\ \\ \\[\\b\\] \\\n \\c\\) =\n ((\\b \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c\\ \\\n \\[\\b \\<^bold>\\\\<^bold>\\\\<^bold>\\\n c\\]) \\\n (\\ \\ \\b \\<^bold>\\ c\\)) \\\n \\[\\, \\b\\, \\c\\]"} {"_id": "502938", "text": "proof (prove)\ngoal (2 subgoals):\n 1. to_fun x \\\\<^bsub>function_ring (carrier R) R\\<^esub> to_fun y\n \\ carrier (function_ring (carrier R) R)\n 2. \\a.\n a \\ carrier R \\\n to_fun (x \\\\<^bsub>P\\<^esub> y) a =\n (to_fun x \\\\<^bsub>function_ring (carrier R) R\\<^esub>\n to_fun y)\n a"} {"_id": "502939", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i < dim_row A; j < dim_col A\\\n \\ (- A) $$ (i, j) = - A $$ (i, j)) &&&\n dim_row (- A) = dim_row A &&& dim_col (- A) = dim_col A"} {"_id": "502940", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n \\closed_csubspace x; closed_csubspace y;\n x \\ y\\\n \\ x +\\<^sub>M orthogonal_complement x \\ y = y"} {"_id": "502941", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card\n {xs.\n (\\i\\X. xs ! i) \\\n (\\i\\Y. \\ xs ! i) \\ length xs = m} =\n 2 ^ (m - card X - card Y)"} {"_id": "502942", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local_ring stfx.carrier_stalk stfx.add_stalk stfx.mult_stalk\n (stfx.zero_stalk V) (stfx.one_stalk V)"} {"_id": "502943", "text": "proof (prove)\ngoal (1 subgoal):\n 1. simple_distributed M (\\x. (X x, Y x)) Pxy"} {"_id": "502944", "text": "proof (prove)\ngoal (3 subgoals):\n 1. Eq p \\ set L\n 2. MPoly_Type.degree p var = 1 \\ MPoly_Type.degree p var = 2\n 3. \\x.\n insertion (nth_default 0 (xs @ x # \\))\n (isolate_variable_sparse p var 2) \\\n 0 \\\n insertion (nth_default 0 (xs @ x # \\))\n (isolate_variable_sparse p var 1) \\\n 0 \\\n insertion (nth_default 0 (xs @ x # \\))\n (isolate_variable_sparse p var 0) \\\n 0"} {"_id": "502945", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\a b c d.\n \\wf_jvm_prog\\<^bsub>\\\\<^esub> P;\n \\ is_class P Start;\n P \\ C sees M, Static : []\\Void = (a, b, c,\n d) in C;\n M \\ clinit; \\' Start start_m = start_\\\\<^sub>m;\n is_class P Object;\n \\Mm. P \\ Object sees_methods Mm;\n m = (a, b, c, d)\\\n \\ class_add P\n (Start, Object, [],\n [start_method C M,\n (clinit, Static, [], Void, Suc 0, 0,\n [Push Unit, Return],\n [])]) \\ start_heap P \\\n 2. \\a b c d.\n \\wf_jvm_prog\\<^bsub>\\\\<^esub> P;\n \\ is_class P Start;\n P \\ C sees M, Static : []\\Void = (a, b, c,\n d) in C;\n M \\ clinit; \\' Start start_m = start_\\\\<^sub>m;\n is_class P Object;\n \\Mm. P \\ Object sees_methods Mm;\n m = (a, b, c, d)\\\n \\ class_add P\n (Start, Object, [],\n [start_method C M,\n (clinit, Static, [], Void, Suc 0, 0,\n [Push Unit, Return],\n [])]),start_heap\n P \\\\<^sub>s [Start \\\n (Map.empty, Done)] \\ \\\n conf_clinit\n (class_add P\n (Start, Object, [],\n [start_method C M,\n (clinit, Static, [], Void, Suc 0, 0,\n [Push Unit, Return], [])]))\n [Start \\ (Map.empty, Done)]\n [([], [], Start, start_m, 0, No_ics)] \\\n (\\b Ts T mxs mxl\\<^sub>0 is.\n (\\xt.\n class_add P\n (Start, Object, [],\n [start_method C M,\n (clinit, Static, [], Void, Suc 0, 0,\n [Push Unit, Return],\n [])]) \\ Start sees start_m, b : Ts\\T = (mxs,\n mxl\\<^sub>0, is, xt) in Start) \\\n conf_f\n (class_add P\n (Start, Object, [],\n [start_method C M,\n (clinit, Static, [], Void, Suc 0, 0,\n [Push Unit, Return], [])]))\n (start_heap P)\n [Start \\ (Map.empty, Done)] ([], [])\n is ([], [], Start, start_m, 0, No_ics))"} {"_id": "502946", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a < (0::'a); b < (0::'a)\\\n \\ (0::'a) < a * b"} {"_id": "502947", "text": "proof (prove)\nusing this:\n local.path g ns \\ n = hd ns \\ m = last ns\n local.path g ms \\ m = hd ms \\ l = last ms\n last (ns @ tl ms) = last ms\n\ngoal (1 subgoal):\n 1. local.path g (ns @ tl ms) \\\n n = hd (ns @ tl ms) \\ l = last (ns @ tl ms)"} {"_id": "502948", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Key K \\ analz H; Key (invKey K) \\ analz H\\\n \\ (Crypt K X \\ synth (analz H)) =\n (X \\ synth (analz H))"} {"_id": "502949", "text": "proof (prove)\nusing this:\n f \\ 0 \\\n lookup p (t \\ lt f) \\ (0::'b) \\\n p0 = p - monom_mult (lookup p (t \\ lt f) / lc f) t f\n\ngoal (1 subgoal):\n 1. lookup p (t \\ lt f) \\ (0::'b)"} {"_id": "502950", "text": "proof (prove)\nusing this:\n CD_on ds X = X\n\ngoal (1 subgoal):\n 1. Cd d X = dX X d"} {"_id": "502951", "text": "proof (prove)\nusing this:\n SubRel.subpath g (tgt e') es v2 subs\n e \\ e' \\ e \\ set es\n\ngoal (1 subgoal):\n 1. \\es'. e' # es = es' @ [e] \\ e \\ set es'"} {"_id": "502952", "text": "proof (prove)\nusing this:\n \\finite ?L; unifier\\<^sub>l\\<^sub>s ?\\ ?L\\\n \\ \\\\. mgu\\<^sub>l\\<^sub>s \\ ?L\n\ngoal (1 subgoal):\n 1. unification"} {"_id": "502953", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n bd\\<^sub>\\ (\\x. g x \\ h x) (f x) =\n bd\\<^sub>\\ g (f x) \\ bd\\<^sub>\\ h (f x)"} {"_id": "502954", "text": "proof (prove)\ngoal (1 subgoal):\n 1. - 1 AND x = x"} {"_id": "502955", "text": "proof (prove)\ngoal (1 subgoal):\n 1. FiniteGraph.add_edge v v' (list_graph_to_graph G) =\n list_graph_to_graph (FiniteListGraph.add_edge v v' G)"} {"_id": "502956", "text": "proof (prove)\ngoal (1 subgoal):\n 1. divide_poly_main lc\n (smult (lc ^ n) (smult lc q) + monom (lc ^ n * coeff r dr) n)\n (smult (lc ^ n) (smult lc r) - monom (lc ^ n * coeff r dr) n * d) d\n (dr - 1) n =\n divide_poly_main lc (smult (lc ^ Suc n) q) (smult (lc ^ Suc n) r) d dr\n (Suc n)"} {"_id": "502957", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gen_refine.mds\\<^sub>A_of x = x"} {"_id": "502958", "text": "proof (prove)\nusing this:\n Ipol ls (Commutative_Ring.pow (k div 2) (sqr P)) =\n Ipol ls (sqr P) [^] (k div 2)\n in_carrier ls\n odd k\n\ngoal (1 subgoal):\n 1. Ipol ls (Commutative_Ring.pow k P) = Ipol ls P [^] k"} {"_id": "502959", "text": "proof (prove)\ngoal (1 subgoal):\n 1. count_terminals a \\ count_terminals b"} {"_id": "502960", "text": "proof (prove)\ngoal (1 subgoal):\n 1. known_ptr\n (cast\\<^sub>s\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>_\\<^sub>r\\<^sub>o\\<^sub>o\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n new_shadow_root_ptr)"} {"_id": "502961", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bd\\<^sub>\\ R X = {y. \\x\\X. s2p R (x, y)}"} {"_id": "502962", "text": "proof (prove)\ngoal (4 subgoals):\n 1. degree u \\ d\n 2. degree [:k:] \\ n - d\n 3. vec n c \\ carrier_vec (d + (n - d))\n 4. poly_of_vec (vec_first (vec n c) (n - d)) * u +\n poly_of_vec (vec_last (vec n c) d) * [:k:] =\n q * u + smult k r"} {"_id": "502963", "text": "proof (prove)\nusing this:\n {t. h ?n (g t) = ?y} \\ S \\ sets lebesgue\n S \\ sets lebesgue\n\ngoal (1 subgoal):\n 1. (\\x. indicat_real {x. h n x = y} (g x))\n \\ borel_measurable (lebesgue_on S)"} {"_id": "502964", "text": "proof (chain)\npicking this:\n front_tickFree a\n (a, b) \\ F P\n [] \\ D P"} {"_id": "502965", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sorted (rev xs); i \\ j; j < length xs\\\n \\ xs ! j \\ xs ! i"} {"_id": "502966", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (degeneralize_ext_impl Gi ecnvi, igb_graph.degeneralize_ext G ecnv)\n \\ bg_impl_rel_ext Re' (Rv \\\\<^sub>r nat_rel)"} {"_id": "502967", "text": "proof (prove)\nusing this:\n v \\ set vs\n\ngoal (1 subgoal):\n 1. v \\ f ` set (remdups_wrt_rev f (x # xs) vs)"} {"_id": "502968", "text": "proof (state)\ngoal (4 subgoals):\n 1. uniformity = (INF e\\{0<..}. principal {(x, y). dist x y < e})\n 2. \\U.\n open U =\n (\\x\\U.\n \\\\<^sub>F (x', y) in uniformity.\n x' = x \\ y \\ U)\n 3. \\x y. (dist x y = 0) = (x = y)\n 4. \\x y z. dist x y \\ dist x z + dist y z"} {"_id": "502969", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\a b.\n quotient_of p = (a, b) \\ thesis) \\\n thesis"} {"_id": "502970", "text": "proof (prove)\nusing this:\n f \\\n restrict1 (canonical_retraction A D \\ canonical_retraction B C)\n (\\ A)\n g \\\n restrict1 (canonical_retraction A C \\ canonical_retraction B' D)\n (\\ A)\n\ngoal (1 subgoal):\n 1. (ChamberComplexFolding A f &&& ChamberComplexFolding A g) &&&\n f ` D = C &&& g ` C = D"} {"_id": "502971", "text": "proof (prove)\nusing this:\n A_BxA.seq Fg' Fg\n \\A_BxA.seq ?g ?f;\n \\A_B.seq (fst ?g) (fst ?f); A.seq (snd ?g) (snd ?f)\\\n \\ ?T\\\n \\ ?T\n\ngoal (1 subgoal):\n 1. A_B.arr (fst Fg')"} {"_id": "502972", "text": "proof (prove)\ngoal (1 subgoal):\n 1. y\\<^sup>\\ * x * y\\<^sup>\\\n \\ x * y\\<^sup>\\ \\ y\\<^sup>\\ * bot"} {"_id": "502973", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\C.ide f;\n \\\\ : G f \\\\<^sub>D G\n f'\\;\n \\D.inv (\\.\\ f') \\\\<^sub>D\n (\\'.map\\<^sub>0 (trg\\<^sub>C f) \\\\<^sub>D\n \\ \\\\<^sub>D\n \\\\<^sub>0 (src\\<^sub>C f)) \\\\<^sub>D\n \\.\\\n f : F f \\\\<^sub>D F f'\\;\n \\\\ : f \\\\<^sub>C f'\\;\n F \\ =\n D.inv (\\.\\ f') \\\\<^sub>D\n (\\'.map\\<^sub>0 (trg\\<^sub>C f) \\\\<^sub>D\n \\ \\\\<^sub>D \\\\<^sub>0 (src\\<^sub>C f)) \\\\<^sub>D\n \\.\\ f\\\n \\ D.cod\n (G \\ \\\\<^sub>D\n \\\\<^sub>0 (src\\<^sub>C f)) =\n D.cod\n (\\ \\\\<^sub>D\n \\\\<^sub>0 (src\\<^sub>C f))"} {"_id": "502974", "text": "proof (prove)\ngoal (1 subgoal):\n 1. degree (pdev_upd X n x) = degree X"} {"_id": "502975", "text": "proof (state)\nthis:\n list_all2 (\\x y. x = norm_instr y) xs (rewrite ys pc instr)\n\ngoal (1 subgoal):\n 1. \\x1a x2a y1a y2a.\n \\fd1 = Fundef x1a x2a; fd2 = Fundef y1a y2a;\n list_all2 (\\x y. x = norm_instr y) x1a y1a;\n x2a = y2a\\\n \\ rel_fundef (\\x y. x = norm_instr y) fd1\n (rewrite_fundef_body fd2 pc instr)"} {"_id": "502976", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l3_inv2 \\ l3_inv3 \\ l3_inv5"} {"_id": "502977", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((=) ===>\n rel_prod2 (=) ===> (=) ===> rel_spmf (rel_prod (=) (rel_prod2 (=))))\n local.ind_cca.oracle_decrypt oracle_decrypt0'"} {"_id": "502978", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (u, v)\n \\ {(u, v). c (u, v) \\ (0::'capacity)} \\\n (0::'capacity) < c (u, v)"} {"_id": "502979", "text": "proof (prove)\ngoal (1 subgoal):\n 1. closed_map X (prod_topology Y X) (\\x. (f x, x))"} {"_id": "502980", "text": "proof (prove)\ngoal (1 subgoal):\n 1. factor_preserving_hom poly_x_minus_y"} {"_id": "502981", "text": "proof (prove)\ngoal (1 subgoal):\n 1. locale_poly ((\\) 0)"} {"_id": "502982", "text": "proof (prove)\nusing this:\n subcocycle u\n 0 \\ c\n - \\ < subcocycle_avg_ereal u\n subcocycle_avg_ereal ?u < \\\n\ngoal (1 subgoal):\n 1. - \\ < subcocycle_avg_ereal (\\n x. c * u n x) &&&\n subcocycle_avg (\\n x. c * u n x) = c * subcocycle_avg u"} {"_id": "502983", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_ran (\\x. lift_transform t (ae x)) (restrictA S \\) =\n restrictA S (map_ran (\\x. lift_transform t (ae x)) \\)"} {"_id": "502984", "text": "proof (prove)\nusing this:\n support_flow (\\g\\Y. (\\e. f e - g e))\n \\ support_flow f\n\ngoal (1 subgoal):\n 1. countable (support_flow (\\g\\Y. (\\e. f e - g e)))"} {"_id": "502985", "text": "proof (prove)\nusing this:\n local.eval p x = local.eval r x\n a \\ carrier R\n\ngoal (1 subgoal):\n 1. a = \\"} {"_id": "502986", "text": "proof (prove)\nusing this:\n \\Exec_movet ci P t e h (stk, loc, pc, xcp)\n (stk'', loc'', pc'', xcp'')\n\ngoal (1 subgoal):\n 1. Q stk'' loc'' pc'' xcp''"} {"_id": "502987", "text": "proof (prove)\nusing this:\n t \\ U (Suc h)\n\ngoal (1 subgoal):\n 1. ins x t \\ T (Suc h)"} {"_id": "502988", "text": "proof (prove)\ngoal (1 subgoal):\n 1. color (paint Black t) = Black"} {"_id": "502989", "text": "proof (prove)\ngoal (1 subgoal):\n 1. i < length (xs \\ ys) \\\n (xs \\ ys) ! i = ys ! i"} {"_id": "502990", "text": "proof (prove)\nusing this:\n (f \\ g) (i, j) =\n sup_monoid.sum (\\k. f (i, k)) {k. True} \\ g (i, j)\n\ngoal (1 subgoal):\n 1. (f \\ g) (i, j) = f (i, i) \\ g (i, j)"} {"_id": "502991", "text": "proof (prove)\ngoal (1 subgoal):\n 1. SPN_fgh.\\.leg0 \\ SPN_fgh.Prj\\<^sub>0\\<^sub>1 =\n SPN_fgh.\\.leg1 \\ SPN_fgh.Prj\\<^sub>0"} {"_id": "502992", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\w \\\\<^sub>n \\[M]\\<^sub>\\\\<^sub>2;\n M \\ M'\\\n \\ w \\\\<^sub>n \\[M']\\<^sub>\\\\<^sub>2"} {"_id": "502993", "text": "proof (state)\ngoal (1 subgoal):\n 1. continuous_on s g \\\n (\\S. finite S \\ g C1_differentiable_on s - S)"} {"_id": "502994", "text": "proof (prove)\nusing this:\n \\col_fst (U * |v\\)|col_fst (U * |v\\)\\ =\n (\\|v\\| * U\\<^sup>\\ * (U * |v\\)) $$ (0, 0)\n dim_col \\|v\\| = dim_vec v\n dim_row U\\<^sup>\\ = dim_vec v\n \\?A \\ carrier_mat ?n\\<^sub>1 ?n\\<^sub>2;\n ?B \\ carrier_mat ?n\\<^sub>2 ?n\\<^sub>3;\n ?C \\ carrier_mat ?n\\<^sub>3 ?n\\<^sub>4\\\n \\ ?A * ?B * ?C = ?A * (?B * ?C)\n\ngoal (1 subgoal):\n 1. \\col_fst (U * |v\\)|col_fst (U * |v\\)\\ =\n (\\|v\\| * (U\\<^sup>\\ * U) * |v\\) $$\n (0, 0)"} {"_id": "502995", "text": "proof (prove)\nusing this:\n Rel = indRelRTPO TRel\n rel_weakly_preserves_barb_set TRel TWB {success}\n rel_weakly_reflects_barb_set TRel TWB {success}\n \\S x.\n x \\ {success} \\\n ((\\Sa. Sa \\S SourceTerm S \\ Sa\\x) \\\n (\\T.\n T \\T SourceTerm S \\ T\\x)) \\\n x \\ {success} \\\n ((\\Sa.\n Sa \\S TargetTerm (\\S\\) \\\n Sa\\x) \\\n (\\T.\n T \\T TargetTerm (\\S\\) \\ T\\x))\n \\S x.\n x \\ {success} \\\n ((\\Sa.\n Sa \\S TargetTerm (\\S\\) \\\n Sa\\x) \\\n (\\T.\n T \\T TargetTerm (\\S\\) \\\n T\\x)) \\\n x \\ {success} \\\n ((\\Sa. Sa \\S SourceTerm S \\ Sa\\x) \\\n (\\T. T \\T SourceTerm S \\ T\\x))\n\ngoal (1 subgoal):\n 1. rel_weakly_respects_barb_set Rel (STCalWB SWB TWB) {success}"} {"_id": "502996", "text": "proof (prove)\ngoal (1 subgoal):\n 1. llexord r (Lazy_llist xs) (Lazy_llist ys) =\n (case xs () of None \\ True\n | Some (x, xs') \\\n case ys () of None \\ False\n | Some (y, ys') \\\n r x y \\ x = y \\ llexord r xs' ys')"} {"_id": "502997", "text": "proof (prove)\nusing this:\n T \\ B\n disjoint B\n\ngoal (1 subgoal):\n 1. T \\ \\ (B - {T}) = {}"} {"_id": "502998", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x\\space N.\n real_cond_exp N (G 0) (discounted_value r (\\m. der) matur)\n x =\n integral\\<^sup>L N (discounted_value r (\\m. der) matur)"} {"_id": "502999", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\t guest room key1 key2.\n \\\\state.owns = Trace.owns t, currk = Trace.currk t,\n issued = Trace.issued t, cards = Trace.cards t,\n roomk = Trace.roomk t, isin = Trace.isin t,\n safe =\n \\r.\n Trace.safe t r \\ Trace.owns t r = None\\\n \\ reach;\n inj initk; hotel t; (key1, key2) \\ Trace.cards t guest;\n Trace.roomk t room = key1 \\ Trace.roomk t room = key2\\\n \\ \\state.owns =\n Trace.owns\n (Enter guest room (key1, key2) # t),\n currk =\n Trace.currk\n (Enter guest room (key1, key2) # t),\n issued = Trace.issued t,\n cards =\n Trace.cards\n (Enter guest room (key1, key2) # t),\n roomk =\n Trace.roomk\n (Enter guest room (key1, key2) # t),\n isin =\n Trace.isin\n (Enter guest room (key1, key2) # t),\n safe =\n \\r.\n Trace.safe\n (Enter guest room (key1, key2) # t)\n r \\\n Trace.owns t r = None\\\n \\ reach"} {"_id": "503000", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\has_dist_on G m n V; p \\ path_set_on G m n V\\\n \\ dist_on G m n V \\ path_weight G p"} {"_id": "503001", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\m. n = m + 2 \\ thesis) \\ thesis"} {"_id": "503002", "text": "proof (prove)\ngoal (1 subgoal):\n 1. U.wtE \\ \\ F.wtE \\"} {"_id": "503003", "text": "proof (prove)\nusing this:\n \\m. i \\# m \\ P m = \\\n \\m. i \\# m \\ Q m = \\\n\ngoal (1 subgoal):\n 1. \\m. i \\# m \\ P m \\ Q m = \\"} {"_id": "503004", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ p \\ u \\ \\;\n supp p \\* u\\\n \\ supp (map fst \\) \\* map snd \\"} {"_id": "503005", "text": "proof (prove)\ngoal (3 subgoals):\n 1. countable UNIV\n 2. countable (Collect TE_wtFsym)\n 3. countable {p. wtPsym p}"} {"_id": "503006", "text": "proof (prove)\nusing this:\n S.is_left_adjoint (S.UP f \\\\<^sub>S S.UP f')\n S.isomorphic (S.UP f \\\\<^sub>S S.UP f') (S.UP (f \\ f'))\n\ngoal (1 subgoal):\n 1. S.is_left_adjoint (S.UP (f \\ f'))"} {"_id": "503007", "text": "proof (prove)\nusing this:\n env = Env b es\n es x = Some e'\n xs = []\n e = e'\n\ngoal (1 subgoal):\n 1. lookup (update ((x # xs) @ y # ys) opt env) (x # xs) =\n Some (update (y # ys) opt e)"} {"_id": "503008", "text": "proof (prove)\ngoal (1 subgoal):\n 1. orthogonal x a"} {"_id": "503009", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bounded (range f)"} {"_id": "503010", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (if k = 0 then ([], map_poly hom p)\n else if map_poly hom (root_unity k) dvd map_poly hom p\n then if map_poly hom p = 0 then ([], 0)\n else map_prod ((#) k) id\n (decompose_prod_root_unity_main\n (map_poly hom p div map_poly hom (root_unity k)) k)\n else decompose_prod_root_unity_main (map_poly hom p) (k - 1)) =\n (if k = 0 then map_prod id (map_poly hom) ([], p)\n else if root_unity k dvd p\n then if p = 0 then map_prod id (map_poly hom) ([], 0)\n else map_prod id (map_poly hom)\n (map_prod ((#) k) id\n (decompose_prod_root_unity_main (p div root_unity k)\n k))\n else map_prod id (map_poly hom)\n (decompose_prod_root_unity_main p (k - 1)))"} {"_id": "503011", "text": "proof (prove)\nusing this:\n winning_path p (LCons v0 P)\n \\winning_path p (LCons v0 P); \\ lnull (LCons v0 P);\n \\ lnull (ltl (LCons v0 P))\\\n \\ winning_path p (ltl (LCons v0 P))\n\ngoal (1 subgoal):\n 1. winning_path p P"} {"_id": "503012", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map (\\t. vars_term_ms (t \\ \\)) ts =\n map (\\t. \\x\\#vars_term_ms t. vars_term_ms (\\ x))\n ts"} {"_id": "503013", "text": "proof (prove)\nusing this:\n \\.token_run x (Suc n) \\ {q. \\ \\\\<^sub>P Rep q}\n x \\ n\n ?x \\ ?n \\\n \\.token_run ?x (Suc ?n) = \\step (\\.token_run ?x ?n) (w ?n)\n\ngoal (1 subgoal):\n 1. \\ \\\\<^sub>P Rep (\\step (\\.token_run x n) (w n))"} {"_id": "503014", "text": "proof (state)\nthis:\n t \\ inverse B\n\ngoal (1 subgoal):\n 1. (\\et ex B L.\n \\0 < et; 0 < ex; cball t0 et \\ T;\n cball x0 ex \\ X;\n unique_on_cylinder t0 (cball t0 et) x0 ex f B L\\\n \\ thesis) \\\n thesis"} {"_id": "503015", "text": "proof (prove)\nusing this:\n stfx.mult_stalk C C' = stfx.class_of (U \\ U') mult\n\ngoal (1 subgoal):\n 1. stfx.class_of (U \\ U') mult \\ stfx.carrier_stalk"} {"_id": "503016", "text": "proof (prove)\nusing this:\n \\Q (smap state ?xa7);\n smap abss (smap state ?xa7) = smap state (smap absc ?xa7);\n smap state ?xa7 \\ Y\\\n \\ smap state (smap absc ?xa7) \\ X \\\n P (smap state (smap absc ?xa7))\n\ngoal (1 subgoal):\n 1. stream_all2 (\\s t. t = absc s) x y \\\n (x \\ {y. smap state y \\ Y \\ Q (smap state y)}) =\n (y \\ {y. smap state y \\ X \\ P (smap state y)})"} {"_id": "503017", "text": "proof (prove)\nusing this:\n Finca_get F2 f = Some fd2\n pc < length (body fd2)\n body fd2 ! pc = Inca.instr.IPush d\n Frame f pc \\ # st' = st\n Fstd_get F1 f = Some fd1\n rel_fundef (\\x y. x = norm_instr y) fd1 fd2\n\ngoal (1 subgoal):\n 1. Sstd.step (State F1 H st)\n (State F1 H (Frame f (Suc pc) (d # \\) # st'))"} {"_id": "503018", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ a \\' \\ l = \\ b"} {"_id": "503019", "text": "proof (state)\nthis:\n solve_exec_code cs = Inl I\n\ngoal (2 subgoals):\n 1. solve_exec_code cs = Inl I \\\n set I \\ fst ` set cs \\\n (\\v.\n (set I, v) \\\\<^sub>i\\<^sub>c\\<^sub>s set cs) \\\n (distinct_indices cs \\\n (\\J\\set I.\n \\v. (J, v) \\\\<^sub>i\\<^sub>c\\<^sub>s set cs))\n 2. solve_exec_code cs = Inr v \\\n \\v\\ \\\\<^sub>c\\<^sub>s (snd ` set cs)"} {"_id": "503020", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_append_child_wf\\<^sub>C\\<^sub>o\\<^sub>r\\<^sub>e\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n Shadow_DOM.get_owner_document Shadow_DOM.get_parent\n Shadow_DOM.get_parent_locs Shadow_DOM.remove_child\n Shadow_DOM.remove_child_locs get_disconnected_nodes\n get_disconnected_nodes_locs set_disconnected_nodes\n set_disconnected_nodes_locs Shadow_DOM.adopt_node\n Shadow_DOM.adopt_node_locs ShadowRootClass.known_ptr\n ShadowRootClass.type_wf Shadow_DOM.get_child_nodes\n Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs\n Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs\n Shadow_DOM.remove get_ancestors_si Shadow_DOM.insert_before\n Shadow_DOM.insert_before_locs Shadow_DOM.append_child\n Shadow_DOM.heap_is_wellformed"} {"_id": "503021", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x = rat_of_int a / rat_of_int b &&& y = rat_of_int c / rat_of_int d"} {"_id": "503022", "text": "proof (chain)\npicking this:\n finite_ReZ_segments (subpath 0 s g) z"} {"_id": "503023", "text": "proof (prove)\nusing this:\n bi \\ 0\n bj \\ 0\n\ngoal (1 subgoal):\n 1. (bi - 1 = bj - 1) = (bi = bj)"} {"_id": "503024", "text": "proof (prove)\ngoal (1 subgoal):\n 1. top_pres (\\

fb\\<^sup>-\\<^sub>\\)"} {"_id": "503025", "text": "proof (prove)\nusing this:\n positive (tensor_mat (A + - m1) m2)\n tensor_mat (- m1) m2 = - tensor_mat m1 m2\n tensor_mat (A + - m1) m2 = tensor_mat A m2 + tensor_mat (- m1) m2\n\ngoal (1 subgoal):\n 1. tensor_mat (A + - m1) m2 = tensor_mat A m2 - tensor_mat m1 m2"} {"_id": "503026", "text": "proof (prove)\nusing this:\n f = Empty\n length ss = arity f\n\ngoal (1 subgoal):\n 1. real (cost f ss) + \\ (exec f ss) - sum_list (map \\ ss)\n \\ U f ss"} {"_id": "503027", "text": "proof (prove)\nusing this:\n x \\ {0..1} - S\n\ngoal (1 subgoal):\n 1. ((\\x. - \\ x) has_vector_derivative\n - vector_derivative \\ (at x within cbox 0 1))\n (at x within cbox 0 1)"} {"_id": "503028", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n ((``) (R ; S) \\ \\) x =\n (\\ \\ \\

\\<^sub>0[\\.leg0, HH\\\\\\.leg1] : H\\HH\\\\\\.chine \\\\<^sub>C HH\\\\\\.chine\\\n 2. \\\\

\\<^sub>1[\\.leg0, H\\H\\\\.leg1] : H\\H\\H\\\\.chine \\\\<^sub>C \\.apex\\\n 3. \\\\

\\<^sub>1[assoc\\\\\\.cod.leg0, \\.cod.leg1] : HH\\H\\\\\\.chine \\\\<^sub>C H\\H\\\\.chine\\\n 4. \\\\

\\<^sub>0[HH\\\\\\.leg0, \\.leg1] : HHH\\\\\\\\.chine \\\\<^sub>C \\.chine\\\n 5. \\\\

\\<^sub>1[HH\\\\\\.leg0, \\.leg1] : HHH\\\\\\\\.chine \\\\<^sub>C HH\\\\\\.chine\\\n 6. \\\\

\\<^sub>1[\\\\.leg0 \\\n \\_\\\\.prj\\<^sub>0, \\.leg1] : HH\\H\\\\\\.chine \\\\<^sub>C H\\H\\\\.chine\\\n 7. \\\\

\\<^sub>1[assoc\\\\\\.dom.leg0, \\.leg1] : HHH\\\\\\\\.chine \\\\<^sub>C HH\\\\\\.chine\\\n 8. \\\\_\\\\.prj\\<^sub>0 : H\\H\\\\.chine \\\\<^sub>C \\\\.apex\\"} {"_id": "503384", "text": "proof (chain)\npicking this:\n 0 < ?e \\\n \\f\\R.\n f ` S \\ {0..1} \\\n (\\t\\S \\ V. f t < ?e) \\\n (\\t\\S - U. 1 - ?e < f t)\n t0 \\ V\n S \\ V \\ U"} {"_id": "503385", "text": "proof (state)\nthis:\n l \\ 1\n\ngoal (2 subgoals):\n 1. ?P \\ c i j + c i' j' \\ c i j' + c i' j\n 2. \\ ?P \\ c i j + c i' j' \\ c i j' + c i' j"} {"_id": "503386", "text": "proof (state)\ngoal (3 subgoals):\n 1. \\i tsa p is \\ sb \\ \\ \\ p' is' ma \\''.\n \\(ts, m, \\) = (tsa, ma, \\'');\n (ts', m', \\') =\n (tsa[i := (p', is @ is', \\, sb, \\, \\, \\)], ma,\n \\'');\n i < length tsa; tsa ! i = (p, is, \\, sb, \\, \\, \\);\n \\\\ p \\\\<^sub>p (p', is')\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 2. \\i tsa p is \\ sb \\ \\ \\ ma \\'' is' \\' sb'\n m'a \\' \\' \\' \\'''.\n \\(ts, m, \\) = (tsa, ma, \\'');\n (ts', m', \\') =\n (tsa[i := (p, is', \\', sb', \\', \\', \\')], m'a,\n \\''');\n i < length tsa; tsa ! i = (p, is, \\, sb, \\, \\, \\);\n (is, \\, sb, ma, \\, \\, \\,\n \\'') \\\\<^sub>s\\<^sub>b (is', \\', sb', m'a,\n \\', \\', \\', \\''')\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' \n 3. \\i tsa p is \\ sb \\ \\ \\ ma \\'' m'a sb' \\'\n \\' \\'''.\n \\(ts, m, \\) = (tsa, ma, \\'');\n (ts', m', \\') =\n (tsa[i := (p, is, \\, sb', \\, \\', \\')], m'a,\n \\''');\n i < length tsa; tsa ! i = (p, is, \\, sb, \\, \\, \\);\n (ma, sb, \\, \\,\n \\'') \\\\<^sub>w (m'a, sb', \\', \\',\n \\''')\\\n \\ \\ts\\<^sub>h' \\\\<^sub>h'.\n (ts\\<^sub>h, m,\n \\\\<^sub>h) \\\\<^sub>s\\<^sub>b\\<^sub>h\\<^sup>* (ts\\<^sub>h',\n m', \\\\<^sub>h') \\\n ts' \\\\<^sub>h ts\\<^sub>h' "} {"_id": "503387", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x\n |{i. qinp i \\ None}| \n liftAll qGoodAbs qbinp \\ |{i. qbinp i \\ None}| KA.span\n ((\\v. v @\\<^sub>v 0\\<^sub>v m) ` baseB \\\n (@\\<^sub>v) (0\\<^sub>v n) ` baseD)"} {"_id": "503390", "text": "proof (prove)\nusing this:\n i < k\n x\\<^sub>1 < x\n x\\<^sub>0 \\ bs ! i * x + (hs ! i) x\n\ngoal (1 subgoal):\n 1. P (bs ! i * x + (hs ! i) x)"} {"_id": "503391", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a < b; A \\ B\\\n \\ add_mset a A < add_mset b B"} {"_id": "503392", "text": "proof (prove)\nusing this:\n Nml (u \\<^bold>\\ v)\n\ngoal (1 subgoal):\n 1. Nml v \\\n \\ is_Prim\\<^sub>0 v \\\n \\ is_Vcomp v \\ \\ is_Lunit' v \\ \\ is_Runit' v"} {"_id": "503393", "text": "proof (prove)\nusing this:\n (\\s vl s1 vl1.\n \\ s vl s1 vl1 \\\n \\ (?s4 s vl s1 vl1) (?vl4 s vl s1 vl1) (?s1.4 s vl s1 vl1)\n (?vl1.4 s vl s1 vl1)) \\\n match\n (\\s vl s1 vl1.\n \\' (?s4 s vl s1 vl1) (?vl4 s vl s1 vl1) (?s1.4 s vl s1 vl1)\n (?vl1.4 s vl s1 vl1))\n s s1 vl1 a ou s' vl'\n\ngoal (1 subgoal):\n 1. match \\' s s1 vl1 a ou s' vl' \\\n ignore \\' s s1 vl1 a ou s' vl'"} {"_id": "503394", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\queue.\n queue_invar queue \\\n queue_\\ (local.enqueue_new_threads queue []) =\n Round_Robin.enqueue_new_threads (queue_\\ queue) []\n 2. \\queue.\n queue_invar queue \\\n queue_invar (local.enqueue_new_threads queue [])\n 3. \\a ntas queue.\n \\\\queue.\n queue_invar queue \\\n queue_\\ (local.enqueue_new_threads queue ntas) =\n Round_Robin.enqueue_new_threads (queue_\\ queue)\n ntas;\n \\queue.\n queue_invar queue \\\n queue_invar (local.enqueue_new_threads queue ntas);\n queue_invar queue\\\n \\ queue_\\\n (local.enqueue_new_threads queue (a # ntas)) =\n Round_Robin.enqueue_new_threads\n (queue_\\ queue) (a # ntas)\n 4. \\a ntas queue.\n \\\\queue.\n queue_invar queue \\\n queue_\\ (local.enqueue_new_threads queue ntas) =\n Round_Robin.enqueue_new_threads (queue_\\ queue)\n ntas;\n \\queue.\n queue_invar queue \\\n queue_invar (local.enqueue_new_threads queue ntas);\n queue_invar queue\\\n \\ queue_invar\n (local.enqueue_new_threads queue (a # ntas))"} {"_id": "503395", "text": "proof (prove)\nusing this:\n x \\ left_app ` pos_of t1\n\ngoal (1 subgoal):\n 1. (\\q.\n \\x = indices.Left # q; position_in q t1\\\n \\ thesis) \\\n thesis"} {"_id": "503396", "text": "proof (prove)\nusing this:\n equator \\ meridian\n proj2_incident K2_centre equator\n proj2_incident K2_centre meridian\n proj2_incident c (apply_cltn2_line equator J)\n proj2_incident c (apply_cltn2_line meridian J)\n\ngoal (1 subgoal):\n 1. apply_cltn2 K2_centre J = c"} {"_id": "503397", "text": "proof (prove)\ngoal (1 subgoal):\n 1. outs_\\ \\ \\\\<^sub>R res \\ res'"} {"_id": "503398", "text": "proof (prove)\ngoal (1 subgoal):\n 1. v $ (j - i) \\ \\x\\ j"} {"_id": "503399", "text": "proof (state)\nthis:\n ?k3 < n \\ norm (v1 ?k3 * v2 ?k3) \\ b1 * b2\n\ngoal (1 subgoal):\n 1. \\i j.\n \\i < dim_row (A1 * A2); j < dim_col (A1 * A2)\\\n \\ norm ((A1 * A2) $$ (i, j)) \\ b1 * b2 * real n"} {"_id": "503400", "text": "proof (prove)\ngoal (1 subgoal):\n 1. compare_1_rat y x = compare (real_of_rat x) (real_of_1 y)"} {"_id": "503401", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\prf \\ proof; \\ \\ fmla; Fvars \\ = {};\n isTrue (PPf (encPf prf) \\\\\\)\\\n \\ bprv (PPf (encPf prf) \\\\\\)"} {"_id": "503402", "text": "proof (state)\nthis:\n set prevs' =\n set [] \\\n \\\n {next_subseqs2_set tail v [0.. set prevs} \\\n set out =\n set [] \\\n ((f head \\ snd) ` set prevs \\\n \\ {out_subseqs2_set tail v [0.. set prevs})\n\ngoal (2 subgoals):\n 1. set prevs' =\n {uu_.\n \\v i j.\n uu_ = (j, f (tail !! j) v) \\\n (i, v) \\ set prevs \\ j < i}\n 2. set out = (f head \\ snd) ` set prevs \\ snd ` set prevs'"} {"_id": "503403", "text": "proof (prove)\nusing this:\n lookup (row_to_poly (pps_to_list S) (row A i)) u \\ (0::'b)\n\ngoal (1 subgoal):\n 1. (\\j.\n \\j < length (pps_to_list S); u = pps_to_list S ! j\\\n \\ thesis) \\\n thesis"} {"_id": "503404", "text": "proof (prove)\nusing this:\n order.greater_eq\n (\\a\\fundfoldpairs.\n case a of (f, g) \\ {Abs_induced_automorph f g})\n (set ss) \\\n gallery (fold fst (map Spair ss) \\ Cs)\n order.greater_eq\n (\\a\\fundfoldpairs.\n case a of (f, g) \\ {Abs_induced_automorph f g})\n (set (ss @ [s]))\n\ngoal (1 subgoal):\n 1. gallery\n (fst (Spair s) \\ (fold fst (map Spair ss) \\ Cs))"} {"_id": "503405", "text": "proof (prove)\nusing this:\n ?A1 \\ set As \\ \\ INTERP N \\ C_of ?A1\n\ngoal (1 subgoal):\n 1. \\ Multiset.Bex (\\\\<^sub># (mset (map C_of As)))\n ((\\l) (INTERP N))"} {"_id": "503406", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\c1 c2.\n \\k.\n norm_bound (of_real_hom.mat_hom A ^\\<^sub>m k)\n (c1 + c2 * real k ^ (d - 1))"} {"_id": "503407", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cos ` set_of (real_interval x)\n \\ set_of (real_interval (cos_float_interval p x))"} {"_id": "503408", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Limit.cone J' C (D \\ F) a (\\ \\ F)"} {"_id": "503409", "text": "proof (prove)\nusing this:\n weak_barbed_bisimulation Rel CWB\n\ngoal (1 subgoal):\n 1. weak_barbed_bisimulation (Rel\\<^sup>=) CWB"} {"_id": "503410", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dsn =\n (case dests dip of None \\ \\\\<^sub>2 (the (rt dip))\n | Some rsn \\ rsn) \\\n dsk = \\\\<^sub>3 (the (rt dip)) \\\n flag =\n (if dests dip = None then \\\\<^sub>4 (the (rt dip))\n else Aodv_Basic.inv) \\\n hops = \\\\<^sub>5 (the (rt dip)) \\\n nhip = \\\\<^sub>6 (the (rt dip)) \\\n pre = \\\\<^sub>7 (the (rt dip))"} {"_id": "503411", "text": "proof (prove)\nusing this:\n list_all carrier_coeff []\n list_all (\\Q. index_free Q i) []\n\ngoal (1 subgoal):\n 1. \\m.\n count m i = length [] + k \\\n indexed_eval_aux ([] @ [P]) i m \\ \\"} {"_id": "503412", "text": "proof (prove)\nusing this:\n weak_unit a\n a \\ a'\n natural_isomorphism Right_a.comp Right_a.comp Right_a.R Right_a.map\n Right_a.\\\n naturally_isomorphic ?A ?B ?F ?G =\n (\\\\. natural_isomorphism ?A ?B ?F ?G \\)\n\ngoal (1 subgoal):\n 1. naturally_isomorphic Right_a.comp Right_a.comp Right_a.R Right_a.map"} {"_id": "503413", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Gauss_Jordan_in_ij A Greatest_plus_one (mod_type_class.from_nat k) $\n Greatest_plus_one $\n mod_type_class.from_nat k =\n (1::'a)"} {"_id": "503414", "text": "proof (prove)\nusing this:\n set ?fs \\ set (deg_shifts ?d ?fs)\n\ngoal (1 subgoal):\n 1. ideal (set fs) \\ ideal (set (deg_shifts d fs))"} {"_id": "503415", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fst ` edges (the_lcg sel Rules (0, {})) \\ L"} {"_id": "503416", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\fun rbt.\n ord.is_rbt cless rbt \\ ID ccompare = None \\\n ord.is_rbt cless\n (rbtreeify (filter fun (RBT_Impl.entries rbt))) \\\n ID ccompare = None"} {"_id": "503417", "text": "proof (prove)\nusing this:\n \\t' p.\n subterm t1 p t' \\ root_term S t' \\ trm_rep t' S \\ t'\n\ngoal (1 subgoal):\n 1. (\\p t'.\n \\subterm t1 p t'; root_term S t';\n trm_rep t' S \\ t'\\\n \\ thesis) \\\n thesis"} {"_id": "503418", "text": "proof (prove)\nusing this:\n Right.ide f\n Right.ide ?f \\ Right.iso (Right.runit ?f)\n Right.arr ?\\ \\ Right.iso ?\\ = local.iso ?\\\n\ngoal (1 subgoal):\n 1. local.iso (Right.runit f)"} {"_id": "503419", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prod f {m..n} =\n (if n < m then 1::'a\n else if n = 0 then f 0 else f n * prod f {m..n - 1})"} {"_id": "503420", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\ca mu c'a E.\n \\While e c = cxt_to_stmt E (ca@[mu]); c' = cxt_to_stmt E c'a;\n \\ca, update_modes mu\n mds, mem\\\\<^sub>w \\\\<^sub>w\n \\c'a, mds', mem'\\\\<^sub>w\\\n \\ c' = Stmt.If e (c ;; While e c) Stop \\\n mds' = mds \\ mem' = mem\n 2. \\b c\\<^sub>1.\n \\While e c = Await b c\\<^sub>1; eval\\<^sub>B mem b;\n no_await c\\<^sub>1;\n \\c\\<^sub>1, mds, mem\\\\<^sub>w \\\\<^sub>w\\<^sup>+ \\c', mds', mem'\\\\<^sub>w;\n is_final c'\\\n \\ c' = Stmt.If e (c ;; While e c) Stop \\\n mds' = mds \\ mem' = mem"} {"_id": "503421", "text": "proof (prove)\nusing this:\n [:d:] * factor = [:c:] * local.g\n\ngoal (1 subgoal):\n 1. smult d factor = [:c:] * local.g"} {"_id": "503422", "text": "proof (prove)\nusing this:\n 0 < e_orig\n\ngoal (1 subgoal):\n 1. 0 < e"} {"_id": "503423", "text": "proof (chain)\npicking this:\n pres_type step (length is) A\n PROP ?psi \\ PROP ?psi\n PROP ?psi \\ PROP ?psi\n wtl (take n is) c 0 s \\ A\n n < length is"} {"_id": "503424", "text": "proof (prove)\nusing this:\n ?x \\ carrier G \\\n h (?x [^] ?n) = h ?x [^]\\<^bsub>H\\<^esub> ?n\n\ngoal (1 subgoal):\n 1. x \\ carrier G \\\n h (x [^] n) = h x [^]\\<^bsub>H\\<^esub> n"} {"_id": "503425", "text": "proof (prove)\nusing this:\n isVal (Lam [y]. e') \\\n \\ : e \\\\<^bsub>upds_list\n (Arg x # S)\\<^esub> \\' : Lam [y]. e'\n\ngoal (1 subgoal):\n 1. \\ : e \\\\<^bsub>upds_list S\\<^esub> \\' : Lam [y]. e'"} {"_id": "503426", "text": "proof (prove)\nusing this:\n i < length a\n big_d' a b y i \\ []\n length y \\ length b\n finite (set (big_d' a b ?y ?i))\n \\finite ?A; ?A \\ {}\\\n \\ Min ?A \\ ?A\n\ngoal (1 subgoal):\n 1. Min (set (big_d' a b y i)) \\ set (big_d' a b y i)"} {"_id": "503427", "text": "proof (state)\nthis:\n cor (1 - Z) = 2 / cor (1 + (cmod z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\k.\n k \\ 0 \\\n (cor X + \\ * cor Y, cor (1 - Z)) = k *\\<^sub>s\\<^sub>v (z1, z2)"} {"_id": "503428", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Collect ((\\) i) \\ Collect ((\\) j) \\\n Collect ((\\) j) \\ Collect ((\\) i)) =\n (i = j) &&&\n \\ (Collect ((\\) i) \\ - Collect ((\\) j) \\\n - Collect ((\\) j) \\ Collect ((\\) i)) &&&\n \\ (- Collect ((\\) i) \\ Collect ((\\) j) \\\n Collect ((\\) j) \\ - Collect ((\\) i))"} {"_id": "503429", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly_deg g = Suc (deg_pm t')"} {"_id": "503430", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\x xa. normalization_euclidean_semiring_class.gcd x xa = gcd x xa"} {"_id": "503431", "text": "proof (prove)\ngoal (1 subgoal):\n 1. countable\n ((\\(x, y). ball x y) `\n (\\ (f ` (\\ \\ {0<..})) \\ \\))"} {"_id": "503432", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xT \\ yU; fresh yU u\\\n \\ subst xT u (close_Var i yU t) =\n close_Var i yU (subst xT u t)"} {"_id": "503433", "text": "proof (prove)\nusing this:\n a \\\\<^sub>\\ \\\\Obj\\\n b \\\\<^sub>\\ \\\\Obj\\\n\ngoal (1 subgoal):\n 1. v11 (vid_on\n (\\\\Arr\\) \\\\<^sup>l\\<^sub>\\\n Hom \\ a b)"} {"_id": "503434", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sim_val SPR.MC spr_simMC spr_sim"} {"_id": "503435", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i.\n \\i < DIM('a); DIM('a) - Suc 0 \\ i\\\n \\ b \\ Basis_list ! (DIM('a) - Suc 0) =\n b \\ Basis_list ! i"} {"_id": "503436", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\b\\set (proj l B).\n \\a\\set (proj l A).\n \\\\.\n b =\n a \\\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\\<^sub>p\n \\ \\\n wt\\<^sub>s\\<^sub>u\\<^sub>b\\<^sub>s\\<^sub>t \\ \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s (subst_range \\)"} {"_id": "503437", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\Group_Theory.monoid (f ` I) addB zeroB; x \\ I\\\n \\ \\v\\I.\n addB (f x) (f v) = zeroB \\\n addB (f v) (f x) = zeroB"} {"_id": "503438", "text": "proof (prove)\nusing this:\n \\\\<^sub>\\ (fflip (fflip f)) = \\\\<^sub>\\ f\n fbrelation (\\\\<^sub>\\ f)\n x \\\\<^sub>\\ \\\\<^sub>\\ (fflip (fflip f))\n\ngoal (1 subgoal):\n 1. (\\a b.\n x = [a, b]\\<^sub>\\ \\ thesis) \\\n thesis"} {"_id": "503439", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Derivation u D []; t \\ set u\\\n \\ is_nonterminal t"} {"_id": "503440", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite X; X \\ {}\\\n \\ \\x\\X. Max X = x"} {"_id": "503441", "text": "proof (prove)\nusing this:\n (cnj (y' * cnj z' - cnj y' * z') = y' * cnj z' - cnj y' * z') =\n is_real (y' * cnj z' - cnj y' * z')\n\ngoal (1 subgoal):\n 1. is_real (\\ * (y' * cnj z' - cnj y' * z'))"} {"_id": "503442", "text": "proof (state)\nthis:\n w = ls' ! last_index Vs V\n\ngoal (1 subgoal):\n 1. \\a.\n l V = \\a\\ \\\n [Vs [\\] ls'](V := l V) \\\\<^sub>m\n [Vs [\\] ls']"} {"_id": "503443", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\A \\ {};\n \\!m. m \\ A \\ (\\x\\A. m \\ x)\\\n \\ Nleast A \\ A \\\n (\\x\\A. Nleast A \\ x)"} {"_id": "503444", "text": "proof (prove)\ngoal (1 subgoal):\n 1. atm_of ` set_mset {#L#} = {atm_of L}"} {"_id": "503445", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a word, set_bit_class)"} {"_id": "503446", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (range A) \\ b \\ smallest_ccdi_sets \\ M"} {"_id": "503447", "text": "proof (prove)\nusing this:\n hindrance_by (\\\\weight := weight \\ - u\\) f\n \\\n\ngoal (1 subgoal):\n 1. (\\a.\n \\a \\ A \\;\n a \\ \\\\<^bsub>\\\\weight := weight \\ - u\\\\<^esub>\n (TER\\<^bsub>\\\\weight := weight \\ - u\\\\<^esub>\n f);\n d_OUT f a < weight \\ a;\n \\ < weight \\ a - d_OUT f a;\n 0 \\ \\\\\n \\ thesis) \\\n thesis"} {"_id": "503448", "text": "proof (prove)\ngoal (1 subgoal):\n 1. the (enum_alt (THE n. enum_alt n = Some x)) = x"} {"_id": "503449", "text": "proof (prove)\nusing this:\n a = snd (splitAt ram2 vs)\n before vs ram1 ram2\n pre_between vs ram1 ram2\n vs =\n (hd vs # b) @\n [ram1] @\n fst (splitAt ram2 (snd (splitAt ram1 vs))) @\n [ram2] @ snd (splitAt ram2 vs)\n\ngoal (1 subgoal):\n 1. vs = [hd vs] @ b @ [ram1] @ between vs ram1 ram2 @ [ram2] @ a"} {"_id": "503450", "text": "proof (prove)\nusing this:\n finite A\n\ngoal (1 subgoal):\n 1. distinct (sorted_list_of_set A)"} {"_id": "503451", "text": "proof (state)\ngoal (1 subgoal):\n 1. P S"} {"_id": "503452", "text": "proof (prove)\ngoal (1 subgoal):\n 1. spmf (completeness_game h w e) True = spmf (return_spmf True) True"} {"_id": "503453", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n A x y \\\n (\\xa ya.\n (\\t.\n cin t xa \\\n (\\u. cin u ya \\ A t u)) \\\n (\\t.\n cin t ya \\\n (\\u. cin u xa \\ A u t)) \\\n (\\t.\n cin t (cinsert x xa) \\\n (\\u. cin u (cinsert y ya) \\ A t u)) \\\n (\\t.\n cin t (cinsert y ya) \\\n (\\u. cin u (cinsert x xa) \\ A u t)))"} {"_id": "503454", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Crypt (pubK TTP)\n \\Agent S, Number AO, Key K, Agent R, hs\\\n \\ used evs;\n Key K \\ analz (knows Spy evs); evs \\ certified_mail\\\n \\ \\m ctxt q.\n hs =\n Hash\n \\Number ctxt, Nonce q, response S R q,\n Crypt K (Number m)\\ \\\n Says S R\n \\Agent S, Agent TTP, Crypt K (Number m),\n Number AO, Number ctxt, Nonce q,\n Crypt (pubK TTP)\n \\Agent S, Number AO, Key K, Agent R,\n hs\\\\\n \\ set evs"} {"_id": "503455", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a br.\n br \\ {(b, y). ok2 y} \\\n (case case br of (m, r) \\ (False, shift2 m r a) of\n (m, r) \\ L_a (m, strip2 r)) =\n Deriv a (case br of (m, r) \\ L_a (m, strip2 r))"} {"_id": "503456", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (THE pa. lift_spmf p = lift_spmf pa) = p"} {"_id": "503457", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pmf z (x, y) = enn2real (h x y)"} {"_id": "503458", "text": "proof (state)\nthis:\n connected (real_of ` inside)\n\ngoal (1 subgoal):\n 1. (\\inside outside.\n \\inside \\ {}; open inside; connected inside;\n outside \\ {}; open outside; connected outside;\n bounded inside; \\ bounded outside;\n inside \\ outside = {};\n inside \\ outside = - path_image c;\n frontier inside = path_image c;\n frontier outside = path_image c\\\n \\ thesis) \\\n thesis"} {"_id": "503459", "text": "proof (state)\nthis:\n (transform a (Terms.Let \\ body))[x::=y] =\n transform a (Terms.Let \\ body)[x::=y]\n\ngoal (5 subgoals):\n 1. \\var x y a.\n (transform a (Var var))[x::=y] = transform a (Var var)[x::=y]\n 2. \\exp var x y a.\n (\\b ba a.\n (transform a exp)[b::=ba] =\n transform a exp[b::=ba]) \\\n (transform a (App exp var))[x::=y] = transform a (App exp var)[x::=y]\n 3. \\var exp x y a.\n \\atom var \\ x; atom var \\ y;\n \\b ba a.\n (transform a exp)[b::=ba] = transform a exp[b::=ba]\\\n \\ (transform a (Lam [var]. exp))[x::=y] =\n transform a (Lam [var]. exp)[x::=y]\n 4. \\b x y a.\n (transform a (Bool b))[x::=y] = transform a (Bool b)[x::=y]\n 5. \\scrut e1 e2 x y a.\n \\\\b ba a.\n (transform a scrut)[b::=ba] = transform a scrut[b::=ba];\n \\b ba a. (transform a e1)[b::=ba] = transform a e1[b::=ba];\n \\b ba a.\n (transform a e2)[b::=ba] = transform a e2[b::=ba]\\\n \\ (transform a (scrut ? e1 : e2))[x::=y] =\n transform a (scrut ? e1 : e2)[x::=y]"} {"_id": "503460", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w \\ top * e\\<^sup>T * w\\<^sup>T\\<^sup>\\ \\\n F * e\\<^sup>T * top \\\n top * e * - F \\\n F\n \\ F * e\\<^sup>T * top \\ F"} {"_id": "503461", "text": "proof (prove)\nusing this:\n conc.globally_sound_mode_use gc\\<^sub>C\n list_all (\\cm. conc.locally_sound_mode_use (cm, snd gc\\<^sub>C))\n (fst gc\\<^sub>C)\n\ngoal (1 subgoal):\n 1. case gc\\<^sub>C of\n (cms, mem) \\\n list_all (\\cm. conc.locally_sound_mode_use (cm, mem))\n cms \\\n conc.globally_sound_mode_use (cms, mem)"} {"_id": "503462", "text": "proof (prove)\nusing this:\n f_range_on A f\n C \\ A \\ B - C \\ A \\\n f (C \\ (B - C)) \\ f C \\ f (B - C)\n B \\ A\n C \\ B\n\ngoal (1 subgoal):\n 1. f B \\ C \\ f C"} {"_id": "503463", "text": "proof (prove)\ngoal (1 subgoal):\n 1. DList_set dxs1 \\ DList_set dxs2 =\n (case ID CEQ('c) of\n None \\\n Code.abort STR ''union DList_set DList_set: ceq = None''\n (\\_. DList_set dxs1 \\ DList_set dxs2)\n | Some x \\ DList_set (DList_Set.union dxs1 dxs2))"} {"_id": "503464", "text": "proof (prove)\nusing this:\n \\i. f i \\ []\n f \\ SEQ (lists A)\n\ngoal (1 subgoal):\n 1. f' \\ SEQ (lists A)"} {"_id": "503465", "text": "proof (prove)\nusing this:\n S = Some S'\n s \\ \\\\<^sub>f S'\n s x \\ \\ (lookup S' x)\n s x \\ \\ a\n s \\ \\\\<^sub>o S\n\ngoal (1 subgoal):\n 1. s \\ \\\\<^sub>o (afilter (V x) a S)"} {"_id": "503466", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a x y.\n \\a \\ carrier R; x \\ carrier M;\n y \\ carrier M\\\n \\ a \\\\<^bsub>M\\<^esub>\n x \\\\<^bsub>M\\<^esub>\n y =\n a \\\\<^bsub>M\\<^esub>\n (x \\\\<^bsub>M\\<^esub> y)"} {"_id": "503467", "text": "proof (prove)\nusing this:\n x \\ path_image \\\n\ngoal (1 subgoal):\n 1. x \\ inside (path_image \\)"} {"_id": "503468", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P \\ C sees M, b : Ts\\T = m in D \\\n P \\ D sees M, b : Ts\\T = m in D"} {"_id": "503469", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rec_eval rec_imp [x, y] = (if 0 < x \\ y = 0 then 0 else 1)"} {"_id": "503470", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (map prot_atom.Atom enum_class.enum) \\\n set [Value, SetType, AttackType, Bottom, OccursSecType] =\n {}"} {"_id": "503471", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\A \\\\<^sub>\\ B;\n B \\\\<^sub>\\ C\\\n \\ A \\\\<^sub>\\ C"} {"_id": "503472", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eval\n (Cn 1 r_ifz\n [r_e2stack, Cn 1 r_prod_encode [Z, r_e2rv],\n Cn 1 r_ifz\n [r_e2i, Cn 1 r_prod_encode [r_e2tail, r_const 1],\n Cn 1 r_ifeq\n [r_e2i, r_const 1,\n Cn 1 r_prod_encode\n [r_e2tail, Cn 1 S [Cn 1 S [Cn 1 r_hd [r_e2xs]]]],\n Cn 1 r_ifeq\n [Cn 1 r_kind [r_e2i], r_const 2,\n Cn 1 r_prod_encode\n [r_e2tail,\n Cn 1 S [Cn 1 r_nth [r_e2xs, Cn 1 r_pdec22 [r_e2i]]]],\n Cn 1 r_ifeq\n [Cn 1 r_kind [r_e2i], r_const 3, r_step_Cn,\n Cn 1 r_ifeq\n [Cn 1 r_kind [r_e2i], r_const 4, r_step_Pr,\n Cn 1 r_ifeq\n [Cn 1 r_kind [r_e2i], r_const 5, r_step_Mn, Z]]]]]]])\n [e] \\=\n (if e2stack e = 0 then prod_encode (0, e2rv e)\n else if e2i e = 0 then prod_encode (e2tail e, 1)\n else if e2i e = 1\n then prod_encode (e2tail e, Suc (Suc (e_hd (e2xs e))))\n else if encode_kind (e2i e) = 2\n then prod_encode\n (e2tail e, Suc (e_nth (e2xs e) (pdec22 (e2i e))))\n else if encode_kind (e2i e) = 3 then estep_Cn e\n else if encode_kind (e2i e) = 4 then estep_Pr e\n else if encode_kind (e2i e) = 5\n then estep_Mn e else 0)"} {"_id": "503473", "text": "proof (prove)\nusing this:\n ID cbl \\ None\n\ngoal (1 subgoal):\n 1. keys t = dom (lookup t)"} {"_id": "503474", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n (\\ \\ \\) x \\ (\\ \\ \\

\\) x"} {"_id": "503475", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\\\\\<^sub>D\\<^sup>-\\<^sup>1[G f \\\\<^sub>D\n G.map\\<^sub>0\n (F.map\\<^sub>0\n b)] : G f \\\\<^sub>D\n G.map\\<^sub>0\n (F.map\\<^sub>0\n b) \\\\<^sub>D G.map\\<^sub>0\n (F.map\\<^sub>0 b') \\\\<^sub>D\n G f \\\\<^sub>D G.map\\<^sub>0 (F.map\\<^sub>0 b)\\"} {"_id": "503476", "text": "proof (prove)\ngoal (1 subgoal):\n 1. apply_chart d1 y \\ X"} {"_id": "503477", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. \\ i x \\"} {"_id": "503478", "text": "proof (state)\ngoal (1 subgoal):\n 1. gen_model M t = gen_model M' t"} {"_id": "503479", "text": "proof (prove)\nusing this:\n ?x \\ sets ?M \\ ?x \\ space ?M\n\ngoal (1 subgoal):\n 1. {uu_.\n \\S N N'.\n uu_ = S \\ N \\\n S \\ sets M \\ N' \\ null_sets M \\ N \\ N'}\n \\ Pow (space M)"} {"_id": "503480", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sin (2 * pi * k) \\ sin (2 * pi * n)"} {"_id": "503481", "text": "proof (prove)\nusing this:\n invariant_1_2 (p1, l1, r1)\n\ngoal (1 subgoal):\n 1. poly (real_of_int_poly p1) (real_of_1 (p1, l1, r1)) = 0 \\\n p1 \\ 0"} {"_id": "503482", "text": "proof (prove)\nusing this:\n c \\ P[- X]\n\ngoal (1 subgoal):\n 1. monomial c t = focus X (monomial (1::'b) t * c)"} {"_id": "503483", "text": "proof (prove)\ngoal (1 subgoal):\n 1. z * z \\ y * y\\<^sup>T \\ (1::'a) '"} {"_id": "503484", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\v\\<^sub>e\\<^sup>2 / (2 * a\\<^sub>o - 2 * a\\<^sub>e) +\n v\\<^sub>o\\<^sup>2 / (2 * a\\<^sub>o) +\n v\\<^sub>e * \\\n < s\\<^sub>o - s\\<^sub>e;\n other.t_stop \\ 0; u_max < s\\<^sub>o; \\ \\ \\ 0;\n other.p_max \\ u_max\\\n \\ v\\<^sub>e\\<^sup>2 / (2 * a\\<^sub>o - 2 * a\\<^sub>e)\n < s\\<^sub>o + v\\<^sub>o\\<^sup>2 / (2 * a\\<^sub>o) -\n s\\<^sub>e -\n v\\<^sub>e * \\"} {"_id": "503485", "text": "proof (prove)\nusing this:\n ts_tuple_rel (set auxlist) =\n {as \\ ts_tuple_rel\n (set (linearize data_prev) \\ set (linearize data_in)).\n valid_tuple tuple_since as}\n\ngoal (1 subgoal):\n 1. ts_tuple_rel\n (set (filter (\\(t, rel). enat (nt - t) \\ right I)\n auxlist)) =\n {as \\ ts_tuple_rel\n (set (linearize data_prev') \\\n set (linearize data_in')).\n valid_tuple tuple_since as}"} {"_id": "503486", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\m n.\n \\prime p; x * x = real p; 0 \\ x; n \\ 0;\n \\x\\ = real m / real n; coprime m n\\\n \\ False"} {"_id": "503487", "text": "proof (prove)\nusing this:\n intact slices W I\\<^sub>1\n intact slices W I\\<^sub>2\n\ngoal (1 subgoal):\n 1. i1.orig.quorum (I\\<^sub>1 \\ I\\<^sub>2)"} {"_id": "503488", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite UNIV"} {"_id": "503489", "text": "proof (prove)\ngoal (1 subgoal):\n 1. statically_complete_calculus Bot_FL Inf_FL (\\\\\\L)\n Red_I Red_F"} {"_id": "503490", "text": "proof (prove)\ngoal (1 subgoal):\n 1. principalideal {x * a |x. True} \\"} {"_id": "503491", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\g' \\ set (next_plane0\\<^bsub>p\\<^esub> g);\n Invariants.inv g; Invariants.inv g';\n set (ExcessTable g (vertices g))\n \\ set (ExcessTable g' (vertices g'))\\\n \\ ExcessNotAt g None \\ ExcessNotAt g' None"} {"_id": "503492", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subgroup {a [^] i |i. i \\ {0..ord a - 1}} G"} {"_id": "503493", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\q.\n \\q \\ W; Q' = Q \\ br_dsq \\ q (Q, W);\n W' = W - {q} \\ (br_dsq \\ q (Q, W) - Q)\\\n \\ thesis) \\\n thesis"} {"_id": "503494", "text": "proof (state)\nthis:\n list_all2 (rel_prod yun_erel (=)) fs Fs\n\ngoal (2 subgoals):\n 1. square_free_heuristic f = None \\\n square_free_factorization f (1, fs) \\\n (\\fi i.\n (fi, i) \\ set fs \\\n content fi = 1 \\ 0 < lead_coeff fi) \\\n distinct (map snd fs)\n 2. \\a.\n square_free_heuristic f = Some a \\\n square_free_factorization f (1, fs) \\\n (\\fi i.\n (fi, i) \\ set fs \\\n content fi = 1 \\ 0 < lead_coeff fi) \\\n distinct (map snd fs)"} {"_id": "503495", "text": "proof (prove)\nusing this:\n inj_on (inv_into (path_components_of X) h) (path_components_of Y) \\\n inv_into (path_components_of X) h ` path_components_of Y =\n path_components_of X\n\ngoal (1 subgoal):\n 1. path_components_of Y \\ {a} \\\n path_components_of X \\ {inv_into (path_components_of X) h a}"} {"_id": "503496", "text": "proof (prove)\ngoal (1 subgoal):\n 1. linear_orderable_1 x \\\n linear_orderable_2 (x \\ - (1::'a))"} {"_id": "503497", "text": "proof (prove)\nusing this:\n ((\\x. enn2ereal (ennreal (f x))) \\\n enn2ereal (ennreal x))\n F =\n ((\\x. ereal (f x)) \\ enn2ereal (ennreal x)) F\n\ngoal (1 subgoal):\n 1. (f \\ x) F"} {"_id": "503498", "text": "proof (prove)\nusing this:\n i \\ block_nominals (ps, a)\n j \\ block_nominals (ps, a)\n sub_block id ?block = ?block\n ?i \\ block_nominals ?block \\\n sub_block (?f(?i := ?j)) ?block = sub_block ?f ?block\n\ngoal (1 subgoal):\n 1. sub_block (id(i := j, j := i)) (ps, a) = (ps, a)"} {"_id": "503499", "text": "proof (prove)\nusing this:\n \\ \\\n (foldl (\\s' a. if \\ s' then s' else assert a s')\n (init t) as')\n P (set as') (assert_all_state t as') \\\n index_valid (set as') (assert_all_state t as')\n\ngoal (1 subgoal):\n 1. P (set (as' @ [a])) (assert_all_state t (as' @ [a])) \\\n index_valid (set (as' @ [a])) (assert_all_state t (as' @ [a]))"} {"_id": "503500", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 1 \\ j"} {"_id": "503501", "text": "proof (prove)\ngoal (1 subgoal):\n 1. z \\ a' \\ b\\<^sup>\\\n \\ a \\ z \\ b\\<^sup>\\"} {"_id": "503502", "text": "proof (prove)\ngoal (1 subgoal):\n 1. matrix_to_iarray (echelon_form_of_upt_k A (ncols A - 1) bezout) =\n echelon_form_of_upt_k_iarrays (matrix_to_iarray A)\n (ncols_iarray (matrix_to_iarray A) - 1) bezout"} {"_id": "503503", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p = 0\\<^sub>p"} {"_id": "503504", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set X \\ range_vars (rm_vars (set X) \\) = {}"} {"_id": "503505", "text": "proof (prove)\nusing this:\n ?e2 \\ parcs G \\\n \\u p v. gen_iapath (pverts G) u p v \\ ?e2 \\ set p\n gen_iapath (pverts G) ?u2 ?p2 ?v2 \\ ?p2 = [(?u2, ?v2)]\n\ngoal (1 subgoal):\n 1. (\\v\\pverts G.\n v \\ pverts G \\\n in_degree (with_proj G) v \\ 2 \\\n (\\x p y.\n gen_iapath (pverts G) x p y \\\n v \\ set (pawalk_verts x p))) \\\n (\\e\\parcs G.\n fst e \\ snd e \\\n (\\x p y.\n gen_iapath (pverts G) x p y \\ e \\ set p)) \\\n (\\u v p q.\n gen_iapath (pverts G) u p v \\\n gen_iapath (pverts G) u q v \\\n p = q) \\\n pverts G \\ pverts G"} {"_id": "503506", "text": "proof (prove)\nusing this:\n X \\ \\\n X \\ \\\n\ngoal (1 subgoal):\n 1. X \\ \\<\\>"} {"_id": "503507", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(b, b') \\ stutter_extend_edges b.V b.E;\n (a, b) \\ R\\\n \\ \\a'.\n (a, a') \\ stutter_extend_edges a.V a.E \\\n (a', b') \\ R"} {"_id": "503508", "text": "proof (prove)\ngoal (1 subgoal):\n 1. d' dvd d"} {"_id": "503509", "text": "proof (prove)\nusing this:\n (\\ eventually ?p ?f \\\n eventually ?pa ?f \\ ?p (rr ?p ?pa)) \\\n (\\ eventually ?p ?fa \\\n \\ ?pa (rr ?p ?pa) \\ eventually ?pa ?fa)\n ?r \\ existence_ivl ?ra ?rb \\ ?rb \\ X\n \\\\<^sub>F i in at_right (y t0). t \\ existence_ivl t0 i\n\ngoal (1 subgoal):\n 1. \\\\<^sub>F i in at_right (y t0). i \\ X"} {"_id": "503510", "text": "proof (prove)\nusing this:\n punit.lt (local.punit.monom_mult ?c ?t ?p) \\ ?t + punit.lt ?p\n t + punit.lt (rep_list f) \\ punit.lt (rep_list p)\n\ngoal (1 subgoal):\n 1. punit.lt\n (local.punit.monom_mult\n (lookup (rep_list p) (t + punit.lt (rep_list f)) /\n punit.lc (rep_list f))\n t (rep_list f)) \\\n punit.lt (rep_list p)"} {"_id": "503511", "text": "proof (prove)\nusing this:\n \\p\\P - {p}. env p < length \\l\n \\p\\P - {p}. env1 p < length \\l1\n\ngoal (1 subgoal):\n 1. (\\pa.\n pa \\ P \\\n (\\l @ [\\]) ! (env(p := length \\l)) pa =\n (\\l1 @ [\\]) ! (env1(p := length \\l1)) pa) =\n (\\pa.\n pa \\ P - {p} \\\n \\l ! env pa = \\l1 ! env1 pa)"} {"_id": "503512", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\x xa.\n \\F1set1 x \\ UNIV;\n F1set2 x \\ min_alg1 s1 s2;\n F1set3 x \\ min_alg2 s1 s2; xa \\ Field SucFbd;\n F1set2 x \\ fst (min_algs s1 s2 xa)\\\n \\ |F1set3 x| \\o ?r'52 x xa\n 2. \\x xa.\n \\F1set1 x \\ UNIV;\n F1set2 x \\ min_alg1 s1 s2;\n F1set3 x \\ min_alg2 s1 s2; xa \\ Field SucFbd;\n F1set2 x \\ fst (min_algs s1 s2 xa)\\\n \\ ?r'52 x xa \\o F1bd' +c F2bd'\n 3. \\x xa xb.\n \\F1set1 x \\ UNIV;\n F1set2 x \\ min_alg1 s1 s2;\n F1set3 x \\ min_alg2 s1 s2; xa \\ Field SucFbd;\n F1set2 x \\ fst (min_algs s1 s2 xa); xb \\ Field SucFbd;\n F1set3 x \\ snd (min_algs s1 s2 xb)\\\n \\ s1 x \\ min_alg1 s1 s2\n 4. \\y\\F2in UNIV (min_alg1 s1 s2) (min_alg2 s1 s2).\n s2 y \\ min_alg2 s1 s2"} {"_id": "503513", "text": "proof (prove)\nusing this:\n (map_of ts(k \\ delete_trie ks t)) k' = Some t'\n\ngoal (1 subgoal):\n 1. (map_of ts(k \\ delete_trie ks t)) k' = Some t'"} {"_id": "503514", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\distinct p; all_in_list p l; singleCombinators p;\n wellformed_policy1_strong p; wellformed_policy3 p;\n allNetsDistinct p\\\n \\ C (list2FWpolicy (FWNormalisationCore.sort p l))\n x =\n C (list2FWpolicy p) x"} {"_id": "503515", "text": "proof (prove)\nusing this:\n share sb\n ((\\ \\\\<^bsub>W'\\<^esub> R' \\\\<^bsub>A'\\<^esub> L') \\\\<^bsub>W\\<^esub> R \\\\<^bsub>A\\<^esub> L) =\n share sb\n (\\ \\\\<^bsub>W'\\<^esub> R' \\\\<^bsub>A'\\<^esub> L') \\\\<^bsub>W\\<^esub> R \\\\<^bsub>A\\<^esub> L\n\ngoal (1 subgoal):\n 1. share sb\n ((\\ \\\\<^bsub>W'\\<^esub> R' \\\\<^bsub>A'\\<^esub> L') \\\\<^bsub>W\\<^esub> R \\\\<^bsub>A\\<^esub> L) =\n share sb\n (\\ \\\\<^bsub>W'\\<^esub> R' \\\\<^bsub>A'\\<^esub> L') \\\\<^bsub>W\\<^esub> R \\\\<^bsub>A\\<^esub> L"} {"_id": "503516", "text": "proof (prove)\nusing this:\n \\, \\ \\\\<^bsub>/F\\<^esub> P\\<^sub>1 c\\<^sub>1 Q, A\n \\s t c'.\n \\ \\ (c\\<^sub>1, s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` pre P\\<^sub>1 \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` Q \\\n c' = Throw \\ t \\ Normal ` A\n \\, \\ \\\\<^bsub>/F\\<^esub> P\\<^sub>2 c\\<^sub>2 Q, A\n \\s t c'.\n \\ \\ (c\\<^sub>2, s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` pre P\\<^sub>2 \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` Q \\\n c' = Throw \\ t \\ Normal ` A\n r \\ b \\ pre P\\<^sub>1\n r \\ - b \\ pre P\\<^sub>2\n\ngoal (1 subgoal):\n 1. \\s t c'.\n \\ \\ (Cond b c\\<^sub>1 c\\<^sub>2,\n s) \\\\<^sup>*\n (c', t) \\\n final (c', t) \\\n s \\ Normal ` pre (AnnBin r P\\<^sub>1 P\\<^sub>2) \\\n t \\ Fault ` F \\\n c' = Skip \\ t \\ Normal ` Q \\\n c' = Throw \\ t \\ Normal ` A"} {"_id": "503517", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (s * l - b1) * (k * l * t - t + a1 * k) +\n (t * l + a1) * (s - k * l * s + b1 * k) =\n s * l * (k * l * t - t + a1 * k) - b1 * (k * l * t - t + a1 * k) +\n t * l * (s - k * l * s + b1 * k) +\n a1 * (s - k * l * s + b1 * k)"} {"_id": "503518", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\n (get_offending_flows (get_ACS M)\n \\nodes = V,\n edges = E \\ backflows (stateful \\ {x})\\)\n \\ backflows\n (filternew_flows_state\n \\hosts = V, flows_fix = E,\n flows_state =\n stateful \\ {x}\\) \\\n \\\n (get_offending_flows (get_ACS M)\n \\nodes = V,\n edges =\n E \\ (stateful \\ {x}) \\\n backflows (stateful \\ {x})\\)\n \\ backflows\n (filternew_flows_state\n \\hosts = V, flows_fix = E,\n flows_state = stateful \\ {x}\\)"} {"_id": "503519", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a. has_as_product J D a"} {"_id": "503520", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\morphism_presheaves_of_rings Spec is_zariski_open\n im_sheafXS.im_sheaf im_sheafXS.im_sheaf_morphisms b\n im_sheafXS.add_im_sheaf im_sheafXS.mult_im_sheaf\n im_sheafXS.zero_im_sheaf im_sheafXS.one_im_sheaf sheaf_spec\n sheaf_spec_morphisms \\b add_sheaf_spec mult_sheaf_spec\n zero_sheaf_spec one_sheaf_spec \\;\n \\U.\n is_zariski_open U \\\n (\\x\\im_sheafXS.im_sheaf U.\n \\\\<^sub>f U (\\ U x) = x) \\\n (\\x\\\\ U. \\ U (\\\\<^sub>f U x) = x);\n homeomorphism X is_open Spec is_zariski_open f\\\n \\ morphism_presheaves_of_rings Spec is_zariski_open\n sheaf_spec sheaf_spec_morphisms \\b add_sheaf_spec\n mult_sheaf_spec zero_sheaf_spec one_sheaf_spec\n im_sheafXS.im_sheaf im_sheafXS.im_sheaf_morphisms b\n im_sheafXS.add_im_sheaf im_sheafXS.mult_im_sheaf\n im_sheafXS.zero_im_sheaf im_sheafXS.one_im_sheaf\n \\\\<^sub>f"} {"_id": "503521", "text": "proof (prove)\nusing this:\n finite A\n OrderingSetIso dual_order.greater_eq dual_order.greater\n dual_order.greater_eq dual_order.greater (Pow A) ((`) f)\n ?f \\ Pow ?A = Pow (?f ` ?A)\n\ngoal (1 subgoal):\n 1. order.simplex_like (Pow A)"} {"_id": "503522", "text": "proof (prove)\nusing this:\n \\y. y \\ ?x\n\ngoal (1 subgoal):\n 1. (\\z.\n z \\ ord.min (\\) z1 z2 \\\n thesis) \\\n thesis"} {"_id": "503523", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\nodes edges.\n \\valid_reqs (get_IFS M);\n wf_graph \\nodes = nodes, edges = edges\\;\n all_security_requirements_fulfilled (get_IFS M)\n \\nodes = nodes, edges = edges\\;\n set edgesList \\ edges;\n G = \\nodes = nodes, edges = edges\\\\\n \\ all_security_requirements_fulfilled (get_IFS M)\n (stateful_policy_to_network_graph\n \\hosts = nodes, flows_fix = edges,\n flows_state =\n set (filter_IFS_no_violations_accu\n \\nodes = nodes, edges = edges\\ M [] edgesList)\\)"} {"_id": "503524", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Limit \\ \\\n \\\\ = \\ (Aleph ` elts \\)"} {"_id": "503525", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\v\\<^sub>1 v\\<^sub>2.\n \\v = v\\<^sub>1 @ [a] @ v\\<^sub>2;\n a \\ set v\\<^sub>1\\\n \\ thesis) \\\n thesis"} {"_id": "503526", "text": "proof (prove)\ngoal (1 subgoal):\n 1. trms\\<^sub>s\\<^sub>t (a # A \\\\<^sub>s\\<^sub>t \\) =\n trms\\<^sub>s\\<^sub>t (a # A) \\\\<^sub>s\\<^sub>e\\<^sub>t \\"} {"_id": "503527", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x\\set (Guards t).\n \\y\\set (Guards t'). mutex x y \\\n \\i r. \\ apply_guards (Guards t @ Guards t') (join_ir i r)\n 2. \\ (\\x\\set (map (\\(x, y). mutex x y)\n (List.product (Guards t) (Guards t'))).\n x) \\\n choice_alt t t' =\n (if \\(x, y)\\set (List.product (Guards t) (Guards t')).\n mutex x y\n then False\n else if Guards t = Guards t'\n then satisfiable (fold gAnd (rev (Guards t)) (Bc True))\n else satisfiable\n (fold gAnd (rev (Guards t @ Guards t')) (Bc True)))"} {"_id": "503528", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. - - - x = - x"} {"_id": "503529", "text": "proof (prove)\nusing this:\n finite \\\n \\A\\\\. term_ok \\ A\n \\A\\\\. typ_of A = Some propT\n\ngoal (1 subgoal):\n 1. \\,\\ \\ mk_eq t u"} {"_id": "503530", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Domainp HMA_M3 x"} {"_id": "503531", "text": "proof (prove)\ngoal (1 subgoal):\n 1. appFLv f args \\ R"} {"_id": "503532", "text": "proof (prove)\nusing this:\n Inv2a_inner s' q\n bal (dblock s' q) = 0\n\ngoal (1 subgoal):\n 1. inp (dblock s' q) = NotAnInput"} {"_id": "503533", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\p\\ \\ F \\q\\\\<^sub>u"} {"_id": "503534", "text": "proof (prove)\nusing this:\n KD_Tree.complete (build ?k ps)\n 0 < length ps \\\n KD_Tree.size_kdt (build ?ks ps) = length ps\n\ngoal (1 subgoal):\n 1. h = height (build k ps)"} {"_id": "503535", "text": "proof (state)\nthis:\n leaves i l \\ leaves (k + 1) r = {}\n\ngoal (1 subgoal):\n 1. \\t1 x2 t2 i j.\n \\\\i j.\n inorder t1 = [i..j] \\\n wpl i j t1 = Wpl i t1;\n \\i j.\n inorder t2 = [i..j] \\ wpl i j t2 = Wpl i t2;\n inorder \\t1, x2, t2\\ = [i..j]\\\n \\ wpl i j \\t1, x2, t2\\ =\n Wpl i \\t1, x2, t2\\"} {"_id": "503536", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\t. \\ t s) \\ Sols (\\t. f) U S 0 s"} {"_id": "503537", "text": "proof (prove)\ngoal (2 subgoals):\n 1. ?A3 \\ check_extension B \\' i' x' r'\n 2. (let b = i \\# dom_m A \\\n x \\ \\ \\ ([x], - 1) \\ set r\n in if \\ b\n then SPEC (\\_. True) \\\n (\\c. RETURN (error_msg i c))\n else let p' = remove1 ([x], - 1) r;\n b = vars_llist p' \\ \\\n in if \\ b\n then SPEC (\\_. True) \\\n (\\c. RETURN (error_msg i c))\n else mult_poly_full p' p' \\\n (\\p2.\n let p' = map (\\(a, b). (a, - b)) p'\n in add_poly_l (p2, p') \\\n (\\q.\n weak_equality_l q [] \\\n (\\eq.\n if eq then RETURN CSUCCESS\n else SPEC (\\_. True) \\\n (\\c. RETURN (error_msg i c))))))\n \\ \\\n {(st, b).\n (\\ is_cfailed st) = b \\\n (is_cfound st \\ spec = r)}\n ?A3"} {"_id": "503538", "text": "proof (prove)\ngoal (1 subgoal):\n 1. - x \\ (x \\ y) = \\"} {"_id": "503539", "text": "proof (prove)\nusing this:\n finite (J ?n)\n J ?n \\ I\n B ?n \\ sets (Pi\\<^sub>M (J ?n) (\\i. borel))\n incseq J\n\ngoal (1 subgoal):\n 1. emeasure (P (J n)) (B n) = emeasure (P' n) (fm n ` B n)"} {"_id": "503540", "text": "proof (prove)\nusing this:\n random_variable MX X\n\ngoal (1 subgoal):\n 1. prob_space (distr M MX X)"} {"_id": "503541", "text": "proof (prove)\ngoal (1 subgoal):\n 1. infs (gaccepting A)\n (gtrace (stake n (fromN 1 ||| r) @- sdrop n (fromN 1 ||| r)) (0, p))"} {"_id": "503542", "text": "proof (prove)\ngoal (1 subgoal):\n 1. to_bl (NOT w) = map Not (to_bl w)"} {"_id": "503543", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fset_of_fmap (fmap_of_list x) = fset_of_list (AList.clearjunk x)"} {"_id": "503544", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_emb P xs xs"} {"_id": "503545", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xa.\n \\\\xa\\set (Abs_State.dom y_).\n lookup x_ xa \\ fun y_ xa;\n \\x\\set (Abs_State.dom z_).\n lookup y_ x \\ fun z_ x;\n xa \\ set (Abs_State.dom z_)\\\n \\ lookup x_ xa \\ fun z_ xa"} {"_id": "503546", "text": "proof (prove)\nusing this:\n filter_pdevs P b = filter_pdevs Q d\n filter_pdevs (\\a b. \\ P a b) b =\n filter_pdevs (\\a b. \\ Q a b) d\n\ngoal (1 subgoal):\n 1. summarize_pdevs p P a b = summarize_pdevs q Q c d"} {"_id": "503547", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a\\ \\\\<^sub>r abs_interval A"} {"_id": "503548", "text": "proof (prove)\nusing this:\n h x \\ h ` (a_kernel R S h +> x)\n\ngoal (1 subgoal):\n 1. the_elem (h ` (a_kernel R S h +> x)) = h x"} {"_id": "503549", "text": "proof (prove)\nusing this:\n low_equal s1 s2\n low_equal ?s1.0 ?s2.0 \\\n get_operand2 ?op_list ?s1.0 = get_operand2 ?op_list ?s2.0\n\ngoal (1 subgoal):\n 1. \\is. get_operand2 is s1 = get_operand2 is s2"} {"_id": "503550", "text": "proof (prove)\nusing this:\n G Mod kernel G (H\\carrier := h ` carrier G\\) h \\ H\n \\carrier := h ` carrier G\\\n\ngoal (1 subgoal):\n 1. card (h ` carrier G) =\n order (G Mod kernel G (H\\carrier := h ` carrier G\\) h)"} {"_id": "503551", "text": "proof (state)\nthis:\n \\\\<^sub>F x in at_top. (powser cs \\ g) x = (f \\ g) x\n\ngoal (1 subgoal):\n 1. is_expansion_aux (powser_ms_aux cs G) (powser cs \\ g) basis"} {"_id": "503552", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fst ` (set gs \\ set bs) \\ dgrad_p_set d m"} {"_id": "503553", "text": "proof (prove)\nusing this:\n check_eqv as r s\n\ngoal (1 subgoal):\n 1. (\\ps.\n \\closure as ([(norm r, norm s)], []) = Some ([], ps);\n atoms r \\ atoms s \\ set as\\\n \\ thesis) \\\n thesis"} {"_id": "503554", "text": "proof (prove)\ngoal (1 subgoal):\n 1. product_prob_space (\\x. measure_pmf (K x))"} {"_id": "503555", "text": "proof (prove)\ngoal (1 subgoal):\n 1. get_S (cpu_reg_val PSR s') = 0"} {"_id": "503556", "text": "proof (prove)\nusing this:\n Option.is_none (devBC t n nid (n\\<^sub>s + Suc (n'''' - n\\<^sub>s)))\n\ngoal (1 subgoal):\n 1. devExt t n nid n\\<^sub>s (Suc (n'''' - n\\<^sub>s)) =\n devExt t n nid n\\<^sub>s (n'''' - n\\<^sub>s)"} {"_id": "503557", "text": "proof (prove)\nusing this:\n is_sum2sq_nat ?n =\n (\\k.\n prime (4 * k + 3) \\\n even (multiplicity (4 * k + 3) ?n))\n (\\p. prime p \\ [p = 3] (mod 4) \\ ?P p) =\n (\\k. prime (4 * k + 3) \\ ?P (4 * k + 3))\n\ngoal (1 subgoal):\n 1. is_sum2sq_nat n =\n (\\p.\n prime p \\ [p = 3] (mod 4) \\\n even (multiplicity p n))"} {"_id": "503558", "text": "proof (prove)\nusing this:\n min_gallery (C # Cs @ [D])\n Bs = Fs @ [F]\n As = []\n C # Cs @ [D] = As @ [A, B] @ Bs\n D \\ f \\ \\\n\ngoal (1 subgoal):\n 1. H \\ walls_betw C D"} {"_id": "503559", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Data (DataValue D) (Data.DataSpace D)"} {"_id": "503560", "text": "proof (prove)\ngoal (1 subgoal):\n 1. DD ars r\n (\\, (s, [\\_step]), \\', fst \\,\n snd \\ @ snd \\')"} {"_id": "503561", "text": "proof (prove)\ngoal (1 subgoal):\n 1. r = AllowPortFromTo a b po \\ r = DenyAllFromTo a b \\ r = DenyAll"} {"_id": "503562", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ = gen (\\) Xs"} {"_id": "503563", "text": "proof (prove)\nusing this:\n sup_continuous\n (\\F x.\n case x of\n (t, s, (t', s') ## \\) \\\n if t < t' then s \\ ss else F (t, s', \\))\n\ngoal (1 subgoal):\n 1. Measurable.pred\n (borel \\\\<^sub>M\n count_space UNIV \\\\<^sub>M stream_space S)\n (\\(t, s, \\). trace_in ss t s \\)"} {"_id": "503564", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\\\ \\ es' : [] _> ts \\\n map typeof vs = map typeof vs'"} {"_id": "503565", "text": "proof (prove)\nusing this:\n pred_stream (\\x. x \\ S) (smap state \\)\n\ngoal (1 subgoal):\n 1. pred_stream (\\u. u \\ V) (smap (snd \\ state) \\)"} {"_id": "503566", "text": "proof (prove)\ngoal (1 subgoal):\n 1. run_nondet (Monad_Overloading.bind m f) =\n Monad_Overloading.bind (run_nondet m)\n (\\A. mUnionMT TYPE('a) (image_mset (run_nondet \\ f) A))"} {"_id": "503567", "text": "proof (prove)\nusing this:\n edges g = {}\n\ngoal (1 subgoal):\n 1. is_tree g"} {"_id": "503568", "text": "proof (prove)\nusing this:\n \\k. a = gcd a b * k\n\ngoal (1 subgoal):\n 1. (\\c. a = gcd a b * c \\ thesis) \\\n thesis"} {"_id": "503569", "text": "proof (prove)\nusing this:\n \\ide t\\<^sub>0; ide \\.chine;\n ide r\\<^sub>0s\\<^sub>1.p\\<^sub>1; src t\\<^sub>0 = trg \\.chine;\n src \\.chine = trg r\\<^sub>0s\\<^sub>1.p\\<^sub>1\\\n \\ \\\\\\<^sup>-\\<^sup>1[t\\<^sub>0, \\.chine, r\\<^sub>0s\\<^sub>1.p\\<^sub>1] : src\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1 \\ trg t\\<^sub>0\\\n \\ide t\\<^sub>0; ide \\.chine;\n ide r\\<^sub>0s\\<^sub>1.p\\<^sub>1; src t\\<^sub>0 = trg \\.chine;\n src \\.chine = trg r\\<^sub>0s\\<^sub>1.p\\<^sub>1\\\n \\ \\\\\\<^sup>-\\<^sup>1[t\\<^sub>0, \\.chine, r\\<^sub>0s\\<^sub>1.p\\<^sub>1] : local.dom\n t\\<^sub>0 \\\n local.dom \\.chine \\\n local.dom\n r\\<^sub>0s\\<^sub>1.p\\<^sub>1 \\ (cod t\\<^sub>0 \\\n cod \\.chine) \\\n cod r\\<^sub>0s\\<^sub>1.p\\<^sub>1\\\n \\ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\\\n \\ local.iso \\\\<^sup>-\\<^sup>1[?f, ?g, ?h]\n (?a \\ ?a') =\n (\\f.\n \\f : ?a \\ ?a'\\ \\\n local.iso f)\n trg r\\<^sub>0s\\<^sub>1.p\\<^sub>1 = src r\\<^sub>0\n\ngoal (1 subgoal):\n 1. t\\<^sub>0 \\\n \\.chine \\ r\\<^sub>0s\\<^sub>1.p\\<^sub>1 \\\n (t\\<^sub>0 \\ \\.chine) \\ r\\<^sub>0s\\<^sub>1.p\\<^sub>1"} {"_id": "503570", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mcont Sup (\\) Sup (\\) (\\x. max x y)"} {"_id": "503571", "text": "proof (prove)\nusing this:\n src f = trg g\n src g = trg h\n trg fg\\<^sub>0h\\<^sub>1.p\\<^sub>0 = src (tab\\<^sub>1 h)\n trg fg\\<^sub>0h\\<^sub>1.p\\<^sub>1 =\n src (tab\\<^sub>0 g \\ f\\<^sub>0g\\<^sub>1.p\\<^sub>0)\n \\f\\<^sub>0g\\<^sub>1.\\ : src\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0 \\ trg\n (tab\\<^sub>1 g)\\\n \\f\\<^sub>0g\\<^sub>1.\\ : tab\\<^sub>0 f \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>1 \\ tab\\<^sub>1 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0\\\n \\arr f; arr f\\<^sub>0g\\<^sub>1.\\;\n arr fg\\<^sub>0h\\<^sub>1.p\\<^sub>1; src f = trg f\\<^sub>0g\\<^sub>1.\\;\n src f\\<^sub>0g\\<^sub>1.\\ =\n trg fg\\<^sub>0h\\<^sub>1.p\\<^sub>1\\\n \\ \\[cod f, cod f\\<^sub>0g\\<^sub>1.\\, cod\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n ((f \\ f\\<^sub>0g\\<^sub>1.\\) \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1) =\n (f \\\n f\\<^sub>0g\\<^sub>1.\\ \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1) \\\n \\[local.dom\n f, local.dom\n f\\<^sub>0g\\<^sub>1.\\, local.dom\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1]\n\ngoal (1 subgoal):\n 1. (f \\\n f\\<^sub>0g\\<^sub>1.\\ \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1) \\\n \\[f, tab\\<^sub>0 f \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>1, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] =\n \\[f, tab\\<^sub>1 g \\\n f\\<^sub>0g\\<^sub>1.p\\<^sub>0, fg\\<^sub>0h\\<^sub>1.p\\<^sub>1] \\\n ((f \\ f\\<^sub>0g\\<^sub>1.\\) \\\n fg\\<^sub>0h\\<^sub>1.p\\<^sub>1)"} {"_id": "503572", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F (img x) \\ local.set b"} {"_id": "503573", "text": "proof (prove)\nusing this:\n chain (\\x. S x ?xa)\n\ngoal (1 subgoal):\n 1. Lub S c = lim_proc (range (\\x. S x c))"} {"_id": "503574", "text": "proof (prove)\nusing this:\n (\\xa y.\n xa \\ y \\\n xa + weight \\ x \\ y + weight \\ x) \\\n d_OUT cap \\x\\ \\ weight \\ x + weight \\ x\n\ngoal (1 subgoal):\n 1. d_OUT cap \\x\\ \\ 2 * weight \\ x"} {"_id": "503575", "text": "proof (prove)\nusing this:\n \\p\\X. (a = p \\ p = c) \\ [[a p c]]\n\ngoal (1 subgoal):\n 1. a = x \\ c = z \\ a = z \\ c = x"} {"_id": "503576", "text": "proof (prove)\nusing this:\n x \\ (0::'a)\n\ngoal (1 subgoal):\n 1. \\\\<^sub># (prime_factorization x) = normalize x"} {"_id": "503577", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_fun\n (rel_fun (=)\\\\\\\\\n (rel_fun (BNF_Def.Grp UNIV g)\\\\\n (BNF_Def.Grp UNIV (map_spmf (map_prod h id))))\\\\)\n (rel_fun (BNF_Def.Grp UNIV (map_gpv' f g h))\n (rel_fun (=) (rel_spmf (BNF_Def.Grp UNIV (map_prod f id)))))\n exec_gpv exec_gpv \\\n exec_gpv callee (map_gpv' f g h gpv) s =\n map_spmf (map_prod f id)\n (exec_gpv (map_fun id (map_fun g (map_spmf (map_prod h id))) callee)\n gpv s)"} {"_id": "503578", "text": "proof (prove)\ngoal (1 subgoal):\n 1. is_unit n \\ squarefree_part n = n"} {"_id": "503579", "text": "proof (prove)\nusing this:\n Max {dist (u k (from_nat n)) (x (from_nat n)) |n. n \\ N} \\ e / 4\n 4 / e < 2 ^ N\n ?x < ?y \\ ?x \\ ?y\n\ngoal (2 subgoals):\n 1. 2 * Max {dist (u k (from_nat n)) (x (from_nat n)) |n. n \\ N}\n \\ 2 * e / 4\n 2. (1 / 2) ^ N \\ e / 4"} {"_id": "503580", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\r\\R. - a = r \\ x"} {"_id": "503581", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\m \\\\<^sub>n SPEC \\; m \\ SPEC gen_rwof'\\\n \\ m \\ SPEC (\\s. gen_rwof' s \\ \\ s)"} {"_id": "503582", "text": "proof (prove)\ngoal (1 subgoal):\n 1. case partition_by_biflows E of\n (biflows, uniflows) \\\n set biflows \\ set (backlinks biflows) \\ set uniflows =\n set E"} {"_id": "503583", "text": "proof (prove)\nusing this:\n wf R\n\ngoal (1 subgoal):\n 1. s \\ terminates_on g"} {"_id": "503584", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ x \\ P x) = ushEx P"} {"_id": "503585", "text": "proof (prove)\nusing this:\n P,E,hp s \\ map Val vs [:] Ts'\n P \\ Ts' [\\] Ts\n\ngoal (1 subgoal):\n 1. length vs = length Ts"} {"_id": "503586", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a A.\n bounded_clinear A \\\n (\\x. a *\\<^sub>C A x)\\<^sup>\\ =\n (\\x. cnj a *\\<^sub>C (A\\<^sup>\\) x)"} {"_id": "503587", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>\\ (\\ (\\\\<^sup>\\ ` X)) =\n \\ (\\\\<^sup>\\ ` X)"} {"_id": "503588", "text": "proof (prove)\nusing this:\n LeftDerives1 (Derive a (take (N - 1) D)) (fst (D ! (N - 1)))\n (snd (D ! (N - 1))) (\\_1 @ [X] @ \\_2)\n splits_at (Derive a (take (N - 1) D)) (fst (D ! (N - 1))) \\_1 Y\n \\_2\n\ngoal (1 subgoal):\n 1. \\u v.\n \\_1 = \\_1 @ u \\\n \\_2 = v @ \\_2 \\\n snd (snd (D ! (N - 1))) = u @ [X] @ v"} {"_id": "503589", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xa.\n RETURN xa \\ f x \\\n WHILE\\<^sub>T\\<^bsup>I\\<^esup> b f xa\n \\ WHILE\\<^bsup>I\\<^esup> b f xa"} {"_id": "503590", "text": "proof (state)\nthis:\n cfg \\ cfg_on_div s\n cfg \\ Collect alternating\n\ngoal (2 subgoals):\n 1. \\i.\n \\i \\ cfg_on_div s;\n i \\ {cfg. alternating cfg}\\\n \\ ?j11 i\n \\ R_G_cfg_on_div s' \\\n {cfg. alternating cfg}\n 2. \\i.\n \\i \\ cfg_on_div s;\n i \\ {cfg. alternating cfg}\\\n \\ emeasure (MDP.T i)\n {x \\ space MDP.St.\n (holds \\ suntil holds \\) (s ## x)} =\n emeasure (R_G.T (?j11 i))\n {x \\ space R_G.St.\n (holds \\' suntil holds \\') (s' ## x)}"} {"_id": "503591", "text": "proof (prove)\nusing this:\n negligible S\n\ngoal (1 subgoal):\n 1. f absolutely_integrable_on S"} {"_id": "503592", "text": "proof (prove)\nusing this:\n distinct xs\n\ngoal (1 subgoal):\n 1. size (bst_of_list xs) = length xs"} {"_id": "503593", "text": "proof (prove)\nusing this:\n \\ln rn.\n b = Oc # ires \\\n tl (Bk # ba) = Bk \\ ln @ Bk # Bk # Oc \\ Suc rs @ Bk \\ rn\n Bk # b = aa \\ list = ba\n c = Bk # ba\n\ngoal (1 subgoal):\n 1. (\\ln rn.\n \\b = Oc # ires;\n tl (Bk # ba) =\n Bk \\ ln @ Bk # Bk # Oc \\ Suc rs @ Bk \\ rn\\\n \\ thesis) \\\n thesis"} {"_id": "503594", "text": "proof (prove)\ngoal (1 subgoal):\n 1. XD = {} \\ XH = {Xd1, Xd1', Xd2, Xd2'} \\\n \\ StableNoDecomp.stable_on UNIV (XD \\ XH)"} {"_id": "503595", "text": "proof (prove)\ngoal (1 subgoal):\n 1. adjoint P * P = 1\\<^sub>m n"} {"_id": "503596", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\R \\ {region X I J r |I J r. valid_region X k I J r};\n v \\ R; R' \\ {region X I J r |I J r. valid_region X k I J r};\n R \\ R'\\\n \\ v \\ R'"} {"_id": "503597", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\h.\n \\sorted_less (separators ts);\n h \\\n is_pfa c tsi (a, n) *\n list_assn (A \\\\<^sub>a id_assn) ts tsi\\\n \\ \\<^sub>a id_assn) ts\n tsi> split_fun (a, n)\n p <\\r.\n is_pfa c tsi (a, n) * list_assn (A \\\\<^sub>a id_assn) ts tsi *\n \\ (split_relation ts (abs_split ts p) r)>\\<^sub>t"} {"_id": "503598", "text": "proof (prove)\ngoal (1 subgoal):\n 1. m1 \\ (\\h\\kernel G H f.\n {h \\\\<^bsub>G\\<^esub> a}) &&&\n m2 \\ (\\h\\kernel G H f. {h \\\\<^bsub>G\\<^esub> a})"} {"_id": "503599", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\p \\ carrier R; deg R S X p \\ an (Suc d); Subring R S;\n Ring S; p = \\\\\n \\ deg R S X (ldeg_p R S X d p) \\ an d\n 2. \\p \\ carrier R; deg R S X p \\ an (Suc d); Subring R S;\n Ring S; p \\ \\\\\n \\ deg R S X (ldeg_p R S X d p) \\ an d"} {"_id": "503600", "text": "proof (prove)\ngoal (1 subgoal):\n 1. constant_rep_design \\ (multiple_blocks n) (\\ * int n)"} {"_id": "503601", "text": "proof (prove)\nusing this:\n ko_ntm.oalist_inv (sort_oalist_ko_ntm' ox xs', ox)\n\ngoal (1 subgoal):\n 1. list_of_oalist_ntm (OAlist_ntm xs) = sort_oalist_ko_ntm xs"} {"_id": "503602", "text": "proof (prove)\nusing this:\n f (\\ (range A)) = f (\\ (range A) - A i) + f (A i)\n\ngoal (1 subgoal):\n 1. f (\\ (range A) - A i) = f (\\ (range A)) - f (A i)"} {"_id": "503603", "text": "proof (prove)\nusing this:\n \\[?a] \\\n \\Chn = \\

\\<^sub>0[Chn ?a, Chn ?a], Dom = Dom (?a \\ ?a),\n Cod = Cod ?a\\\n src ?\\ \\\n if arr ?\\ then mkObj (C.cod (Leg0 (Dom ?\\))) else null\n trg ?\\ \\\n if arr ?\\ then mkObj (C.cod (Leg1 (Dom ?\\))) else null\n ?\\ \\ ?\\ \\\n if arr ?\\ \\ arr ?\\ \\ src ?\\ = trg ?\\\n then \\Chn = chine_hcomp ?\\ ?\\,\n Dom =\n \\Leg0 =\n Leg0 (Dom ?\\) \\\n \\

\\<^sub>0[Leg0 (Dom ?\\), Leg1 (Dom ?\\)],\n Leg1 =\n Leg1 (Dom ?\\) \\\n \\

\\<^sub>1[Leg0 (Dom ?\\), Leg1 (Dom ?\\)]\\,\n Cod =\n \\Leg0 =\n Leg0 (Cod ?\\) \\\n \\

\\<^sub>0[Leg0 (Cod ?\\), Leg1 (Cod ?\\)],\n Leg1 =\n Leg1 (Cod ?\\) \\\n \\

\\<^sub>1[Leg0\n (Cod ?\\), Leg1\n (Cod ?\\)]\\\\\n else null\n arrow_of_spans (\\) (src f)\n arr ?\\ = arrow_of_spans (\\) ?\\\n arrow_of_spans (\\) f\n \\C.arr ?f; C.cod ?f = ?b\\\n \\ ?b \\ ?f = ?f\n\ngoal (1 subgoal):\n 1. \\[src f] =\n \\Chn = \\

\\<^sub>1[f.dsrc, f.dsrc],\n Dom =\n \\Leg0 = \\

\\<^sub>1[f.dsrc, f.dsrc],\n Leg1 = \\

\\<^sub>1[f.dsrc, f.dsrc]\\,\n Cod = \\Leg0 = f.dsrc, Leg1 = f.dsrc\\\\"} {"_id": "503604", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_optionT (BNF_Def.Grp A f) =\n BNF_Def.Grp {x. set_optionT x \\ A} (map_optionT f)"} {"_id": "503605", "text": "proof (prove)\nusing this:\n {i. f 0 \\ i \\ i < f n \\ hamlet (Rep_run r i c)} =\n f ` {i. 0 \\ i \\ i < n \\ hamlet (Rep_run r (f i) c)}\n f 0 = 0\n\ngoal (1 subgoal):\n 1. {i. i < f n \\ hamlet (Rep_run r i c)} =\n f ` {i. i < n \\ hamlet (Rep_run r (f i) c)}"} {"_id": "503606", "text": "proof (prove)\nusing this:\n is_minimal_basis (set (a # xs @ ys))\n a \\ set (xs @ ys)\n\ngoal (1 subgoal):\n 1. is_minimal_basis (set (trd (xs @ ys) a # xs @ ys))"} {"_id": "503607", "text": "proof (state)\nthis:\n (Q, W, \\d) \\ frp_invar T1 T2 \\\n (Q, W, \\d) \\ frp_cond\n\ngoal (1 subgoal):\n 1. \\s a b c.\n \\(a, b, c) \\ frp_invar T1 T2 \\\n (a, b, c) \\ frp_cond;\n s = (a, b, c)\\\n \\ c =\n reduce_rules\n (\\_prod (ta_rules T1) (ta_rules T2))\n (f_accessible\n (\\_prod (ta_rules T1) (ta_rules T2))\n (ta_initial T1 \\ ta_initial T2))"} {"_id": "503608", "text": "proof (state)\nthis:\n graph_homomorphism (LG (edges (fst R)) (vertices (fst R))) (chain_sup S) f\n\ngoal (1 subgoal):\n 1. consequence_graph Rs (chain_sup S)"} {"_id": "503609", "text": "proof (prove)\nusing this:\n compile_fundef \\ (\\ g) fd1 = Ok fd2\n rel_option (\\fd1 fd2. compile_fundef \\ (\\ f) fd1 = Ok fd2)\n (map_of xs f) (map_of ys' f)\n\ngoal (1 subgoal):\n 1. rel_option (\\fd1 fd2. compile_fundef \\ (\\ f) fd1 = Ok fd2)\n (map_of ((g, fd1) # xs) f) (map_of ((g, fd2) # ys') f)"} {"_id": "503610", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Pow (Set t) = fold_keys (\\x A. A \\ insert x ` A) t {{}}"} {"_id": "503611", "text": "proof (prove)\nusing this:\n (\\n. g n * (a * z) ^ n) \\ 0\n 1 - z \\ \\\\<^sub>\\\\<^sub>0\n 1 + z \\ \\\\<^sub>\\\\<^sub>0\n\ngoal (1 subgoal):\n 1. isCont\n (\\z.\n a *\n ((\\n. g n * (a * z) ^ n) *\n (\\n. diffs f n * (a * z) ^ n) -\n (\\n. f n * (a * z) ^ n) *\n (\\n. diffs g n * (a * z) ^ n)) /\n (\\n. g n * (a * z) ^ n)\\<^sup>2 +\n Polygamma 1 (1 + z) +\n Polygamma 1 (1 - z))\n z"} {"_id": "503612", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (c *\\<^sub>R a \\ c *\\<^sub>R b) =\n ((0 < c \\ a \\ b) \\\n (c < 0 \\ b \\ a))"} {"_id": "503613", "text": "proof (prove)\nusing this:\n \\0 < ?e2; ?t2 \\ S \\ V\\\n \\ 1 - q (NN ?e2) ?t2 < ?e2\n \\0 < ?e2; ?t2 \\ S - U\\\n \\ q (NN ?e2) ?t2 < ?e2\n\ngoal (1 subgoal):\n 1. \\e.\n 0 < e \\\n \\f\\R.\n f ` S \\ {0..1} \\\n (\\t\\S \\ V. f t < e) \\\n (\\t\\S - U. 1 - e < f t)"} {"_id": "503614", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F.unit a \\\\<^sub>D \\\\<^sub>D[F.map\\<^sub>0 a] =\n (F \\\\<^sub>C[a] \\\\<^sub>D \\ (a, a)) \\\\<^sub>D\n (F.unit a \\\\<^sub>D F.unit a)"} {"_id": "503615", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bij_betw Suc {..n'} {Suc 0..n}"} {"_id": "503616", "text": "proof (prove)\nusing this:\n f \\ g \\ g \\ f\\<^sup>\\ \\ g \\ g\n\ngoal (1 subgoal):\n 1. h + f \\ g \\ g \\\n f\\<^sup>\\ \\ h \\ g"} {"_id": "503617", "text": "proof (prove)\nusing this:\n subterms\\<^sub>s\\<^sub>e\\<^sub>t\n (ik\\<^sub>s\\<^sub>t (unlabel \\) \\\n assignment_rhs\\<^sub>s\\<^sub>t (unlabel \\))\n \\ subterms\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (ik\\<^sub>s\\<^sub>t ` unlabel ` \\

) \\\n \\ (assignment_rhs\\<^sub>s\\<^sub>t ` unlabel ` \\

))\n wf\\<^sub>l\\<^sub>s\\<^sub>t\\<^sub>s \\

\\\n tfr\\<^sub>s\\<^sub>e\\<^sub>t\n (\\ (trms\\<^sub>l\\<^sub>s\\<^sub>t ` \\

)) \\\n wf\\<^sub>t\\<^sub>r\\<^sub>m\\<^sub>s\n (\\ (trms\\<^sub>l\\<^sub>s\\<^sub>t ` \\

)) \\\n (\\\\\\\\

.\n list_all tfr\\<^sub>s\\<^sub>t\\<^sub>p (unlabel \\)) \\\n (\\f T K M \\.\n Fun f T \\\\<^sub>s\\<^sub>e\\<^sub>t\n \\ (ik\\<^sub>s\\<^sub>t ` unlabel ` \\

) \\\n \\\n (assignment_rhs\\<^sub>s\\<^sub>t ` unlabel ` \\

) \\\n Ana (Fun f T) = (K, M) \\\n Ana (Fun f T \\ \\) =\n (K \\\\<^sub>l\\<^sub>i\\<^sub>s\\<^sub>t \\,\n M \\\\<^sub>l\\<^sub>i\\<^sub>s\\<^sub>t \\))\n \\ \\ \\

\n ?A \\ ?B \\ SMP ?A \\ SMP ?B\n\ngoal (1 subgoal):\n 1. \\f T K M \\.\n Fun f T \\\\<^sub>s\\<^sub>e\\<^sub>t\n ik\\<^sub>s\\<^sub>t (unlabel \\) \\\n assignment_rhs\\<^sub>s\\<^sub>t (unlabel \\) \\\n Ana (Fun f T) = (K, M) \\\n Ana (Fun f T \\ \\) =\n (K \\\\<^sub>l\\<^sub>i\\<^sub>s\\<^sub>t \\,\n M \\\\<^sub>l\\<^sub>i\\<^sub>s\\<^sub>t \\)"} {"_id": "503618", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\, fst \\)\n \\ oreachable A (otherwith S {i} (orecvmsg R)) (other U {i}) \\\n snd \\ \\ reachable qmsg (recvmsg (R \\)) \\\n (\\m\\set (fst (snd \\)). R \\ m)"} {"_id": "503619", "text": "proof (prove)\nusing this:\n f \\ Ar\n g \\ Ar\n\ngoal (1 subgoal):\n 1. set_func\n (Hom(A,_) \\<^bsub>\\\\<^esub> g \\\n Hom(A,_) \\<^bsub>\\\\<^esub> f) =\n compose (Hom A (Dom f)) (restrict ((\\) g) (Hom A (Dom g)))\n (restrict ((\\) f) (Hom A (Dom f)))"} {"_id": "503620", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\s.\n \\s \\ br'_invar \\; s \\ br'_cond\\\n \\ br'_\\ s \\ br_cond\n 2. \\s s'.\n \\s \\ br'_invar \\; s \\ br'_cond;\n (s, s') \\ br'_step \\\\\n \\ (br'_\\ s, br'_\\ s')\n \\ br_step \\\n 3. br'_\\ ` br'_initial \\ \\ {br_initial \\}\n 4. br'_\\ ` br'_invar \\ \\ br_invar \\\n 5. \\a aa b.\n (a, aa, b) \\ br'_invar \\ \\\n br'_\\ (a, aa, b) \\ br_cond \\\n (a, aa, b) \\ br'_cond"} {"_id": "503621", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ : f \\\\<^sub>B\n src\\<^sub>B\n f \\\\<^sub>B f' \\\\<^sub>B\n src\\<^sub>B f'\\"} {"_id": "503622", "text": "proof (prove)\nusing this:\n design_isomorphism \\' \\' \\ \\ (inv_into \\ \\)\n \\design_isomorphism ?\\ ?\\ ?\\' ?\\' ?bij_map;\n ?x \\ incidence_system.intersection_numbers ?\\\\\n \\ ?x \\ incidence_system.intersection_numbers ?\\'\n ?x \\ source.intersection_numbers \\\n ?x \\ target.intersection_numbers\n\ngoal (1 subgoal):\n 1. source.intersection_numbers = target.intersection_numbers"} {"_id": "503623", "text": "proof (state)\nthis:\n arr g \\\n arr f \\\n (f = Zero \\ g \\ One \\ f \\ Zero \\ g = One)\n\ngoal (2 subgoals):\n 1. seq g f \\\n arr g \\\n arr f \\\n (f = Zero \\ g \\ One \\ f \\ Zero \\ g = One)\n 2. arr g \\\n arr f \\\n (f = Zero \\ g \\ One \\\n f \\ Zero \\ g = One) \\\n seq g f"} {"_id": "503624", "text": "proof (prove)\nusing this:\n finite V\n\ngoal (1 subgoal):\n 1. wf (inv_image\n (less_than_bool <*lex*>\n greater_bounded (max_dist src + 1) <*lex*> finite_psubset)\n (\\(f, PRED, C, N, d). (\\ f, d, C)))"} {"_id": "503625", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\P pre rely guar mid Q post.\n \\\\ P sat [pre, rely, guar, mid];\n \\ P sat [pre, rely, guar, mid];\n \\ Q sat [mid, rely, guar, post];\n \\ Q sat [mid, rely, guar, post]\\\n \\ \\ Seq P Q sat [pre, rely, guar, post]\n 2. \\pre rely P1 b guar post P2.\n \\stable pre rely;\n \\ P1 sat [pre \\ b, rely, guar, post];\n \\ P1 sat [pre \\ b, rely, guar, post];\n \\ P2 sat [pre \\ - b, rely, guar, post];\n \\ P2 sat [pre \\ - b, rely, guar, post];\n \\s. (s, s) \\ guar\\\n \\ \\ Cond b P1\n P2 sat [pre, rely, guar, post]\n 3. \\pre rely b post P guar.\n \\stable pre rely; pre \\ - b \\ post;\n stable post rely;\n \\ P sat [pre \\ b, rely, guar, pre];\n \\ P sat [pre \\ b, rely, guar, pre];\n \\s. (s, s) \\ guar\\\n \\ \\ While b P sat [pre, rely, guar, post]\n 4. \\pre rely post P b guar.\n \\stable pre rely; stable post rely;\n \\V.\n \\ P sat [pre \\ b \\\n {V}, {(x, y).\n x =\n y}, UNIV, {s.\n (V, s) \\ guar} \\\n post] \\\n \\ P sat [pre \\ b \\\n {V}, {(x, y).\n x =\n y}, UNIV, {s.\n (V, s) \\ guar} \\\n post]\\\n \\ \\ Await b P sat [pre, rely, guar, post]\n 5. \\pre pre' rely rely' guar' guar post' post P.\n \\pre \\ pre'; rely \\ rely';\n guar' \\ guar; post' \\ post;\n \\ P sat [pre', rely', guar', post'];\n \\ P sat [pre', rely', guar', post']\\\n \\ \\ P sat [pre, rely, guar, post]"} {"_id": "503626", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x.\n 2 * f (g1 (2 * x)) *\n vector_derivative g1 (at (2 * x))) has_integral\n i1)\n {0..1 / 2} &&&\n ((\\x.\n 2 * f (g2 (2 * x - 1)) *\n vector_derivative g2 (at (2 * x - 1))) has_integral\n i2)\n {1 / 2..1}"} {"_id": "503627", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ Abs_atMost ` {..class ?P Object = \\(?D, ?fs, ?ms)\\;\n ?Mm = map_option (\\m. (m, Object)) \\ map_of ?ms\\\n \\ ?P \\ Object sees_methods ?Mm\n is_class P Object\n\ngoal (1 subgoal):\n 1. (\\Mm.\n P \\ Object sees_methods Mm \\\n thesis) \\\n thesis"} {"_id": "503629", "text": "proof (chain)\npicking this:\n has_products (Collect J.ide)\n has_products (Collect J.arr)"} {"_id": "503630", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y. of_hypreal (x / y) = of_hypreal x / of_hypreal y"} {"_id": "503631", "text": "proof (prove)\nusing this:\n n = Actual_out (m, x)\n targetnode a = m\n valid_edge a\n V \\ set (ParamDefs (targetnode a))\n\ngoal (1 subgoal):\n 1. V \\ Def (parent_node n)"} {"_id": "503632", "text": "proof (prove)\ngoal (1 subgoal):\n 1. transform_syn 1 (let y be App (Var f) g in let x be e in Var x ) =\n let y be Lam [z]. App (App (Var f) g)\n z in let x be Lam [z]. App e z in Var x "} {"_id": "503633", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (P \\ true) = P"} {"_id": "503634", "text": "proof (prove)\nusing this:\n is_quantum_predicate (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1))\n is_quantum_predicate (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1))\n Utrans_P vars1 ?A = Utrans (tensor_P ?A (1\\<^sub>m K))\n is_quantum_predicate ?P \\\n \\\\<^sub>p {?P} SKIP {?P}\n is_quantum_predicate ?P \\\n \\\\<^sub>p {adjoint ?U * ?P * ?U} Utrans ?U {?P}\n \\is_quantum_predicate ?P; is_quantum_predicate ?Q;\n is_quantum_predicate ?R; \\\\<^sub>p {?P} ?S1.0 {?Q};\n \\\\<^sub>p {?Q} ?S2.0 {?R}\\\n \\ \\\\<^sub>p {?P} ?S1.0;; ?S2.0 {?R}\n \\\\k. k < ?n \\ is_quantum_predicate (?P k);\n is_quantum_predicate ?Q;\n \\k.\n k < ?n \\\n \\\\<^sub>p {?P k} ?S ! k {?Q}\\\n \\ \\\\<^sub>p {matrix_sum d\n (\\k. adjoint (?M k) * ?P k * ?M k) ?n}\n Measure ?n ?M ?S {?Q}\n \\is_quantum_predicate ?P; is_quantum_predicate ?Q;\n \\\\<^sub>p {?Q} ?S\n {adjoint (?M 0) * ?P * ?M 0 +\n adjoint (?M 1) * ?Q * ?M 1}\\\n \\ \\\\<^sub>p {adjoint (?M 0) * ?P * ?M 0 +\n adjoint (?M 1) * ?Q * ?M 1}\n While ?M ?S {?P}\n \\is_quantum_predicate ?P; is_quantum_predicate ?Q;\n is_quantum_predicate ?P'; is_quantum_predicate ?Q'; ?P \\\\<^sub>L ?P';\n \\\\<^sub>p {?P'} ?S {?Q'}; ?Q' \\\\<^sub>L ?Q\\\n \\ \\\\<^sub>p {?P} ?S {?Q}\n\ngoal (1 subgoal):\n 1. \\\\<^sub>p {adjoint (tensor_P mat_Ph (1\\<^sub>m K)) *\n (adjoint (exexH_k (n - 1)) * Q2 *\n exexH_k (n - 1)) *\n tensor_P mat_Ph (1\\<^sub>m K)}\n Utrans_P vars1 mat_Ph\n {adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)}"} {"_id": "503635", "text": "proof (prove)\ngoal (1 subgoal):\n 1. of_int_poly rg * of_int_poly sh =\n smult (inverse (r * s)) (of_int_poly p)"} {"_id": "503636", "text": "proof (state)\nthis:\n \n\ngoal (1 subgoal):\n 1. list_all (\\m. oiface (fst (snd m)) = ifaceAny)\n (map (apfst of_nat) (annotate_rlen (lr_of_tran_fbs rt fw ifs)))"} {"_id": "503637", "text": "proof (prove)\nusing this:\n irreducible (\\\\<^sub># A)\n \\\\<^sub># A dvd \\\\<^sub># B * \\\\<^sub># C\n\ngoal (1 subgoal):\n 1. \\\\<^sub># A dvd \\\\<^sub># B \\\n \\\\<^sub># A dvd \\\\<^sub># C"} {"_id": "503638", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A B OS P Q \\ C B OS P Q"} {"_id": "503639", "text": "proof (prove)\nusing this:\n \\arev b \\ b; b \\ arev a; b \\ a\\\n \\ perm_restrict arev (arcs G - {a, arev a}) (arev b) = b\n\ngoal (1 subgoal):\n 1. perm_restrict arev (arcs G - {a, arev a})\n (perm_restrict arev (arcs G - {a, arev a}) b) =\n b"} {"_id": "503640", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ A B C CongA A' B' C'"} {"_id": "503641", "text": "proof (prove)\nusing this:\n \\finite (Set.insert (priority t) (set (bts_dfs priority rs)));\n Set.insert (priority t) (set (bts_dfs priority rs)) \\ {};\n finite (set (bt_dfs priority r));\n set (bt_dfs priority r) \\ {}\\\n \\ Min (Set.insert (priority t)\n (set (bts_dfs priority rs)) \\\n set (bt_dfs priority r)) =\n ord_class.min\n (Min (Set.insert (priority t)\n (set (bts_dfs priority rs))))\n (Min (set (bt_dfs priority r)))\n Min (set (bt_dfs priority t)) =\n Min (Set.insert (priority t) (set (bts_dfs priority rs)) \\\n set (bt_dfs priority r))\n\ngoal (1 subgoal):\n 1. Min (set (bt_dfs priority t)) =\n ord_class.min\n (Min (Set.insert (priority t) (set (bts_dfs priority rs))))\n (Min (set (bt_dfs priority r)))"} {"_id": "503642", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ ++\\<^bsub>domA\n \\\\<^esub> \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>\\\\<^sub>1\\<^esub>)|\\<^sup>\\\\<^bsub>r\\<^esub> =\n (\\|\\<^sup>\\\\<^bsub>r\\<^esub> ++\\<^bsub>domA\n \\\\<^esub> \\<^bold>\\\\ \\ \\<^bold>\\\\<^bsub>\\\\<^sub>2\\<^esub>)|\\<^sup>\\\\<^bsub>r\\<^esub>"} {"_id": "503643", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hom_induced p (subtopology X S) {} X T id\n \\ Group.iso (homology_group p (subtopology X S))\n (relative_homology_group p X T) &&&\n hom_induced p (subtopology X T) {} X S id\n \\ Group.iso (homology_group p (subtopology X T))\n (relative_homology_group p X S)"} {"_id": "503644", "text": "proof (prove)\nusing this:\n \\ supports \\\n\ngoal (1 subgoal):\n 1. \\ supports \\ \\\\<^sub>s \\ &&&\n (\\ supports \\ \\\n \\ supports \\ \\\\<^sub>s \\)"} {"_id": "503645", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coprime = algebraic_semidom_class.coprime"} {"_id": "503646", "text": "proof (prove)\nusing this:\n Inf_from (Liminf_llist Ns - Red_F (Liminf_llist Ns))\n \\ Inf_from (Liminf_llist Ns - Sup_llist (lmap Red_F Ns))\n Inf_from (Liminf_llist Ns - Sup_llist (lmap Red_F Ns))\n \\ Sup_llist (lmap Red_I Ns)\n\ngoal (1 subgoal):\n 1. Inf_from (Liminf_llist Ns - Red_F (Liminf_llist Ns))\n \\ Sup_llist (lmap Red_I Ns)"} {"_id": "503647", "text": "proof (state)\nthis:\n (\\x. min (max (- h x) (f ?k x)) (h x)) integrable_on S\n\ngoal (3 subgoals):\n 1. h integrable_on S\n 2. \\k x.\n x \\ S \\\n norm (min (max (- h x) (f k x)) (h x)) \\ h x\n 3. \\x.\n x \\ S \\\n (\\k. min (max (- h x) (f k x)) (h x))\n \\ g x"} {"_id": "503648", "text": "proof (prove)\ngoal (1 subgoal):\n 1. subst x (F x) (subst y (F y) t) = subst y (F y) (subst x (F x) t)"} {"_id": "503649", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\x it \\ x' it' \\'.\n \\it \\ S; \\ ` it \\ \\ ` S;\n \\ it \\; \\' (\\ ` it) \\';\n \\x\\S - it. \\y\\it. RR x y;\n \\x\\\\ ` S - \\ ` it.\n \\xa\\it. RR' x (\\ xa);\n x' = \\ x; it' = \\ ` it;\n ((it, \\), \\ ` it, \\') \\ R; x \\ it;\n \\y\\it - {x}. RR x y;\n \\xa\\\\ ` it - {\\ x}.\n RR' (\\ x) xa\\\n \\ f x \\\n \\ \\\n {(\\, \\').\n ((it - {x}, \\),\n \\ ` it - {\\ x}, \\')\n \\ R}\n (f' (\\ x) \\')\n 2. \\\\ \\'.\n \\\\ {} \\; \\' {} \\';\n (({}, \\), {}, \\') \\ R\\\n \\ (\\, \\') \\ R'\n 3. \\it \\ it' \\'.\n \\it \\ S; it' \\ S'; \\ it \\;\n \\' it' \\';\n \\x\\S - it. \\y\\it. RR x y;\n \\x\\S' - it'. \\y\\it'. RR' x y;\n it' = \\ ` it; ((it, \\), it', \\') \\ R;\n it \\ {}; it' \\ {}; \\ True; \\ True\\\n \\ (\\, \\') \\ R'"} {"_id": "503650", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\spmf (R1 m1 m2 \\ D) True -\n spmf\n (funct m1 m2 \\\n (\\(out1, out2).\n S1 m1 out1 \\\n (\\sview.\n D sview \\\n (\\b'. return_spmf (\\ b')))))\n False\\ =\n \\spmf (R1 m1 m2 \\ D) True -\n spmf (funct m1 m2 \\ (\\(o1, o2). S1 m1 o1 \\ D))\n True\\"} {"_id": "503651", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pn.eq_m (pn.Mp C) (prod_list Gs) \\\n mset Fs = mset (map Mp Gs) \\\n (\\G.\n G \\ set Gs \\ monic G \\ pn.Mp G = G)"} {"_id": "503652", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x |\\| A \\ x \\<^bold>* F A = F A"} {"_id": "503653", "text": "proof (prove)\ngoal (1 subgoal):\n 1. emeasure lborel ((\\x. (fst x, - snd x)) -` A) =\n emeasure (distr lborel lborel (\\(x, y). (x, - y))) A"} {"_id": "503654", "text": "proof (state)\ngoal (2 subgoals):\n 1. \\\\<^sub>F i in sequentially.\n G x \\ Gamma_series x i * (1 + x / real i)\n 2. sequentially \\ bot"} {"_id": "503655", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fst (strip_comb (app t u)) = fst (strip_comb t)"} {"_id": "503656", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isodefl (cast\\t) t"} {"_id": "503657", "text": "proof (prove)\ngoal (1 subgoal):\n 1. G|-{P}. the (body pn) .{Q} \\\n insert ({P}. BODY pn .{Q}) G|-{P}. the (body pn) .{Q}"} {"_id": "503658", "text": "proof (prove)\ngoal (1 subgoal):\n 1. currentLevel (prefixToLevel_aux M l i) \\ l - i"} {"_id": "503659", "text": "proof (prove)\nusing this:\n (\\n''\\n.\n \\ t n'' \\\n (\\n'\\n. n' < n'' \\ \\' t n')) \\\n (\\n'\\n. \\' t n')\n ?\\' \\\\<^sub>c ?\\ \\\n ?\\' \\\\<^sub>c ?\\ \\\\<^sup>c \\\\<^sub>c?\\'\n\ngoal (1 subgoal):\n 1. (\\' \\\\<^sub>c \\) t n"} {"_id": "503660", "text": "proof (state)\nthis:\n graph H h = graph H' h'\n\ngoal (1 subgoal):\n 1. graph H h = graph H' h' \\ False"} {"_id": "503661", "text": "proof (prove)\nusing this:\n dgrad_set_le d (pp_of_term ` Keys F) (pp_of_term ` Keys G)\n dgrad_set_le d (pp_of_term ` Keys G) (pp_of_term ` Keys H)\n\ngoal (1 subgoal):\n 1. dgrad_set_le d (pp_of_term ` Keys F) (pp_of_term ` Keys H)"} {"_id": "503662", "text": "proof (prove)\ngoal (1 subgoal):\n 1. k \\ 0 \\\n (\\x. (f' \\

k) x = f x) =\n (\\i.\n (f i = NoMsg \\\n (\\j\n (f i \\ NoMsg \\\n (\\n\n (\\j f' (i * k + j) = NoMsg))))"} {"_id": "503663", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\a. Pre FIG.Out (Sim a) \\ Pre DFIG.Out a\n 2. \\a b.\n \\Pre FIG.Out (Sim a); DFIG.Out (a, b)\\\n \\ FIG.Out (Sim a, Sim b)"} {"_id": "503664", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. x \\ {(s, a). stk_strict_subs s}"} {"_id": "503665", "text": "proof (prove)\ngoal (1 subgoal):\n 1. let s' = \\_update (pivot_tableau x\\<^sub>i x\\<^sub>j (\\ s)) s\n in \\ s' = \\ s \\\n \\\\<^sub>i s' = \\\\<^sub>i s \\\n \\ s' = \\ s \\ \\\\<^sub>c s' = \\\\<^sub>c s"} {"_id": "503666", "text": "proof (prove)\nusing this:\n y \\\\<^sub>\\ \\\\<^sub>\\ vreal_uminus\n\ngoal (1 subgoal):\n 1. (\\x.\n \\x \\\\<^sub>\\ \\\\<^sub>\\ vreal_uminus;\n y = -\\<^sub>\\ x\\\n \\ thesis) \\\n thesis"} {"_id": "503667", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pathsCard\n (paths\n (substitute (FBinds \\\\ae) (thunks \\)\n (substitute (FBinds \\\\(Aheap \\ e\\a))\n (thunks \\) (Texp e\\a) \\\\\n Fstack as S))) f|`\n (- domA \\) =\n pathsCard\n (paths\n (substitute (FBinds \\\\ae) (thunks \\)\n (ttree_restr (- domA \\)\n (substitute (FBinds \\\\(Aheap \\ e\\a))\n (thunks \\) (Texp e\\a)) \\\\\n Fstack as S)))"} {"_id": "503668", "text": "proof (prove)\nusing this:\n \\N \\ M; M - M \\# N = {#}\\\n \\ \\y. count M y < count N y\n\ngoal (1 subgoal):\n 1. multeqp P N M = (multp P N M \\ N = M)"} {"_id": "503669", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. value (t x))\n \\ restrict_space B\n (t -`\n trees_cyl\n (map_tree (\\_. space M)\n \\ls, u, rs\\)) \\\\<^sub>M\n M"} {"_id": "503670", "text": "proof (prove)\ngoal (1 subgoal):\n 1. countable A \\ countable (trees A)"} {"_id": "503671", "text": "proof (prove)\ngoal (1 subgoal):\n 1. complex_of_real ((norm A)\\<^sup>2 * \\) =\n complex_of_real ((norm (A*) * sqrt \\)\\<^sup>2)"} {"_id": "503672", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dnf_to_bool \\ (dnf_and d1 d2) =\n (dnf_to_bool \\ d1 \\ dnf_to_bool \\ d2)"} {"_id": "503673", "text": "proof (prove)\nusing this:\n f : a' \\\\<^bsub>\\\\<^esub> b'\n a = \\\\Dom\\\\f\\\n b = \\\\Cod\\\\f\\\n\ngoal (1 subgoal):\n 1. g \\\\<^sub>\\ \\\\ArrMap\\ `\\<^sub>\\\n Hom \\ a b"} {"_id": "503674", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x esf xa esfa.\n \\envObsC esf = envObsC esfa;\n x \\ set (listToFuns (snd envInitBits) agents);\n esf \\ set (fst envInitBits);\n s =\n \\es = esf, ps = x,\n pubActs = (default, \\_. default)\\;\n xa \\ set (listToFuns (snd envInitBits) agents);\n esfa \\ set (fst envInitBits);\n s' =\n \\es = esfa, ps = xa,\n pubActs = (default, \\_. default)\\\\\n \\ \\es = esf, ps = x(a := xa a),\n pubActs = (default, \\_. default)\\\n \\ (\\esf.\n \\es = esf, ps = x(a := xa a),\npubActs = (default, \\_. default)\\) `\n set (fst envInitBits)\n 2. \\x esf xa esfa.\n \\envObsC esf = envObsC esfa;\n x \\ set (listToFuns (snd envInitBits) agents);\n esf \\ set (fst envInitBits);\n s =\n \\es = esf, ps = x,\n pubActs = (default, \\_. default)\\;\n xa \\ set (listToFuns (snd envInitBits) agents);\n esfa \\ set (fst envInitBits);\n s' =\n \\es = esfa, ps = xa,\n pubActs = (default, \\_. default)\\\\\n \\ x(a := xa a)\n \\ set (listToFuns (snd envInitBits) agents)"} {"_id": "503675", "text": "proof (prove)\nusing this:\n Ide t \\\n Arr t \\\n Dom t = t \\\n Cod t = t \\\n ide \\t\\ \\\n ide \\\\<^bold>\\t\\<^bold>\\\\ \\\n \\t\\<^bold>\\\\\n \\ hom \\t\\\n \\\\<^bold>\\t\\<^bold>\\\\\n Arr ?t \\\n \\<^bold>\\Src ?t\\<^bold>\\ =\n Src \\<^bold>\\?t\\<^bold>\\\n Arr ?t \\ \\<^bold>\\Src ?t\\<^bold>\\ = Src ?t\n obj ?a \\ \\[?a] = \\[?a]\n obj ?a \\ \\[?a] = \\[?a]\n C.obj ?a \\ obj (E ?a)\n\ngoal (1 subgoal):\n 1. \\x1 x1a x1b.\n \\C.obj x1; x1b = x1;\n \\<^bold>\\Src t\\<^bold>\\ =\n \\<^bold>\\x1a\\<^bold>\\\\<^sub>0;\n ide (E x1) \\\n \\\\t\\<^bold>\\\\ : \\t\\ \\ E\n x1\\;\n \\a. obj a \\ \\[a] = \\[a];\n \\a. obj a \\ \\[a] = \\[a];\n \\f. ide f \\ \\ f = \\[f];\n \\f. ide f \\ \\ f = \\[f];\n \\t.\n Arr t \\\n Arr \\<^bold>\\t\\<^bold>\\ \\\n Src \\<^bold>\\t\\<^bold>\\ = Src t \\\n Trg \\<^bold>\\t\\<^bold>\\ = Trg t;\n \\t.\n Arr t \\\n Dom \\<^bold>\\t\\<^bold>\\ =\n \\<^bold>\\Dom t\\<^bold>\\ \\\n Cod \\<^bold>\\t\\<^bold>\\ =\n \\<^bold>\\Cod t\\<^bold>\\;\n \\<^bold>\\t\\<^bold>\\ =\n \\<^bold>\\x1\\<^bold>\\\\<^sub>0;\n Ide t \\\n Arr t \\\n Dom t = t \\ Cod t = t \\ ide \\t\\;\n \\t. is_Prim\\<^sub>0 (Src t)\\\n \\ \\ (E x1a) = \\[E x1]"} {"_id": "503676", "text": "proof (prove)\nusing this:\n indexed_eval_aux (Qs @ [P]) i m \\ carrier R\n count m i = length Qs + k\n indexed_eval_aux (Qs @ [P]) i m \\ \\\n\ngoal (1 subgoal):\n 1. (if i \\# m + {#i#}\n then indexed_eval_aux (Qs @ [P]) i (m + {#i#} - {#i#}) else \\)\n \\ carrier R - {\\}"} {"_id": "503677", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cxt_to_stmt E c = Skip \\ c = Skip \\ E = []"} {"_id": "503678", "text": "proof (prove)\nusing this:\n prefix_of \\' \\ \\ i < progress \\'\n prefix_of \\ \\\n \\prefix_of ?\\ ?\\; prefix_of ?\\' ?\\\\\n \\ ?\\ \\ ?\\' \\ ?\\' \\ ?\\\n \\ i < progress \\\n ?\\ \\ ?\\' \\\n progress ?\\ \\ progress ?\\'\n\ngoal (1 subgoal):\n 1. \\ \\ \\'"} {"_id": "503679", "text": "proof (prove)\ngoal (1 subgoal):\n 1. t \\ Zero \\\n pnPlus (Plus r s) t = pnPlus r (pnPlus s t)"} {"_id": "503680", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f \\ \\ \\ kstar f \\ \\"} {"_id": "503681", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\k. f k ^ lookup x k) = (\\k\\keys x. f k ^ lookup x k)"} {"_id": "503682", "text": "proof (prove)\nusing this:\n F \\\\<^sub>\\ (\\ \\\\<^sub>D\\<^sub>G\\<^sub>3 \\ \\\\<^sub>D\\<^sub>G\\<^sub>3 \\)\\Arr\\\n\ngoal (1 subgoal):\n 1. (\\f f' f''.\n \\F = [f, f', f'']\\<^sub>\\;\n f \\\\<^sub>\\ \\\\Arr\\;\n f' \\\\<^sub>\\ \\\\Arr\\;\n f'' \\\\<^sub>\\ \\\\Arr\\\\\n \\ thesis) \\\n thesis"} {"_id": "503683", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ y \\ oLog b x \\ oLog b y"} {"_id": "503684", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P \\ h \\; h a = \\(C, fs)\\;\n P,h \\ (C, fs') \\\\\n \\ P \\ h(a \\ (C, fs')) \\"} {"_id": "503685", "text": "proof (prove)\nusing this:\n finite gen\n\ngoal (1 subgoal):\n 1. finite ((*\\<^sub>v) B ` gen)"} {"_id": "503686", "text": "proof (prove)\nusing this:\n prv (dsj (eql t zer) (exi x' (eql t (suc (Var x')))))\n prv (imp (exi x' (eql t (suc (Var x')))) (exi x (eql t (suc (Var x)))))\n\ngoal (1 subgoal):\n 1. prv (dsj (eql t zer) (exi x (eql t (suc (Var x)))))"} {"_id": "503687", "text": "proof (prove)\nusing this:\n 1 < length l\n\ngoal (1 subgoal):\n 1. hd (tl l) = l ! 1"} {"_id": "503688", "text": "proof (prove)\nusing this:\n \\C.arr f\n local.map ?f \\\n if \\C.arr ?f then \\\\C.rep ?f\\ else D.null\n Arr ?t \\\n \\\\<^bold>\\?t\\<^bold>\\\\ =\n \\?t\\\n \\C.arr ?f \\\n \\C.rep (\\C.dom ?f) =\n \\<^bold>\\\\C.DOM ?f\\<^bold>\\\n Arr ?t \\ Ide (Dom ?t)\n\ngoal (1 subgoal):\n 1. D.dom (local.map f) = local.map (\\C.dom f)"} {"_id": "503689", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((A ===> A ===> A) ===> A ===> (A ===> A ===> A) ===> A ===> (=))\n (comm_semiring_1_ow (Collect (Domainp A))) class.comm_semiring_1"} {"_id": "503690", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (P x y \\ Q x y)\\(x, y)\\v\\ =\n (P x y\\(x, y)\\v\\ \\\n Q x y\\(x, y)\\v\\)"} {"_id": "503691", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((((((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n set_iterator (foldri (gen_wf_\\n gen_cfg_wf))\n (nodes (gen_wf.\\ gen_cfg_wf)))) \\\n ((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (\\g v.\n True \\\n set_iterator (foldri (gen_wf_inEdges' gen_cfg_wf v))\n (pred (gen_wf.\\ gen_cfg_wf) v))) \\\n (\\g.\n gen_wf_Entry gen_cfg_wf\n \\ set (gen_wf_\\n gen_cfg_wf)) \\\n (\\g.\n True \\\n graph_path_base.inEdges (\\_. gen_wf_inEdges' gen_cfg_wf)\n g (gen_wf_Entry gen_cfg_wf) =\n []) \\\n (\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n True \\\n (\\ns.\n graph_path_base.path2\n (\\_. gen_wf_\\n gen_cfg_wf) (\\_. True)\n (\\_. gen_wf_inEdges' gen_cfg_wf) g\n (gen_wf_Entry gen_cfg_wf) ns n))) \\\n ((\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n gen_wf_defs gen_cfg_wf n \\ gen_wf_uses gen_cfg_wf n =\n {}) \\\n (\\g n. finite (gen_wf_defs gen_cfg_wf n))) \\\n (\\v g n.\n v \\ gen_wf_uses gen_cfg_wf n \\\n n \\ set (gen_wf_\\n gen_cfg_wf)) \\\n (\\g n. finite (gen_wf_uses gen_cfg_wf n)) \\\n (\\g. True)) \\\n (\\g.\n \\m\\set (gen_wf_\\n gen_cfg_wf).\n \\v\\gen_wf_uses gen_cfg_wf m.\n CFG_base.defAss' (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. True) (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf_Entry gen_cfg_wf)\n (\\_. gen_wf_defs gen_cfg_wf) g m v)) \\\n (((((((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n set_iterator (foldri (gen_wf_\\n gen_cfg_wf))\n (nodes (gen_wf.\\ gen_cfg_wf)))) \\\n ((\\g.\n True \\\n finite (nodes (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n finite (edges (gen_wf.\\ gen_cfg_wf))) \\\n (\\g.\n True \\\n valid_graph (gen_wf.\\ gen_cfg_wf))) \\\n (\\g v.\n True \\\n set_iterator (foldri (gen_wf_inEdges' gen_cfg_wf v))\n (pred (gen_wf.\\ gen_cfg_wf) v))) \\\n (\\g.\n gen_wf_Entry gen_cfg_wf\n \\ set (gen_wf_\\n gen_cfg_wf)) \\\n (\\g.\n True \\\n graph_path_base.inEdges (\\_. gen_wf_inEdges' gen_cfg_wf)\n g (gen_wf_Entry gen_cfg_wf) =\n []) \\\n (\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n True \\\n (\\ns.\n graph_path_base.path2\n (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. True) (\\_. gen_wf_inEdges' gen_cfg_wf)\n g (gen_wf_Entry gen_cfg_wf) ns n))) \\\n ((\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n gen_wf_defs' gen_cfg_wf n \\ gen_wf.uses' gen_cfg_wf n =\n {}) \\\n (\\g n. finite (gen_wf_defs' gen_cfg_wf n))) \\\n (\\v g n.\n v \\ gen_wf.uses' gen_cfg_wf n \\\n n \\ set (gen_wf_\\n gen_cfg_wf)) \\\n (\\g n. finite (gen_wf.uses' gen_cfg_wf n)) \\\n (\\g. True)) \\\n ((\\g. finite (dom (gen_wf.phis' gen_cfg_wf))) \\\n (\\g n v vs.\n gen_wf.phis' gen_cfg_wf (n, v) = Some vs \\\n n \\ set (gen_wf_\\n gen_cfg_wf))) \\\n (\\g n v args.\n gen_wf.phis' gen_cfg_wf (n, v) = Some args \\\n length\n (graph_path_base.predecessors\n (\\_. gen_wf_inEdges' gen_cfg_wf) g n) =\n length args) \\\n (\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n gen_wf_defs' gen_cfg_wf n \\\n CFG_SSA_base.phiDefs (\\_. gen_wf.phis' gen_cfg_wf) g n =\n {}) \\\n (\\n g m.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n m \\ set (gen_wf_\\n gen_cfg_wf) \\\n n \\ m \\\n CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g n \\\n CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g m =\n {})) \\\n (\\v g n.\n v \\ CFG_SSA_base.allUses\n (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf.uses' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g\n n \\\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n CFG_SSA_base.defAss (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. True) (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf_Entry gen_cfg_wf)\n (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g n v) \\\n (\\g v.\n gen_wf.phis' gen_cfg_wf (gen_wf_Entry gen_cfg_wf, v) =\n None)) \\\n ((\\g n.\n gen_wf_defs gen_cfg_wf n =\n gen_wf_var gen_cfg_wf ` gen_wf_defs' gen_cfg_wf n) \\\n (\\n g.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n gen_wf_uses gen_cfg_wf n =\n gen_wf_var gen_cfg_wf ` gen_wf.uses' gen_cfg_wf n)) \\\n (\\g n ns m v x v'.\n graph_path_base.path2 (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. True) (\\_. gen_wf_inEdges' gen_cfg_wf) g n ns\n m \\\n n \\ set (tl ns) \\\n v \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g n \\\n v \\ CFG_SSA_base.allUses\n (\\_. gen_wf_\\n gen_cfg_wf)\n (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf.uses' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g m \\\n x \\ set (tl ns) \\\n v' \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g\n x \\\n gen_wf_var gen_cfg_wf v' \\ gen_wf_var gen_cfg_wf v) \\\n (\\g n v vs v'.\n gen_wf.phis' gen_cfg_wf (n, v) = Some vs \\\n v' \\ set vs \\\n gen_wf_var gen_cfg_wf v' = gen_wf_var gen_cfg_wf v) \\\n (\\n g v v'.\n n \\ set (gen_wf_\\n gen_cfg_wf) \\\n v \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g n \\\n v' \\ CFG_SSA_base.allDefs (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf) g\n n \\\n v \\ v' \\\n gen_wf_var gen_cfg_wf v' \\ gen_wf_var gen_cfg_wf v)"} {"_id": "503692", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\c x.\n complex_of_real ((c *\\<^sub>R x) $\\<^sub>e 0) +\n \\ * complex_of_real ((c *\\<^sub>R x) $\\<^sub>e Suc 0) =\n c *\\<^sub>R\n (complex_of_real (x $\\<^sub>e 0) +\n \\ * complex_of_real (x $\\<^sub>e Suc 0))"} {"_id": "503693", "text": "proof (prove)\nusing this:\n a \\ {0..1}\n\ngoal (1 subgoal):\n 1. x_ \\ {0..1 - a} -\n ({1 - a} \\ (\\x. x - a) ` S) \\\n f (shiftpath a g x_) * vector_derivative (shiftpath a g) (at x_) =\n f (g (a + x_)) * vector_derivative g (at (a + x_))"} {"_id": "503694", "text": "proof (prove)\ngoal (1 subgoal):\n 1. openin (final_topology X Y f) =\n (\\U.\n U \\ X \\\n (\\i. openin (Y i) (f i -` U \\ topspace (Y i))))"} {"_id": "503695", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cyclic_group (\\ \\)"} {"_id": "503696", "text": "proof (prove)\nusing this:\n equiv ?A ?r \\ equiv (?A \\ ?B) (Restr ?r ?B)\n equiv non_zero_vectors proportionality\n\ngoal (1 subgoal):\n 1. equiv (non_zero_vectors \\ Collect invertible)\n (Restr proportionality (Collect invertible))"} {"_id": "503697", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((map (\\x. x \\ id RRR) [1, 2, 3, 4, 6] =\n [False, False, True, False, True] &&&\n map (\\x. x \\ - RRR) [1, 2, 3, 4, 6] =\n [True, True, False, True, False] &&&\n map (\\x. x \\ closure RRR) [1, 2, 3, 4, 6] =\n [True, True, True, False, True]) &&&\n (map (\\x. x \\ (closure \\ uminus) RRR)\n [1, 2, 3, 4, 6] =\n [True, True, True, True, True] &&&\n map (\\x. x \\ (uminus \\ closure) RRR)\n [1, 2, 3, 4, 6] =\n [False, False, False, True, False]) &&&\n map (\\x. x \\ (uminus \\ closure \\ uminus) RRR)\n [1, 2, 3, 4, 6] =\n [False, False, False, False, False] &&&\n map (\\x. x \\ (closure \\ uminus \\ closure) RRR)\n [1, 2, 3, 4, 6] =\n [False, True, True, True, False]) &&&\n (map (\\x.\n x \\ (uminus \\ closure \\ uminus \\ closure)\n RRR)\n [1, 2, 3, 4, 6] =\n [True, False, False, False, True] &&&\n map (\\x.\n x \\ (closure \\ uminus \\ closure \\ uminus)\n RRR)\n [1, 2, 3, 4, 6] =\n [True, True, False, False, False] &&&\n map (\\x.\n x \\ (uminus \\ closure \\ uminus \\\n closure \\\n uminus)\n RRR)\n [1, 2, 3, 4, 6] =\n [False, False, True, True, True]) &&&\n (map (\\x.\n x \\ (closure \\ uminus \\ closure \\\n uminus \\\n closure)\n RRR)\n [1, 2, 3, 4, 6] =\n [True, True, False, False, True] &&&\n map (\\x.\n x \\ (uminus \\ closure \\ uminus \\\n closure \\\n uminus \\\n closure)\n RRR)\n [1, 2, 3, 4, 6] =\n [False, False, True, True, False]) &&&\n map (\\x.\n x \\ (closure \\ uminus \\ closure \\\n uminus \\\n closure \\\n uminus)\n RRR)\n [1, 2, 3, 4, 6] =\n [False, True, True, True, True] &&&\n map (\\x.\n x \\ (uminus \\ closure \\ uminus \\\n closure \\\n uminus \\\n closure \\\n uminus)\n RRR)\n [1, 2, 3, 4, 6] =\n [True, False, False, False, False]"} {"_id": "503698", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bijective (inv hom)"} {"_id": "503699", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\finite \\; open S;\n \\T. T \\ \\ \\ \\a b. T = cbox a b;\n \\T.\n T \\ \\ \\ S \\ interior T = {}\\\n \\ S \\ interior (\\ \\) = {}"} {"_id": "503700", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly.coeff Q' k = (if n < k then 0 else (- 1) ^ (n - k) * e' (n - k))"} {"_id": "503701", "text": "proof (prove)\ngoal (1 subgoal):\n 1. n dvd j"} {"_id": "503702", "text": "proof (prove)\ngoal (1 subgoal):\n 1. fix\\ValK_copy_rec = (ID, ID)"} {"_id": "503703", "text": "proof (prove)\nusing this:\n i < length (map (\\i. S (Suc i)) [0..i. S (Suc i)) [0..i. S (Suc i)) [0.. R"} {"_id": "503704", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x Y.\n \\cont g; chain Y; chain (\\i. Y i `;` g x)\\\n \\ ((\\i. Y i) `;` g x) \\\n (\\i. Y i `;` g x)"} {"_id": "503705", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.Sup_lattice smap sleq"} {"_id": "503706", "text": "proof (prove)\nusing this:\n \\\\' \\\\<^sub>S\n S.UP\n f : (S.UP f \\\\<^sub>S S.UP g) \\\\<^sub>S\n S.UP\n f \\\\<^sub>S S.trg\n (S.UP f) \\\\<^sub>S\n S.UP f\\\n \\S.\\' (S.UP f) (S.UP g)\n (S.UP\n f) : S.UP f \\\\<^sub>S\n S.UP g \\\\<^sub>S\n S.UP\n f \\\\<^sub>S (S.UP f \\\\<^sub>S\n S.UP g) \\\\<^sub>S\n S.UP f\\\n\ngoal (1 subgoal):\n 1. (\\' \\\\<^sub>S S.UP f) \\\\<^sub>S\n S.\\' (S.UP f) (S.UP g) (S.UP f) =\n \\' \\\\<^sub>S S.UP f"} {"_id": "503707", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a_ \\ U; b_ \\ U\\\n \\ (b_ <\\<^sub>o\\<^sub>w a_) =\n (a_ =\n (if a_ \\\\<^sub>o\\<^sub>w b_ then b_\n else a_) \\\n a_ \\ b_)"} {"_id": "503708", "text": "proof (prove)\nusing this:\n ?A \\ carrier_mat ?n ?n \\\n \\as.\n char_poly ?A = (\\a\\as. [:- a, 1:]) \\\n length as = ?n\n A \\ carrier_mat n n\n\ngoal (1 subgoal):\n 1. (\\es.\n char_poly A = (\\e\\es. [:- e, 1:]) \\\n thesis) \\\n thesis"} {"_id": "503709", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\evs3 T1 Kas' Ta.\n \\A \\ bad; evs3 \\ kerbIV;\n Says A Kas \\Agent A, Agent Tgs, Number T1\\\n \\ set evs3;\n Says Kas' A\n (Crypt (shrK A)\n \\Key authK, Agent Tgs, Number Ta, authTicket\\)\n \\ set evs3;\n valid Ta wrt T1; Key authK \\ analz (knows Spy evs3);\n authTicket \\ parts (knows Spy evs3); T2 = CT evs3;\n Says A Tgs\n \\authTicket,\n Crypt authK \\Agent A, Number (CT evs3)\\,\n Agent B\\\n \\ set evs3;\n \\B X.\n Says A Tgs\n \\X,\n Crypt authK \\Agent A, Number (CT evs3)\\,\n Agent B\\\n \\ set evs3 \\\n Says A B\n \\X,\n Crypt authK\n \\Agent A, Number (CT evs3)\\\\\n \\ set evs3\\\n \\ False"} {"_id": "503710", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {s'. (s, s') \\ E} \\ S"} {"_id": "503711", "text": "proof (prove)\nusing this:\n infer ?\\ P = Some ?\\ \\\n ?\\ \\ P : ?\\\n infer \\ (Snd P) = Some \\\n\ngoal (1 subgoal):\n 1. \\ \\ Snd P : \\"} {"_id": "503712", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ys clist.\n \\xs \\ []; length clist = length xs;\n (xs, s) #\n ys \\ map (\\i. (fst i, s) # snd i) (zip xs clist);\n \\i cptn\\\n \\ (xs, s) # ys \\ par_cptn"} {"_id": "503713", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rell_FGf L1' L2' x y"} {"_id": "503714", "text": "proof (prove)\nusing this:\n p \\ carrier P\n ?p \\ carrier P \\ coeff P ?p ?n \\ carrier R\n ?p \\ carrier P \\\n \\\\<^bsub>P\\<^esub> ?p \\ carrier P\n\ngoal (1 subgoal):\n 1. coeff P (\\\\<^bsub>P\\<^esub> p) n =\n \\ coeff P p n \\\n (coeff P p n \\ coeff P (\\\\<^bsub>P\\<^esub> p) n)"} {"_id": "503715", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eval_fps (G1 * G2) (s - 1) = g s"} {"_id": "503716", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lookup (update_by_fun k f xs) k' =\n (if k = k' then f else id) (lookup xs k')"} {"_id": "503717", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (z \\ 0 \\ arg z = pi / 2) = (is_imag z \\ 0 < Im z)"} {"_id": "503718", "text": "proof (state)\nthis:\n 0 \\ ?x \\ ?x * A ?x - T ?x \\ {0..c1 * ?x}\n\ngoal (4 subgoals):\n 1. \\c\\0. \\x\\0. x * A x - T x \\ {0..c * x}\n 2. (\\x. A x - ln x) \\ O(\\_. 1)\n 3. \\c>0. \\x\\1 / c. c * x \\ S x\n 4. S \\ \\(\\x. x)"} {"_id": "503719", "text": "proof (prove)\ngoal (1 subgoal):\n 1. delete_index n (x # xs) =\n (if n = 0 then xs else x # delete_index (n - 1) xs)"} {"_id": "503720", "text": "proof (prove)\ngoal (1 subgoal):\n 1. divisor_count n = divisor_sigma 0 n"} {"_id": "503721", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P \\ \\C\\\\<^sub>sM(map Val vs),\n (h\\<^sub>1, l\\<^sub>1, sh\\<^sub>1),\n False\\ \\\n \\INIT D ([D],False) \\ C\\\\<^sub>sM(map\n Val vs),\n (h\\<^sub>1, l\\<^sub>1, sh\\<^sub>1),False\\"} {"_id": "503722", "text": "proof (prove)\nusing this:\n S \\ {0..card Y}\n \\?A \\ ?B; finite ?B\\ \\ finite ?A\n\ngoal (1 subgoal):\n 1. finite S"} {"_id": "503723", "text": "proof (prove)\ngoal (2 subgoals):\n 1. rs, \\\\<^sub>1 \\\\<^sub>v Sconst name \\ ?v\\<^sub>1\n 2. \\\\<^sub>v ?v\\<^sub>1 \\ val\\<^sub>2"} {"_id": "503724", "text": "proof (prove)\nusing this:\n a \\ b \\ c =\n \\<^bold>g [^] n \\ \\<^bold>g [^] k \\ \\<^bold>g [^] j\n\ngoal (1 subgoal):\n 1. \\<^bold>g [^] n \\ \\<^bold>g [^] k \\ \\<^bold>g [^] j =\n \\<^bold>g [^] (n + (k + j))"} {"_id": "503725", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.iso (Left_a.lunit a') \\\n \\Left_a.lunit\n a' : a \\ a' \\ a'\\"} {"_id": "503726", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Ball (Set_Monad xs) P = list_all P xs &&&\n Ball (DList_set dxs) P' =\n (case ID CEQ('b) of\n None \\\n Code.abort STR ''Ball DList_set: ceq = None''\n (\\_. Ball (DList_set dxs) P')\n | Some x \\ DList_Set.dlist_all P' dxs) &&&\n Ball (RBT_set rbt) P'' =\n (case ID ccompare of\n None \\\n Code.abort STR ''Ball RBT_set: ccompare = None''\n (\\_. Ball (RBT_set rbt) P'')\n | Some x \\ RBT_Set2.all P'' rbt)"} {"_id": "503727", "text": "proof (prove)\nusing this:\n x \\ {x. A *v x = b}\n A *v p = b\n\ngoal (1 subgoal):\n 1. A *v x - A *v p = 0"} {"_id": "503728", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\factor\\\\<^sup>2 *\n \\local.g\\\\<^sup>2) ^\n j'\n \\ (\\factor\\\\<^sup>2 *\n (2 ^ j' * \\factor\\\\<^sup>2)) ^\n j'"} {"_id": "503729", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ \\\n \\ \\\n {S \\ \\.\n \\U. U \\ \\ \\ S \\ U}"} {"_id": "503730", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\wf_graph \\nodes = N, edges = E\\;\n E' \\ E\\\n \\ num_reachable_norefl\n \\nodes = N, edges = E'\\ v\n \\ num_reachable_norefl\n \\nodes = N, edges = E\\ v"} {"_id": "503731", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prob_space (exponential (escape_rate s))"} {"_id": "503732", "text": "proof (prove)\ngoal (1 subgoal):\n 1. natural_transformation C'.VV.comp (\\\\<^sub>D)\n H\\<^sub>D\\<^sub>'oFF.map FoH\\<^sub>C\\<^sub>'.map cmp"} {"_id": "503733", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inf x_ y_ \\ x_"} {"_id": "503734", "text": "proof (prove)\nusing this:\n fv (\\, scrut, Alts e\\<^sub>1 e\\<^sub>2 # S)\n \\ set L \\ domA \\\n fv (\\, scrut)\n \\ set L \\ domA \\ \\\n fv (\\, Bool b) \\ set L \\ domA \\\n domA \\ \\ domA \\\n\ngoal (1 subgoal):\n 1. fv (\\, if b then e\\<^sub>1 else e\\<^sub>2, S)\n \\ set L \\ domA \\"} {"_id": "503735", "text": "proof (prove)\nusing this:\n prob_space N \\\n (\\asset\\stocks Mkt.\n Filtration.filtration N F \\\n (\\t.\n integrable N (discounted_value r (prices Mkt asset) t)) \\\n borel_adapt_stoch_proc F\n (discounted_value r (prices Mkt asset)) \\\n (\\t s.\n t \\ s \\\n AEeq N\n (real_cond_exp N (F t) (discounted_value r (prices Mkt asset) s))\n (discounted_value r (prices Mkt asset) t)))\n trading_strategy pf\n space M = space N \\ events \\ sets N\n support_set pf \\ stocks Mkt\n \\n.\n \\asset\\support_set pf.\n integrable N\n (\\w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)\n\ngoal (1 subgoal):\n 1. \\n. sigma_finite_subalgebra N (F n)"} {"_id": "503736", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (sign l\\<^sub>1 \\ sign l\\<^sub>2 \\\n get_pred l\\<^sub>1 = get_pred l\\<^sub>2 \\\n get_terms l\\<^sub>1 = get_terms l\\<^sub>2) =\n (l\\<^sub>2 = l\\<^sub>1\\<^sup>c)"} {"_id": "503737", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i = 0.. (\\i = 0.."} {"_id": "503739", "text": "proof (prove)\nusing this:\n ?y \\ set_spmf p \\ spmf (f ?y) x \\ r\n 0 \\ r\n emeasure (measure_spmf ?p) (space (measure_spmf ?p)) \\ 1\n\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. ennreal r \\measure_spmf p\n \\ ennreal r"} {"_id": "503740", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l = liminf (\\n. ereal (f n / f (Suc n) - 1)) + 1"} {"_id": "503741", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.iso f"} {"_id": "503742", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f ` carrier G \\ kernel H K g = {\\\\<^bsub>H\\<^esub>} &&&\n f ` carrier G <#>\\<^bsub>H\\<^esub> kernel H K g = carrier H"} {"_id": "503743", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mutex x y \\ gval (gAnd y x) s \\ true"} {"_id": "503744", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (out \\ outs'_rpv rpv) =\n (\\input. out \\ outs'_gpv (rpv input))"} {"_id": "503745", "text": "proof (prove)\nusing this:\n a \\ carrier R\n a \\ \\\n f = monom P a n\n g \\ carrier P\n a \\ g (deg R g) [^] n \\ \\\n\ngoal (1 subgoal):\n 1. to_polynomial R a (deg R (to_polynomial R a)) \\\n (g [^]\\<^bsub>P\\<^esub> n) (deg R (g [^]\\<^bsub>P\\<^esub> n)) \\\n \\"} {"_id": "503746", "text": "proof (prove)\nusing this:\n prv (neg (neg (PP \\\\G\\)))\n \\?\\ \\ fmla; Fvars ?\\ = {};\n prv (neg (neg (PP \\?\\\\)))\\\n \\ prv (PP \\?\\\\)\n\ngoal (1 subgoal):\n 1. prv (PP \\\\G\\)"} {"_id": "503747", "text": "proof (prove)\nusing this:\n \\ \\ s\n min_lvar_not_in_bounds s = Some x\\<^sub>i\n \\\\<^sub>l\\<^sub>b\n (lt (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative))\n (\\\\ s\\ x\\<^sub>i)\n (LB (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n s x\\<^sub>i)\n\ngoal (2 subgoals):\n 1. \\I.\n \\min_rvar_incdec\n (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n s x\\<^sub>i =\n Inl I;\n check'\n (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n x\\<^sub>i s =\n set_unsat I s\\\n \\ P (check (set_unsat I s))\n 2. \\x\\<^sub>j l\\<^sub>i.\n \\min_rvar_incdec\n (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n s x\\<^sub>i =\n Inr x\\<^sub>j;\n l\\<^sub>i =\n the (LB (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n s x\\<^sub>i);\n check'\n (if \\\\ s\\ x\\<^sub>i <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i\n then Positive else Negative)\n x\\<^sub>i s =\n pivot_and_update x\\<^sub>i x\\<^sub>j l\\<^sub>i s\\\n \\ P (check\n (pivot_and_update x\\<^sub>i x\\<^sub>j l\\<^sub>i\n s))"} {"_id": "503748", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Bernstein_changes_01 p P =\n changes (coeffs (reciprocal_poly p P \\\\<^sub>p [:1, 1:]))"} {"_id": "503749", "text": "proof (chain)\npicking this:\n \\ u \\\\<^sub>\\ ?u';\n ?u' \\ RED \\\\\n \\ App t ?u' \\ RED \\\n u' \\ RED \\"} {"_id": "503750", "text": "proof (state)\nthis:\n 0 = (b1 - B1) * x1 + (b2 - B2) * y1 + c - C\n\ngoal (1 subgoal):\n 1. (x, y) = (x1, y1) \\ (x, y) = (x2, y2)"} {"_id": "503751", "text": "proof (prove)\nusing this:\n - (1::'a) \\ (0::'a) \\\n poly_roots (Polynomial.smult (- (1::'a)) p) = poly_roots p\n\ngoal (1 subgoal):\n 1. poly_roots (- p) = poly_roots p"} {"_id": "503752", "text": "proof (prove)\ngoal (1 subgoal):\n 1. EVAL (E.Src (Dom a)) =\n EVAL \\<^bold>\\Map a\\<^bold>\\\\<^sub>0"} {"_id": "503753", "text": "proof (prove)\nusing this:\n f \\ A \\\\<^sub>E B\n (\\x. f' (inv p\\<^sub>A x)) \\ A \\\\<^sub>E B\n finite B\n \\f \\ A \\\\<^sub>E B;\n (\\x. f' (inv p\\<^sub>A x)) \\ A \\\\<^sub>E B;\n finite B\\\n \\ \\p.\n p permutes B \\\n (\\x\\A. f x = p (f' (inv p\\<^sub>A x)))\n\ngoal (1 subgoal):\n 1. (\\p\\<^sub>B.\n \\p\\<^sub>B permutes B;\n \\x\\A. f x = p\\<^sub>B (f' (inv p\\<^sub>A x))\\\n \\ thesis) \\\n thesis"} {"_id": "503754", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. f (g x) i) \\ L \\\\<^sub>M M i"} {"_id": "503755", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x ^ o * \\ \\ (1::'a) = x ^ o"} {"_id": "503756", "text": "proof (state)\nthis:\n 0 < r\n\ngoal (1 subgoal):\n 1. \\r.\n 0 < r \\\n \\k.\n \\n\\k. \\inverse (X n) - inverse (Y n)\\ < r"} {"_id": "503757", "text": "proof (prove)\nusing this:\n A.ide a\n A.ide ?x \\\n transformation_by_components.map (\\\\<^sub>A) (\\\\<^sub>A)\n FG.map \\o ?x =\n F (G ?x)\n \\ \\\n transformation_by_components.map (\\\\<^sub>A) (\\\\<^sub>A)\n FG.map \\o\n\ngoal (1 subgoal):\n 1. \\ a = FG.map a"} {"_id": "503758", "text": "proof (prove)\ngoal (1 subgoal):\n 1. param_opt i t e = None \\\n \\y. lowest_tops [i, t, e] = Some y"} {"_id": "503759", "text": "proof (prove)\nusing this:\n a = SK_EV_SIGNAL x31_ x32_\n\ngoal (1 subgoal):\n 1. atomic_step_invariant\n (case a of\n SK_IPC dir stage partner page \\\n atomic_step_ipc (current s) dir stage partner page s\n | SK_EV_WAIT EV_FINISH EV_CONSUME_ALL \\\n atomic_step_ev_wait_all (current s) s\n | SK_EV_WAIT EV_FINISH EV_CONSUME_ONE \\\n atomic_step_ev_wait_one (current s) s\n | SK_EV_WAIT _ consume \\ s\n | SK_EV_SIGNAL EV_SIGNAL_PREP partner \\ s\n | SK_EV_SIGNAL EV_SIGNAL_FINISH partner \\\n atomic_step_ev_signal (current s) partner s\n | NONE \\ s)"} {"_id": "503760", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\\\ = map_tree f t) = (t = \\\\)"} {"_id": "503761", "text": "proof (prove)\ngoal (1 subgoal):\n 1. for_rec3 f a n n n n =\n nfoldli [0..x. True)\n (\\k.\n nfoldli [0..x. True)\n (\\i.\n nfoldli [0..x. True)\n (\\j a. f a k i j)))\n a"} {"_id": "503762", "text": "proof (chain)\npicking this:\n x = \\x1, x2\\"} {"_id": "503763", "text": "proof (prove)\ngoal (1 subgoal):\n 1. isolate_variable_sparse p x (i + 1) \\ 0 \\\n MPoly_Type.degree (f i) x = i"} {"_id": "503764", "text": "proof (prove)\ngoal (1 subgoal):\n 1. move =\n filter (\\(t, X). left I \\ nt - t) (linearize data_prev)"} {"_id": "503765", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\type_wf h; document_ptr |\\| document_ptr_kinds h\\\n \\ h \\ ok get_disconnected_nodes document_ptr"} {"_id": "503766", "text": "proof (state)\nthis:\n N' = fmdrop C N\n\ngoal (1 subgoal):\n 1. ran_m (fmupd C C' N) =\n add_mset C' (remove1_mset (the (fmlookup N C)) (ran_m N))"} {"_id": "503767", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Group.group (G \\\\ H)"} {"_id": "503768", "text": "proof (state)\ngoal (1 subgoal):\n 1. similar M1 M3"} {"_id": "503769", "text": "proof (prove)\ngoal (1 subgoal):\n 1. r \\ field_char_0.Rats (/) (1::'a) (+) (0::'a) (-) \\\n \\n.\n r =\n field_char_0.of_rat (/) (1::'a) (+) (0::'a) (-) (nat_to_rat_surj n)"} {"_id": "503770", "text": "proof (prove)\ngoal (1 subgoal):\n 1. arity (generalize_fundef fd2) = arity fd2"} {"_id": "503771", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\h2.\n \\h2 \\ F; hn_invalid (list_assn P) [] [] = true;\n p = []; p' = []\\\n \\ hn_refine F f1' \\' R f1\n 2. \\h2 a list aa lista.\n \\h2 \\ F;\n hn_invalid (list_assn P) (a # list) (aa # lista) = true;\n p = a # list; p' = aa # lista\\\n \\ hn_refine\n (hn_ctxt P a aa *\n hn_ctxt (list_assn P) list lista *\n F)\n (f2' aa lista)\n (hn_ctxt P' a aa *\n hn_ctxt (list_assn P') list lista *\n \\')\n R (f2 a list)"} {"_id": "503772", "text": "proof (prove)\ngoal (1 subgoal):\n 1. oracle_flag key = eval_oracle key ()"} {"_id": "503773", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P (sum_case f g a) =\n (\\ ((\\x.\n a = ZFC_Cardinals.Inl x \\ \\ P (f x)) \\\n (\\y.\n a = ZFC_Cardinals.Inr y \\ \\ P (g y)) \\\n \\ is_sum a \\ \\ P undefined))"} {"_id": "503774", "text": "proof (prove)\nusing this:\n Suc 0 < f (Suc 0)\n mono f\n\ngoal (1 subgoal):\n 1. \\i. Suc 0 < f (Suc i)"} {"_id": "503775", "text": "proof (prove)\nusing this:\n h \\ attach_shadow_root element_ptr shadow_root_mode\n \\\\<^sub>r new_shadow_root_ptr\n h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>h h2\n \\?h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>h ?h';\n ?h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>r ?new_shadow_root_ptr\\\n \\ object_ptr_kinds ?h' =\n object_ptr_kinds ?h |\\|\n {|cast\\<^sub>s\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>_\\<^sub>r\\<^sub>o\\<^sub>o\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n ?new_shadow_root_ptr|}\n h \\ new\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>R\\<^sub>o\\<^sub>o\\<^sub>t_M\n \\\\<^sub>r new_shadow_root_ptr\n h \\ attach_shadow_root element_ptr shadow_root_mode\n \\\\<^sub>r result\n\ngoal (1 subgoal):\n 1. object_ptr_kinds h2 =\n {|cast\\<^sub>s\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>_\\<^sub>r\\<^sub>o\\<^sub>o\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n result|} |\\|\n object_ptr_kinds h"} {"_id": "503776", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\c.\n singular_chain p\n (subtopology (powertop_real UNIV) (standard_simplex p))\n c \\\n chain_boundary p (chain_map p f c) =\n chain_map (p - Suc 0) f (chain_boundary p c)) \\\n singular_subdivision (p - Suc 0)\n (chain_map (p - Suc 0) f\n (chain_boundary p (frag_of (restrict id (standard_simplex p))))) =\n frag_extend\n (\\f.\n chain_map (p - Suc 0) f\n (simplicial_subdivision (p - Suc 0)\n (frag_of (restrict id (standard_simplex (p - Suc 0))))))\n (chain_boundary p (frag_of f))"} {"_id": "503777", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\countable A; inj f\\\n \\ countable {x. f x \\ A}"} {"_id": "503778", "text": "proof (state)\nthis:\n dim_row C = n\n\ngoal (1 subgoal):\n 1. jordan_nf A (triangular_to_jnf_vector A)"} {"_id": "503779", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n f s.\n \\ideal R A; H \\ carrier M; r \\ carrier R;\n H \\ {}; f \\ {j. j \\ n} \\ H;\n s \\ {j. j \\ n} \\ A;\n x =\n \\\\<^sub>e M (\\j. s j \\\\<^sub>s f j) n\\\n \\ \\na.\n \\fa\\{j. j \\ na} \\ H.\n \\sa\n \\{j. j \\ na} \\ A.\n r \\\\<^sub>s\n \\\\<^sub>e M (\\j.\n s j \\\\<^sub>s f j) n =\n \\\\<^sub>e M (\\j.\n sa j \\\\<^sub>s fa j) na"} {"_id": "503780", "text": "proof (prove)\nusing this:\n P \\ \\blocks\n (this # pns, Class D # Ts, Addr a # vs, body),\n s\\ \\\n \\e',s'\\\n fv body \\ {this} \\ set pns\n s = (h\\<^sub>2, l\\<^sub>2)\n s' = (h\\<^sub>3, l\\<^sub>3)\n length Ts = length pns\n length vs = length pns\n\ngoal (1 subgoal):\n 1. l\\<^sub>3 = l\\<^sub>2"} {"_id": "503781", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (i < DIM('a) \\\n isDERIV i fa (list_of_eucl x @ params)) &&&\n interpret_floatarith fa (list_of_eucl x @ params) \\ \\"} {"_id": "503782", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l \\ u \\\n (Abstract_Rigorous_Numerics.cfuncset l u X = {}) = (X = {})"} {"_id": "503783", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lex_ord_pair ko f (sort_oalist_aux ko xs) (sort_oalist_aux ko ys) =\n Some Eq"} {"_id": "503784", "text": "proof (prove)\ngoal (1 subgoal):\n 1. M = N"} {"_id": "503785", "text": "proof (prove)\nusing this:\n evaluate True ?env ?st ?e ?r = (?r = eval ?env ?e ?st)\n evaluate_list True ?env ?st ?es ?r' = (?r' = eval_list ?env ?es ?st)\n evaluate_match True ?env ?st ?v ?pes ?v' ?r =\n (?r = eval_match ?env ?v ?pes ?v' ?st)\n evaluate True ?env ?st ?e ?r = (?r = eval' ?env ?e ?st)\n evaluate_list True ?env ?st ?es ?r' = (?r' = eval_list' ?env ?es ?st)\n evaluate_match True ?env ?st ?v ?pes ?v' ?r =\n (?r = eval_match' ?env ?v ?pes ?v' ?st)\n\ngoal (1 subgoal):\n 1. \\x xa xb. eval' x xa xb = eval x xa xb"} {"_id": "503786", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\(term, i)\\a. insertion f term * f v ^ i) = 0"} {"_id": "503787", "text": "proof (prove)\ngoal (1 subgoal):\n 1. content (ball c r) = unit_ball_vol (real DIM('a)) * r ^ DIM('a)"} {"_id": "503788", "text": "proof (prove)\nusing this:\n At = LeqUni p\n \\x.\n aEvalUni (LeqUni p) x = aEvalUni (EqUni p) x \\\n aEvalUni (LessUni p) x\n aEvalUni (LessUni (?a, ?b, ?c)) ?x = (?a * ?x\\<^sup>2 + ?b * ?x + ?c < 0)\n aEvalUni (LeqUni (?a, ?b, ?c)) ?x =\n (?a * ?x\\<^sup>2 + ?b * ?x + ?c \\ 0)\n p = (a__, b__, c__)\n (?x \\ ?y) = (?x < ?y \\ ?x = ?y)\n\ngoal (1 subgoal):\n 1. (\\y'>(A + B * sqrt C) / D.\n \\x\\{(A + B * sqrt C) / D<..y'}. aEvalUni At x) =\n (\\y'>(A + B * sqrt C) / D.\n \\x\\{(A + B * sqrt C) / D<..y'}.\n aEvalUni (EqUni p) x \\ aEvalUni (LessUni p) x)"} {"_id": "503789", "text": "proof (prove)\nusing this:\n p \\# prime_factorization N\n x \\# prime_factorization (of_nat p)\n prime_factors (of_nat p) = {of_nat p}\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "503790", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M {..x} \\ Sigma_Algebra.measure M (space M)"} {"_id": "503791", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A * Ba + B * Aa < A * B + Aa * Ba"} {"_id": "503792", "text": "proof (prove)\nusing this:\n Per A' B' C' \\ A B C LtA A' B' C'\n Acute ?A ?B ?C \\ ?A \\ ?B \\ ?C \\ ?B\n \\Acute ?A ?B ?C; ?D \\ ?E; ?E \\ ?F;\n Per ?D ?E ?F\\\n \\ ?A ?B ?C LtA ?D ?E ?F\n Acute A B C\n \\ ?A ?B ?C LtA ?A ?B ?C\n\ngoal (1 subgoal):\n 1. \\ Per A B C"} {"_id": "503793", "text": "proof (state)\nthis:\n on_triple f `` E \\ edges (S (max i j))\n f `` V \\ vertices (S (max i j))\n\ngoal (1 subgoal):\n 1. \\i. graph_homomorphism (LG E V) (S i) f"} {"_id": "503794", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Cong A Q' X' P'"} {"_id": "503795", "text": "proof (prove)\ngoal (1 subgoal):\n 1. update_field_1 (\\_. y) (make_foo a b) = make_foo y b"} {"_id": "503796", "text": "proof (prove)\nusing this:\n t \\ t'\n \\ \\ t : T\n\ngoal (1 subgoal):\n 1. \\ \\ t' : T"} {"_id": "503797", "text": "proof (prove)\ngoal (1 subgoal):\n 1. interaction_bound consider (gpv_stop gpv) =\n interaction_bound consider gpv"} {"_id": "503798", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\x y xa.\n x < y \\\n uniformly_continuous_on (?s3 x y xa) (cts_step x y)\n 2. \\x y xa. x < y \\ xa \\ interior (?s3 x y xa)\n 3. \\x y xa.\n x < y \\ \\cts_step x y xa\\ \\ 1"} {"_id": "503799", "text": "proof (prove)\ngoal (1 subgoal):\n 1. redundant g = redundant_code g"} {"_id": "503800", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\arb' x'.\n \\rpre arb' x';\n (x', x_)\n \\ inv_image (finite_psupset local.reachable)\n fst\\\n \\ f x' \\ SPEC (rpost arb' x');\n rpre S x_;\n REC\\<^sub>T\n (\\dfs (V, v).\n if v = tgt then RETURN (V, True)\n else let V = insert v V\n in FOREACH\\<^sub>C (E `` {v}) (\\(V, brk). \\ brk)\n (\\v' (V, brk).\n if v' \\ V then dfs (V, v')\n else RETURN (V, False))\n (V, False)) =\n f;\n x_ = (V, v); v \\ tgt\\\n \\ FOREACH\\<^sub>C (E `` {v})\n (\\(V, brk). \\ brk)\n (\\v' (V, brk).\n if v' \\ V then f (V, v')\n else RETURN (V, False))\n (insert v V, False)\n \\ SPEC (rpost S x_)"} {"_id": "503801", "text": "proof (prove)\nusing this:\n \\rel (gen_cong R) ?x2 ?y2; R \\ ?R'2; local.cong ?R'2\\\n \\ ?R'2 (eval ?x2) (eval ?y2)\n\ngoal (1 subgoal):\n 1. local.cong (gen_cong R)"} {"_id": "503802", "text": "proof (prove)\nusing this:\n g \\ injectionsUniverse\n x \\ Domain g\n\ngoal (1 subgoal):\n 1. g ,, x \\ g `` {x}"} {"_id": "503803", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ts t = None \\ init_fin_descend_thr ts t = None"} {"_id": "503804", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ generat.\n integral\\<^sup>N\n (measure_spmf\n (map_spmf (bad \\ snd)\n (callee s (generat.output generat))))\n (indicator {True}) +\n ennreal k *\n ennreal_of_enat\n (if consider (generat.output generat) then n - 1\n else n) *\n integral\\<^sup>N\n (measure_spmf\n (map_spmf (bad \\ snd)\n (callee s (generat.output generat))))\n (indicator {False})\n \\restrict_space (measure_spmf (the_gpv gpv))\n {IO out c |out c. True} =\n \\\\<^sup>+ generat.\n ennreal\n (spmf\n (map_spmf (bad \\ snd)\n (callee s (generat.output generat)))\n True) +\n ennreal k *\n ennreal_of_enat\n (if consider (generat.output generat) then n - 1\n else n) *\n ennreal\n (spmf\n (map_spmf (bad \\ snd)\n (callee s (generat.output generat)))\n False)\n \\restrict_space (measure_spmf (the_gpv gpv))\n {IO out c |out c. True}"} {"_id": "503805", "text": "proof (prove)\nusing this:\n \\\\ \\_ \\\\\\<^sub>T\\<^sub>E\\<^sub>S\\<^sub>L =\n \\\\ remdups\n \\_ \\\\\\<^sub>T\\<^sub>E\\<^sub>S\\<^sub>L\n\ngoal (1 subgoal):\n 1. \\\\ a_ #\n \\_ \\\\\\<^sub>T\\<^sub>E\\<^sub>S\\<^sub>L =\n \\\\ remdups\n (a_ #\n \\_) \\\\\\<^sub>T\\<^sub>E\\<^sub>S\\<^sub>L"} {"_id": "503806", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\valuation K v;\n x \\ vp K v \\<^bsup>\\Vr K v n\\<^esup>; 0 < an n;\n 0 < n\\\n \\ (an n = 0 \\\n x \\ carrier (Vr K v)) \\\n (an n \\ 0 \\\n (an n = \\ \\\n x = \\\\<^bsub>Vr K v\\<^esub>) \\\n (an n \\ \\ \\\n x \\ vp K v\n \\<^bsup>\\Vr K\n v na (an n)\\<^esup>))\n 2. \\valuation K v;\n x \\ vp K v \\<^bsup>\\Vr K v n\\<^esup>; 0 \\ an n;\n an n \\ n_val K v x\\\n \\ an n \\ n_val K v x"} {"_id": "503807", "text": "proof (prove)\nusing this:\n uniformly_continuous_on (range to_metric_completion) I\n\ngoal (1 subgoal):\n 1. uniformly_continuous_on (range to_metric_completion)\n (\\x. f (I x))"} {"_id": "503808", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_reflects_binary_pred TRel (\\b. hasBarb b TWB)"} {"_id": "503809", "text": "proof (prove)\nusing this:\n v \\ t\n isPath v p2 t\n\ngoal (1 subgoal):\n 1. (\\w p2'.\n \\p2 = (v, w) # p2'; (v, w) \\ E; isPath w p2' t\\\n \\ thesis) \\\n thesis"} {"_id": "503810", "text": "proof (prove)\nusing this:\n x = s2 !! n\n\ngoal (1 subgoal):\n 1. sinterleave s1 s2 !! (2 * n + 1) = x"} {"_id": "503811", "text": "proof (prove)\nusing this:\n i \\ n\n\ngoal (1 subgoal):\n 1. X i -` {X i x} =\n (\\z\\comp_proj_i X n i (X i x). proj_stoch_proc X n -` {z})"} {"_id": "503812", "text": "proof (prove)\nusing this:\n expr_sem_rf \\ e' \\ space (stock_measure t')\n \\ \\ space (state_measure V' \\)\n\ngoal (1 subgoal):\n 1. dens_ctxt_measure \\ \\ \\\n (\\\\.\n expr_sem \\ e \\\n (\\x.\n return_val (op_sem oper <|x, expr_sem_rf \\ e'|>))) =\n dens_ctxt_measure \\ \\ \\\n (\\\\.\n expr_sem \\ e \\\n (\\x. return_val (op_sem oper <|x, expr_sem_rf \\ e'|>)))"} {"_id": "503813", "text": "proof (prove)\nusing this:\n x < y\n\ngoal (1 subgoal):\n 1. \\z>x. z < y"} {"_id": "503814", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\enn2ereal\n (emeasure (measure_spmf (map_spmf (\\x. (f x, bad x)) p))\n (A \\ {False}) +\n \\\\<^sup>+ x. indicator (A \\ {True})\n (f x, bad x)\n \\measure_spmf p) -\n enn2ereal\n (emeasure (measure_spmf (map_spmf (\\x. (f x, bad x)) q))\n (A \\ {False}) +\n \\\\<^sup>+ x. indicator (A \\ {True})\n (f x, bad x)\n \\measure_spmf q)\\ =\n \\enn2ereal\n (\\\\<^sup>+ x. indicator (A \\ {True})\n (f x, bad x)\n \\measure_spmf p) -\n enn2ereal\n (\\\\<^sup>+ x. indicator (A \\ {True})\n (f x, bad x)\n \\measure_spmf q)\\"} {"_id": "503815", "text": "proof (prove)\ngoal (1 subgoal):\n 1. OFCLASS('a fps, euclidean_ring_gcd_class)"} {"_id": "503816", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {SV_inv3} TS.trans sv_TS {> SV_inv3}"} {"_id": "503817", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ S \\ connected_component S x x"} {"_id": "503818", "text": "proof (prove)\ngoal (1 subgoal):\n 1. while\\<^sup>\\ false do P od = II"} {"_id": "503819", "text": "proof (prove)\nusing this:\n CARD('b) \\ CARD('a)\n\ngoal (1 subgoal):\n 1. @@x = x"} {"_id": "503820", "text": "proof (prove)\nusing this:\n i < Monitor.progress \\ (formula.Until \\1 I \\2)\n (plen \\)\n\ngoal (1 subgoal):\n 1. (\\b. right I = enat b \\ thesis) \\\n thesis"} {"_id": "503821", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\_trivial (\\\\<^sub>1 \\\\<^sub>\\ \\\\<^sub>2) =\n (\\_trivial \\\\<^sub>1 \\ \\_trivial \\\\<^sub>2)"} {"_id": "503822", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite\n (m' `\n (\\x\\\\\n (intersecting `\n V `\n \\ (intersecting ` V ` intersecting (V k))).\n M (V x)))"} {"_id": "503823", "text": "proof (prove)\nusing this:\n gb_schema_aux_dom (data, bs, ps)\n\ngoal (1 subgoal):\n 1. P data bs ps (gb_schema_aux data bs ps)"} {"_id": "503824", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bdd_below\n ((\\t. infdist (d t) {c A--c B}) -` {..D0} \\ {ty..B})"} {"_id": "503825", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bst t \\ bst (delete x t)"} {"_id": "503826", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.aligned\\<^sub>d (Suc e) (local.encode (Suc e) (n div base))"} {"_id": "503827", "text": "proof (prove)\nusing this:\n p \\ PrimRec1\n\ngoal (1 subgoal):\n 1. (\\x y. p x) \\ PrimRec2"} {"_id": "503828", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sin (z + of_real pi + of_real pi) = sin z"} {"_id": "503829", "text": "proof (prove)\nusing this:\n S.ide ?f \\ S.lunit' ?f = ?f\n S.ide ?f \\ S.runit ?f = ?f\n\ngoal (1 subgoal):\n 1. S.lunit' (S.UP f) \\\\<^sub>S S.runit (S.UP f) = S.UP f"} {"_id": "503830", "text": "proof (prove)\nusing this:\n \\ 0 \\ x\n\ngoal (1 subgoal):\n 1. \\ln (1 + x) - x\\ \\ 2 * x\\<^sup>2"} {"_id": "503831", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nondetM return bind merge cempty csingle cUn"} {"_id": "503832", "text": "proof (state)\nthis:\n t0 \\ rt\n rt \\ t1\n flow0 x rt \\ P\n \\flow0 x ?t' \\ P; t0 \\ ?t'; ?t' \\ t1\\\n \\ rt \\ ?t'\n\ngoal (1 subgoal):\n 1. \\!t.\n 0 < t \\\n t \\ existence_ivl0 x \\\n flow0 x t \\ P \\\n (\\s\\{0<.. P)"} {"_id": "503833", "text": "proof (prove)\nusing this:\n Gateway req dt a stop lose d ack i vc\n\ngoal (1 subgoal):\n 1. GatewayReq req dt a stop lose d ack i vc"} {"_id": "503834", "text": "proof (prove)\nusing this:\n s' = fx0.tortoise_hare f x arbitrary\n properties f x ?lambda ?mu \\\n \\\\s. True\\ fx0.tortoise_hare f x\n \\\\s.\n nu s \\ {0<..?lambda + ?mu} \\\n l s = ?lambda \\ m s = ?mu\\\n\ngoal (1 subgoal):\n 1. fx0.properties f x (l s') (m s')"} {"_id": "503835", "text": "proof (prove)\ngoal (1 subgoal):\n 1. real_of_int q * pa *\n (\\j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1)))\n \\ real_of_int q * pa * 2 * real_of_int (b (n + 1)) /\n real_of_int (a (n + 1))"} {"_id": "503836", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. - g x \\ - |R\\ (\\y. - f y) x) =\n (\\x. |R\\ (\\y. - f y) x \\ g x)"} {"_id": "503837", "text": "proof (prove)\ngoal (1 subgoal):\n 1. diamond_y_gen \\ x_coord piecewise_C1_differentiable_on {0..1}"} {"_id": "503838", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ (x, e1) # \\ \\\\ \\\n \\ (x, e2) # \\ \\\\"} {"_id": "503839", "text": "proof (prove)\nusing this:\n finite I\n I \\ {}\n i \\ I\n \\\\i. i \\ I \\ 0 < l i;\n \\i.\n i \\ I \\\n distributed M lborel (X i)\n (\\x. ennreal (exponential_density (l i) x));\n indep_vars (\\i. borel) X I\\\n \\ distributed M lborel (\\x. MIN i\\I. X i x)\n (\\x. ennreal (exponential_density (sum l I) x))\n ?i12 \\ insert i I \\ 0 < l ?i12\n ?i12 \\ insert i I \\\n distributed M lborel (X ?i12)\n (\\x. ennreal (exponential_density (l ?i12) x))\n indep_vars (\\i. borel) X (insert i I)\n\ngoal (1 subgoal):\n 1. distributed M lborel (\\x. min (X i x) (MIN i\\I. X i x))\n (\\x. ennreal (exponential_density (l i + sum l I) x))"} {"_id": "503840", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n. (\\i = 1..n. 1 / (real i)\\<^sup>2) - T (real n))\n \\ \\(\\n.\n - S n -\n EM_remainder (2 * Suc N + 1)\n (\\x.\n - fact (2 * Suc N + 2) / x ^ (2 * Suc N + 3))\n (int n))"} {"_id": "503841", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\y\\set (Abs_State.dom S2) - set (Abs_State.dom S1).\n m_ivl (fun S2 y)) =\n 0"} {"_id": "503842", "text": "proof (prove)\ngoal (1 subgoal):\n 1. impl_of_RBT_HM hm_empty_const = HashMap_Impl.empty ()"} {"_id": "503843", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa.\n \\K1.enabled x1 x; K2.enabled x2 xa\\\n \\ indicat_real\n {\\\n \\ space (stream_space (count_space UNIV)).\n P1 (smap fst \\) \\\n P2 (smap snd \\)}\n (szip\\<^sub>E x1 x2 (x, xa)) =\n indicat_real\n {\\\n \\ space (stream_space (count_space UNIV)).\n P1 \\}\n x *\n indicat_real\n {\\\n \\ space (stream_space (count_space UNIV)).\n P2 \\}\n xa"} {"_id": "503844", "text": "proof (prove)\nusing this:\n w \\ (0::'w)\n S_reduced_for w ss\n S_reduced_for w ts\n x \\ set ts\n S_reduced_for ?a ss \\ ss \\ lists S\n \\w \\ (0::'w); S_reduced_for w ss;\n S_reduced_for w ?ts\\\n \\ \\xss. flip_altsublist_chain (ss # xss @ [?ts])\n \\?x \\ lists S;\n flip_altsublist_chain (?x # ?zs @ [?y])\\\n \\ set ?y = set ?x\n\ngoal (1 subgoal):\n 1. x \\ set ss"} {"_id": "503845", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C.seq \\\\.prj\\<^sub>1\n (\\_\\\\.prj\\<^sub>0 \\\n \\\\_\\_\\\\.Prj\\<^sub>0\\<^sub>1 \\\\.leg0, \\\\.leg1\\ \\_\\_\\\\.Prj\\<^sub>0\\)"} {"_id": "503846", "text": "proof (prove)\ngoal (1 subgoal):\n 1. snd (Rep_segment x) = fst (Rep_segment y) \\\n {ya.\n fst (Rep_segment x) \\ ya \\\n ya \\ fst (Rep_segment y)} \\\n {ya. fst (Rep_segment y) \\ ya \\ ya \\ snd (Rep_segment y)} =\n {ya.\n fst (Rep_segment\n (Abs_segment (fst (Rep_segment x), snd (Rep_segment y))))\n \\ ya \\\n ya \\ snd (Rep_segment\n (Abs_segment\n (fst (Rep_segment x), snd (Rep_segment y))))}"} {"_id": "503847", "text": "proof (prove)\nusing this:\n ind_cca'.game \\ = PRF.game_0 (reduction_prf \\)\n game2 \\ = PRF.game_1 (reduction_prf \\)\n\ngoal (1 subgoal):\n 1. \\spmf (ind_cca'.game \\) True - spmf (game2 \\) True\\ =\n PRF.advantage (reduction_prf \\)"} {"_id": "503848", "text": "proof (prove)\nusing this:\n p = return_spmf (Inl (Some (a, m)), (), Store m)\n q = return_spmf (Store m, None, [m \\ a])\n length m = id' \\\n query \\ local.valid_insecQ <+> nlists UNIV (id' \\) <+> UNIV\n state \\ set_spmf q\n (answer, state')\n \\ set_spmf\n ((local.lazy_channel_insec \\\\<^sub>O\n lazy_channel_send \\\\<^sub>O local.lazy_channel_recv_f)\n state query)\n query = Inl adv_query\n adv_query = ForwardOrEdit foe\n foe = Some am'\n\ngoal (1 subgoal):\n 1. local.S'\n (cond_spmf_fst\n (p \\\n (\\s.\n (callee_auth_channel sim \\\\<^sub>O\n \\\\channel.send_oracle \\\\<^sub>O\n \\\\channel.recv_oracle)\n s query))\n answer)\n (cond_spmf_fst\n (q \\\n (\\s.\n (local.lazy_channel_insec \\\\<^sub>O\n lazy_channel_send \\\\<^sub>O local.lazy_channel_recv_f)\n s query))\n answer)"} {"_id": "503849", "text": "proof (prove)\nusing this:\n cmod (lead_coeff f) *\n (\\a\\complex_roots_complex f. max 1 (cmod a))\n \\ l2norm_complex f\n\ngoal (1 subgoal):\n 1. mahler_measure_poly f \\ l2norm_complex f"} {"_id": "503850", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. f (\\ x) = \\ (f ` x) \\\n \\x.\n (\\xa. f (fpower f xa x)) = (\\xa. fpower f xa (f x))"} {"_id": "503851", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\# product_mset A B) = (x \\# A \\# B)"} {"_id": "503852", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Bseq (\\n. \\i\\n. 1 + norm (f i - (1::'a)))"} {"_id": "503853", "text": "proof (prove)\nusing this:\n is_aligned w n\n\ngoal (1 subgoal):\n 1. to_bl w = take (LENGTH('a) - n) (to_bl w) @ replicate n False"} {"_id": "503854", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\n.\n \\i\\\\<^bsub>R,n\\<^esub> J =\n R \\\\<^sub>p mprod_expR R e f n;\n f \\ {j. j \\ Suc n} \\ carrier R;\n \\k\\Suc n.\n J k =\n R \\\\<^sub>p f k \\<^bsup>\\R e k\\<^esup>;\n f \\ {j. j \\ n} \\ carrier R;\n f (Suc n) \\ carrier R;\n R \\\\<^sub>p f (Suc n)\n \\<^bsup>\\R e (Suc n)\\<^esup> =\n R \\\\<^sub>p\n (f (Suc n)^\\<^bsup>R e (Suc n)\\<^esup>)\\\n \\ mprod_expR R e f n \\ carrier R\n 2. \\n.\n \\i\\\\<^bsub>R,n\\<^esub> J =\n R \\\\<^sub>p mprod_expR R e f n;\n f \\ {j. j \\ Suc n} \\ carrier R;\n \\k\\Suc n.\n J k =\n R \\\\<^sub>p f k \\<^bsup>\\R e k\\<^esup>;\n f \\ {j. j \\ n} \\ carrier R;\n f (Suc n) \\ carrier R;\n R \\\\<^sub>p f (Suc n)\n \\<^bsup>\\R e (Suc n)\\<^esup> =\n R \\\\<^sub>p\n (f (Suc n)^\\<^bsup>R e (Suc n)\\<^esup>)\\\n \\ f (Suc n)^\\<^bsup>R e (Suc n)\\<^esup>\n \\ carrier R\n 3. \\n.\n \\i\\\\<^bsub>R,n\\<^esub> J =\n R \\\\<^sub>p mprod_expR R e f n;\n f \\ {j. j \\ Suc n} \\ carrier R;\n \\k\\Suc n.\n J k =\n R \\\\<^sub>p f k \\<^bsup>\\R e k\\<^esup>;\n f \\ {j. j \\ n} \\ carrier R;\n f (Suc n) \\ carrier R;\n R \\\\<^sub>p f (Suc n)\n \\<^bsup>\\R e (Suc n)\\<^esup> =\n R \\\\<^sub>p\n (f (Suc n)^\\<^bsup>R e (Suc n)\\<^esup>)\\\n \\ R \\\\<^sub>p\n mprod_expR R e f n \\\\<^sub>r\n R \\\\<^sub>p f (Suc n)\n \\<^bsup>\\R e (Suc n)\\<^esup> =\n R \\\\<^sub>p\n mprod_expR R e f n \\\\<^sub>r\n R \\\\<^sub>p\n (f (Suc n)^\\<^bsup>R e (Suc n)\\<^esup>)"} {"_id": "503855", "text": "proof (prove)\ngoal (1 subgoal):\n 1. composite_functor A B C F G"} {"_id": "503856", "text": "proof (prove)\nusing this:\n unitary_schur_decomposition A (e # es) = (C, P, Q)\n\ngoal (1 subgoal):\n 1. C = four_block_mat A1 (A2 * P') A0 B &&&\n P =\n W *\n four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 (n - 1)) (0\\<^sub>m (n - 1) 1)\n P' &&&\n Q =\n four_block_mat (1\\<^sub>m 1) (0\\<^sub>m 1 (n - 1)) (0\\<^sub>m (n - 1) 1)\n Q' *\n W'"} {"_id": "503857", "text": "proof (prove)\ngoal (1 subgoal):\n 1. p \\ (\\x. P x) = (\\x. (p \\ P) x)"} {"_id": "503858", "text": "proof (prove)\nusing this:\n edge \\ x y\n path \\ y p z\n p \\ [] \\\n path \\ (y, hd p) (zip p (tl p)) (last (y # butlast p), z)\n y # p \\ []\n\ngoal (1 subgoal):\n 1. path \\ (x, hd (y # p)) (zip (y # p) (tl (y # p)))\n (last (x # butlast (y # p)), z)"} {"_id": "503859", "text": "proof (prove)\nusing this:\n ?t1 \\ tag ` \\ \\\n Sigma_Algebra.measure lebesgue (BOX2 (\\ ?t1) ?t1) * 3 ^ DIM('a) =\n Sigma_Algebra.measure lebesgue (BOX (\\ ?t1) ?t1) * (2 * 3) ^ DIM('a)\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure lebesgue\n (\\t\\tag ` \\. BOX2 (\\ t) t) *\n 3 ^ DIM('a) =\n Sigma_Algebra.measure lebesgue\n (\\t\\tag ` \\. BOX (\\ t) t) *\n 6 ^ DIM('a)"} {"_id": "503860", "text": "proof (prove)\ngoal (1 subgoal):\n 1. internals sig \\ externals sig = {}"} {"_id": "503861", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ldistinct ys \\ ldistinct (lzip xs ys)"} {"_id": "503862", "text": "proof (prove)\nusing this:\n finite X\n card X \\ 1\n F \\ P[X]\n f \\ ideal F\n f \\ 0\n g \\ punit.reduced_GB F\n\ngoal (1 subgoal):\n 1. poly_deg g \\ poly_deg f"} {"_id": "503863", "text": "proof (prove)\nusing this:\n singleCombinators p \\\n distinct p \\\n allNetsDistinct p \\\n wellformed_policy1 p \\\n wellformed_policy2Pr p \\ wellformed_policy3Pr p\n\ngoal (1 subgoal):\n 1. singleCombinators (a # p) \\\n distinct (a # p) \\\n allNetsDistinct (a # p) \\\n wellformed_policy1 (a # p) \\\n wellformed_policy2Pr (a # p) \\\n wellformed_policy3Pr (a # p)"} {"_id": "503864", "text": "proof (prove)\nusing this:\n igFresh MOD xs y X\n igFresh MOD xs y X'\n igSwap MOD xs y x X = igSwap MOD xs y x' X'\n igWls MOD s X\n igWls MOD s X'\n \\xs x x' y s X X'.\n isInBar (xs, s) \\\n igWls MOD s X \\ igWls MOD s X' \\\n igFresh MOD xs y X \\\n igFresh MOD xs y X' \\\n igSwap MOD xs y x X = igSwap MOD xs y x' X' \\\n igAbs MOD xs x X = igAbs MOD xs x' X'\n isInBar (xs, s)\n\ngoal (1 subgoal):\n 1. eAbs MOD xs x (OK X) = eAbs MOD xs x' (OK X')"} {"_id": "503865", "text": "proof (prove)\nusing this:\n fst e' = last_state e\n\ngoal (1 subgoal):\n 1. last_state (append_exec e e') = last_state e'"} {"_id": "503866", "text": "proof (prove)\nusing this:\n f \\ diff_fun_space\n g \\ diff_fun_space\n\ngoal (1 subgoal):\n 1. f + g \\ diff_fun_space"} {"_id": "503867", "text": "proof (prove)\nusing this:\n finite X\n\ngoal (1 subgoal):\n 1. MiscTools.sum' f (X \\ Domain f) = MiscTools.sum' f X"} {"_id": "503868", "text": "proof (prove)\nusing this:\n v0 \\ hd R'\n v0 \\ set R'\n\ngoal (1 subgoal):\n 1. v0 \\(v0 # R')\\ z"} {"_id": "503869", "text": "proof (prove)\nusing this:\n {inv_begin1 x} tcopy_begin |+| tcopy_loop {inv_loop0 x}\n {inv_end1 x} tcopy_end {inv_end0 x}\n inv_loop0 x = inv_end1 x\n\ngoal (1 subgoal):\n 1. {inv_begin1 x} tcopy_begin |+| tcopy_loop |+| tcopy_end {inv_end0 x}"} {"_id": "503870", "text": "proof (prove)\ngoal (1 subgoal):\n 1. var_monom x * var_monom x \\ 1"} {"_id": "503871", "text": "proof (prove)\ngoal (1 subgoal):\n 1. exp (fds_nth f (Suc 0)) * eval_fds (fds_exp f') s = exp (eval_fds f s)"} {"_id": "503872", "text": "proof (state)\nthis:\n lnull (lscan f w a)\n\ngoal (2 subgoals):\n 1. \\w a. lnull (lscan f w a) \\ lfinite w\n 2. \\w a.\n \\lfinite (lscan f w a); \\ lnull (lscan f w a);\n \\wa aa.\n ltl (lscan f w a) = lscan f wa aa \\\n lfinite wa\\\n \\ lfinite w"} {"_id": "503873", "text": "proof (prove)\ngoal (1 subgoal):\n 1. app xs ys zs \\ zs = xs @ ys"} {"_id": "503874", "text": "proof (prove)\nusing this:\n ?a \\ H \\\n (\\y\\J. {?a \\ y}) \\\n (\\y\\J. {x \\ y}) =\n {}\n\ngoal (1 subgoal):\n 1. \\a b.\n \\a \\ insert x H; b \\ insert x H;\n a \\ b\\\n \\ (\\) a ` J \\ (\\) b ` J = {}"} {"_id": "503875", "text": "proof (prove)\nusing this:\n llast (ltl xs) \\ lset xs\n\ngoal (1 subgoal):\n 1. llast xs \\ lset xs"} {"_id": "503876", "text": "proof (prove)\nusing this:\n \\r af.\n generalized_sfw (lr_of_tran_fbs rt fw ifs) p = Some (r, af) \\\n (if snd af = Drop then [] = [] else [] = [Forward (fst af)])\n\ngoal (1 subgoal):\n 1. (\\r oif.\n generalized_sfw (lr_of_tran_fbs rt fw ifs) p =\n Some (r, oif, Drop) \\\n thesis) \\\n thesis"} {"_id": "503877", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\\\<^sub>7 y = x) = (x = 0) &&&\n (x \\\\<^sub>7 y = y) = (y = 0)"} {"_id": "503878", "text": "proof (prove)\ngoal (1 subgoal):\n 1. g ` underS r a = underS s (g a)"} {"_id": "503879", "text": "proof (prove)\ngoal (1 subgoal):\n 1. f ` S \\ lmeasurable \\\n m * Sigma_Algebra.measure lebesgue S =\n Sigma_Algebra.measure lebesgue (f ` S)"} {"_id": "503880", "text": "proof (prove)\ngoal (1 subgoal):\n 1. free_name (free name) = name &&&\n (is_free t \\ free (free_name t) = t)"} {"_id": "503881", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ DBuffer \\ Buffer inp out"} {"_id": "503882", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xs \\ Lxx x y \\ 2 \\ length xs"} {"_id": "503883", "text": "proof (prove)\nusing this:\n (_Exit_) postdominates n\n n -as'\\\\<^sub>\\* pex\n method_exit pex\n\ngoal (1 subgoal):\n 1. (_Exit_) \\ set (sourcenodes as')"} {"_id": "503884", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sort_key f xs = sort_key f ys"} {"_id": "503885", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (((return \\\\\\ (\\))\n (\\i. show i @ error_msg_notin_dom_err) nat_of_uint64,\n RETURN \\ (\\i. show i @ error_msg_notin_dom_err))\n \\ uint64_nat_assn\\<^sup>k \\\\<^sub>a id_assn &&&\n ((return \\\\\\ (\\)) error_msg_reused_dom\n nat_of_uint64,\n RETURN \\ error_msg_reused_dom)\n \\ uint64_nat_assn\\<^sup>k \\\\<^sub>a id_assn) &&&\n (uncurry\n (\\x. (return \\\\ error_msg) (nat_of_uint64 x)),\n uncurry (RETURN \\\\ error_msg))\n \\ uint64_nat_assn\\<^sup>k *\\<^sub>a\n id_assn\\<^sup>k \\\\<^sub>a status_assn id_assn &&&\n (uncurry (return \\\\ error_msg),\n uncurry (RETURN \\\\ error_msg))\n \\ nat_assn\\<^sup>k *\\<^sub>a\n id_assn\\<^sup>k \\\\<^sub>a status_assn id_assn"} {"_id": "503886", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\Ca c.\n \\is_class G C; ws_prog G;\n \\C c. class G C = Some c \\ unique (cfields c);\n class G Ca = Some c;\n Ca \\ Object \\\n G\\Ca\\\\<^sub>C1super c \\\n unique (DeclConcepts.fields G (super c)) \\\n is_class G (super c)\\\n \\ ws_prog G\n 2. \\Ca c.\n \\is_class G C; ws_prog G;\n \\C c. class G C = Some c \\ unique (cfields c);\n class G Ca = Some c;\n Ca \\ Object \\\n G\\Ca\\\\<^sub>C1super c \\\n unique (DeclConcepts.fields G (super c)) \\\n is_class G (super c)\\\n \\ unique\n (map (\\(fn, y). ((fn, Ca), y))\n (cfields c) @\n (if Ca = Object then []\n else DeclConcepts.fields G (super c)))"} {"_id": "503887", "text": "proof (prove)\nusing this:\n (1 - u) *\\<^sub>R x + u *\\<^sub>R y \\ S\n\ngoal (1 subgoal):\n 1. False"} {"_id": "503888", "text": "proof (prove)\nusing this:\n bdd_below\n {T\\<^sub>p init qs as |as. length as = length qs} \\\n \\ {T\\<^sub>p init qs as |as. length as = length qs}\n \\ T\\<^sub>p init qs\n (take x Strat @\n (fst (Strat ! x),\n take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) #\n drop (Suc x) Strat)\n\ngoal (1 subgoal):\n 1. \\ {T\\<^sub>p init qs as |as. length as = length qs}\n \\ T\\<^sub>p init qs\n (take x Strat @\n (fst (Strat ! x),\n take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) #\n drop (Suc x) Strat)"} {"_id": "503889", "text": "proof (prove)\nusing this:\n es \\ 0\n\ngoal (1 subgoal):\n 1. e_length es = Suc (pdec1 (es - 1))"} {"_id": "503890", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bdd_below ((\\x. x \\ a) ` s)"} {"_id": "503891", "text": "proof (prove)\nusing this:\n HNext s s'\n HInv1 s \\ HInv2 s \\ HInv2 s' \\ HInv4 s\n\ngoal (1 subgoal):\n 1. HInv4b s' p"} {"_id": "503892", "text": "proof (prove)\ngoal (1 subgoal):\n 1. slow.check_eqv n \\ \\ \\\n \\.lang\\<^sub>W\\<^sub>S\\<^sub>1\\<^sub>S n \\ =\n \\.lang\\<^sub>W\\<^sub>S\\<^sub>1\\<^sub>S n \\"} {"_id": "503893", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\[[\\]]\\ \\ \\; \\ REP \\'\\\n \\ \\\\'.\n (\\ REP \\') \\\n ([\\]\\' \\ \\')"} {"_id": "503894", "text": "proof (prove)\ngoal (1 subgoal):\n 1. prime_nat p = (1 < p \\ (\\n\\{1<.. n dvd p))"} {"_id": "503895", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Func_option A B \\ Pfunc A B"} {"_id": "503896", "text": "proof (prove)\nusing this:\n natural_transformation A B F G \\\n Go\\.A.ide a\n \\natural_transformation ?A ?B ?F ?G ?\\;\n \\ partial_magma.arr ?A ?f\\\n \\ ?\\ ?f = partial_magma.null ?B\n \\natural_transformation ?A ?B ?F ?G ?\\;\n partial_magma.arr ?A ?f\\\n \\ ?B (?G ?f) (?\\ (partial_magma.dom ?A ?f)) =\n ?\\ ?f\n Go\\.A.ide ?a \\ Go\\.map ?a = B (G ?a) (\\ ?a)\n \\Go\\.B.arr ?f; Go\\.B.cod ?f = ?b\\\n \\ B ?b ?f = ?f\n\ngoal (1 subgoal):\n 1. Go\\.map a = \\ a"} {"_id": "503897", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mds_yields_stable_types m \\\n tyenv_wellformed m (\\_of_mds m) (\\_of_mds m) {}"} {"_id": "503898", "text": "proof (prove)\nusing this:\n iface_packet_check ifl pi \\ None\n\ngoal (1 subgoal):\n 1. (\\i3.\n iface_packet_check ifl pi = Some i3 \\\n thesis) \\\n thesis"} {"_id": "503899", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\xa.\n fmlookup xm k = Some x \\\n fmlookup xm xa = fmlookup (fmdrop k xm(k \\\\<^sub>f x)) xa"} {"_id": "503900", "text": "proof (prove)\nusing this:\n finite UNIV\n\ngoal (1 subgoal):\n 1. finite UNIV"} {"_id": "503901", "text": "proof (prove)\ngoal (1 subgoal):\n 1. a = e \\ a = b \\ a \\ e"} {"_id": "503902", "text": "proof (prove)\nusing this:\n A \\ B\n A B Le C D\n ?A ?B Le ?C ?C \\ ?A = ?B\n\ngoal (1 subgoal):\n 1. C \\ D"} {"_id": "503903", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w ?! i = w ?! nth_least s k"} {"_id": "503904", "text": "proof (prove)\nusing this:\n class.linorder (\\x y. y \\ x) (\\x y. y < x)\n\ngoal (1 subgoal):\n 1. class.wellorder (\\x y. y \\ x) (\\x y. y < x)"} {"_id": "503905", "text": "proof (prove)\ngoal (1 subgoal):\n 1. linorder_rank R (insert y A) x =\n (if x \\ y \\ (y, x) \\ R then 1 else 0) +\n linorder_rank R A x"} {"_id": "503906", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\A \\ bad; B \\ bad\\\n \\ extr bad IK (insert (Confid A B X) CH) =\n extr bad IK CH"} {"_id": "503907", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_of_oalist_tc (OAlist_tc_except_min xs) = tl (list_of_oalist_tc xs)"} {"_id": "503908", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x s.\n \\0 < return_time p2 x - return_time p1 x; 0 < s;\n s \\ return_time p1 x; flow0 x s \\ p2\\\n \\ False"} {"_id": "503909", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cotrace (coS t final_i) (Co_Snapshot.ldrop final_i t)"} {"_id": "503910", "text": "proof (prove)\nusing this:\n L.seq \\ \\\n\ngoal (1 subgoal):\n 1. L.in_hom \\ (L.cod \\) (L.cod \\)"} {"_id": "503911", "text": "proof (prove)\nusing this:\n Finite_Set.fold lcm (1::'a) (Set t) = fold_keys lcm t (1::'a)\n\ngoal (1 subgoal):\n 1. Lcm\\<^sub>f\\<^sub>i\\<^sub>n (Set t) = fold_keys lcm t (1::'a)"} {"_id": "503912", "text": "proof (prove)\nusing this:\n ?j \\ set js \\ i \\ ?j\n k < length buf\n buf ! k = []\n length Ps = length js \\\n length js = length buf \\\n (\\ia js ! ia \\ list_all2 (Ps ! ia) [i..s. proper_it (succ_it s v) (succ_it s v)"} {"_id": "503914", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\i j. (CHR ''a'' ^^^ i @ CHR ''b'' ^^^ j \\ A) = (i = j)"} {"_id": "503915", "text": "proof (prove)\nusing this:\n A a' t \\ log \\ (real (size1 t) / 2) + 1\n\ngoal (1 subgoal):\n 1. A a t \\ log \\ (real (size1 t)) + 1"} {"_id": "503916", "text": "proof (prove)\ngoal (1 subgoal):\n 1. continuous_on {a..b} f \\\n f absolutely_integrable_on {a..b}"} {"_id": "503917", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\F\\set Fs\\<^sub>1. distinct F;\n \\F\\set Fs\\<^sub>2. distinct F; distinct Fs\\<^sub>1;\n inj_on (\\xs. {xs} // {\\}) (set Fs\\<^sub>1);\n distinct Fs\\<^sub>2;\n inj_on (\\xs. {xs} // {\\}) (set Fs\\<^sub>2)\\\n \\ pr_iso_test0 Map.empty Fs\\<^sub>1 Fs\\<^sub>2 =\n (\\\\.\n is_pr_iso \\ Fs\\<^sub>1 Fs\\<^sub>2)"} {"_id": "503918", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\U = A \\ B; A \\ X; B \\ Y;\n X \\ U; X \\ Y = {}\\\n \\ A = X"} {"_id": "503919", "text": "proof (prove)\nusing this:\n x \\ mds\\<^sub>2 GuarNoReadOrWrite\n\ngoal (1 subgoal):\n 1. x \\ mds GuarNoReadOrWrite"} {"_id": "503920", "text": "proof (prove)\nusing this:\n C.obj a\n D.isomorphic ?a ?a' =\n (\\f.\n \\f : ?a \\\\<^sub>D ?a'\\ \\\n D.iso f)\n D.isomorphic ?f ?g \\ D.isomorphic ?g ?f\n D.iso (unit a)\n\ngoal (1 subgoal):\n 1. D.isomorphic (map\\<^sub>0 a \\\\<^sub>D \\\\<^sub>0 a)\n (F.map\\<^sub>0 a)"} {"_id": "503921", "text": "proof (state)\nthis:\n incseq u\n c < ereal (u ?i)\n ereal (u ?i) < b\n (\\x. ereal (u x)) \\ b\n\ngoal (1 subgoal):\n 1. (\\l u.\n \\einterval a b = (\\i. {l i..u i}); incseq u;\n decseq l; \\i. l i < u i; \\i. a < ereal (l i);\n \\i. ereal (u i) < b;\n (\\x. ereal (l x)) \\ a;\n (\\x. ereal (u x)) \\ b\\\n \\ thesis) \\\n thesis"} {"_id": "503922", "text": "proof (prove)\nusing this:\n u \\ limit r\n range r \\ reachable\n limit ?r \\ range ?r\n\ngoal (1 subgoal):\n 1. u \\ reachable"} {"_id": "503923", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_all tfr\\<^sub>s\\<^sub>t\\<^sub>p A''"} {"_id": "503924", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (row C1 (i - nr1) @\\<^sub>v row D1 (i - nr1)) \\\n (col A2 j @\\<^sub>v col C2 j) =\n row C1 (i - nr1) \\ col A2 j +\n row D1 (i - nr1) \\ col C2 j"} {"_id": "503925", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 1 * x = x"} {"_id": "503926", "text": "proof (prove)\nusing this:\n Formula.sat \\ V (map the v) i (formula.Agg y \\ b f \\)\n \\ fv \\ \\ {0..zs.\n \\length zs = b;\n Formula.sat \\ V (zs @ map the v) i \\\\\n \\ thesis) \\\n thesis"} {"_id": "503927", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\Xa Ya.\n \\overrider Xa; idem_overrider Xa; overrider Ya;\n idem_overrider Ya;\n \\s\\<^sub>1 s\\<^sub>2 s\\<^sub>3.\n Ya (Xa s\\<^sub>1 s\\<^sub>2) s\\<^sub>3 =\n Xa (Ya s\\<^sub>1 s\\<^sub>3) s\\<^sub>2;\n idem_scene X; idem_scene Y; X \\\\<^sub>S Y;\n \\s\\<^sub>1 s\\<^sub>2.\n Ya s\\<^sub>2 (Xa s\\<^sub>2 s\\<^sub>1) =\n Xa s\\<^sub>2 (Ya s\\<^sub>2 s\\<^sub>1)\\\n \\ (\\x y. Xa x (Ya x y)) = (\\x y. x)\n 2. \\Xa Ya s\\<^sub>1 s\\<^sub>2.\n \\overrider Xa; idem_overrider Xa; overrider Ya;\n idem_overrider Ya;\n \\s\\<^sub>1 s\\<^sub>2 s\\<^sub>3.\n Ya (Xa s\\<^sub>1 s\\<^sub>2) s\\<^sub>3 =\n Xa (Ya s\\<^sub>1 s\\<^sub>3) s\\<^sub>2;\n idem_scene X; idem_scene Y; X \\\\<^sub>S Y;\n Ya s\\<^sub>2 (Xa s\\<^sub>2 s\\<^sub>1) \\\n Xa s\\<^sub>2 (Ya s\\<^sub>2 s\\<^sub>1)\\\n \\ (\\x y. y) = (\\x y. x)"} {"_id": "503928", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 smod x = 0"} {"_id": "503929", "text": "proof (prove)\ngoal (1 subgoal):\n 1. $x\\ \\ $y"} {"_id": "503930", "text": "proof (prove)\nusing this:\n set_of (mkt_bal_r (fst (delete_max l)) (snd (delete_max l)) r) =\n set_of l \\ set_of r\n set_of (delete_root t) =\n set_of (mkt_bal_r (fst (delete_max l)) (snd (delete_max l)) r)\n set_of t - {n} = set_of l \\ set_of r\n t = MKT n (MKT ln ll lr lh) (MKT rn rl rr rh) h\n t = MKT n l r h\n\ngoal (1 subgoal):\n 1. set_of (delete_root t) = set_of t - {n}"} {"_id": "503931", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (f1 \\[F] g1) = (f2 \\[F] g2)"} {"_id": "503932", "text": "proof (prove)\nusing this:\n \\ (\\\\<^bsub>the_singleton\\<^esub>t n)\n\ngoal (1 subgoal):\n 1. eval the_singleton t t' n [\\]\\<^sub>b"} {"_id": "503933", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\C. farkas_coefficients_ns (snd ` set ns) C"} {"_id": "503934", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. Lambert_W (f x)) has_real_derivative\n f' / (f x + exp (Lambert_W (f x))))\n (at x within A)"} {"_id": "503935", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ell.\n ell \\ set (rows_to_keep A) \\ ell < dim_row A"} {"_id": "503936", "text": "proof (prove)\nusing this:\n wf_set_bits_int f\n\ngoal (1 subgoal):\n 1. bit (BITS a. f a) 0 = f 0"} {"_id": "503937", "text": "proof (prove)\nusing this:\n x \\ Sigma I \\\n y \\ Sigma I \\\n z \\ I\n z \\ fst x\n z \\ fst y\n z' \\ I\n z' \\ fst x\n z' \\ fst y\n\ngoal (1 subgoal):\n 1. (z, \\\\<^bsub>z\\<^esub> (\\ (fst x) z (snd x))\n (\\ (fst y) z (snd y)))\n \\ Sigma I \\ &&&\n (z',\n \\\\<^bsub>z'\\<^esub> (\\ (fst x) z' (snd x))\n (\\ (fst y) z' (snd y)))\n \\ Sigma I \\"} {"_id": "503938", "text": "proof (prove)\nusing this:\n (\\\\<^bsub>V\\<^esub>v\\B'. a' v \\\\<^bsub>V\\<^esub> v) =\n \\\\<^bsub>V\\<^esub>\n a' = (\\v. if v \\ A then a v else \\\\<^bsub>K\\<^esub>)\n b \\ A\n b \\ carrier V\n\ngoal (1 subgoal):\n 1. a' b \\\\<^bsub>V\\<^esub> b \\\\<^bsub>V\\<^esub>\n (\\\\<^bsub>V\\<^esub>a\\B'. a' a \\\\<^bsub>V\\<^esub> a) =\n \\\\<^bsub>V\\<^esub>"} {"_id": "503939", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_iarray f as = IArray (map f (IArray.list_of as))"} {"_id": "503940", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\States SA \\ \\ (States ` F) = {};\n \\x\\F. S \\ States x; dom G = \\ (States ` F);\n \\ (ran G) = F - {Root F G}; RootEx F G; OneAncestor F G;\n NoCycles F G; SA = Root F G\\\n \\ States (Root F G) \\ \\ (States ` F)"} {"_id": "503941", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Q a"} {"_id": "503942", "text": "proof (prove)\nusing this:\n X xs\n\ngoal (1 subgoal):\n 1. lsorted xs"} {"_id": "503943", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\A = x"} {"_id": "503944", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (ubounds as) =\n {uu_.\n \\i k ks.\n uu_ = (i, ks) \\ Le i (k # ks) \\ set as \\ k < 0}"} {"_id": "503945", "text": "proof (prove)\nusing this:\n nodes H = V \\ edges H \\ E\n\ngoal (1 subgoal):\n 1. nodes H = V"} {"_id": "503946", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a d.\n dnf_to_bool \\ (dnf_not d) =\n (\\ dnf_to_bool \\ d) \\\n dnf_to_bool \\ (listprepend (map invert a) (dnf_not d)) =\n (\\ cnf_to_bool \\ a \\ \\ dnf_to_bool \\ d)"} {"_id": "503947", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s' =\n snd (fst (mul_instr_sub1 (fst instr)\n (ucast\n (if fst instr = arith_type UMUL \\\n fst instr = arith_type UMULcc\n then word_of_int\n (uint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n uint (get_operand2 (snd instr) s))\n else word_of_int\n (sint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n sint (get_operand2 (snd instr) s))))\n (snd (fst (write_reg\n (if get_operand_w5 (snd instr ! 3) \\ 0\n then ucast\n (if fst instr = arith_type UMUL \\\nfst instr = arith_type UMULcc\n then word_of_int\n (uint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n uint (get_operand2 (snd instr) s))\n else word_of_int\n (sint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n sint (get_operand2 (snd instr) s)))\n else user_reg_val\n (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! 3))\n (snd (fst\n (write_cpu\n (ucast\n ((if fst instr = arith_type UMUL \\ fst instr = arith_type UMULcc\n then word_of_int\n (uint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n uint (get_operand2 (snd instr) s))\n else word_of_int\n (sint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n sint (get_operand2 (snd instr) s))) >>\n 32))\n Y (snd (fst (get_curr_win () s)))))))\n (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! 3))\n (snd (fst (write_cpu\n (ucast\n ((if fst instr = arith_type UMUL \\ fst instr = arith_type UMULcc\n then word_of_int\n (uint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n uint (get_operand2 (snd instr) s))\n else word_of_int\n (sint\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n sint (get_operand2 (snd instr) s))) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))) \\\n get_S (cpu_reg_val PSR s) = 0 \\\n (get_operand_w5 (snd instr ! 3) \\ 0 \\\n (fst instr = arith_type UMUL \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (arith_type UMUL)\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (ucast\n (ucast\n (user_reg_val\n (fst (fst (get_curr_win () s))) (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! 3))\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0) \\\n (fst instr = arith_type UMULcc \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (arith_type UMULcc)\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (ucast\n (ucast\n (user_reg_val\n (fst (fst (get_curr_win () s))) (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! 3))\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0) \\\n (fst instr \\ arith_type UMUL \\\n fst instr \\ arith_type UMULcc \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (fst instr)\n (ucast\n (scast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (ucast\n (scast\n (user_reg_val\n (fst (fst (get_curr_win () s))) (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s)))\n (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! 3))\n (snd (fst\n (write_cpu\n (ucast\n (scast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0)) \\\n (get_operand_w5 (snd instr ! 3) = 0 \\\n (fst instr = arith_type UMUL \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (arith_type UMUL)\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (user_reg_val\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s)))))))\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0) \\\n (fst instr = arith_type UMULcc \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (arith_type UMULcc)\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (user_reg_val\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s)))))))\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (ucast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n ucast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0) \\\n (fst instr \\ arith_type UMUL \\\n fst instr \\ arith_type UMULcc \\\n get_S\n (cpu_reg_val PSR\n (snd (fst (mul_instr_sub1 (fst instr)\n (ucast\n (scast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s)))\n (snd (fst (write_reg\n (user_reg_val\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (scast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s)))))))\n (fst (fst (get_curr_win () s))) 0\n (snd (fst\n (write_cpu\n (ucast\n (scast\n (user_reg_val (fst (fst (get_curr_win () s)))\n (get_operand_w5 (snd instr ! Suc 0))\n (snd (fst (get_curr_win () s)))) *\n scast (get_operand2 (snd instr) s) >>\n 32))\n Y (snd (fst (get_curr_win () s))))))))))))) =\n 0))"} {"_id": "503948", "text": "proof (state)\nthis:\n j = i\n\ngoal (1 subgoal):\n 1. \\x.\n \\x < length ns; 0 < x;\n 0 < offs_num (length ns) xs index key mi ma x; x \\ i\\\n \\ False"} {"_id": "503949", "text": "proof (prove)\nusing this:\n 0 < fst w\n ts = h # a # ts'\n Ball (set ts) local.valid\n a \\ Leaf \\\n r a = snd (fst w, l h - 1) \\\n (case ts' of [] \\ l a = fst (fst w, l h - 1)\n | u # us \\\n adjacent (fst (fst w, l h - 1), r u) ts' \\ l a = Suc (r u))\n local.valid z\n l z =\n (case a # ts' of [] \\ fst w\n | t\\<^sub>1 # ts'' \\ Suc (r t\\<^sub>1))\n\ngoal (1 subgoal):\n 1. local.valid (combine a z)"} {"_id": "503950", "text": "proof (prove)\nusing this:\n class.linorder (\\c d. of_char c \\ of_char d)\n (\\c d. of_char c < of_char d)\n\ngoal (1 subgoal):\n 1. class.linorder less_eq_char less_char"} {"_id": "503951", "text": "proof (prove)\nusing this:\n vpeq (current s) s t \\ vpeq u s t\n current s = current t\n\ngoal (1 subgoal):\n 1. vpeq u (snd (snd (CISK_control s (current s) (execs (current s)))))\n (snd (snd (CISK_control t (current t) (execs (current t)))))"} {"_id": "503952", "text": "proof (prove)\nusing this:\n \\finite {i}; finite r; {i} \\ r = {}\\\n \\ mset_ran a {i} + mset_ran a r =\n mset_ran a ({i} \\ r)\n\ngoal (1 subgoal):\n 1. \\finite r; i \\ r\\\n \\ add_mset (a i) (mset_ran a r) =\n mset_ran a (insert i r)"} {"_id": "503953", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A E'' TS C D"} {"_id": "503954", "text": "proof (prove)\ngoal (1 subgoal):\n 1. k \\ X - S"} {"_id": "503955", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\[u] \\\n (trg u \\ \\) \\\n \\[trg f, f, v] \\\n (\\\\<^sup>-\\<^sup>1[f] \\ \\[f] \\ v) \\\n \\\\<^sup>-\\<^sup>1[f, trg v, v] \\\n (f \\ \\\\<^sup>-\\<^sup>1[v]) =\n \\[u] \\\n (trg u \\ \\) \\\n can (\\<^bold>\\trg u\\<^bold>\\\\<^sub>0 \\<^bold>\\\n \\<^bold>\\f\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\v\\<^bold>\\)\n (\\<^bold>\\f\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\v\\<^bold>\\)"} {"_id": "503956", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\a b k.\n k \\ 0 \\\n M = k *\\<^sub>s\\<^sub>m (a, b, cnj b, cnj a) \\\n sgn k' = sgn (Re (mat_det (a, b, cnj b, cnj a)))"} {"_id": "503957", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x\\<^sup>T\\<^sup>+ \\ 1') ; p \\ (- 1'\\<^sup>T \\ 1') ; p"} {"_id": "503958", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\op.\n \\ops' = [op]; op \\ set \\\\\n \\ thesis) \\\n thesis"} {"_id": "503959", "text": "proof (prove)\nusing this:\n finite K\n K \\ I\n Y \\ sets (Pi\\<^sub>M K M)\n finite J\n J \\ I\n X \\ sets (Pi\\<^sub>M J M)\n\ngoal (1 subgoal):\n 1. emeasure (P (K \\ J)) (emb (K \\ J) J X) = emeasure (P J) X"} {"_id": "503960", "text": "proof (prove)\nusing this:\n list_emb\n (\\y z. P y z \\ (\\x. P x y \\ P x z))\n ys zs\n list_emb P xs ys\n\ngoal (1 subgoal):\n 1. list_emb P xs zs"} {"_id": "503961", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ts = butlast ts @ [(sub, sep)]; last ts = (sub, sep);\n t = Leaf\\\n \\ inorder_list (butlast ts) @ inorder sub @ [sep] =\n inorder_list ts"} {"_id": "503962", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A *\\<^sub>v (v + - w) = A *\\<^sub>v v - A *\\<^sub>v w"} {"_id": "503963", "text": "proof (prove)\ngoal (1 subgoal):\n 1. presAtm atm \\ discr (Atm atm)"} {"_id": "503964", "text": "proof (prove)\ngoal (1 subgoal):\n 1. in_traverse (delete_elt_tree (Node lt x m rt)) =\n in_traverse lt @ in_traverse rt"} {"_id": "503965", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Q X \\ R Y"} {"_id": "503966", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (length l < length ps &&& 0 < length l) &&&\n length r < length ps &&& 0 < length r"} {"_id": "503967", "text": "proof (prove)\nusing this:\n smult d f =m smult (c * d) (\\\\<^sub># fs)\n\ngoal (1 subgoal):\n 1. factorization_m (smult d f) (c * d, fs)"} {"_id": "503968", "text": "proof (prove)\nusing this:\n inverse a \\ HFinite\n a \\ \\\n a \\ 0\n a * w \\ a * z\n\ngoal (1 subgoal):\n 1. w \\ z"} {"_id": "503969", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x\\<^sup>-\\<^sup>\\\\<^sup>-\\<^sup>\\ \\\\<^sub>Q x"} {"_id": "503970", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (map (($v) v) [0.. snd (stopcont ivl) &&&\n fst (stopcont ivl)\n \\ sbelow_halfspace rsctn \\ UNIV) &&&\n fst (stopcont ivl) \\ snd (stopcont ivl) &&&\n snd (stopcont ivl) \\ sbelow_halfspace rsctn \\ UNIV &&&\n flowsto ivl {0..} (snd (stopcont ivl))\n (fst (stopcont ivl) \\ trap)"} {"_id": "503972", "text": "proof (prove)\nusing this:\n constructive_security_obsf (real1 ?\\) (ideal1 ?\\) (sim1 ?\\)\n (\\_real1 ?\\) (\\_inner1 ?\\) (\\_common1 ?\\)\n (absorb (\\ ?\\)\n (obsf_converter\n (parallel_wiring \\\n (1\\<^sub>C |\\<^sub>\\\n converter_of_resource\n (sim2 ?\\ |\\<^sub>= 1\\<^sub>C \\ ideal2 ?\\)))))\n (advantage\n (absorb (\\ ?\\)\n (obsf_converter\n (parallel_wiring \\\n (1\\<^sub>C |\\<^sub>\\\n converter_of_resource\n (sim2 ?\\ |\\<^sub>= 1\\<^sub>C \\ ideal2 ?\\)))))\n (obsf_resource\n (sim1 ?\\ |\\<^sub>= 1\\<^sub>C \\ ideal1 ?\\))\n (obsf_resource (real1 ?\\)))\n constructive_security_obsf (real2 ?\\) (ideal2 ?\\) (sim2 ?\\)\n (\\_real2 ?\\) (\\_inner2 ?\\) (\\_common2 ?\\)\n (absorb (\\ ?\\)\n (obsf_converter\n (parallel_wiring \\\n (converter_of_resource (real1 ?\\) |\\<^sub>\\\n 1\\<^sub>C))))\n (advantage\n (absorb (\\ ?\\)\n (obsf_converter\n (parallel_wiring \\\n (converter_of_resource (real1 ?\\) |\\<^sub>\\\n 1\\<^sub>C))))\n (obsf_resource\n (sim2 ?\\ |\\<^sub>= 1\\<^sub>C \\ ideal2 ?\\))\n (obsf_resource (real2 ?\\)))\n\ngoal (1 subgoal):\n 1. constructive_security_obsf\n (parallel_wiring \\ real1 \\ \\ real2 \\)\n (parallel_wiring \\ ideal1 \\ \\ ideal2 \\)\n (sim1 \\ |\\<^sub>= sim2 \\)\n (\\_real1 \\ \\\\<^sub>\\ \\_real2 \\)\n (\\_inner1 \\ \\\\<^sub>\\ \\_inner2 \\)\n (\\_common1 \\ \\\\<^sub>\\ \\_common2 \\)\n (\\ \\)\n (advantage\n (absorb (\\ \\)\n (obsf_converter\n (parallel_wiring \\\n (1\\<^sub>C |\\<^sub>\\\n converter_of_resource\n (sim2 \\ |\\<^sub>= 1\\<^sub>C \\ ideal2 \\)))))\n (obsf_resource\n (sim1 \\ |\\<^sub>= 1\\<^sub>C \\ ideal1 \\))\n (obsf_resource (real1 \\)) +\n advantage\n (absorb (\\ \\)\n (obsf_converter\n (parallel_wiring \\\n (converter_of_resource (real1 \\) |\\<^sub>\\\n 1\\<^sub>C))))\n (obsf_resource\n (sim2 \\ |\\<^sub>= 1\\<^sub>C \\ ideal2 \\))\n (obsf_resource (real2 \\)))"} {"_id": "503973", "text": "proof (prove)\nusing this:\n mode = invmode statM e\n a = Null \\ is_static statM\n\ngoal (1 subgoal):\n 1. mode = IntVir \\ a \\ Null"} {"_id": "503974", "text": "proof (state)\nthis:\n \\' i = \\ i\n \\j\\{i}. \\ j = \\ j\n\ngoal (12 subgoals):\n 1. \\\\' \\ l s\\<^sub>m\\<^sub>s\\<^sub>g p.\n \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}broadcast(s\\<^sub>m\\<^sub>s\\<^sub>g) .\n p),\n broadcast\n (s\\<^sub>m\\<^sub>s\\<^sub>g (\\ i)),\n \\', p)\n \\ oseqp_sos \\ i\n 2. \\\\' \\ l s\\<^sub>i\\<^sub>p\\<^sub>s\n s\\<^sub>m\\<^sub>s\\<^sub>g p.\n \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}groupcast(s\\<^sub>i\\<^sub>p\\<^sub>s,\n s\\<^sub>m\\<^sub>s\\<^sub>g) .\n p),\n groupcast\n (s\\<^sub>i\\<^sub>p\\<^sub>s (\\ i))\n (s\\<^sub>m\\<^sub>s\\<^sub>g (\\ i)),\n \\', p)\n \\ oseqp_sos \\ i\n 3. \\\\' \\ l s\\<^sub>i\\<^sub>p s\\<^sub>m\\<^sub>s\\<^sub>g p\n q. \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}unicast(s\\<^sub>i\\<^sub>p,\n s\\<^sub>m\\<^sub>s\\<^sub>g) .\n p \\ q),\n unicast (s\\<^sub>i\\<^sub>p (\\ i))\n (s\\<^sub>m\\<^sub>s\\<^sub>g (\\ i)),\n \\', p)\n \\ oseqp_sos \\ i\n 4. \\\\' \\ l s\\<^sub>i\\<^sub>p s\\<^sub>m\\<^sub>s\\<^sub>g p\n q. \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}unicast(s\\<^sub>i\\<^sub>p,\n s\\<^sub>m\\<^sub>s\\<^sub>g) .\n p \\ q),\n \\unicast (s\\<^sub>i\\<^sub>p\n (\\ i)),\n \\', q)\n \\ oseqp_sos \\ i\n 5. \\\\' \\ l s\\<^sub>m\\<^sub>s\\<^sub>g p.\n \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\, {l}send(s\\<^sub>m\\<^sub>s\\<^sub>g) .\n p),\n send (s\\<^sub>m\\<^sub>s\\<^sub>g (\\ i)),\n \\', p)\n \\ oseqp_sos \\ i\n 6. \\\\' \\ l s\\<^sub>d\\<^sub>a\\<^sub>t\\<^sub>a p.\n \\\\' i = \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}deliver(s\\<^sub>d\\<^sub>a\\<^sub>t\\<^sub>a) .\n p),\n deliver\n (s\\<^sub>d\\<^sub>a\\<^sub>t\\<^sub>a\n (\\ i)),\n \\', p)\n \\ oseqp_sos \\ i\n 7. \\\\' u\\<^sub>m\\<^sub>s\\<^sub>g msg \\ l p.\n \\\\' i = u\\<^sub>m\\<^sub>s\\<^sub>g msg (\\ i);\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\,\n {l}receive(u\\<^sub>m\\<^sub>s\\<^sub>g) .\n p),\n receive msg, \\', p)\n \\ oseqp_sos \\ i\n 8. \\\\' u \\ l p.\n \\\\' i = u (\\ i);\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\, {l}\\u\\\n p),\n \\, \\', p)\n \\ oseqp_sos \\ i\n 9. \\\\ pn a \\' p'.\n \\((\\, \\ pn), a, \\', p')\n \\ oseqp_sos \\ i;\n \\j\\{i}. \\ j = \\ j \\\n \\\\'.\n (\\j\\{i}. \\' j = \\' j) \\\n ((\\, \\ pn), a, \\', p')\n \\ oseqp_sos \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\, call(pn)), a, \\', p')\n \\ oseqp_sos \\ i\n 10. \\\\ p a \\' p' q.\n \\((\\, p), a, \\', p')\n \\ oseqp_sos \\ i;\n \\j\\{i}. \\ j = \\ j \\\n \\\\'.\n (\\j\\{i}. \\' j = \\' j) \\\n ((\\, p), a, \\', p') \\ oseqp_sos \\ i;\n \\j\\{i}. \\ j = \\ j\\\n \\ \\\\'.\n (\\j\\{i}.\n \\' j = \\' j) \\\n ((\\, p\n \\\n q),\n a, \\', p')\n \\ oseqp_sos \\ i\nA total of 12 subgoals..."} {"_id": "503975", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monoid N (\\) \\"} {"_id": "503976", "text": "proof (prove)\ngoal (1 subgoal):\n 1. good_prob_primality_test (\\n a. strong_fermat_liar a n) n\n (1 / 2)"} {"_id": "503977", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj_on\n (\\s.\n \\(\\(f, g)\\fundfoldpairs.\n {Abs_induced_automorph f g}) -\n {s}\\)\n (\\(f, g)\\fundfoldpairs. {Abs_induced_automorph f g})"} {"_id": "503978", "text": "proof (prove)\ngoal (1 subgoal):\n 1. effect (!i) h\\<^sub>2 h\\<^sub>2 (Ref.get h\\<^sub>2 i) &&&\n Ref.get h\\<^sub>2 i \\ Ref.get h\\<^sub>1 i &&&\n Ref.get h\\<^sub>2 i < Array.length h\\<^sub>2 a"} {"_id": "503979", "text": "proof (prove)\ngoal (1 subgoal):\n 1. possible_allocations_rel G N =\n injectionsUniverse \\\n {a. Domain a \\ all_partitions G \\ Range a \\ N}"} {"_id": "503980", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\' a b c s.\n \\small_step \\' (c, s) = Some (a, b);\n small_steps \\' (a, b) (Some (c', s'));\n small_steps \\ (PScope \\' a, b)\n (Some (PScope \\' c', s'))\\\n \\ small_steps \\ (PScope \\' c, s)\n (Some (PScope \\' c', s'))"} {"_id": "503981", "text": "proof (prove)\nusing this:\n \\ide (tab\\<^sub>0 r); seq (local.inv (rep r)) (rep r)\\\n \\ local.inv (rep r) \\ rep r \\ tab\\<^sub>0 r =\n (local.inv (rep r) \\ tab\\<^sub>0 r) \\\n (rep r \\ tab\\<^sub>0 r)\n ide ?r \\ local.iso (rep ?r)\n ide ?r \\\n \\rep ?r : src ?r \\ trg ?r\\\n ide ?r \\\n \\rep ?r : tab\\<^sub>1 ?r \\\n (tab\\<^sub>0\n ?r)\\<^sup>* \\ ?r\\\n local.iso ?f \\ inverse_arrows ?f (local.inv ?f)\n inverse_arrows ?f ?g \\ ?g \\ ?f = local.dom ?f\n\ngoal (1 subgoal):\n 1. (local.inv (rep r) \\ tab\\<^sub>0 r) \\\n (rep r \\ tab\\<^sub>0 r) =\n (tab\\<^sub>1 r \\ (tab\\<^sub>0 r)\\<^sup>*) \\ tab\\<^sub>0 r"} {"_id": "503982", "text": "proof (prove)\nusing this:\n local.gen_lossless_gpv gpv\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "503983", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\result = i_Exec_Comp_Stream_Init trans_fun input c;\n \\x_n c_n.\n P1 x_n \\ P2 c_n \\ Q (trans_fun x_n c_n);\n \\t\\I. inext t I' = Suc t\\\n \\ \\ t I.\n P1 (input t) \\ P2 (result t) \\\n (\\ t1 t I'. Q (result t1))"} {"_id": "503984", "text": "proof (chain)\npicking this:\n \\(B)\\ \\ \\(B)\\"} {"_id": "503985", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sigma_finite_measure (Pi\\<^sub>M I M)"} {"_id": "503986", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\d>0.\n \\x\\X.\n \\x'\\X.\n dist x' x < d \\ dist (f x') (f x) < e"} {"_id": "503987", "text": "proof (prove)\nusing this:\n cyclic_on f S\n\ngoal (1 subgoal):\n 1. cyclic_on (perm_swap x y f) ((x \\\\<^sub>F y) ` S)"} {"_id": "503988", "text": "proof (prove)\nusing this:\n M $$ (i, j) =\n mat_mult (mult.row_length (mat_to_cols_list A)) (mat_to_cols_list A)\n (mat_to_cols_list B) !\n j !\n i\n Matrix_Legacy.mat (mult.row_length (mat_to_cols_list A)) (dim_col A)\n (mat_to_cols_list A)\n Matrix_Legacy.mat (dim_col A) (dim_col B) (mat_to_cols_list B)\n 0 < dim_col A\n i < dim_row M\n j < dim_col M\n 0 < dim_col ?A \\\n mult.row_length (mat_to_cols_list ?A) = dim_row ?A\n \\plus_mult (1::?'a1) (*) (0::?'a1) (+) ?inver;\n Matrix_Legacy.mat (mult.row_length ?m1.0) ?n ?m1.0;\n Matrix_Legacy.mat ?n ?nc ?m2.0; ?i < mult.row_length ?m1.0;\n ?j < ?nc\\\n \\ mat_multI (0::?'a1) (+) (*) (mult.row_length ?m1.0)\n ?m1.0 ?m2.0 !\n ?j !\n ?i =\n plus_mult.scalar_product (*) (0::?'a1) (+)\n (Matrix_Legacy.row ?m1.0 ?i)\n (Matrix_Legacy.col ?m2.0 ?j)\n plus_mult 1 (*) 0 (+) (a_inv cpx_rng)\n\ngoal (1 subgoal):\n 1. M $$ (i, j) =\n plus_mult.scalar_product (*) 0 (+)\n (Matrix_Legacy.row (mat_to_cols_list A) i)\n (Matrix_Legacy.col (mat_to_cols_list B) j)"} {"_id": "503989", "text": "proof (prove)\nusing this:\n p = 0\n\ngoal (1 subgoal):\n 1. singular_subdivision (Suc (p - Suc 0))\n (k (p - Suc 0) (chain_boundary p c)) =\n singular_subdivision p (k (p - Suc 0) (chain_boundary p c))"} {"_id": "503990", "text": "proof (prove)\nusing this:\n b = - 1\n\ngoal (1 subgoal):\n 1. a = (0::'b)"} {"_id": "503991", "text": "proof (prove)\ngoal (1 subgoal):\n 1. const_list ($* [e])"} {"_id": "503992", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\host shadow_root_ptra.\n \\\\host shadow_root_ptra.\n h' \\ get_shadow_root host\n \\\\<^sub>r Some\n shadow_root_ptra \\\n shadow_root_ptra = shadow_root_ptr \\\n shadow_root_ptra |\\| shadow_root_ptr_kinds h';\n h' \\ get_shadow_root host\n \\\\<^sub>r Some shadow_root_ptra\\\n \\ shadow_root_ptra |\\| shadow_root_ptr_kinds h'"} {"_id": "503993", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\n.\n \\\\n. A (Suc n) \\ A n;\n normal_set (A (Suc n))\\\n \\ (LEAST z. z \\ A (Suc n)) \\ A (Suc n)\n 2. \\y.\n \\\\n. normal_set (A n);\n \\n. A (Suc n) \\ A n;\n y \\ \\ (range A)\\\n \\ oLimit (\\n. OrdinalVeblen.ordering (A n) 0)\n \\ y"} {"_id": "503994", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y. - - x \\ - (- x \\ y)"} {"_id": "503995", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Predicate.the A =\n singleton\n (\\x.\n Code.abort STR ''not_unique'' (\\_. Predicate.the A))\n A"} {"_id": "503996", "text": "proof (prove)\ngoal (1 subgoal):\n 1. card (set (ap gs bs (ps -- sps) [] data')) \\ card (set (ps -- sps))"} {"_id": "503997", "text": "proof (prove)\ngoal (1 subgoal):\n 1. coreflexive\n ((x\\<^sup>\\ \\ x\\<^sup>T\\<^sup>\\) \\\n x\\<^sup>T * x)"} {"_id": "503998", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bprv\n (imp (subst \\ \\\\\\ xx)\n (subst \\ \\\\\\ xx))"} {"_id": "503999", "text": "proof (prove)\nusing this:\n extended_ord.is_hom_GB_bound (homogenize None ` extend_indets ` F) b\n ?f7 \\ homogenize None ` extend_indets ` F \\\n homogeneous ?f7\n g' \\ extended_ord.punit.reduced_GB\n (homogenize None ` extend_indets ` F)\n\ngoal (1 subgoal):\n 1. poly_deg g' \\ b"} {"_id": "504000", "text": "proof (prove)\nusing this:\n atom i \\ (x, y)\n atom j \\ (i, x, y)\n atom k \\ (i, j, y)\n\ngoal (1 subgoal):\n 1. atom j \\ (i, Var i EQ y IFF Var i SUBS y AND y SUBS Var i)"}

((``) S \\ \\) \\\n ((``) R \\ \\))\n x"} {"_id": "503029", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ top"} {"_id": "503030", "text": "proof (prove)\nusing this:\n c N = round (real_of_int (B * b N) / real_of_int (a N))\n real_of_int (c (N + 1)) / real_of_int (a N) =\n real_of_int (c N) - real_of_int (B * b N) / real_of_int (a N)\n \\of_int (round ?x) - ?x\\ \\ (1::?'a) / (2::?'a)\n\ngoal (1 subgoal):\n 1. \\real_of_int (c (N + 1)) / real_of_int (a N)\\ \\ 1 / 2"} {"_id": "503031", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C1 = C2"} {"_id": "503032", "text": "proof (prove)\nusing this:\n t \\ t'\n {t..t'} \\ existence_ivl0 x\n (?A \\ {}) = (?A = {})\n\ngoal (1 subgoal):\n 1. x \\ X"} {"_id": "503033", "text": "proof (state)\nthis:\n reduc_saturated N\n\ngoal (2 subgoals):\n 1. reduc_saturated N \\ reduc_calc.saturated N\n 2. reduc_calc.saturated N \\ reduc_saturated N"} {"_id": "503034", "text": "proof (prove)\ngoal (1 subgoal):\n 1. P p (q + r)"} {"_id": "503035", "text": "proof (prove)\nusing this:\n P \\ \\e_,s\\<^sub>0_\\ \\\n \\addr a_,(h_, l_, sh_)\\\n \\iconf (shp s\\<^sub>0_) e_;\n P,shp s\\<^sub>0_ \\\\<^sub>b (e_,?b) \\\\\n \\ P \\ \\e_,s\\<^sub>0_,\n ?b\\ \\*\n \\addr a_,(h_, l_, sh_),\n False\\\n h_ a_ = \\(D_, fs_)\\\n (D_, C_) \\ (subcls1 P)\\<^sup>*\n iconf (shp s\\<^sub>0_) (Cast C_ e_)\n P,shp s\\<^sub>0_ \\\\<^sub>b (Cast C_ e_,b) \\\n\ngoal (1 subgoal):\n 1. P \\ \\Cast C_ e_,s\\<^sub>0_,b\\ \\*\n \\THROW ClassCast,(h_, l_, sh_),False\\"} {"_id": "503036", "text": "proof (prove)\nusing this:\n finite A\n a \\ A\n f \\ A \\ carrier_vec n \\\n dim_vec (finsum V f A) = n\n f \\ insert a A \\ carrier_vec n\n\ngoal (1 subgoal):\n 1. dim_vec (f a + finsum V f A) = n"} {"_id": "503037", "text": "proof (prove)\nusing this:\n user_reg_val curr_win (rd OR 1) y = user_reg_val curr_win (rd OR 1) ya\n\ngoal (1 subgoal):\n 1. low_equal yb yc"} {"_id": "503038", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Variable (Suc (Suc 0)) \\ var &&&\n out \\ Variable (Suc (Suc 0)) \\\n Variable (Suc (Suc 0)) \\ out) &&&\n inp \\ Variable (Suc (Suc 0)) \\\n Variable (Suc (Suc 0)) \\ inp &&&\n Variable (Suc (Suc 0)) \\ Fvars \\ &&&\n Variable (Suc (Suc 0)) \\ Fvars \\"} {"_id": "503039", "text": "proof (prove)\nusing this:\n p \\ span (insert r B)\n r \\ span B\n\ngoal (1 subgoal):\n 1. p \\ span B"} {"_id": "503040", "text": "proof (prove)\nusing this:\n has_unique_favorites R\n\ngoal (1 subgoal):\n 1. pref_profile_unique_favorites agents alts R"} {"_id": "503041", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\simple_match_valid ac; \\ simple_matches ac p;\n match_iface (oiface ac) (p_oiface p);\n OF_match_linear OF_match_fields_safe\n (map (\\x.\n split3 OFEntry\n (x1, x,\n case ba of Accept \\ [Forward ad]\n | Drop \\ []))\n (simple_match_to_of_match ac ifs))\n p \\\n NoAction\\\n \\ False"} {"_id": "503042", "text": "proof (chain)\npicking this:\n weak_conv_m (\\ \\ s \\ r) (interval_measure F)"} {"_id": "503043", "text": "proof (prove)\nusing this:\n map_of subs = map_of ?subs \\\n raw_match_term f f' (map_of ?subs) =\n map_option map_of (assoc_match_term f f' ?subs)\n raw_match_term f f' (map_of subs) = Some (map_of ?subs) \\\n raw_match_term u u' (map_of ?subs) =\n map_option map_of (assoc_match_term u u' ?subs)\n\ngoal (1 subgoal):\n 1. Option.bind (map_option map_of (assoc_match_term f f' subs))\n (raw_match_term u u') =\n map_option map_of\n (Option.bind (assoc_match_term f f' subs) (assoc_match_term u u'))"} {"_id": "503044", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sa.is_run (snd \\ r) \\\n local.sa.L \\ snd \\ r \\ local.igba.lang"} {"_id": "503045", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (n \\ length kvs \\\n inv2 (fst (rbtreeify_f n kvs))) &&&\n (n \\ Suc (length kvs) \\\n inv2 (fst (rbtreeify_g n kvs)))"} {"_id": "503046", "text": "proof (prove)\ngoal (1 subgoal):\n 1. TOTAL_wrt \\ = TOTAL"} {"_id": "503047", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Y xs.\n \\chain Y; chain (\\i. ccTTree (Y i) G);\n xs \\ valid_lists (\\ (range Y)) G\\\n \\ xs = [] \\\n (\\x. xs \\ valid_lists (Y x) G)"} {"_id": "503048", "text": "proof (state)\nthis:\n \\ \\ f \\ local.inv \\ = f'\n\ngoal (1 subgoal):\n 1. \\' \\\n (\\ \\ f \\ local.inv \\ \\\n w' \\ \\ \\ w) =\n \\' \\ (f' \\ \\)"} {"_id": "503049", "text": "proof (prove)\ngoal (1 subgoal):\n 1. derives1 a (drop (length w) b)"} {"_id": "503050", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vars_term (scf_term scf (Fun f ss)) =\n (\\x\\set (scf_list (scf (f, length ss)) ss).\n vars_term (scf_term scf x))"} {"_id": "503051", "text": "proof (prove)\ngoal (2 subgoals):\n 1. wb_lens X \\\n X ;\\<^sub>L fst\\<^sub>L \\ Y ;\\<^sub>L snd\\<^sub>L\n 2. wb_lens X \\\n x ;\\<^sub>L (Y ;\\<^sub>L snd\\<^sub>L) =\n x ;\\<^sub>L (Y ;\\<^sub>L snd\\<^sub>L)"} {"_id": "503052", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\ (range c)) \\ 0"} {"_id": "503053", "text": "proof (prove)\ngoal (1 subgoal):\n 1. snd (snd (foldl echelon_form_of_column_k_iarrays\n (matrix_to_iarray A, 0, bezout) [0.. log 2 x - real_of_int \\log 2 x\\\n\ngoal (1 subgoal):\n 1. 0 < truncate_down p x"} {"_id": "503055", "text": "proof (prove)\nusing this:\n u \\ inversion u\n v \\ inversion u\n {u, v} \\ circline_set l1 \\ circline_set l2\n \\u \\ inversion u; u \\ v;\n inversion u \\ v\\\n \\ \\!H.\n u \\ circline_set H \\\n inversion u \\ circline_set H \\\n v \\ circline_set H\n u \\ v\n \\is_poincare_line l1; u \\ circline_set l1\\\n \\ inversion u \\ circline_set l1\n \\is_poincare_line l2; u \\ circline_set l2\\\n \\ inversion u \\ circline_set l2\n is_poincare_line l1\n is_poincare_line l2\n\ngoal (1 subgoal):\n 1. l1 = l2"} {"_id": "503056", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nfa.init (Reverse_nfa MS) = dfa.final MS"} {"_id": "503057", "text": "proof (prove)\nusing this:\n pointermap_p_valid n (dpm bdd) \\\n (case pm_pth (dpm bdd) n of\n (nv, nt, ne) \\\n nv = v \\ Rmi_g nt t bdd \\ Rmi_g ne e bdd)\n pointermap_p_valid ?p (dpm bdd) \\\n pointermap_p_valid ?p (dpm s')\n Rmi_g x1a t bdd \\ Rmi_g x1a t s'\n Rmi_g x2a e bdd \\ Rmi_g x2a e s'\n\ngoal (1 subgoal):\n 1. pointermap_p_valid n (dpm s') \\\n (case pm_pth (dpm s') n of\n (nv, nt, ne) \\\n nv = v \\ Rmi_g nt t s' \\ Rmi_g ne e s')"} {"_id": "503058", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ptr' disc_nodes.\n h \\ get_tag_name ptr' \\\\<^sub>r disc_nodes =\n h' \\ get_tag_name ptr' \\\\<^sub>r disc_nodes"} {"_id": "503059", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (p \\ - (1::'a) \\ (1::'a))\\<^sup>T *\n (p \\ - (1::'a) \\ (1::'a)) =\n (p \\ - (1::'a) \\\n (p \\ - (1::'a))\\<^sup>T)\\<^sup>\\"} {"_id": "503060", "text": "proof (prove)\nusing this:\n Can t\n Nml t\n \\Can ?t; Nml ?t\\ \\ Ide ?t\n Ide ?t \\ ?t \\ HHom (Src ?t) (Trg ?t)\n Ide ?t \\ ?t \\ VHom ?t ?t\n\ngoal (1 subgoal):\n 1. t \\<^bold>\\\\<^bold>\\\\<^bold>\\ Inv t = Cod t"} {"_id": "503061", "text": "proof (prove)\ngoal (1 subgoal):\n 1. exec_gpv callee (pauses xs) s =\n map_spmf (Pair ())\n (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) callee)\n (return_spmf s) xs)"} {"_id": "503062", "text": "proof (prove)\ngoal (2 subgoals):\n 1. 16 \\ 128 - n \\\n length (dropWhile Not (to_bl (w && (mask 16 << n) >> n))) \\ 16\n 2. 16 \\ 128 - n \\\n w && (mask 16 << n) >> n << n = w && (mask 16 << n)"} {"_id": "503063", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n\\<^sub>1.\n (\\m q.\n n\\<^sub>1 \\ m \\\n (Bex (configuration q m) token_succeeds \\\n q \\ \\\\<^sub>R m) \\\n (q \\ \\\\<^sub>R m \\\n Ball (configuration q m) token_succeeds)) \\\n thesis) \\\n thesis"} {"_id": "503064", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map (case_rule (\\m'. Rule (MatchAnd m m'))) (Rule m' a' # rs) =\n Rule (MatchAnd m m') a' #\n map (case_rule (\\m'. Rule (MatchAnd m m'))) rs"} {"_id": "503065", "text": "proof (prove)\nusing this:\n Norm s0\\\\(G, L)\n \\prg = G, cls = accC,\n lcl = L\\\\In1r (Expr e)\\T\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom (locals\n (snd (Norm\n s0))) \\In1r (Expr e)\\ A\n\ngoal (1 subgoal):\n 1. (\\E.\n \\prg = G, cls = accC,\n lcl =\n L\\\\ dom (locals\n (snd (Norm\n s0))) \\In1l e\\ E \\\n thesis) \\\n thesis"} {"_id": "503066", "text": "proof (prove)\nusing this:\n x \\ Extend (set ((\\<^bold>\\ p) # G))\n (mk_finite_char\n (close\n {set G |G. \\ \\ imply G \\<^bold>\\}))\n from_nat\n\ngoal (1 subgoal):\n 1. Kripke pi\n (reach\n (mk_finite_char\n (close\n {set G |G.\n \\ \\ imply G\n \\<^bold>\\}))), Extend\n (set ((\\<^bold>\\ p) # G))\n (mk_finite_char\n (close\n {set G |G.\n \\ \\ imply G \\<^bold>\\}))\n from_nat \\ x"} {"_id": "503067", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x dvd prod_mset A"} {"_id": "503068", "text": "proof (prove)\ngoal (1 subgoal):\n 1. akra_bazzi_lower x\\<^sub>0 x\\<^sub>1 k as bs ts f integrable integral g\n g'"} {"_id": "503069", "text": "proof (prove)\nusing this:\n unstream (maps_trans f g) (s', None) =\n List.maps\n ((\\a. case a of (a, b) \\ unstream a b) \\ f)\n (unstream g s')\n generator g s = Yield x s'\n\ngoal (1 subgoal):\n 1. unstream (maps_trans f g) (s, None) =\n List.maps\n ((\\a. case a of (a, b) \\ unstream a b) \\ f)\n (unstream g s)"} {"_id": "503070", "text": "proof (prove)\ngoal (1 subgoal):\n 1. two_chain_vertical_boundary {rot_circle_cube} -\n {rot_circle_top_edge, rot_circle_bot_edge} =\n {rot_circle_left_edge, rot_circle_right_edge}"} {"_id": "503071", "text": "proof (prove)\nusing this:\n \\prg = G, cls = accC,\n lcl = L\\\\In1l (Cast castT e)\\T\n\ngoal (1 subgoal):\n 1. (\\eT.\n \\\\prg = G, cls = accC,\n lcl = L\\\\e\\-eT;\n G\\eT\\? castT; T = Inl castT\\\n \\ thesis) \\\n thesis"} {"_id": "503072", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\x.\n \\R module N; f \\ mHom R M N; bijec\\<^bsub>M,N\\<^esub> f;\n invmfun R M N f \\ aHom N M; x \\ carrier N;\n f (invmfun R M N f x) = f \\\\\n \\ x = \\\\<^bsub>N\\<^esub>\n 2. \\R module N; f \\ mHom R M N; bijec\\<^bsub>M,N\\<^esub> f;\n invmfun R M N f \\ aHom N M\\\n \\ {\\\\<^bsub>N\\<^esub>}\n \\ ker\\<^bsub>N,M\\<^esub> invmfun R M N f\n 3. \\R module N; f \\ mHom R M N; bijec\\<^bsub>M,N\\<^esub> f;\n invmfun R M N f \\ mHom R N M\\\n \\ surjec\\<^bsub>N,M\\<^esub> invmfun R M N f"} {"_id": "503073", "text": "proof (prove)\ngoal (1 subgoal):\n 1. span_in_category (\\)\n \\Leg0 =\n \\\\tab\\<^sub>0\n (local.dom \\)\\\\,\n Leg1 =\n \\\\tab\\<^sub>1\n (local.dom \\)\\\\\\"} {"_id": "503074", "text": "proof (prove)\nusing this:\n \\?A \\ carrier V; ?v \\ span ?A\\\n \\ span ?A = span (?A \\ {?v})\n gen_set {\\\\<^bsub>V\\<^esub>}\n \\finite ?A; ?A \\ carrier V;\n \\a.\n \\a \\ ?A \\ carrier K;\n lincomb a ?A = \\\\<^bsub>V\\<^esub>\\\n \\ \\v\\?A.\n a v = \\\\<^bsub>K\\<^esub>\\\n \\ lin_indpt ?A\n \\\\<^bsub>V\\<^esub> \\ span ?U\n\ngoal (1 subgoal):\n 1. lin_indpt {} \\ gen_set {} \\ {} \\ carrier V"} {"_id": "503075", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w =\n (\\x\\T. c x *\\<^sub>R x) + (\\x\\S - T. c x *\\<^sub>R x)"} {"_id": "503076", "text": "proof (prove)\nusing this:\n finite A\n \\i\\A. finite (B i)\n\ngoal (1 subgoal):\n 1. finite (sum B A)"} {"_id": "503077", "text": "proof (prove)\ngoal (6 subgoals):\n 1. \\w parent owner_document document_ptr h h' x.\n \\w \\ Shadow_DOM.adopt_node_locs parent owner_document\n document_ptr;\n h \\ w \\\\<^sub>h h';\n x |\\| object_ptr_kinds h\\\n \\ x |\\| object_ptr_kinds h'\n 2. \\w parent owner_document document_ptr h h' x.\n \\w \\ Shadow_DOM.adopt_node_locs parent owner_document\n document_ptr;\n h \\ w \\\\<^sub>h h';\n x |\\| object_ptr_kinds h'\\\n \\ x |\\| object_ptr_kinds h\n 3. \\w parent owner_document document_ptr h h'.\n \\w \\ Shadow_DOM.adopt_node_locs parent owner_document\n document_ptr;\n h \\ w \\\\<^sub>h h';\n ShadowRootClass.type_wf h\\\n \\ ShadowRootClass.type_wf h'\n 4. \\w parent owner_document document_ptr h h'.\n \\w \\ Shadow_DOM.adopt_node_locs parent owner_document\n document_ptr;\n h \\ w \\\\<^sub>h h';\n ShadowRootClass.type_wf h'\\\n \\ ShadowRootClass.type_wf h\n 5. \\h document_ptr child.\n h \\ ok Shadow_DOM.adopt_node document_ptr\n child \\\n child |\\| node_ptr_kinds h\n 6. \\h owner_document node h' ptr children children' x.\n \\h \\ Shadow_DOM.adopt_node owner_document node\n \\\\<^sub>h h';\n h \\ Shadow_DOM.get_child_nodes ptr\n \\\\<^sub>r children;\n h' \\ Shadow_DOM.get_child_nodes ptr\n \\\\<^sub>r children';\n ShadowRootClass.known_ptrs h; ShadowRootClass.type_wf h;\n x \\ set children'\\\n \\ x \\ set children"} {"_id": "503078", "text": "proof (prove)\ngoal (1 subgoal):\n 1. loc (map ENV w) = []"} {"_id": "503079", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hindered \\ \\\n \\\\>0. hindered_by \\ \\"} {"_id": "503080", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa.\n (\\x xa. is_cc_lub x xa = (xa = coCallsLub x)) \\\n is_cc_lub x xa = x <<| xa"} {"_id": "503081", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ai sctni. RETURN (above_sctn_impl ai sctni), above_sctn)\n \\ appr_rel \\\n \\lv_rel\\sctn_rel \\\n \\bool_rel\\nres_rel"} {"_id": "503082", "text": "proof (prove)\nusing this:\n llength x = llength a\n\ngoal (1 subgoal):\n 1. llist_all2 S x a = llist_all2 T y b"} {"_id": "503083", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\sorted1 (xs @ (a, a') # ys @ (b, b') # zs @ (c, c') # us);\n x < c\\\n \\ del_list x\n (xs @ (a, a') # ys @ (b, b') # zs @ (c, c') # us) =\n del_list x (xs @ (a, a') # ys @ (b, b') # zs) @\n (c, c') # us"} {"_id": "503084", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x y.\n rel_prod R S (BNF_Def.convol f1 g1 x) (BNF_Def.convol f2 g2 y)) =\n inf (BNF_Def.vimage2p f1 f2 R) (BNF_Def.vimage2p g1 g2 S)"} {"_id": "503085", "text": "proof (state)\ngoal (4 subgoals):\n 1. list.set (hd \\s # tl \\s) \\ ON\n 2. list.set ms \\ {0<..}\n 3. length (tl \\s) = length ms\n 4. sorted_wrt (\\x y. y < x) (hd \\s # tl \\s)"} {"_id": "503086", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Dynkin_system \\ {A \\ sigma_sets \\ G. P A}"} {"_id": "503087", "text": "proof (prove)\ngoal (1 subgoal):\n 1. safe_distance_normal.safe_distance_1r a\\<^sub>e v\\<^sub>e \\ =\n safe_distance_1r a\\<^sub>e v\\<^sub>e \\"} {"_id": "503088", "text": "proof (prove)\nusing this:\n \\t\\fst ` set (linearize data_prev').\n t \\ nt \\ nt - t < left I\n {tas \\ ts_tuple_rel (set (linearize data_in)).\n valid_tuple tuple_since tas \\ ?as = snd tas} =\n {tas \\ ts_tuple_rel (set (linearize data_in)).\n valid_tuple tuple_since' tas \\ ?as = snd tas}\n\ngoal (1 subgoal):\n 1. valid_mmsaux args cur\n (add_new_table_mmsaux args X\n (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since))\n (case auxlist of [] \\ [(cur, X)]\n | (t, y) # ts \\\n if t = cur then (t, y \\ X) # ts else (cur, X) # auxlist)"} {"_id": "503089", "text": "proof (state)\nthis:\n P,t \\ (xcp, h', frs) -ta-jvm\\ \\'\n\ngoal (1 subgoal):\n 1. \\ta x' m' s.\n execd_mthr.r_syntax P t (xcp, frs) h' ta x' m' \\\n exec_mthr.actions_ok s t ta"} {"_id": "503090", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel (s + u) (t + u) = rel s t"} {"_id": "503091", "text": "proof (prove)\nusing this:\n inverse_arrows \\ (MkArr (Cod \\) (Dom \\) (B.inv (Map \\)))\n\ngoal (1 subgoal):\n 1. local.iso \\"} {"_id": "503092", "text": "proof (prove)\ngoal (11 subgoals):\n 1. \\x y. dist x y = \\x - y\\\n 2. \\a x y. a *\\<^sub>R (x + y) = a *\\<^sub>R x + a *\\<^sub>R y\n 3. \\a b x. (a + b) *\\<^sub>R x = a *\\<^sub>R x + b *\\<^sub>R x\n 4. \\a b x. a *\\<^sub>R b *\\<^sub>R x = (a * b) *\\<^sub>R x\n 5. \\x. 1 *\\<^sub>R x = x\n 6. \\x. sgn x = inverse (\\x\\) *\\<^sub>R x\n 7. uniformity = (INF e\\{0<..}. principal {(x, y). dist x y < e})\n 8. \\U.\n open U =\n (\\x\\U.\n \\\\<^sub>F (x', y) in uniformity.\n x' = x \\ y \\ U)\n 9. \\x. (\\x\\ = 0) = (x = 0)\n 10. \\x y.\n \\x + y\\\n \\ (\\x\\) + (\\y\\)\nA total of 11 subgoals..."} {"_id": "503093", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\invc t; invh t; invpst t; color t = Black\\\n \\ invc (paint Black (del k t)) \\\n invh (paint Black (del k t)) \\\n invpst (del k t) \\\n color (paint Black (del k t)) = Black"} {"_id": "503094", "text": "proof (state)\nthis:\n [\\]\\' \\ r\n\ngoal (4 subgoals):\n 1. \\\\ \\ \\ \\ \\ \\'.\n \\[[\\]]\\ \\ \\;\n \\\\'.\n \\ REP \\' \\\n \\\\'.\n (\\ REP \\') \\\n ([\\]\\' \\ \\');\n [[\\]]\\ \\ \\;\n \\\\'.\n \\ REP \\' \\\n \\\\'.\n (\\ REP \\') \\\n ([\\]\\' \\ \\');\n \\ REP \\'\\\n \\ \\\\'.\n (\\ REP \\') \\\n ([\\ ;;\n \\]\\' \\ \\')\n 2. \\\\ \\ x \\ \\'.\n \\\\ = \\\n (Inr x := snd ([\\]<>\\),\n Inl x := fst ([\\]<>\\));\n \\ REP \\'\\\n \\ \\\\'.\n (\\ REP \\') \\\n ([x :=\n \\]\\' \\ \\')\n 3. \\\\ \\ \\ \\ \\'.\n \\[[\\]]\\ \\ \\;\n \\\\'.\n \\ REP \\' \\\n \\\\'.\n (\\ REP \\') \\\n ([\\]\\' \\ \\');\n \\ REP \\'\\\n \\ \\\\'.\n (\\ REP \\') \\\n ([\\ \\\\\n \\]\\' \\ \\')\n 4. \\\\ \\ \\ \\ \\'.\n \\[[\\]]\\ \\ \\;\n \\\\'.\n \\ REP \\' \\\n \\\\'.\n (\\ REP \\') \\\n ([\\]\\' \\ \\');\n \\ REP \\'\\\n \\ \\\\'.\n (\\ REP \\') \\\n ([\\ \\\\\n \\]\\' \\ \\')"} {"_id": "503095", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_minsky f g (map_minsky f' g' M) =\n map_minsky (f \\ f') (g \\ g') M"} {"_id": "503096", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\monotone ord (ord_spmf (=)) f;\n cont lub ord lub_spmf (ord_spmf (=)) f\\\n \\ monotone ord (\\) (\\p. spmf (f p) x)"} {"_id": "503097", "text": "proof (prove)\ngoal (1 subgoal):\n 1. H\\<^sub>L g (L.comp \\ \\) = g \\ \\ \\ \\"} {"_id": "503098", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xs \\ [] \\\n braun_list t xs =\n (\\l x r.\n t = \\l, x, r\\ \\\n x = hd xs \\\n braun_list l (take_nths 1 1 xs) \\\n braun_list r (take_nths 2 1 xs))"} {"_id": "503099", "text": "proof (prove)\nusing this:\n (P'', Q') \\ Rel\\\n\ngoal (1 subgoal):\n 1. (Q', P'') \\ Rel"} {"_id": "503100", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (F g \\\\<^sub>D \\\\<^sub>D[F f]) \\\\<^sub>D\n \\\\<^sub>D[F g, F f, F.map\\<^sub>0 (src\\<^sub>C f)] =\n \\\\<^sub>D[F g \\\\<^sub>D F f]"} {"_id": "503101", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(n_, dom_) \\ {(n, dom). dom = set [0.. {dom.\n set_mset (dom_m A) \\ dom \\ finite dom}\\\n \\ RETURN (2 ^ 64 \\ n_) \\ SPEC (\\_. True)"} {"_id": "503102", "text": "proof (prove)\ngoal (1 subgoal):\n 1. V \\ {} \\\n (if V\\<^bsub>T\\<^esub> = {} then 0 else max_bag_card - 1) < card V"} {"_id": "503103", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\Ball (set ws) Lyndon \\\n linorder.sorted lexordp_eq (rev ws);\n i \\ j; j < \\<^bold>|ws\\<^bold>|\\\n \\ ws ! j \\lex ws ! i"} {"_id": "503104", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rec_inseparable\n {p. 0 \\ nat_to_ce_set p \\ 1 \\ nat_to_ce_set p}\n {p. 0 \\ nat_to_ce_set p \\ 1 \\ nat_to_ce_set p}"} {"_id": "503105", "text": "proof (prove)\nusing this:\n L.in_hom \\ (L.cod \\) (L.cod \\)\n L.hom ?a ?b = hom ?a ?b \\ Collect (left a)\n\ngoal (1 subgoal):\n 1. \\\\ : L.cod\n \\ \\ L.cod\n \\\\ \\\n arr \\ \\ \\ \\ Collect (left a)"} {"_id": "503106", "text": "proof (prove)\ngoal (1 subgoal):\n 1. total_quasi_ordered_set A r = total_quasi_ordered_set A r'"} {"_id": "503107", "text": "proof (prove)\ngoal (1 subgoal):\n 1. pprefix_closed P"} {"_id": "503108", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Policy.sp_spec_subj_obj (Step.partition tid) (PAGE page) READ"} {"_id": "503109", "text": "proof (prove)\ngoal (1 subgoal):\n 1. w =\n restrict0 diff_fun_space\n (\\g. restrict0 sub.diff_fun_space v (restrict0 sub.carrier g))"} {"_id": "503110", "text": "proof (prove)\nusing this:\n P \\ C sees M, b : Ts\\T = m in D\n P \\ C' \\\\<^sup>* C\n wf_prog wf_md P\n \\D' Ts' T' m'.\n P \\ C' sees M, b : Ts'\\T' = m' in D' \\\n P \\ Ts [\\] Ts' \\ P \\ T' \\ T\n\ngoal (1 subgoal):\n 1. \\D' Ts' T' m'.\n P \\ C' sees M, b : Ts'\\T' = m' in D' \\\n P \\ Ts [\\] Ts' \\\n P \\ T' \\ T \\\n P \\ C' \\\\<^sup>* D' \\\n is_type P T' \\\n (\\T\\set Ts'. is_type P T) \\\n wf_md P D' (M, b, Ts', T', m')"} {"_id": "503111", "text": "proof (prove)\nusing this:\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs e\\<^sub>1,\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1 null,\n (h\\<^sub>1, ls\\<^sub>1,\n sh\\<^sub>1)\\ \\\n l\\<^sub>1 \\\\<^sub>m [Vs [\\] ls\\<^sub>1]\n length ls = length ls\\<^sub>1\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs e\\<^sub>2,\n (h\\<^sub>1, ls\\<^sub>1,\n sh\\<^sub>1)\\ \\\n \\fin\\<^sub>1 (Val v),\n (h\\<^sub>2, ls\\<^sub>2,\n sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [Vs [\\] ls\\<^sub>2]\n P \\ \\e\\<^sub>1,(h, l, sh)\\ \\\n \\null,(h\\<^sub>1, l\\<^sub>1, sh\\<^sub>1)\\\n \\fv e\\<^sub>1 \\ set ?Vs;\n l \\\\<^sub>m [?Vs [\\] ?ls];\n length ?Vs + max_vars e\\<^sub>1 \\ length ?ls\\\n \\ \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 ?Vs\n e\\<^sub>1,\n (h, ?ls, sh)\\ \\\n \\fin\\<^sub>1 null,\n (h\\<^sub>1, ls', sh\\<^sub>1)\\ \\\n l\\<^sub>1 \\\\<^sub>m [?Vs [\\] ls']\n P \\ \\e\\<^sub>2,\n (h\\<^sub>1, l\\<^sub>1, sh\\<^sub>1)\\ \\\n \\Val v,(h\\<^sub>2, l\\<^sub>2, sh\\<^sub>2)\\\n \\fv e\\<^sub>2 \\ set ?Vs;\n l\\<^sub>1 \\\\<^sub>m [?Vs [\\] ?ls];\n length ?Vs + max_vars e\\<^sub>2 \\ length ?ls\\\n \\ \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 ?Vs\n e\\<^sub>2,\n (h\\<^sub>1, ?ls, sh\\<^sub>1)\\ \\\n \\fin\\<^sub>1 (Val v),\n (h\\<^sub>2, ls', sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [?Vs [\\] ls']\n fv (e\\<^sub>1\\F{D} := e\\<^sub>2) \\ set Vs\n l \\\\<^sub>m [Vs [\\] ls]\n length Vs + max_vars (e\\<^sub>1\\F{D} := e\\<^sub>2) \\ length ls\n\ngoal (1 subgoal):\n 1. \\ls'.\n compP\\<^sub>1\n P \\\\<^sub>1 \\compE\\<^sub>1 Vs\n (e\\<^sub>1\\F{D} := e\\<^sub>2),\n (h, ls, sh)\\ \\\n \\fin\\<^sub>1 (THROW NullPointer),\n (h\\<^sub>2, ls', sh\\<^sub>2)\\ \\\n l\\<^sub>2 \\\\<^sub>m [Vs [\\] ls']"} {"_id": "503112", "text": "proof (prove)\nusing this:\n \\i.\n dterm_sem I (if i = vid1 then \\' else Const 0) (a, b) =\n dterm_sem I (if i = vid1 then \\ else Const 0) (a, b)\n\ngoal (1 subgoal):\n 1. Functions I var\n (\\i.\n dterm_sem I (if i = vid1 then \\ else Const 0) (a, b)) =\n Functions I var\n (\\i. dterm_sem I (if i = vid1 then \\' else Const 0) (a, b))"} {"_id": "503113", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^bsub>\\\\<^esub> c \\ Strongly_Secure [c]"} {"_id": "503114", "text": "proof (prove)\nusing this:\n Cop_S.inverse_arrows\n (Cop_S.MkArr (\\'.Y a) (\\'.Y a') \\\\'.map)\n (Cop_S.MkArr (\\'.Y a') (\\'.Y a) \\'\\.map)\n\ngoal (1 subgoal):\n 1. Cop_S.iso (Cop_S.MkArr (\\'.Y a) (\\'.Y a') \\\\'.map)"} {"_id": "503115", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s.\n s \\ UNIV1 \\\n Timess (pderivs s r1) r2 \\ pderivs_lang (PSuf s) r2\n \\ Timess (pderivs_lang UNIV1 r1) r2 \\\n pderivs_lang UNIV1 r2"} {"_id": "503116", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (|vars xs| finite (vars xs)) \\\n (|\\ (range terms)| \n finite (\\ (range terms))) \\\n Ball (\\ (range terms)) good \\\n (|\\xs s. abs (xs, s)| \n finite (\\xs s. abs (xs, s))) \\\n Ball (\\xs s. abs (xs, s)) goodAbs \\\n (|envs| finite envs) \\ Ball envs goodEnv"} {"_id": "503117", "text": "proof (prove)\ngoal (1 subgoal):\n 1. op_dghm \\ :\n op_dg\n \\ \\\\\\<^sub>D\\<^sub>G\\<^sub>.\\<^sub>t\\<^sub>m\\<^bsub>\\\\<^esub> op_dg\n \\"} {"_id": "503118", "text": "proof (prove)\nusing this:\n system_step q sh' sh\n atCs (cPGM (GST sh p)) \\ {{}}\n\ngoal (1 subgoal):\n 1. atCs (cPGM (GST sh' p)) \\ {{}}"} {"_id": "503119", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\group_hom (relative_homology_group p X S)\n (homology_group (p - 1) (subtopology X S))\n (hom_boundary p X S) \\\n hom_induced (p - 1) (subtopology X S) {} X {} id\n \\ hom (homology_group (p - 1) (subtopology X S))\n (homology_group (p - 1) X) \\\n (\\x.\n (x \\ carrier\n (homology_group (p - 1) (subtopology X S)) \\\n (\\xa.\n (xa \\ hom_induced (p - 1) (subtopology X S) {} X\n {} id x) =\n (xa \\ \\\\<^bsub>homology_group (p - 1) X\\<^esub>))) =\n (x \\ hom_boundary p X S `\n carrier (relative_homology_group p X S)));\n hom_boundary p X S C\n \\ carrier (homology_group (p - 1) (subtopology X S))\\\n \\ hom_boundary p X S C\n \\ hom_boundary p X S `\n carrier (relative_homology_group p X S)"} {"_id": "503120", "text": "proof (prove)\nusing this:\n ?\\ closes (a, \\) # \\ \\\n ?\\ \\ RED \\\n a \\ \\\n a \\ \\\n \\ closes \\\n ?a \\ ?\\ \\\n \\\\. (?a, \\) \\ set ?\\\n\ngoal (1 subgoal):\n 1. \\s\\RED \\. ((a, s) # \\) \\ RED \\"} {"_id": "503121", "text": "proof (prove)\ngoal (1 subgoal):\n 1. natural_transformation (\\\\<^sub>C) (\\\\<^sub>B) F'\n (FG.GF.map \\ F') (FG.\\ \\ F')"} {"_id": "503122", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dist (center + radius *\\<^sub>R of_radiant \\)\n (center + radius *\\<^sub>R of_radiant \\) =\n sqrt\n (radius\\<^sup>2 *\n ((cos \\ - cos \\)\\<^sup>2 +\n (sin \\ - sin \\)\\<^sup>2))"} {"_id": "503123", "text": "proof (state)\nthis:\n prefix as_1 as\n exec_plan s as_1 = last slist_1\n length slist_1 = Suc (length as_1)\n\ngoal (1 subgoal):\n 1. \\as1 as2 as3.\n as1 @ as2 @ as3 = as \\\n exec_plan s (as1 @ as2) = exec_plan s as1 \\ as2 \\ []"} {"_id": "503124", "text": "proof (prove)\ngoal (4 subgoals):\n 1. \\x1 x2 x y.\n atom y \\ x1 EQ x2 \\\n (x \\ y) \\ x1 EQ x2 = (x1 EQ x2)(x::=Var y)\n 2. \\x1 x2 x y.\n \\\\b ba.\n atom ba \\ x1 \\\n (b \\ ba) \\ x1 = x1(b::=Var ba);\n \\b ba.\n atom ba \\ x2 \\\n (b \\ ba) \\ x2 = x2(b::=Var ba);\n atom y \\ x1 OR x2\\\n \\ (x \\ y) \\ x1 OR x2 =\n (x1 OR x2)(x::=Var y)\n 3. \\x xa y.\n \\\\b ba.\n atom ba \\ x \\\n (b \\ ba) \\ x = x(b::=Var ba);\n atom y \\ neg x\\\n \\ (xa \\ y) \\ neg x =\n (neg x)(xa::=Var y)\n 4. \\x1 x2 x y.\n \\atom x1 \\ x; atom x1 \\ y;\n \\b ba.\n atom ba \\ x2 \\\n (b \\ ba) \\ x2 = x2(b::=Var ba);\n atom y \\ exi x1 x2\\\n \\ (x \\ y) \\ exi x1 x2 =\n (exi x1 x2)(x::=Var y)"} {"_id": "503125", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.dioid_one_zero (\\) (\\) \\ \\\n mat_less_eq mat_less"} {"_id": "503126", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (as, x) \\lst (=) supp p (as', x')"} {"_id": "503127", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ord_option ord (lub_option lub Y) y"} {"_id": "503128", "text": "proof (prove)\nusing this:\n lookup (MP_oalist ys) v = (0::'b)\n lookup (MP_oalist xs) v \\ (0::'b)\n lt_of_nat_term_order cmp_term v ?u \\\n lookup (MP_oalist ys) ?u = lookup (MP_oalist xs) ?u\n\ngoal (1 subgoal):\n 1. ord_strict_p (MP_oalist ys) (MP_oalist xs)"} {"_id": "503129", "text": "proof (prove)\ngoal (1 subgoal):\n 1. run_gpv hash.oracle game0 Map.empty =\n map_spmf fst\n (sample_uniform (order \\) \\\n (\\x. sample_uniform (order \\) \\ game1 x))"} {"_id": "503130", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\f'.\n \\continuous_on U f'; f' ` U \\ C;\n \\x.\n x \\ T \\ f' x = (g \\ f) x\\\n \\ thesis) \\\n thesis"} {"_id": "503131", "text": "proof (state)\nthis:\n aS\\<^bsub>T\\<^esub> = iS\\<^bsub>CI T\\<^esub>\n\ngoal (1 subgoal):\n 1. \\ \\ Axioms.Theory T"} {"_id": "503132", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\i c\\<^sub>C mds\\<^sub>C mem\\<^sub>C mem\\<^sub>C'.\n \\i < length cms; cms\\<^sub>C ! i = (c\\<^sub>C, mds\\<^sub>C);\n conc.low_mds_eq mds\\<^sub>C mem\\<^sub>C mem\\<^sub>C'\\\n \\ (\\fst (cms\\<^sub>A !\n i), snd (cms\\<^sub>A ! i), mem\\<^sub>A_of mem\\<^sub>C\\\\<^sub>A,\n \\fst (cms\\<^sub>A !\n i), snd (cms\\<^sub>A ! i), mem\\<^sub>A_of mem\\<^sub>C'\\\\<^sub>A)\n \\ \\\\<^sub>A_rel (cms ! i) \\\n (\\c\\<^sub>C, mds\\<^sub>C, mem\\<^sub>C\\\\<^sub>C,\n \\c\\<^sub>C, mds\\<^sub>C, mem\\<^sub>C'\\\\<^sub>C)\n \\ P_rel (cms ! i)\n 2. \\mem.\n True \\ conc.sound_mode_use (cms\\<^sub>C, mem)"} {"_id": "503133", "text": "proof (prove)\ngoal (1 subgoal):\n 1. term_variants_pred P (Fun f (S @ t # T)) (Fun g (S @ s # T))"} {"_id": "503134", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\H \\ t EQ t'; H \\ u EQ u'\\\n \\ H \\ pls t u EQ pls t' u'"} {"_id": "503135", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A' \\# remove1_mset a (remdups_mset Aa) = A' \\# Aa"} {"_id": "503136", "text": "proof (state)\nthis:\n space (K x) \\ {}\n\ngoal (3 subgoals):\n 1. sets\n (K x \\\n (\\y.\n distr (lim_sequence y) (Pi\\<^sub>M UNIV (\\j. M))\n (case_nat y))) =\n sets (Pi\\<^sub>M UNIV (\\j. M))\n 2. \\A J.\n \\finite J; J \\ UNIV;\n \\i. i \\ J \\ A i \\ sets M\\\n \\ emeasure (lim_sequence x)\n (prod_emb UNIV (\\j. M) J\n (Pi\\<^sub>E J A)) =\n emeasure\n (K x \\\n (\\y.\n distr (lim_sequence y)\n (Pi\\<^sub>M UNIV (\\j. M))\n (case_nat y)))\n (prod_emb UNIV (\\j. M) J (Pi\\<^sub>E J A))\n 3. finite_measure (lim_sequence x)"} {"_id": "503137", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\single_valued R; x \\ Domain R\\\n \\ (x, construct_fun_from_rel R x) \\ R"} {"_id": "503138", "text": "proof (prove)\ngoal (1 subgoal):\n 1. B \\\\<^bsub>relative_homology_group p X S\\<^esub>\n inv\\<^bsub>relative_homology_group p X S\\<^esub> hom_induced p X T X S\n id D =\n hom_induced p X T X S id (hom_induced p X {} X T id W) \\\n B =\n hom_induced p X T X S id\n D \\\\<^bsub>relative_homology_group p X S\\<^esub>\n hom_induced p X T X S id (hom_induced p X {} X T id W)"} {"_id": "503139", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\x\\cnf. sat_clause \\ x; x \\ cnf;\n (l\\<^sub>1, l\\<^sub>2) \\ edges_of_clause x;\n sat_lit \\ l\\<^sub>1\\\n \\ sat_lit \\ l\\<^sub>2"} {"_id": "503140", "text": "proof (prove)\nusing this:\n ldistinct (lmap f xs)\n\ngoal (1 subgoal):\n 1. ldistinct xs"} {"_id": "503141", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (mat_rel R ===> mat_rel R ===> (=)) (=) (=)"} {"_id": "503142", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s t ss tt n na.\n \\secure c;\n \\s ss.\n s \\ ss \\ evalB b s = evalB b ss;\n s \\ t ; s , While b c \\\\<^sub>n ss ;\n t , While b c \\\\<^sub>na tt \\\n \\ ss \\ tt "} {"_id": "503143", "text": "proof (prove)\nusing this:\n ?m \\ ?n \\ f ?n \\ f ?m\n\ngoal (1 subgoal):\n 1. f \\ Inf (range f)"} {"_id": "503144", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\left_inverse_matrix A Y; right_inverse_matrix A X\\\n \\ Y * A * X = X"} {"_id": "503145", "text": "proof (state)\nthis:\n even n\n\ngoal (2 subgoals):\n 1. real_of_float x = 0 \\\n cos (real_of_float x)\n \\ {real_of_float\n (lb_sin_cos_aux prec (get_even n) 1 1\n (x *\n x))..real_of_float\n (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}\n 2. real_of_float x \\ 0 \\\n cos (real_of_float x)\n \\ {real_of_float\n (lb_sin_cos_aux prec (get_even n) 1 1\n (x *\n x))..real_of_float\n (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"} {"_id": "503146", "text": "proof (prove)\nusing this:\n winning_path p (ldropn n P)\n ltl (ldropn n P) = ldropn (Suc n) P\n \\ lnull (ldropn n P)\n \\winning_path p (ldropn n P); \\ lnull (ldropn n P);\n \\ lnull (ltl (ldropn n P))\\\n \\ winning_path p (ltl (ldropn n P))\n enat (Suc n) < llength P\n\ngoal (1 subgoal):\n 1. winning_path p (ldropn (Suc n) P)"} {"_id": "503147", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly_mult (local.monom a n) p =\n local.normalize (map ((\\) a) p @ replicate n \\)"} {"_id": "503148", "text": "proof (prove)\nusing this:\n arr f\n x \\ Dom f\n arr ?f \\ ide (cod ?f)\n arr f \\\n Fun f \\ extensional (Dom f) \\ (Dom f \\ Cod f)\n ide ?a \\ local.set ?a \\ Univ\n\ngoal (1 subgoal):\n 1. Fun f x \\ Univ"} {"_id": "503149", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\B.\n \\x \\ roofed_gen G B S; path G x p y; y \\ B\\\n \\ (\\z\\set p. z \\ S) \\ x \\ S"} {"_id": "503150", "text": "proof (chain)\npicking this:\n 2 ^ (k + 1) = 2 *n 2 ^ k"} {"_id": "503151", "text": "proof (prove)\nusing this:\n 0 + int (card X) - int j - 1 gchoose (card X - j - 1) =\n 0 - int (poly_deg f) + int (card X) - int j - 1 gchoose (card X - j - 1) +\n (0 - int (poly_deg f) + int (card X) - int j gchoose (card X - j)) +\n (0 + int (card X) - int j gchoose (card X - j)) -\n 2 -\n (\\i = Suc j..card X.\n 0 - int (aa i) + int i - int j - 1 gchoose (i - j) +\n (0 - int (bb i) + int i - int j - 1 gchoose (i - j)))\n\ngoal (1 subgoal):\n 1. 1 =\n (- 1) ^ (card X - Suc j) *\n (int (poly_deg f) - 1 gchoose (card X - Suc j)) +\n (- 1) ^ (card X - j) * (int (poly_deg f) - 1 gchoose (card X - j)) -\n 1 -\n (\\i = Suc j..card X.\n (- 1) ^ (i - j) *\n (int (aa i) gchoose (i - j) + (int (bb i) gchoose (i - j))))"} {"_id": "503152", "text": "proof (prove)\ngoal (2 subgoals):\n 1. uint asi = 15 \\\n fst (case read_data_cache s addr of None \\ (None, s)\n | Some w \\ (Some w, s)) =\n fst (case read_data_cache\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr of\n None \\\n (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n | Some w \\\n (Some w, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))\n 2. \\uint asi \\ 1; uint asi \\ 2;\n uint asi \\ {8, 9}; uint asi \\ {10, 11};\n uint asi \\ 13; uint asi \\ 15\\\n \\ (uint addr = 12 \\\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem s addr (word_of_int 8)\n in if r1 = None\n then let r2 = load_word_mem s addr (word_of_int 10)\n in if r2 = None then (None, s) else (r2, s)\n else (r1, s)\n else if asi_int = 2\n then (Some (sys_reg_val DCCR s), s)\n else if asi_int \\ {8, 9}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_instr_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int \\ {10, 11}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_data_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int = 13\n then Let (read_instr_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int = 15\n then Let (read_data_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int \\ {16, 17} then (None, s)\n else if asi_int \\ {20, 21} then (None, s)\n else if asi_int = 24 then (None, s)\n else if asi_int = 25\nthen (mmu_reg_val (mmu s) addr, s)\nelse if asi_int = 28 then (mem_val_w32 asi (ucast addr) s, s)\n else if asi_int = 29 then (None, s) else (None, s)) =\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 8)\n in if r1 = None\n then let r2 = load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 10)\n in if r2 = None\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r2, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r1, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 2\n then (Some\n (sys_reg_val DCCR\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int \\ {8, 9}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int \\ {10, 11}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 13\n then Let (read_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)))\n else if asi_int = 15\n then Let (read_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int \\ {16, 17}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {20, 21}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 24\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 25\nthen (mmu_reg_val\n (mmu (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))\n addr,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\nelse if asi_int = 28\n then (mem_val_w32 asi (ucast addr)\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 29\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))) \\\n (uint addr \\ 12 \\\n (uint addr = 8 \\\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem s addr (word_of_int 8)\n in if r1 = None\n then let r2 = load_word_mem s addr (word_of_int 10)\n in if r2 = None then (None, s) else (r2, s)\n else (r1, s)\n else if asi_int = 2\n then (Some (sys_reg_val ICCR s), s)\n else if asi_int \\ {8, 9}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_instr_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int \\ {10, 11}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_data_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int = 13\n then Let (read_instr_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int = 15\n then Let (read_data_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int \\ {16, 17} then (None, s)\n else if asi_int \\ {20, 21} then (None, s)\n else if asi_int = 24 then (None, s)\n else if asi_int = 25\n then (mmu_reg_val (mmu s) addr, s)\n else if asi_int = 28 then (mem_val_w32 asi (ucast addr) s, s)\n else if asi_int = 29 then (None, s) else (None, s)) =\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 8)\n in if r1 = None\n then let r2 = load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 10)\n in if r2 = None\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r2, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r1, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 2\n then (Some\n (sys_reg_val ICCR\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int \\ {8, 9}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int \\ {10, 11}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 13\n then Let (read_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int = 15\n then Let (read_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int \\ {16, 17}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {20, 21}\n then (None, s\n \\cpu_reg := new_cpu_reg,\nuser_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var,\ntraps := new_traps, undef := new_undef\\)\n else if asi_int = 24\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 25\n then (mmu_reg_val\n (mmu (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))\n addr,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 28\n then (mem_val_w32 asi (ucast addr)\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 29\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))) \\\n (uint addr \\ 8 \\\n (uint addr = 0 \\\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem s addr (word_of_int 8)\nin if r1 = None\n then let r2 = load_word_mem s addr (word_of_int 10)\n in if r2 = None then (None, s) else (r2, s)\n else (r1, s)\n else if asi_int = 2\nthen (Some (sys_reg_val CCR s), s)\nelse if asi_int \\ {8, 9}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_instr_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int \\ {10, 11}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w.\n (Some w, add_data_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int = 13\n then Let (read_instr_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int = 15\n then Let (read_data_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int \\ {16, 17} then (None, s)\n else if asi_int \\ {20, 21} then (None, s)\n else if asi_int = 24 then (None, s)\n else if asi_int = 25\n then (mmu_reg_val (mmu s) addr, s)\n else if asi_int = 28 then (mem_val_w32 asi (ucast addr) s, s)\n else if asi_int = 29 then (None, s) else (None, s)) =\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 8)\nin if r1 = None\n then let r2 = load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr (word_of_int 10)\n in if r2 = None\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r2, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r1, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 2\nthen (Some\n (sys_reg_val CCR\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\nelse if asi_int \\ {8, 9}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {10, 11}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 13\n then Let (read_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int = 15\n then Let (read_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int \\ {16, 17}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {20, 21}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 24\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 25\n then (mmu_reg_val\n (mmu (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))\n addr,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 28\n then (mem_val_w32 asi (ucast addr)\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 29\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\))) \\\n (uint addr \\ 0 \\\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem s addr (word_of_int 8)\nin if r1 = None\n then let r2 = load_word_mem s addr (word_of_int 10)\n in if r2 = None then (None, s) else (r2, s)\n else (r1, s)\n else if asi_int = 2 then (None, s)\nelse if asi_int \\ {8, 9}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w. (Some w, add_instr_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int \\ {10, 11}\n then let ccr_val = sys_reg s CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem s addr asi)\n (case_option (None, s)\n (\\w.\n (Some w, add_data_cache s addr w 15)))\n else (load_word_mem s addr asi, s)\n else if asi_int = 13\n then Let (read_instr_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int = 15\n then Let (read_data_cache s addr)\n (case_option (None, s) (\\w. (Some w, s)))\n else if asi_int \\ {16, 17} then (None, s)\n else if asi_int \\ {20, 21} then (None, s)\n else if asi_int = 24 then (None, s)\n else if asi_int = 25\n then (mmu_reg_val (mmu s) addr, s)\n else if asi_int = 28 then (mem_val_w32 asi (ucast addr) s, s)\n else if asi_int = 29 then (None, s) else (None, s)) =\n fst (let asi_int = uint asi\n in if asi_int = 1\n then let r1 =\n load_word_mem\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n addr (word_of_int 8)\nin if r1 = None\n then let r2 = load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr (word_of_int 10)\n in if r2 = None\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r2, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (r1, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 2\nthen (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\nelse if asi_int \\ {8, 9}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {10, 11}\n then let ccr_val =\n sys_reg\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n CCR\n in if ccr_val AND 1 \\ 0\n then Let (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w,\n add_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr w 15)))\n else (load_word_mem\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr asi,\n s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 13\n then Let (read_instr_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite,\n state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int = 15\n then Let (read_data_cache\n (s\\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n addr)\n (case_option\n (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n (\\w.\n (Some w, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)))\n else if asi_int \\ {16, 17}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg,\n dwrite := new_dwrite,\n state_var := new_state_var,\n traps := new_traps,\n undef := new_undef\\)\n else if asi_int \\ {20, 21}\n then (None, s\n \\cpu_reg := new_cpu_reg,\n user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 24\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var, traps := new_traps,\n undef := new_undef\\)\n else if asi_int = 25\n then (mmu_reg_val\n (mmu (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\))\n addr,\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 28\n then (mem_val_w32 asi (ucast addr)\n (s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\),\n s\\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else if asi_int = 29\n then (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)\n else (None, s\n \\cpu_reg := new_cpu_reg, user_reg := new_user_reg,\n dwrite := new_dwrite, state_var := new_state_var,\n traps := new_traps, undef := new_undef\\)))))"} {"_id": "503153", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set ancestor_ancestors \\ set ancestors"} {"_id": "503154", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (gbai_L, gba_L)\n \\ \\Rv, Re,\n Rv0\\gen_gba_impl_rel_ext Rm Rl Racc \\ Rl &&&\n (gen_gba_impl_ext, gba_rec_ext)\n \\ Rl \\\n Rm \\ \\Rm, Rl\\gen_gba_impl_rel_eext"} {"_id": "503155", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite {len m i j xs |xs. set xs \\ {0..k} \\ distinct xs}"} {"_id": "503156", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\atom i \\ (j, A); atom j \\ A\\\n \\ {A} \\ (Zero IN Var j)(j::=Eats Zero Zero)\n 2. \\atom i \\ (j, A); atom j \\ A\\\n \\ {Var i IN Var j} \\ A"} {"_id": "503157", "text": "proof (prove)\nusing this:\n x \\ E\n\ngoal (1 subgoal):\n 1. 0 \\ \\x\\"} {"_id": "503158", "text": "proof (prove)\nusing this:\n order.greater n 0\n order.greater (add_order (s + t)) n \\ add_order (s + t) = 0\n \\order.greater ?k 0; order.greater (add_order ?a) ?k\\\n \\ ?a +^ ?k \\ (0::?'a)\n \\add_order ?a = 0; order.greater ?n 0\\\n \\ ?a +^ ?n \\ (0::?'a)\n sum_list (alternating_list (2 * n) s t) = (0::'w)\n\ngoal (1 subgoal):\n 1. False"} {"_id": "503159", "text": "proof (prove)\nusing this:\n bar []\n\ngoal (1 subgoal):\n 1. \\vs. is_prefix vs f \\ good vs"} {"_id": "503160", "text": "proof (prove)\nusing this:\n x' = Some x''\n r \\ range c\n z \\ results r\n Resumption.resumption.Pause out c = x \\ f\n x = Resumption.resumption.Done x'\n\ngoal (1 subgoal):\n 1. z \\ (\\a\\results x. results (f a))"} {"_id": "503161", "text": "proof (prove)\ngoal (1 subgoal):\n 1. abs_conv_abscissa (f - g)\n \\ max (abs_conv_abscissa f) (abs_conv_abscissa g)"} {"_id": "503162", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {} \\\n {(=) Z} Impl M {\\t. \\n. Z -Impl M-n\\ t}"} {"_id": "503163", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_integrable lborel ({0..1 / 2} \\ {1 / 2..1})\n (\\t. t powr (a - 1) * (1 - t) powr (b - 1))"} {"_id": "503164", "text": "proof (prove)\nusing this:\n a \\ open_segment (a - complex_of_real d) (a + complex_of_real e)\n\ngoal (1 subgoal):\n 1. open_segment (a - complex_of_real d) (a + complex_of_real e)\n \\ inside (path_image p)"} {"_id": "503165", "text": "proof (prove)\ngoal (1 subgoal):\n 1. unit_counit_adjunction (\\\\<^sub>C) (\\\\<^sub>D) F G \\\n \\"} {"_id": "503166", "text": "proof (prove)\ngoal (1 subgoal):\n 1. D InAngle A B C"} {"_id": "503167", "text": "proof (prove)\nusing this:\n ?h \\ Maps.Comp ?G ?F \\\n Maps.Comp ?G ?F = \\?h\\\n \\ide ?f; ide ?g; src ?f = trg ?g\\\n \\ \\\\?f \\ ?g\\\\ =\n Maps.MkArr (src ?g) (trg ?f)\n (Maps.Comp \\?f\\ \\?g\\)\n is_left_adjoint cmp.chine\n rs.cmp = cmp.chine\n\ngoal (1 subgoal):\n 1. Maps.Comp \\tab\\<^sub>0 (r \\ s)\\\n \\rs.cmp\\ =\n \\tab\\<^sub>0 (r \\ s) \\ rs.cmp\\"} {"_id": "503168", "text": "proof (prove)\nusing this:\n [[a x c]]\n [[?a ?b ?c]] \\ \\ [[?b ?a ?c]]\n [[?a ?b ?c]] \\ \\ [[?a ?c ?b]]\n [[?a ?b ?c]] \\ \\ [[?b ?c ?a]]\n [[?a ?b ?c]] \\ \\ [[?c ?a ?b]]\n\ngoal (1 subgoal):\n 1. \\ ([[b a x]] \\ [[b c x]] \\ x = a \\ x = c)"} {"_id": "503169", "text": "proof (prove)\nusing this:\n length w < n\n \\x\\set I.\n case x of Inl p \\ p \\ n\n | Inr P \\ \\p\\P. p \\ n\n\ngoal (1 subgoal):\n 1. sdrop (Suc n) (stream_enc (w, I)) =\n sconst (any, replicate (length I) False)"} {"_id": "503170", "text": "proof (state)\ngoal (1 subgoal):\n 1. isCont (\\x. if x \\ a then g x else f x) a"} {"_id": "503171", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\card A \\ 0;\n \\a b.\n \\a \\ A; b \\ A; a \\ b\\\n \\ coprime a b\\\n \\ Lcm A = normalize (\\A)"} {"_id": "503172", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z.\n - - (x \\ y) \\ - (- x \\ z) =\n - (- x \\ z)"} {"_id": "503173", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a.\n a \\ (0::'a) \\ span S \\ {x. a \\ x = 0}"} {"_id": "503174", "text": "proof (prove)\nusing this:\n awalk u (p @ q) w\n v \\ set (awalk_verts u (p @ q))\n awalk u p v\n awalk v q w\n\ngoal (1 subgoal):\n 1. set (awalk_verts u (p @ q)) =\n set (awalk_verts u p) \\ set (awalk_verts v q)"} {"_id": "503175", "text": "proof (prove)\nusing this:\n \\\\<^sub>l\\<^sub>b (lt dir) (\\\\ s\\ x\\<^sub>i)\n (LB dir s x\\<^sub>i)\n \\ \\\\<^sub>u\\<^sub>b (lt dir)\n (\\\\ s\\ x\\<^sub>i) (UB dir s x\\<^sub>i)\n \\\\ s\\ x\\<^sub>i' <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i' \\\n c = the (\\\\<^sub>l s x\\<^sub>i') \\\n min_rvar_incdec Positive s x\\<^sub>i' = Inr x\\<^sub>j \\\n \\ \\\\ s\\ x\\<^sub>i' <\\<^sub>l\\<^sub>b\n \\\\<^sub>l s x\\<^sub>i' \\\n c = the (\\\\<^sub>u s x\\<^sub>i') \\\n min_rvar_incdec Negative s x\\<^sub>i' = Inr x\\<^sub>j\n x\\<^sub>i = x\\<^sub>i'\n s' = pivot_and_update x\\<^sub>i' x\\<^sub>j c s\n \\ (\\ s)\n \\ s\n x\\<^sub>i' \\ lvars (\\ s)\n x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ s) x\\<^sub>i')\n \\\\ (\\ s); \\ s; x\\<^sub>i \\ lvars (\\ s);\n x\\<^sub>j \\ rvars_eq (eq_for_lvar (\\ s) x\\<^sub>i)\\\n \\ Mapping.lookup\n (\\ (pivot_and_update x\\<^sub>i x\\<^sub>j c s))\n x\\<^sub>i =\n Some c\n dir = Positive \\ dir = Negative\n\ngoal (1 subgoal):\n 1. \\\\ s'\\ x\\<^sub>i = the (LB dir s x\\<^sub>i)"} {"_id": "503176", "text": "proof (prove)\nusing this:\n e \\ set (cycle_edges p)\n h_minus i e = h_plus i e\n cycle \\ p\n\ngoal (1 subgoal):\n 1. h_diff i e = 0"} {"_id": "503177", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a \\ 0 \\ p \\ 0;\n poly_y_x (poly_y_x p) = p\\\n \\ poly_y_x (poly_y_x ([:a:] + pCons 0 p)) =\n [:a:] + pCons 0 p"} {"_id": "503178", "text": "proof (state)\nthis:\n \\tst c1 c2.\n ?c3 = if tst then c1 else c2 \\\n discr c1 \\ discr c2 \\\n discr ?c3\n\ngoal (1 subgoal):\n 1. discr (if tst then c1 else c2)"} {"_id": "503179", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x + y = y + x"} {"_id": "503180", "text": "proof (prove)\nusing this:\n c \\ i\n\ngoal (1 subgoal):\n 1. (case inv_map \\ i of None \\ 0\n | Some a \\ ket a) \\\\<^sub>C\n ket j =\n ket i \\\\<^sub>C\n (case \\ j of None \\ 0 | Some a \\ ket a)"} {"_id": "503181", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mset (zip (take j xsa @ x # drop j xsa) (take j ysa @ y # drop j ysa)) =\n mset (zip (x # xs) (y # ys))"} {"_id": "503182", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dim_row (post_meas0 2 |\\\\<^sub>1\\<^sub>0\\ 1) =\n dim_row |unit_vec 4 0\\"} {"_id": "503183", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n \\spmf\n (connect (\\ x)\n (sim_outer x \\ sim_inner x |\\<^sub>= 1\\<^sub>C \\\n ideal x))\n True -\n ?c12 x\\ +\n \\?c12 x - spmf (connect (\\ x) (real x)) True\\\n \\ \\spmf\n (connect\n (absorb (\\ x) (sim_outer x |\\<^sub>= 1\\<^sub>C))\n (sim_inner x |\\<^sub>= 1\\<^sub>C \\ ideal x))\n True -\n spmf\n (connect\n (absorb (\\ x) (sim_outer x |\\<^sub>= 1\\<^sub>C))\n (middle x))\n True\\ +\n \\spmf\n (connect (\\ x)\n (sim_outer x |\\<^sub>= 1\\<^sub>C \\ middle x))\n True -\n spmf (connect (\\ x) (real x)) True\\"} {"_id": "503184", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\U.\n U \\ top.neighborhoods \\ \\\n ring_homomorphism (local.key_map U) (\\ U) (add_sheaf_spec U)\n (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U)\n pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at\n pi.zero_local_ring_at pi.one_local_ring_at"} {"_id": "503185", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\PAC_Format (\\, A) (\\', B);\n \\ (vars ` set_mset A) \\ \\\\\n \\ {p \\ pac_ideal (set_mset B).\n vars p \\ \\}\n \\ {p \\ pac_ideal (set_mset A).\n vars p \\ \\} \\\n \\ \\ \\' \\\n \\ (vars ` set_mset B) \\ \\'"} {"_id": "503186", "text": "proof (prove)\nusing this:\n List.member (sorted_list_of_set {x. \\q\\set qs. rpoly q x = 0})\n r\n\ngoal (1 subgoal):\n 1. sorted_list_of_set {x. \\q\\set qs. rpoly q x = 0} ! 0 \\ r"} {"_id": "503187", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((ccExp_syn a (Var x) = \\ &&&\n ccExp_syn 0 (Lam [x]. e) = fv (Lam [x]. e)\\<^sup>2) &&&\n ccExp_syn (inc\\a) (Lam [x]. e) = cc_delete x (ccExp_syn a e) &&&\n ccExp_syn a (App e x) =\n ccExp_syn (inc\\a) e \\ {x} G\\ insert x (fv e)) &&&\n ((\\ nonrec \\ \\\n ccExp_syn a (Terms.Let \\ e) =\n CCfix \\\\\n (Afix \\\\\n (Aexp_syn' a e \\\n (\\_. up\\0) f|` thunks \\),\n ccExp_syn a e) G|`\n (- domA \\)) &&&\n (x \\ fv e' \\\n ccExp_syn a (let x be e' in e ) =\n cc_delete x\n (ccBind x e'\\\n (Aheap_nonrec x e'\\(Aexp_syn' a e, ccExp_syn a e),\n ccExp_syn a e) \\\n fv e' G\\\n (ccNeighbors x (ccExp_syn a e) -\n (if isVal e' then {} else {x})) \\\n ccExp_syn a e))) &&&\n ccExp_syn a (Bool b) = \\ &&&\n ccExp_syn a (scrut ? e1 : e2) =\n ccExp_syn 0 scrut \\\n (ccExp_syn a e1 \\ ccExp_syn a e2) \\\n edom (Aexp_syn' 0 scrut) G\\\n (edom (Aexp_syn' a e1) \\ edom (Aexp_syn' a e2))"} {"_id": "503188", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\r x.\n Some r =\n map_fun (\\x. x) id Some\n (\\u s w.\n case u of \\\\ x \\ ConcreteInWorld x w\n | \\\\ \\ \\\n False) \\\n \\w. \\\\ x \\ {x. r (\\\\ x) dj w}"} {"_id": "503189", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Abs_uint64 (numeral n) = numeral n"} {"_id": "503190", "text": "proof (prove)\ngoal (1 subgoal):\n 1. igSwapIGAbsSTR MOD \\ igSwapIGAbs MOD"} {"_id": "503191", "text": "proof (prove)\ngoal (1 subgoal):\n 1. test (x \\ y)"} {"_id": "503192", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lang Allreg = (\\a. {[a]})"} {"_id": "503193", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\x xa.\n \\y\\1.\n (P x \\ Q xa \\ x xa = 0 \\ y = 0) \\\n (P x \\ Q xa \\ x xa = 1 \\ y = 1) \\\n (P x \\ Q xa \\ y = 0 \\\n x xa \\ f x xa) \\\n (P x \\ Q xa \\ y = 1 \\ f x xa \\ x xa)"} {"_id": "503194", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\unfree t = None; unapp t = None; unconst t = None;\n term_cases (\\_. True) (\\_. True)\n (\\t u. no_abs t \\ no_abs u) False t\\\n \\ False"} {"_id": "503195", "text": "proof (state)\nthis:\n A \\ \\l, D\\ \\\\<^bsub>(k \\\n v'),v,n\\<^esub>* \\l', D'\\\n valid_dbm D n\n [D']\\<^bsub>v,n\\<^esub> \\ {}\n\ngoal (1 subgoal):\n 1. (\\k v n.\n A \\ \\l, D\\ \\\\<^bsub>k,v,n\\<^esub>* \\l', D'\\ \\\n valid_dbm D n \\\n [D']\\<^bsub>v,n\\<^esub> \\ {} \\\n (\\Z.\n A \\ \\l, [D]\\<^bsub>v,n\\<^esub>\\ \\* \\l', Z\\ \\\n Z \\ {}) \\\n thesis) \\\n thesis"} {"_id": "503196", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sub>D\\<^sub>G\\<^sub>.\\<^sub>1 \\ \\ :\n \\ \\\\<^sub>D\\<^sub>G\n \\ \\\\\\<^sub>D\\<^sub>G\\<^bsub>\\\\<^esub> \\"} {"_id": "503197", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\m \\ 0; n \\ 0; m < 0; n < 0;\n (\\z. x = ant z) \\ x = \\\\\n \\ m *\\<^sub>a (n *\\<^sub>a x) = (m * n) *\\<^sub>a x\n 2. \\m \\ 0; n \\ 0;\n x = - \\ \\ (\\z. x = ant z) \\ x = \\;\n m < 0; n = 0 \\ 0 < n\\\n \\ m *\\<^sub>a (n *\\<^sub>a x) = (m * n) *\\<^sub>a x\n 3. \\m \\ 0; n \\ 0; n < 0 \\ n = 0 \\ 0 < n;\n x = - \\ \\ (\\z. x = ant z) \\ x = \\;\n m = 0 \\ 0 < m\\\n \\ m *\\<^sub>a (n *\\<^sub>a x) = (m * n) *\\<^sub>a x"} {"_id": "503198", "text": "proof (state)\nthis:\n lin_indpt (set ws)\n\ngoal (1 subgoal):\n 1. lin_indpt (set (ws @ normal_vectors ws))"} {"_id": "503199", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\b1 b2.\n u ` JF1col 0 b1 = JF1col 0 (JF1map u b1) \\\n u ` JF2col 0 b2 = JF2col 0 (JF2map u b2)\n 2. \\n.\n \\b1 b2.\n u ` JF1col n b1 = JF1col n (JF1map u b1) \\\n u ` JF2col n b2 = JF2col n (JF2map u b2) \\\n \\b1 b2.\n u ` JF1col (Suc n) b1 = JF1col (Suc n) (JF1map u b1) \\\n u ` JF2col (Suc n) b2 = JF2col (Suc n) (JF2map u b2)"} {"_id": "503200", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cring (union_ring C)"} {"_id": "503201", "text": "proof (prove)\ngoal (1 subgoal):\n 1. List_Factoring.pairwise BIT"} {"_id": "503202", "text": "proof (prove)\ngoal (1 subgoal):\n 1. n \\ \\x\\ i"} {"_id": "503203", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Key (Auth_ShaKey n) \\ spied s'"} {"_id": "503204", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n f s g t m x h.\n \\x \\ Suc n;\n \\k\\Suc n. t k \\ \\\\<^bsub>R\\<^esub>;\n inj_on f {j. j \\ Suc n}; inj_on g {j. j \\ Suc n}; m = Suc n;\n free_generator R M H; Ring R; aGroup R; H \\ carrier M;\n \\f s g t m.\n f \\ {j. j \\ n} \\ H \\\n inj_on f {j. j \\ n} \\\n s \\ {j. j \\ n} \\ carrier R \\\n g \\ {j. j \\ m} \\ H \\\n inj_on g {j. j \\ m} \\\n t \\ {j. j \\ m} \\ carrier R \\\n l_comb R M n s f = l_comb R M m t g \\\n (\\j\\n. s j \\ \\\\<^bsub>R\\<^esub>) \\\n (\\k\\m.\n t k \\ \\\\<^bsub>R\\<^esub>) \\\n n = m \\\n (\\h.\n h \\ {j. j \\ n} \\ {j. j \\ n} \\\n (\\l\\n. cmp f h l = g l \\ cmp s h l = t l));\n f \\ {j. j \\ Suc n} \\ H;\n s \\ {j. j \\ Suc n} \\ carrier R;\n g \\ {j. j \\ Suc n} \\ H;\n t \\ {j. j \\ Suc n} \\ carrier R;\n l_comb R M (Suc n) (cmp s (transpos x (Suc n)))\n (cmp f (transpos x (Suc n))) =\n l_comb R M (Suc n) t g;\n \\j\\Suc n. s j \\ \\\\<^bsub>R\\<^esub>;\n {y. \\x\\Suc n. y = g x} =\n {y. \\x\\Suc n. y = f x};\n x \\ Suc n; f x = g (Suc n); ideal R (carrier R);\n h \\ {j. j \\ Suc n} \\ {j. j \\ Suc n};\n \\l\\Suc n.\n cmp f (cmp (transpos x (Suc n)) h) l = g l \\\n cmp s (cmp (transpos x (Suc n)) h) l = t l;\n transpos x (Suc n)\n \\ {i. i \\ Suc n} \\ {i. i \\ Suc n}\\\n \\ \\h.\n h \\ {j. j \\ Suc n} \\\n {j. j \\ Suc n} \\\n (\\l\\Suc n.\n cmp f h l = g l \\ cmp s h l = t l)"} {"_id": "503205", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (if encode (Mn n f) = 0 then 0\n else if encode (Mn n f) = 1 then 1 else pdec1 (encode (Mn n f))) =\n 5"} {"_id": "503206", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (Group.group (G\\carrier := P\\) \\\n prime p \\\n order (G\\carrier := P\\) =\n p ^ a * (card P div p ^ a) \\\n finite (carrier (G\\carrier := P\\))) \\\n \\ p dvd card P div p ^ a &&&\n (PcalM \\\n {s. s \\ carrier (G\\carrier := P\\) \\\n card s = p ^ a}) &&&\n PRelM \\\n {(N1, N2).\n N1 \\ PcalM \\\n N2 \\ PcalM \\\n (\\g\\carrier (G\\carrier := P\\).\n N1 = N2 #>\\<^bsub>G\\carrier := P\\\\<^esub> g)}"} {"_id": "503207", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cs\\<^bsup>\\\\<^esup> l = cs\\<^bsup>\\'\\<^esup> l'"} {"_id": "503208", "text": "proof (prove)\nusing this:\n filter_mset ((\\) ps) \\ =\n {#b \\# \\. \\ ` ps \\ \\ ` b#}\n\ngoal (1 subgoal):\n 1. filter_mset ((\\) (\\ ` ps)) \\' =\n image_mset ((`) \\) (filter_mset ((\\) ps) \\)"} {"_id": "503209", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P \\ T1\\bop\\T2 :: T;\n P \\ T1\\bop\\T2 :: T'\\\n \\ T = T'"} {"_id": "503210", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [shd v] \\\\<^sub>F\\<^sub>I shd v ## sdrop 1 v"} {"_id": "503211", "text": "proof (prove)\nusing this:\n Q \\ {H. subgroup H G \\ card H = p ^ a}\n\ngoal (1 subgoal):\n 1. card Q = p ^ a"} {"_id": "503212", "text": "proof (prove)\nusing this:\n load1 p1 s1\n match i s1 s2\n \\b\\<^sub>1 i'.\n L1.sem_behaves s1 b\\<^sub>1 \\\n rel_behaviour (match i') b\\<^sub>1 b2\n\ngoal (1 subgoal):\n 1. \\b1 i. L1.behaves p1 b1 \\ rel_behaviour (match i) b1 b2"} {"_id": "503213", "text": "proof (prove)\nusing this:\n T = Transaction A B C D E F\n transaction_proj n (Transaction A B C D E F) =\n Transaction A (proj n B) (proj n C) (proj n D) (proj n E) (proj n F)\n trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t (proj n ?A)\n \\ trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t ?A\n\ngoal (1 subgoal):\n 1. trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\n (transaction_receive (transaction_proj n T) @\n transaction_selects (transaction_proj n T) @\n transaction_checks (transaction_proj n T) @\n transaction_updates (transaction_proj n T) @\n transaction_send (transaction_proj n T))\n \\ trms\\<^sub>l\\<^sub>s\\<^sub>s\\<^sub>t\n (transaction_receive T @\n transaction_selects T @\n transaction_checks T @\n transaction_updates T @ transaction_send T)"} {"_id": "503214", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ assertion \\\n (x ^ o * \\ \\ (1::'a)) ^ o * \\ \\\n (1::'a) =\n x"} {"_id": "503215", "text": "proof (prove)\nusing this:\n (\\<^bold>g [^] \\0) [^] r \\\n inv ((\\<^bold>g [^] \\1) [^] r) =\n (\\<^bold>g [^] \\0) [^] r \\\n inv (\\<^bold>g [^] \\1) [^] r\n\ngoal (1 subgoal):\n 1. (\\<^bold>g [^] \\0) [^] r \\\n inv ((\\<^bold>g [^] \\1) [^] r) =\n (\\<^bold>g [^] \\0 \\ inv (\\<^bold>g [^] \\1)) [^] r"} {"_id": "503216", "text": "proof (prove)\ngoal (1 subgoal):\n 1. listsp B l \\ listsp (A \\ B) l"} {"_id": "503217", "text": "proof (prove)\ngoal (1 subgoal):\n 1. rel_spmf (\\(a, s1') (b, s2'). a = b \\ X s1' s2')\n (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"} {"_id": "503218", "text": "proof (chain)\npicking this:\n fold_graph f z B y'\n insert x B \\ A"} {"_id": "503219", "text": "proof (prove)\nusing this:\n a' = b'\n finpref A l' \\ finpref A l''\n\ngoal (1 subgoal):\n 1. lnull a = lnull b \\\n (\\ lnull a \\\n \\ lnull b \\\n lhd a = lhd b \\\n (ltl a \\ A\\<^sup>\\ \\\n ltl b \\ A\\<^sup>\\ \\\n finpref A (ltl a) \\ finpref A (ltl b) \\\n ltl a = ltl b))"} {"_id": "503220", "text": "proof (prove)\nusing this:\n r \\ 0 \\\n lookup p (s \\ lt r) \\ (0::'b) \\\n q = p - monom_mult (lookup p (s \\ lt r) / lc r) s r\n\ngoal (1 subgoal):\n 1. lookup p (s \\ lt r) \\ (0::'b) &&&\n q = p - monom_mult (lookup p (s \\ lt r) / lc r) s r"} {"_id": "503221", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\h ptr child.\n \\l_known_ptrs ShadowRootClass.known_ptr\n ShadowRootClass.known_ptrs;\n Shadow_DOM.heap_is_wellformed h; ShadowRootClass.type_wf h;\n ShadowRootClass.known_ptrs h;\n h \\ Shadow_DOM.get_root_node ptr\n \\\\<^sub>r cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n child\\\n \\ h \\ Shadow_DOM.get_parent child\n \\\\<^sub>r None\n 2. \\h child ptr root.\n \\l_known_ptrs ShadowRootClass.known_ptr\n ShadowRootClass.known_ptrs;\n h \\ Shadow_DOM.get_parent child\n \\\\<^sub>r Some ptr;\n h \\ Shadow_DOM.get_root_node\n (cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n child)\n \\\\<^sub>r root\\\n \\ h \\ Shadow_DOM.get_root_node ptr\n \\\\<^sub>r root\n 3. \\h child ptr root.\n \\l_known_ptrs ShadowRootClass.known_ptr\n ShadowRootClass.known_ptrs;\n h \\ Shadow_DOM.get_parent child\n \\\\<^sub>r Some ptr;\n h \\ Shadow_DOM.get_root_node ptr\n \\\\<^sub>r root\\\n \\ h \\ Shadow_DOM.get_root_node\n (cast\\<^sub>n\\<^sub>o\\<^sub>d\\<^sub>e\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\\<^sub>2\\<^sub>o\\<^sub>b\\<^sub>j\\<^sub>e\\<^sub>c\\<^sub>t\\<^sub>_\\<^sub>p\\<^sub>t\\<^sub>r\n child)\n \\\\<^sub>r root"} {"_id": "503222", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\ y = - x \\ y"} {"_id": "503223", "text": "proof (prove)\nusing this:\n C.obj a\n C.obj a'\n \\\\'.map\\<^sub>0 a' \\\\<^sub>D\n g \\\\<^sub>D\n \\\\<^sub>0\n a : F.map\\<^sub>0\n a \\\\<^sub>D F.map\\<^sub>0\n a'\\\n \\\\ : F f \\\\<^sub>D \\'.map\\<^sub>0\n a' \\\\<^sub>D\n g \\\\<^sub>D \\\\<^sub>0 a\\ \\\n D.iso \\\n D.isomorphic ?a ?a' =\n (\\f.\n \\f : ?a \\\\<^sub>D ?a'\\ \\\n D.iso f)\n\ngoal (1 subgoal):\n 1. D.isomorphic\n (\\\\<^sub>0 a' \\\\<^sub>D\n F f \\\\<^sub>D \\'.map\\<^sub>0 a)\n (\\\\<^sub>0 a' \\\\<^sub>D\n (\\'.map\\<^sub>0 a' \\\\<^sub>D\n g \\\\<^sub>D \\\\<^sub>0 a) \\\\<^sub>D\n \\'.map\\<^sub>0 a)"} {"_id": "503224", "text": "proof (state)\nthis:\n \\ \\ \\ cbox (- vec c) (vec c)\n\ngoal (2 subgoals):\n 1. \\ \\ \\ sets lebesgue\n 2. cbox (- vec c) (vec c) \\ lmeasurable"} {"_id": "503225", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eexp (ereal x) = ereal (exp x)"} {"_id": "503226", "text": "proof (prove)\ngoal (1 subgoal):\n 1. stutter_extend_edges UNIV (rel_of_enex enex) =\n rel_of_enex (stutter_extend_enex enex)"} {"_id": "503227", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x31 x32.\n \\a = \\<^bold>\\un_Prim a\\<^bold>\\;\n Nml (x31 \\<^bold>\\ x32); b = x31 \\<^bold>\\ x32\\\n \\ a \\<^bold>\\ x31 \\<^bold>\\ x32 =\n a \\<^bold>\\ x31 \\<^bold>\\ x32"} {"_id": "503228", "text": "proof (prove)\nusing this:\n emeasure (PAR x)\n {p \\ space (PAR x). \\y\\I x. select_first x p y} =\n 1\n countable (I ?x)\n\ngoal (1 subgoal):\n 1. AE p in PAR x. \\y\\I x. select_first x p y"} {"_id": "503229", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sem\\<^sub>e\\<^sub>s\\<^sub>t_d M\\<^sub>0 \\ \\"} {"_id": "503230", "text": "proof (prove)\nusing this:\n (r', s') .=\\<^bsub>rel\\<^esub> (r, s)\n equivalence rel\n equivalence ?S \\\n (\\x.\n x \\ carrier ?S \\ x .=\\<^bsub>?S\\<^esub> x) \\\n (\\x y.\n x .=\\<^bsub>?S\\<^esub> y \\\n x \\ carrier ?S \\\n y \\ carrier ?S \\ y .=\\<^bsub>?S\\<^esub> x) \\\n (\\x y z.\n x .=\\<^bsub>?S\\<^esub> y \\\n y .=\\<^bsub>?S\\<^esub> z \\\n x \\ carrier ?S \\\n y \\ carrier ?S \\\n z \\ carrier ?S \\ x .=\\<^bsub>?S\\<^esub> z)\n\ngoal (1 subgoal):\n 1. (r, s) .=\\<^bsub>rel\\<^esub> (r', s')"} {"_id": "503231", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a. \\ PVar T \\ t \\ a"} {"_id": "503232", "text": "proof (state)\nthis:\n P \\ \\e0,s0,b0\\ \\\n \\e1,s1,b1\\\n P \\ \\e1,s1,b1\\ \\*\n \\Val v',s\\<^sub>1,b\\<^sub>1\\\n \\init_exp_of e1 = \\unit\\;\n INIT_class e1 = \\C\\; \\ sub_RI ?e'\\\n \\ P \\ \\init_switch e1 ?e',s1,\n b1\\ \\*\n \\?e',s\\<^sub>1,\n icheck P (the (INIT_class e1))\n ?e'\\\n init_exp_of e0 = \\unit\\\n INIT_class e0 = \\C\\\n \\ sub_RI e'\n\ngoal (1 subgoal):\n 1. \\a aa b ab ac ba e'.\n \\P \\ \\a,aa,b\\ \\\n \\ab,ac,ba\\;\n P \\ \\ab,ac,ba\\ \\*\n \\Val v',s\\<^sub>1,b\\<^sub>1\\;\n \\e'.\n \\init_exp_of ab = \\unit\\;\n INIT_class ab = \\C\\; \\ sub_RI e'\\\n \\ P \\ \\init_switch ab e',ac,\n ba\\ \\*\n \\e',s\\<^sub>1,icheck P (the (INIT_class ab)) e'\\;\n init_exp_of a = \\unit\\;\n INIT_class a = \\C\\; \\ sub_RI e'\\\n \\ P \\ \\init_switch a e',aa,\n b\\ \\*\n \\e',s\\<^sub>1,icheck P (the (INIT_class a)) e'\\"} {"_id": "503233", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\\\<^sub>e a = x /\\<^sub>L a"} {"_id": "503234", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 2 ^ N\\<^sup>2 * B2_LLL F ^ (2 * N) \\ 1"} {"_id": "503235", "text": "proof (prove)\ngoal (1 subgoal):\n 1. favorites R i \\ alts"} {"_id": "503236", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a b.\n \\Monotone \\; FIX \\ (a, b)\\\n \\ \\ (FIX \\) (a, b)"} {"_id": "503237", "text": "proof (prove)\nusing this:\n risk_neutral_prob N\n filt_equiv F M N\n trading_strategy pf\n self_financing Mkt pf\n stock_portfolio Mkt pf\n \\n.\n \\asset\\support_set pf.\n integrable N (\\w. prices Mkt asset n w * pf asset (Suc n) w)\n \\n.\n \\asset\\support_set pf.\n integrable N\n (\\w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)\n\ngoal (1 subgoal):\n 1. \\x\\support_set pf.\n AEeq N\n (real_cond_exp N (F n)\n (discounted_value r (\\m y. prices Mkt x m y * pf x m y)\n (Suc n)))\n (discounted_value r\n (\\m y. prices Mkt x m y * pf x (Suc m) y) n)"} {"_id": "503238", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ring_hom_ring R (S Quot P) ((+>\\<^bsub>S\\<^esub>) P \\ h)"} {"_id": "503239", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ccomp c \\ cfs\\ \\\n cpred c (cfs ! 0) (cfs ! (length cfs - 1))"} {"_id": "503240", "text": "proof (prove)\nusing this:\n (?x = ?y) = (\\z. P z ?x = P z ?y)\n \\z. P z x = P z y\n\ngoal (1 subgoal):\n 1. x = y"} {"_id": "503241", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hd\\<^sub>L \\ tl\\<^sub>L"} {"_id": "503242", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inv_fg (inv_fg l) = l"} {"_id": "503243", "text": "proof (prove)\ngoal (2 subgoals):\n 1. (x \\ set (insort y [])) = (x = y \\ x \\ set [])\n 2. \\a xs.\n (x \\ set (insort y xs)) =\n (x = y \\ x \\ set xs) \\\n (x \\ set (insort y (a # xs))) =\n (x = y \\ x \\ set (a # xs))"} {"_id": "503244", "text": "proof (state)\nthis:\n bv_to_int w2 = 0\n\ngoal (2 subgoals):\n 1. bv_to_int w2 = 0 \\\n length (bv_sub w1 w2) \\ Suc (max (length w1) (length w2))\n 2. bv_to_int w2 \\ 0 \\\n length (bv_sub w1 w2) \\ Suc (max (length w1) (length w2))"} {"_id": "503245", "text": "proof (state)\nthis:\n a = 2 * m\n\ngoal (2 subgoals):\n 1. a = 2 * m \\\n dist (geod (Suc n) a) (geod (Suc n) (Suc a)) = dist x0 y0 / (2 * 2 ^ n)\n 2. a = Suc (2 * m) \\\n dist (geod (Suc n) a) (geod (Suc n) (Suc a)) = dist x0 y0 / (2 * 2 ^ n)"} {"_id": "503246", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_attach_shadow_root\\<^sub>S\\<^sub>h\\<^sub>a\\<^sub>d\\<^sub>o\\<^sub>w\\<^sub>_\\<^sub>D\\<^sub>O\\<^sub>M\n ShadowRootClass.known_ptr set_shadow_root set_shadow_root_locs set_mode\n set_mode_locs attach_shadow_root ShadowRootClass.type_wf get_tag_name\n get_tag_name_locs get_shadow_root get_shadow_root_locs"} {"_id": "503247", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x1a x2a bv S1 S2.\n S1 \\ S2 \\\n \\x1 x2.\n inv_less' bv (aval'' x1a S1) (aval'' x2a S1) =\n (x1, x2) \\\n (\\x1b x2b.\n inv_less' bv (aval'' x1a S2) (aval'' x2a S2) =\n (x1b, x2b) \\\n inv_aval' x1a x1 (inv_aval' x2a x2 S1)\n \\ inv_aval' x1a x1b (inv_aval' x2a x2b S2))"} {"_id": "503248", "text": "proof (prove)\ngoal (1 subgoal):\n 1. class.bounded_lattice (\\) (\\x y. y \\ x)\n (\\x y. y < x) (\\) \\ \\"} {"_id": "503249", "text": "proof (prove)\nusing this:\n Determinant.det\n (Matrix.mat 2 2\n (\\(i, j).\n if i = 0 \\ j = 0 then s\n else if i = 0 \\ j = 1 then - b1\n else if i = 1 \\ j = 0 then t else a1)) =\n (1::'a)\n invertible_mat\n (Matrix.mat 2 2\n (\\(i, j).\n if i = 0 \\ j = 0 then s\n else if i = 0 \\ j = 1 then - b1\n else if i = 1 \\ j = 0 then t else a1)) =\n (Determinant.det\n (Matrix.mat 2 2\n (\\(i, j).\n if i = 0 \\ j = 0 then s\n else if i = 0 \\ j = 1 then - b1\n else if i = 1 \\ j = 0 then t else a1)) dvd\n (1::'a))\n\ngoal (1 subgoal):\n 1. invertible_mat\n (Matrix.mat 2 2\n (\\(i, j).\n if i = 0 \\ j = 0 then s\n else if i = 0 \\ j = 1 then - b1\n else if i = 1 \\ j = 0 then t else a1))"} {"_id": "503250", "text": "proof (state)\nthis:\n c \\ insert c' (set cl)\n\ngoal (2 subgoals):\n 1. cl[n := c'] = [] \\ False\n 2. \\c. c \\ set (cl[n := c']) \\ proper c"} {"_id": "503251", "text": "proof (prove)\nusing this:\n (q, \\, q') \\ \\\\<^sub>k ` limit \\\n (q, \\, q') \\ S\n (?m, ?\\, ?m') \\ limit \\ \\\n \\p p'. ?m k = Some p \\ ?m' k = Some p'\n\ngoal (1 subgoal):\n 1. (\\m m'.\n \\m k = Some q; m' k = Some q';\n (m, \\, m') \\ limit \\\\\n \\ thesis) \\\n thesis"} {"_id": "503252", "text": "proof (state)\ngoal (2 subgoals):\n 1. \\i x.\n \\i < length (sturm p) - 2;\n poly (sturm p ! (i + 1)) x = 0\\\n \\ poly (sturm p ! (i + 2)) x * poly (sturm p ! i) x\n < 0\n 2. \\x\\<^sub>0.\n poly p x\\<^sub>0 = 0 \\\n \\\\<^sub>F x in at x\\<^sub>0.\n sgn (poly (p * sturm p ! 1) x) =\n (if x\\<^sub>0 < x then 1 else - 1)"} {"_id": "503253", "text": "proof (prove)\nusing this:\n z = x\n\ngoal (1 subgoal):\n 1. z \\ s"} {"_id": "503254", "text": "proof (prove)\ngoal (1 subgoal):\n 1. x \\\\<^sub>\\ set (list.set xs)"} {"_id": "503255", "text": "proof (state)\nthis:\n l = l1\n (e, a) = n_unwrap nd\n r = r1\n\ngoal (2 subgoals):\n 1. ft_invar l\n 2. ft_invar r"} {"_id": "503256", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj h \\\n inj_on (\\(k, v). (k, h v)) (fset (fset_of_fmap m))"} {"_id": "503257", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {l.. then snd (x0, x1) else fst (x0, x1)) =\n (case (x0, x1) of\n (x0, x1) \\\n sample_uniform (order \\) \\\n (\\r.\n sample_uniform (order \\) \\\n (\\\\0.\n sample_uniform (order \\) \\\n (\\\\1.\n assert_spmf\n (\\<^bold>g [^] r = \\<^bold>g [^] r \\\n (\\<^bold>g [^] \\0) [^] r \\\n \\<^bold>g [^] (if \\ then 1 else 0) \\\n inv ((\\<^bold>g [^] \\1) [^] r \\\n \\<^bold>g [^] (if \\ then 1 else 0)) =\n (\\<^bold>g [^] \\0 \\\n inv (\\<^bold>g [^] \\1)) [^]\n r) \\\n (\\_.\n sample_uniform (order \\) \\\n (\\u0.\n sample_uniform (order \\) \\\n (\\u1.\n sample_uniform (order \\) \\\n (\\v0.\n sample_uniform (order \\) \\\n (\\v1.\n return_spmf\n ((),\n if \\\n then ((\\<^bold>g [^] \\1) [^] r \\\n \\<^bold>g [^] (if \\ then 1 else 0) \\\n inv \\<^bold>g) [^]\n u1 \\\n (\\<^bold>g [^] \\1) [^] v1 \\\n x1 \\\n inv (((\\<^bold>g [^] r) [^] u1 \\ \\<^bold>g [^] v1) [^]\n \\1)\n else ((\\<^bold>g [^] \\0) [^] r \\\n \\<^bold>g [^] (if \\ then 1 else 0)) [^]\n u0 \\\n (\\<^bold>g [^] \\0) [^] v0 \\\n x0 \\\n inv (((\\<^bold>g [^] r) [^] u0 \\ \\<^bold>g [^] v0) [^]\n \\0)))))))))))"} {"_id": "503259", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\t t' u u'.\n \\atom t \\ (v, tm, y, y', t', u, u');\n atom t' \\ (v, tm, y, y', u, u');\n atom u \\ (v, tm, y, y', u');\n atom u' \\ (v, tm, y, y')\\\n \\ thesis) \\\n thesis"} {"_id": "503260", "text": "proof (prove)\ngoal (1 subgoal):\n 1. All local.power_p32_dom"} {"_id": "503261", "text": "proof (prove)\nusing this:\n (\\, p) \\ oreachable A S U\n\ngoal (1 subgoal):\n 1. (\\, p) \\ oreachable A S' U'"} {"_id": "503262", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Abs ((p1 + p2) \\ Rep a) =\n Abs (p1 \\ Rep (Abs (p2 \\ Rep a)))"} {"_id": "503263", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\\\<^bold>\\a\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\b\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n \\<^bold>\\c\\<^bold>\\\\ \\\n (\\\\<^bold>\\a\\<^bold>\\\\ \\\n \\\\<^bold>\\b\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\c\\<^bold>\\\\)) \\\n \\[\\\\<^bold>\\a\\<^bold>\\\\, \\\\<^bold>\\b\\<^bold>\\\\, \\\\<^bold>\\c\\<^bold>\\\\]) \\\n ((\\a\\<^bold>\\\\ \\\n \\b\\<^bold>\\\\) \\\n \\c\\<^bold>\\\\) =\n (\\(\\<^bold>\\a\\<^bold>\\ \\<^bold>\\\\<^bold>\\\\<^bold>\\\n \\<^bold>\\b\\<^bold>\\) \\<^bold>\\\n \\<^bold>\\c\\<^bold>\\\\ \\\n (\\\\<^bold>\\a\\<^bold>\\ \\<^bold>\\\n \\<^bold>\\b\\<^bold>\\\\ \\\n \\\\<^bold>\\c\\<^bold>\\\\)) \\\n ((\\a\\<^bold>\\\\ \\\n \\b\\<^bold>\\\\) \\\n \\c\\<^bold>\\\\)"} {"_id": "503264", "text": "proof (prove)\nusing this:\n \\?r \\ ?L;\n \\n. ?r n \\ {x. f x = (0::'b)}\\\n \\ ?L \\ {x. f x = (0::'b)}\n\ngoal (1 subgoal):\n 1. closed (f -` {0::'b})"} {"_id": "503265", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\k.\n Max {k. (T ^^ k) ((T ^^ n) x) \\ A} < k \\\n (T ^^ k) ((T ^^ n) x) \\ A"} {"_id": "503266", "text": "proof (prove)\nusing this:\n subterms fa \\ Mapping.keys M1 \\\n Mapping.keys M \\ Mapping.keys M1 \\\n (\\f\\Mapping.keys M1.\n subterms f \\ Mapping.keys M1 \\\n the (Mapping.lookup M1 f) < length slp1 \\\n interpret_slp slp1 xs ! slp_index_lookup slp1 M1 f =\n interpret_floatarith f xs)\n slp_of_fas' (fa # fas) M slp = (M', slp')\n ?f \\ Mapping.keys M \\\n subterms ?f \\ Mapping.keys M\n ?f \\ Mapping.keys M \\\n the (Mapping.lookup M ?f) < length slp\n ?f \\ Mapping.keys M \\\n interpret_slp slp xs ! slp_index_lookup slp M ?f =\n interpret_floatarith ?f xs\n\ngoal (1 subgoal):\n 1. \\ (subterms ` set fas) \\ Mapping.keys M' \\\n Mapping.keys M1 \\ Mapping.keys M' \\\n (\\f\\Mapping.keys M'.\n subterms f \\ Mapping.keys M' \\\n the (Mapping.lookup M' f) < length slp' \\\n interpret_slp slp' xs ! slp_index_lookup slp' M' f =\n interpret_floatarith f xs)"} {"_id": "503267", "text": "proof (prove)\ngoal (1 subgoal):\n 1. eq\\[::]\\ys =\n (eq\\(slength\\[::])\\\n (slength\\ys) andalso prefix\\[::]\\ys)"} {"_id": "503268", "text": "proof (state)\nthis:\n ?x / ?y = ?x * inverse ?y\n\ngoal (1 subgoal):\n 1. \\a. inverse (a *\\<^sub>C 1) = 1 /\\<^sub>C a"} {"_id": "503269", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\name input xvec Tvec.\n \\\\b ba.\n p \\ subs' input b ba =\n subs' (p \\ input) (p \\ b)\n (p \\ ba);\n p \\ name \\ p \\ xvec;\n p \\ name \\ p \\ Tvec\\\n \\ \\(p \\\n name)subs' (p \\ input)\n (p \\ xvec) (p \\ Tvec) =\n subs' (\\(p \\ name)(p \\ input))\n (p \\ xvec) (p \\ Tvec)"} {"_id": "503270", "text": "proof (state)\ngoal (1 subgoal):\n 1. sum_upto (\\n. real n powr - s)\n \\ O(\\x. x powr max 0 (1 - s))"} {"_id": "503271", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\m. poly_mapping.lookup (p + q) m *\n (\\v. f v ^ poly_mapping.lookup m v)) =\n (\\m. poly_mapping.lookup p m *\n (\\v. f v ^ poly_mapping.lookup m v)) +\n (\\m. poly_mapping.lookup q m *\n (\\v. f v ^ poly_mapping.lookup m v))"} {"_id": "503272", "text": "proof (prove)\ngoal (1 subgoal):\n 1. generate_check a b =\n concat\n (map (\\ys.\n map (\\xs. (xs, ys))\n (filter (suffs (C\\<^sub>1 b (fst ys)) a)\n (alls (B\\<^sub>1 b) a)))\n (filter (suffs C\\<^sub>2 b) (alls B\\<^sub>2 b)))"} {"_id": "503273", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x y.\n \\(Label x, Arity x, Guards x, Outputs x, Updates x, more x)\n < (Label y, Arity y, Guards y, Outputs y, Updates y,\n more y) \\\n x = y;\n (Label y, Arity y, Guards y, Outputs y, Updates y, more y)\n < (Label x, Arity x, Guards x, Outputs x, Updates x, more x) \\\n y = x\\\n \\ x = y\n 2. \\x y.\n ((Label x, Arity x, Guards x, Outputs x, Updates x, more x)\n < (Label y, Arity y, Guards y, Outputs y, Updates y, more y) \\\n x = y) \\\n (Label y, Arity y, Guards y, Outputs y, Updates y, more y)\n < (Label x, Arity x, Guards x, Outputs x, Updates x, more x) \\\n y = x"} {"_id": "503274", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ {(e, f e) |e. f e \\ bot} \\\n (\\y.\n y \\ {(e, f e) |e. f e \\ bot} \\\n (case x of\n (e, fe) \\ \\(d, fd). fe + bot \\ fd + bot)\n y)"} {"_id": "503275", "text": "proof (state)\nthis:\n \\y r c.\n y \\ X \\ r \\ D y \\ r \\ D xa \\ r @ [c] = x\n\ngoal (1 subgoal):\n 1. \\ (\\t.\n t \\ D xa \\\n \\ t < x) \\\n x \\ T u"} {"_id": "503276", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\xs' ox.\n list_of_oalist_ntm xs = (xs', ox) \\\n thesis) \\\n thesis"} {"_id": "503277", "text": "proof (prove)\nusing this:\n I \\ II \\ {}\n\ngoal (1 subgoal):\n 1. \\II. II \\ Q \\ I \\ II"} {"_id": "503278", "text": "proof (prove)\nusing this:\n \\ \\ \\\\P \\ \\ \\ P'\n \\ = \\\n\ngoal (1 subgoal):\n 1. Prop \\ P'"} {"_id": "503279", "text": "proof (state)\nthis:\n xd = (x, d)\n\ngoal (1 subgoal):\n 1. \\a m.\n eval_monom_list g m \\ eval_monom_list f m \\\n (0::'a) \\ eval_monom_list g m \\\n eval_monom_list g (a # m) \\ eval_monom_list f (a # m) \\\n (0::'a) \\ eval_monom_list g (a # m)"} {"_id": "503280", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vars_clause (subst_cls C \\)\n \\ \\ (vars_term ` range \\)"} {"_id": "503281", "text": "proof (prove)\ngoal (1 subgoal):\n 1. correctDeCompositionVARSYSTEM"} {"_id": "503282", "text": "proof (prove)\nusing this:\n q g0 = q' g0\n\ngoal (1 subgoal):\n 1. q g0 \\ punit.dgrad_p_set d m \\\n lt (q g0 \\ g0) \\\\<^sub>t lt p"} {"_id": "503283", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\self res \\.\n ((\\_. res \\) \\\n (\\_.\n (if (\\_.\n (self.boss@pre \\ null)\n \\) then (\\_.\nSet{self.salary@pre}\n \\) else (\\_.\n self.boss@pre.contents@pre()->including\\<^sub>S\\<^sub>e\\<^sub>t(self.salary@pre)\n \\) endif)\n \\))\n \\ =\n ((\\_. res \\) \\\n (\\_.\n (if (\\_.\n (\\_. self \\.boss@pre \\ null)\n \\) then (\\_.\nSet{\\_. self \\.salary@pre}\n \\) else (\\_.\n \\_.\n self\n \\.boss@pre.contents@pre()->including\\<^sub>S\\<^sub>e\\<^sub>t(\\_.\n self \\.salary@pre)\n \\) endif)\n \\))\n \\"} {"_id": "503284", "text": "proof (prove)\nusing this:\n [] = sorted_list_of_set A\n finite A\n mono_on g A\n inj_on g A\n finite ?A \\ (sorted_list_of_set ?A = []) = (?A = {})\n\ngoal (1 subgoal):\n 1. sorted_list_of_set (g ` A) = map g (sorted_list_of_set A)"} {"_id": "503285", "text": "proof (state)\nthis:\n finite (gState_progress_rel prog `` {g})\n\ngoal (1 subgoal):\n 1. gState_inv prog g \\\n finite ((gState_progress_rel prog)\\<^sup>* `` {g})"} {"_id": "503286", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mpoly_to_nested_poly (p * q) v =\n mpoly_to_nested_poly p v * mpoly_to_nested_poly q v"} {"_id": "503287", "text": "proof (prove)\ngoal (1 subgoal):\n 1. t \\ s"} {"_id": "503288", "text": "proof (prove)\nusing this:\n q \\ l qs \\ \\\n\ngoal (1 subgoal):\n 1. f_accessible \\ {qs ! i} \\ f_accessible \\ {q}"} {"_id": "503289", "text": "proof (state)\nthis:\n x \\ y\n\ngoal (1 subgoal):\n 1. \\\\ys. cnt x (remove_cycles xs x ys) \\ max 1 (cnt x ys);\n x \\ y\\\n \\ cnt x (remove_cycles (y # xs) x ys)\n \\ max 1 (cnt x ys)"} {"_id": "503290", "text": "proof (state)\nthis:\n \\?x \\ C; ?y \\ C; ?x < ?y\\\n \\ f' ?x * (?y - ?x) \\ f ?y - f ?x\n \\?x \\ C; ?y \\ C; ?x < ?y\\\n \\ f' ?y * (?x - ?y) \\ f ?x - f ?y\n\ngoal (1 subgoal):\n 1. \\convex C;\n \\x.\n x \\ C \\ (f has_real_derivative f' x) (at x);\n \\x.\n x \\ C \\ (f' has_real_derivative f'' x) (at x);\n \\x. x \\ C \\ 0 \\ f'' x; x \\ C;\n y \\ C\\\n \\ f' x * (y - x) \\ f y - f x"} {"_id": "503291", "text": "proof (prove)\nusing this:\n (\\<^bold>\\ Nom j) at b in' H\n b \\ assign A H a\n (\\<^bold>\\ (\\<^bold>\\ p)) at b in' H\n\ngoal (1 subgoal):\n 1. (\\<^bold>\\ (\\<^bold>@ j p)) at b in' H"} {"_id": "503292", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (((new_array v \\ new_array' v \\ integer_of_nat) &&&\n array_length \\ nat_of_integer \\ array_length') &&&\n (array_get a \\ array_get' a \\ integer_of_nat) &&&\n array_set a \\ array_set' a \\ integer_of_nat) &&&\n ((array_grow a \\ array_grow' a \\ integer_of_nat) &&&\n array_shrink a \\ array_shrink' a \\ integer_of_nat) &&&\n (array_get_oo x a \\ array_get_oo' x a \\ integer_of_nat) &&&\n array_set_oo f a \\ array_set_oo' f a \\ integer_of_nat"} {"_id": "503293", "text": "proof (prove)\ngoal (1 subgoal):\n 1. neg\\(le\\x\\y) = lt\\y\\x"} {"_id": "503294", "text": "proof (state)\nthis:\n sd_chain k = Some q\n\ngoal (2 subgoals):\n 1. \\n. sd_chain n = None \\ thesis\n 2. \\n. \\q. sd_chain n = Some q \\ thesis"} {"_id": "503295", "text": "proof (prove)\nusing this:\n BG \\ Bot_G\n\ngoal (1 subgoal):\n 1. \\BG'\\Bot_G. BG' \\ lifted_q_calc.\\_Fset N"} {"_id": "503296", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s f ts.\n \\Node f ts \\ trees A; s \\ set ts; f \\ A;\n \\t\\set ts. t \\ trees A\\\n \\ size s < Suc (size_list size ts)"} {"_id": "503297", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sbis B1 B2 s1 s2\n (\\ (fst ` {(R1, R2) |R1 R2. sbis B1 B2 s1 s2 R1 R2}))\n (\\ (snd ` {(R1, R2) |R1 R2. sbis B1 B2 s1 s2 R1 R2}))"} {"_id": "503298", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a aa ba ab ac baa.\n \\P \\ \\e,s,b\\ \\*\n \\a,aa,ba\\;\n P \\ \\a,aa,ba\\ \\\n \\ab,ac,baa\\\\\n \\ P \\ \\Val\n v \\bop\\ a,\n aa,ba\\ \\\n \\Val v \\bop\\ ab,ac,baa\\"} {"_id": "503299", "text": "proof (prove)\ngoal (1 subgoal):\n 1. sum_to_tup\\(tup_to_sum\\F) = F"} {"_id": "503300", "text": "proof (prove)\ngoal (1 subgoal):\n 1. A B C CongA A' B' C'"} {"_id": "503301", "text": "proof (prove)\nusing this:\n t \\ ball z d\n\ngoal (1 subgoal):\n 1. t \\ \\\\<^sub>\\\\<^sub>0"} {"_id": "503302", "text": "proof (prove)\ngoal (1 subgoal):\n 1. permutations_of_set A \\ shuffles xs ys = shuffles xs ys"} {"_id": "503303", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ \\cs,css,s\\ \\ t"} {"_id": "503304", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\b. red F a b \\ thesis) \\ thesis"} {"_id": "503305", "text": "proof (prove)\nusing this:\n P ?p \\ f ?p \\ ?p \\ ?p \\ p0 \\\n P (?p \\ f ?p) \\\n f (?p \\ f ?p) \\ ?p \\ f ?p \\\n ?p \\ f ?p \\ p0\n \\\\s.\n \\P s \\ f s \\ s \\ s \\ p0;\n s \\ f s < s\\\n \\ P (s \\ f s) \\\n f (s \\ f s)\n \\ s \\ f s \\\n s \\ f s \\ p0;\n P p0 \\ f p0 \\ p0 \\ p0 \\ p0\\\n \\ P p \\ f p \\ p \\ p \\ p0\n\ngoal (1 subgoal):\n 1. P p \\ f p \\ p"} {"_id": "503306", "text": "proof (prove)\nusing this:\n i \\ j \\ stable_rank y i\n\ngoal (1 subgoal):\n 1. j < i"} {"_id": "503307", "text": "proof (prove)\nusing this:\n \\s.\n \\fmla\n \\set (adds\n (ground_action.effect\n (Ground_Action pre (Effect add del)))).\n \\ is_predAtom fmla \\ s \\ fmla\n fmla\n \\ apply_effect\n (ground_action.effect (Ground_Action pre (Effect add del)))\n M\\<^sub>0\n Ball (close_world M\\<^sub>0) ((\\) (close_eq s\\<^sub>0)) \\\n wm_basic M\\<^sub>0 \\\n (\\\\\\\\. sound_opr \\ (f \\))\n\ngoal (1 subgoal):\n 1. (\\atm. fmla \\ Atom atm) \\\n s \\ fmla"} {"_id": "503308", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\ y = \\z\\) =\n (\\x'.\n x = \\x'\\ \\ y x' = \\z\\)"} {"_id": "503309", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_\\ id s = s"} {"_id": "503310", "text": "proof (prove)\ngoal (1 subgoal):\n 1. monom (1::'a) 1 * pderiv (charpoly A) =\n sum_UNIV_type (\\i. charpoly (erase_mat A i i)) TYPE('n)"} {"_id": "503311", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dim_row (step_3_c x l k i A) = dim_row A &&&\n dim_col (step_3_c x l k i A) = dim_col A"} {"_id": "503312", "text": "proof (state)\nthis:\n e_0 \\ e_1\n\ngoal (1 subgoal):\n 1. \\index_set A; A \\ {}; A \\ UNIV;\n \\n\\A. nat_to_ce_set n = {}\\\n \\ one_reducible_to univ_ce (- A)"} {"_id": "503313", "text": "proof (prove)\ngoal (1 subgoal):\n 1. zeroring R \\ J_rad R = carrier R"} {"_id": "503314", "text": "proof (state)\ngoal (1 subgoal):\n 1. (\\x\\chain.\n case x of\n (k, g) \\ real_of_int k * line_integral F basis g) =\n 0"} {"_id": "503315", "text": "proof (prove)\nusing this:\n \\s X. igWls MOD s X \\ igWls MOD' s (h X)\n \\xs x s X.\n isInBar (xs, s) \\ igWls MOD s X \\\n hA (igAbs MOD xs x X) = igAbs MOD' xs x (h X)\n \\xs x s X.\n isInBar (xs, s) \\ igWls MOD' s X \\\n hA' (igAbs MOD' xs x X) = igAbs MOD'' xs x (h' X)\n\ngoal (1 subgoal):\n 1. \\xs x s X.\n isInBar (xs, s) \\ igWls MOD s X \\\n (hA' \\ hA) (igAbs MOD xs x X) =\n igAbs MOD'' xs x ((h' \\ h) X)"} {"_id": "503316", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 25001 / 10 ^ 5 + (xs ! 0)\\<^sup>2 + 2 * (xs ! 1)\\<^sup>2 +\n 2 * (xs ! 2)\\<^sup>2 +\n 2 * (xs ! 3)\\<^sup>2 +\n 2 * (xs ! 4)\\<^sup>2 +\n 2 * (xs ! 5)\\<^sup>2 +\n 2 * (xs ! 6)\\<^sup>2 -\n xs ! 0 =\n interpret_floatarith magnetism xs"} {"_id": "503317", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 1 / real (f j') -\n x * (\\j = 0..j = 0.._commit.rel_game \\ =\n TRY sample_uniform (order \\) \\\n (\\w.\n let (h, w) = ((\\<^bold>g [^] w, g' [^] w), w)\n in \\ h \\\n (\\w'.\n return_spmf\n ([w = w'] (mod order \\) \\\n [x *\n w = x * w'] (mod order \\)))) ELSE return_spmf False\n TRY sample_uniform (order \\) \\\n (\\w.\n let (h, w) = ((\\<^bold>g [^] w, g' [^] w), w)\n in \\ h \\\n (\\w'.\n return_spmf\n ([w = w'] (mod order \\) \\\n [x *\n w = x * w'] (mod order \\)))) ELSE return_spmf False =\n local.dis_log_alt.dis_log3 \\\n\ngoal (1 subgoal):\n 1. chaum_ped_\\_commit.rel_advantage \\ =\n local.dis_log_alt.advantage3 \\"} {"_id": "503319", "text": "proof (prove)\nusing this:\n s \\ reachable (closed (pnet np n)) TT\n netgmap sr s = netmask (net_tree_ips n) (\\, \\)\n wf_net_tree n\n (s, a, s') \\ automaton.trans (closed (pnet np n))\n\ngoal (1 subgoal):\n 1. (\\\\' \\'.\n \\((\\, \\), a, \\', \\')\n \\ automaton.trans (oclosed (opnet onp n));\n netgmap sr s' =\n netmask (net_tree_ips n) (\\', \\')\\\n \\ thesis) \\\n thesis"} {"_id": "503320", "text": "proof (prove)\nusing this:\n ide a\n ide b\n ide c\n \\g : local.prod a b \\ c\\\n x \\ local.set (local.prod a b)\n \\?X \\<^bold>\\ ?A; hfun (?A * ?B) ?C ?F\\\n \\ \\!G.\n \\?X, G\\ \\<^bold>\\\n hlam ?A ?B ?C ?F\n \\?X \\<^bold>\\ ?A; hfun (?A * ?B) ?C ?F\\\n \\ happ (hlam ?A ?B ?C ?F) ?X =\n (THE G.\n \\?X, G\\ \\<^bold>\\\n hlam ?A ?B ?C ?F)\n \\?X \\<^bold>\\ ?A; hfun (?A * ?B) ?C ?F\\\n \\ happ (hlam ?A ?B ?C ?F) ?X =\n \\yz \\<^bold>\\ ?B * ?C.\n \\\\?X, hfst yz\\,\n hsnd yz\\ \\<^bold>\\\n ?F\\\n \\?X \\<^bold>\\ ?A; hfun (?A * ?B) ?C ?F;\n ?Y \\<^bold>\\ ?B\\\n \\ happ (happ (hlam ?A ?B ?C ?F) ?X) ?Y =\n happ ?F \\?X, ?Y\\\n\ngoal (1 subgoal):\n 1. UP (happ\n (happ\n (hlam (ide_to_hf a) (ide_to_hf b) (ide_to_hf c) (arr_to_hfun g))\n (hfst (DOWN x)))\n (hsnd (DOWN x))) =\n UP (happ (arr_to_hfun g) (DOWN x))"} {"_id": "503321", "text": "proof (state)\ngoal (1 subgoal):\n 1. cls_val_process Mkt2 (qty_mult_comp pf qty) n\n \\ borel_measurable (F n)"} {"_id": "503322", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\t.\n t \\ {0..1} \\\n p ((k \\ (\\t. (t, undefined))) t) = g t"} {"_id": "503323", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (nf_ACI \\ \\\n \\ = NFOR (norm_ACI ` disjuncts \\)) &&&\n (nf_ACI \\ \\\n \\ = NFAND (norm_ACI ` conjuncts \\))"} {"_id": "503324", "text": "proof (prove)\nusing this:\n ics = Throwing Cs a\n\ngoal (1 subgoal):\n 1. exec_step_ind (exec_step_input P C M pc ics) P h stk loc C M pc ics frs\n sh (xp', h', frs', sh')"} {"_id": "503325", "text": "proof (prove)\ngoal (1 subgoal):\n 1. dom (C DenyAll) \\ {}"} {"_id": "503326", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ [HNext rmhist p]_(c p, r p, m p, rmhist!p) \\\n ImpNext p \\\n \\ unchanged (e p, c p, r p, m p, rmhist!p) \\\n $S5 rmhist p \\\n S6 rmhist p$ \\\n RPCReply crCh rmCh rst p \\\n unchanged (e p, c p, m p) \\\n S6 rmhist p$ \\\n RPCFail crCh rmCh rst p \\ unchanged (e p, c p, m p)"} {"_id": "503327", "text": "proof (prove)\ngoal (1 subgoal):\n 1. C.inv (\\ (C.cod f)) \\\\<^sub>C\n f \\\\<^sub>C \\ (C.dom f) =\n F (G f)"} {"_id": "503328", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\(n, a)\\n_as. [:- a, 1::'a:] ^ n) =\n [:- a, 1::'a:] ^ n *\n (\\(n, a)\\as @ bs. [:- a, 1::'a:] ^ n)"} {"_id": "503329", "text": "proof (state)\nthis:\n r.eq_m\n ((D + Polynomial.smult q B) * (S - Polynomial.smult p A') +\n (H + Polynomial.smult q A) * (T - Polynomial.smult p B'))\n (D * S + H * T + Polynomial.smult q (B * S + A * T) -\n Polynomial.smult p (A' * D + B' * H) -\n Polynomial.smult (p * q) (A * B' + B * A'))\n\ngoal (5 subgoals):\n 1. r.Mp D' = D'\n 2. r.Mp H' = H'\n 3. r.Mp S' = S'\n 4. r.Mp T' = T'\n 5. r.eq_m (D' * S' + H' * T') 1"} {"_id": "503330", "text": "proof (prove)\nusing this:\n valid_edge a\n valid_edge a'\n sourcenode a = Entry\n sourcenode a' = Entry\n targetnode a' = Exit\n targetnode a = Exit\n\ngoal (1 subgoal):\n 1. a = a'"} {"_id": "503331", "text": "proof (prove)\nusing this:\n q \\ (\\a. hom_component a z) ` A\n\ngoal (1 subgoal):\n 1. (\\a2.\n \\a2 \\ A; q = hom_component a2 z\\\n \\ thesis) \\\n thesis"} {"_id": "503332", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite (rs 1)"} {"_id": "503333", "text": "proof (prove)\nusing this:\n flowsto X0 {s + t |s t. s \\ T \\ t \\ {0<..}}\n (CX \\ CY \\ UNIV \\ X1') (fst ` X2 \\ UNIV)\n\ngoal (1 subgoal):\n 1. flowsto X0 {0<..} CZ (fst ` X2 \\ UNIV)"} {"_id": "503334", "text": "proof (prove)\ngoal (1 subgoal):\n 1. blinding_of_on UNIV id (=)"} {"_id": "503335", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\k n1 n2.\n \\B = k + n1; Ba = k + n2;\n head_\\ n1 < head_\\ n2\\\n \\ thesis) \\\n thesis"} {"_id": "503336", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Ifm vs (x # bs) (msubsteq2 c t a b) =\n Ifm vs\n (Itm vs (x # bs) t / \\c\\\\<^sub>p\\<^bsup>vs\\<^esup> # bs)\n (Eq (CNP 0 a b))"} {"_id": "503337", "text": "proof (prove)\nusing this:\n a \\ supp ((p \\ a \\ a) + p)\n\ngoal (1 subgoal):\n 1. supp ((p \\ a \\ a) + p) \\ supp p"} {"_id": "503338", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\l1 l2.\n l1 \\ l2 \\\n GSMP_disjoint (N2 (proj_unl l1 B)) (N2 (proj_unl l2 B))\n (f (set C) - {m. {} \\\\<^sub>c m})"} {"_id": "503339", "text": "proof (prove)\ngoal (1 subgoal):\n 1. False \\<^bold>\\ (\\x. \\<^bold>\\ (E x))"} {"_id": "503340", "text": "proof (prove)\ngoal (1 subgoal):\n 1. tprg\\Ext\\\\<^sub>C Base"} {"_id": "503341", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gba_to_idx_ext ecnv G\n \\ SPEC\n (\\G'.\n igba.lang G' = lang \\\n igba_rec.more G' = ecnv G \\ igba G')"} {"_id": "503342", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Red_F_\\_empty_q q = wf_lift.empty_ord.Red_F_\\"} {"_id": "503343", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {.x, u, v. localPrecS (u, v) (a, b) x.} \\ [: x, u, v \\ y,\n u', v' . localRelS (u, v) (u', v') x y :] \\\n {.x, u, v. localPrecS' (u, v) (c, d) x.} \\\n [: x, u, v \\ y, u', v' . localRelS' (u, v) (u', v') x y :] =\n {.x, u, v. localPrecSS' (u, v) (e, f) x.} \\ [: x, u,\n v \\ y, u', v' . localRelSS' (u, v) (u', v') x y :]"} {"_id": "503344", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_SSup (F \\ G) = map_SSup F \\ map_SSup G"} {"_id": "503345", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (t = Empty) = (toList t = [])"} {"_id": "503346", "text": "proof (prove)\ngoal (1 subgoal):\n 1. scf (prefs_from_table xs) = scf (prefs_from_table ys)"} {"_id": "503347", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\i.\n i \\ is \\\n refl_on A (f i) \\\n (\\x y.\n (x, y) \\ f i \\ (y, x) \\ f i) \\\n trans (f i)) \\\n \\x y.\n (x, y) \\ \\ (f ` is) \\\n (y, x) \\ \\ (f ` is)"} {"_id": "503348", "text": "proof (prove)\ngoal (1 subgoal):\n 1. SN_order_pair (s_mul_ext NS S) (ns_mul_ext NS S)"} {"_id": "503349", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n.\n \\SEQ n s = Some t \\ SEQ (Suc n) s = NEXT t;\n SEQ (Suc n) s = Some t\\\n \\ SEQ (Suc (Suc n)) s = NEXT t"} {"_id": "503350", "text": "proof (prove)\nusing this:\n \\\\Chn = fgh.chine_assoc',\n Dom = Cod (f \\ g \\ h),\n Cod =\n Dom ((f \\ g) \\\n h)\\ : f \\\n g \\ h \\ (f \\ g) \\ h\\\n\ngoal (1 subgoal):\n 1. (if arr \\Chn = fgh.chine_assoc',\n Dom = Cod (f \\ g \\ h),\n Cod = Dom ((f \\ g) \\ h)\\\n then mkObj assoc'_fgh.dtrg else null) =\n \\Chn = assoc'_fgh.dtrg,\n Dom = \\Leg0 = assoc'_fgh.dtrg, Leg1 = assoc'_fgh.dtrg\\,\n Cod =\n \\Leg0 = assoc'_fgh.dtrg,\n Leg1 = assoc'_fgh.dtrg\\\\"} {"_id": "503351", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\A C B.\n \\M = zhmset_of A + C; N = zhmset_of B + C; A < B\\\n \\ thesis) \\\n thesis"} {"_id": "503352", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x. plane_vec_eq x x"} {"_id": "503353", "text": "proof (prove)\nusing this:\n \\n. F1 n \\ F1 (Suc n)\n \\n. F2 n \\ F2 (Suc n)\n\ngoal (1 subgoal):\n 1. \\n.\n F1 n \\ F2 n \\ F1 (Suc n) \\ F2 (Suc n)"} {"_id": "503354", "text": "proof (prove)\nusing this:\n anybranch T (\\p. closed_branch p T Cs)\n\ngoal (1 subgoal):\n 1. \\T.\n anybranch T (\\p. falsifies\\<^sub>c\\<^sub>s p Cs) \\\n anyinternal T (\\p. \\ falsifies\\<^sub>c\\<^sub>s p Cs)"} {"_id": "503355", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\u y k - real k * lim (\\n. u y n / real n)\\\n \\ 2 * deltaG TYPE('a)"} {"_id": "503356", "text": "proof (prove)\ngoal (1 subgoal):\n 1. aB (boolV b) = b"} {"_id": "503357", "text": "proof (prove)\nusing this:\n deg R (Cring_Poly.compose R f g) = deg R f * deg R g\n\ngoal (1 subgoal):\n 1. deg R (Cring_Poly.compose R (Cring_Poly.truncate R f) g)\n < deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g)"} {"_id": "503358", "text": "proof (prove)\nusing this:\n (SUP i. F i ?x) = ennreal (f ?x)\n \\\\<^sup>+ x. ennreal\n (enn2real (F i x) *\n indicat_real\n (box (- (real i *\\<^sub>R One))\n (real i *\\<^sub>R One))\n x)\n \\lborel\n < \\\n\ngoal (1 subgoal):\n 1. (\\x.\n enn2real (F i x) *\n indicat_real (box (- (real i *\\<^sub>R One)) (real i *\\<^sub>R One))\n x) integrable_on\n UNIV"} {"_id": "503359", "text": "proof (prove)\nusing this:\n ps = a # ps'\n (List tl p ps \\\n List tl q qs \\\n set ps \\ set qs = {} \\ rev ps @ qs = rev Ps @ Qs) \\\n p = Ref a\n\ngoal (1 subgoal):\n 1. List (tl(p \\ q)) (tl (addr p)) ps' \\\n List (tl(p \\ q)) p (a # qs) \\\n set ps' \\ set (a # qs) = {} \\ rev ps' @ a # qs = rev Ps @ Qs"} {"_id": "503360", "text": "proof (prove)\nusing this:\n op \\ set (\\\\<^sub>\\)\n\ngoal (1 subgoal):\n 1. v \\ set (\\\\<^sub>\\)"} {"_id": "503361", "text": "proof (prove)\nusing this:\n inv f -` carrier = carrier\n\ngoal (1 subgoal):\n 1. finite_total_preorder_on (inv f -` carrier) (map_relation (inv f) le)"} {"_id": "503362", "text": "proof (prove)\nusing this:\n y \\ A\n\ngoal (1 subgoal):\n 1. orda y (luba A)"} {"_id": "503363", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\unit_vec n i|unit_vec n i\\ = 1"} {"_id": "503364", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id\n (hom_induced (p - 1) (subtopology X T) {} (subtopology X S) {} id z) =\n \\\\<^bsub>relative_homology_group (p - 1) (subtopology X S) T\\<^esub> \\\n x =\n hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id\n (y \\\\<^bsub>homology_group (p - 1) (subtopology X S)\\<^esub>\n inv\\<^bsub>homology_group (p - 1) (subtopology X S)\\<^esub> hom_induced\n (p - 1) (subtopology X T) {} (subtopology X S)\n {} id z)"} {"_id": "503365", "text": "proof (prove)\ngoal (1 subgoal):\n 1. shift_map_keys_punit t f (fmap_of_list xs) =\n fmap_of_list (map (\\(k, v). (t + k, f v)) xs)"} {"_id": "503366", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\x\\X.\n n_ivl (lookup (S1 \\ S2) x) \\ n_ivl (lookup S1 x)\n 2. \\a\\X.\n n_ivl (lookup (S1 \\ S2) a) < n_ivl (lookup S1 a)"} {"_id": "503367", "text": "proof (prove)\ngoal (1 subgoal):\n 1. 0 < m"} {"_id": "503368", "text": "proof (prove)\ngoal (1 subgoal):\n 1. routing_action_next_hop_update h pk =\n routing_action_update (next_hop_update (\\_. Some h)) pk"} {"_id": "503369", "text": "proof (prove)\ngoal (1 subgoal):\n 1. new_configured_SecurityInvariant\n (SINVAR_BLPtrusted.sinvar, SINVAR_BLPtrusted.default_node_properties,\n SINVAR_BLPtrusted.receiver_violation,\n BLPtrusted.node_props\n \\node_properties = BLP_security_levels\\) =\n new_configured_SecurityInvariant\n (SINVAR_BLPtrusted.sinvar, SINVAR_BLPtrusted.default_node_properties,\n SINVAR_BLPtrusted.receiver_violation,\n nm_node_props SINVAR_LIB_BLPtrusted\n \\node_properties = BLP_security_levels\\)"} {"_id": "503370", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\S \\ {};\n \\A.\n A \\ S \\\n A \\ Field r \\\n (\\a\\A. local.under a \\ A)\\\n \\ \\ S \\ Field r \\\n (\\a\\\\ S.\n local.under a \\ \\ S)"} {"_id": "503371", "text": "proof (prove)\ngoal (1 subgoal):\n 1. one_sheaf_spec U \\ \\ U"} {"_id": "503372", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l \\ i = b"} {"_id": "503373", "text": "proof (prove)\nusing this:\n finite K\n inj_on g K\n pdevs_domain x \\ g ` K\n \\?i \\ K; g ?i \\ pdevs_domain x\\\n \\ f ?i = (0::'b)\n \\?i \\ K; g ?i \\ pdevs_domain x\\\n \\ f ?i = e (g ?i) *\\<^sub>R pdevs_apply x (g ?i)\n\ngoal (1 subgoal):\n 1. (\\i\\the_inv_into K g ` pdevs_domain x.\n e (g i) *\\<^sub>R pdevs_apply x (g i)) =\n sum f K"} {"_id": "503374", "text": "proof (state)\nthis:\n {#x \\# hs1. x dvdm f1#} \\ hs1\n\ngoal (1 subgoal):\n 1. {#x \\# hs1. x dvdm f1#} \\ hs1 \\ False"} {"_id": "503375", "text": "proof (prove)\nusing this:\n \\x\\<^sub>j\n \\ rvars_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i);\n lhs (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i)\n \\ rvars_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i)\\\n \\ rvars_eq\n (pivot_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i)\n x\\<^sub>j) =\n {lhs (\\ s !\n eq_idx_for_lvar (\\ s) x\\<^sub>i)} \\\n (rvars_eq\n (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i) -\n {x\\<^sub>j})\n lhs (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i) = x\\<^sub>i\n x\\<^sub>j \\ rvars_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i)\n x\\<^sub>i \\ rvars_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i)\n\ngoal (1 subgoal):\n 1. rvars_eq\n (pivot_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i) x\\<^sub>j) =\n {x\\<^sub>i} \\\n (rvars_eq (\\ s ! eq_idx_for_lvar (\\ s) x\\<^sub>i) - {x\\<^sub>j})"} {"_id": "503376", "text": "proof (prove)\nusing this:\n periodic_orbit x\n closed_orbit ?x \\ ?x \\ X\n periodic_orbit ?x = (closed_orbit ?x \\ 0 < period ?x)\n\ngoal (1 subgoal):\n 1. x \\ X"} {"_id": "503377", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\a g.\n \\ide a;\n \\g : local.prod a\n b \\ c\\\\\n \\ \\!f.\n \\f : a \\ x\\ \\\n g = C e (local.prod f b)"} {"_id": "503378", "text": "proof (prove)\nusing this:\n simply_connected U\n simply_connected ?S =\n (path_connected ?S \\\n (\\p.\n path p \\\n path_image p \\ ?S \\\n pathfinish p = pathstart p \\\n (\\a. a \\ ?S \\ homotopic_loops ?S p (linepath a a))))\n\ngoal (1 subgoal):\n 1. path_connected U"} {"_id": "503379", "text": "proof (prove)\nusing this:\n 0 < stpb \\\n crsp ly (abc_step_l (as', am') (Some ins)) (steps0 (s', l', r') tp stpb)\n ires\n\ngoal (1 subgoal):\n 1. \\stpaa\\Suc stp.\n crsp ly (as, am) (steps0 (Suc 0, l, r) tp stpaa) ires"} {"_id": "503380", "text": "proof (prove)\ngoal (1 subgoal):\n 1. scalar_prod (map (\\k. scalar_prod r ((v # m) ! k)) [0..k. scalar_prod (row (v # m) k) c) [0.. 1\\<^sub>\\ \\ y \\ 1\\<^sub>\\) =\n d (x \\ 1\\<^sub>\\) \\ d (y \\ 1\\<^sub>\\)"} {"_id": "503382", "text": "proof (prove)\nusing this:\n V.in_hom \\ f g\n V.ide ?\\ = (arrow_of_spans (\\) ?\\ \\ C.ide (Chn ?\\))\n V.arr ?\\ = arrow_of_spans (\\) ?\\\n V.dom =\n (\\\\.\n if V.arr \\\n then \\Chn = span_in_category.apex (\\) (Dom \\),\n Dom = Dom \\, Cod = Dom \\\\\n else V.null)\n V.cod =\n (\\\\.\n if V.arr \\\n then \\Chn = span_in_category.apex (\\) (Cod \\),\n Dom = Cod \\, Cod = Cod \\\\\n else V.null)\n\ngoal (1 subgoal):\n 1. \\Chn \\ : Chn f \\ Chn g\\"} {"_id": "503383", "text": "proof (state)\ngoal (8 subgoals):\n 1. \\\\

\\<^sub>0[ra, ru.prj\\<^sub>1 \\ \\.chine]"} {"_id": "501815", "text": "proof (prove)\nusing this:\n enat i \\ llength t' \\\n cotrace (local.cos c' t' i) (Co_Snapshot.ldrop i t')\n enat (Suc i) \\ llength t\n c \\ ev \\ c'\n cotrace c' t'\n\ngoal (1 subgoal):\n 1. cotrace (local.cos c t (Suc i)) (Co_Snapshot.ldrop (Suc i) t)"} {"_id": "501816", "text": "proof (prove)\nusing this:\n rotate2 x = rotate2 y\n\ngoal (1 subgoal):\n 1. rotate2 (rotate2 x) = rotate2 (rotate2 y)"} {"_id": "501817", "text": "proof (prove)\nusing this:\n agree fm fm1\n agree fm fm2\n\ngoal (1 subgoal):\n 1. \\v.\n v \\ fmdom' fm \\\n v \\ fmdom' (fm2 ++\\<^sub>f fm1) \\\n fmlookup fm v = fmlookup (fm2 ++\\<^sub>f fm1) v"} {"_id": "501818", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set_lebesgue_integral M (\\n. (T ^^ n) --` A) f =\n set_lebesgue_integral M (space M) f"} {"_id": "501819", "text": "proof (state)\nthis:\n (ut, mon_w fg w1, P1) \\ S_cs fg k\n (return fg q, mon_w fg w2, P2) \\ S_cs fg k\n\ngoal (2 subgoals):\n 1. \\p.\n \\sh = [return fg p, v]; e = LRet; c' = ch\\\n \\ (v, mon_w fg (w @ [e]), P) \\ S_cs fg k\n 2. \\u p.\n \\sh = [u]; e = LSpawn p; c' = add_mset [entry fg p] ch;\n (u, Spawn p, v) \\ edges fg\\\n \\ (v, mon_w fg (w @ [e]), P) \\ S_cs fg k"} {"_id": "501820", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (mem_val_alt 10 (a AND 68719476732)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None) \\\n (mem_val_alt 10 ((a AND 68719476732) + 1)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None) \\\n (mem_val_alt 10 ((a AND 68719476732) + 2)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None) \\\n (mem_val_alt 10 ((a AND 68719476732) + 3)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None \\\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n None) \\\n ((\\y.\n mem_val_alt 10 (a AND 68719476732)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n ((\\y.\n mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (\\y.\n mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) =\n Some y) \\\n (((ucast\n (case mem_val_alt 10 (a AND 68719476732)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) of\n Some v \\ v) <<\n 24) OR\n (ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 1)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) of\n Some v \\ v) <<\n 16)) OR\n (ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 2)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) of\n Some v \\ v) <<\n 8)) OR\n ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 3)\n (s1\\mem := (mem s1)\n (10 := mem s1 10(addr \\ val),\n 11 := (mem s1 11)(addr := None))\\) of\n Some v \\ v) =\n (((ucast\n (case mem_val_alt 10 (a AND 68719476732)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) of\n Some v \\ v) <<\n 24) OR\n (ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 1)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) of\n Some v \\ v) <<\n 16)) OR\n (ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 2)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) of\n Some v \\ v) <<\n 8)) OR\n ucast\n (case mem_val_alt 10 ((a AND 68719476732) + 3)\n (s2\\mem := (mem s2)\n (10 := mem s2 10(addr \\ val),\n 11 := (mem s2 11)(addr := None))\\) of\n Some v \\ v)))"} {"_id": "501821", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\Cod t\\<^bold>\\\\ \\\n \\t\\ \\ \\u\\ =\n (\\Cod t\\<^bold>\\\\ \\\n \\t\\) \\\n \\u\\"} {"_id": "501822", "text": "proof (prove)\nusing this:\n (\\idx. if idx < r1 then 0 else if idx < r2 then i else l)\n \\ {0.. {0..l}\n \\j\\r2 - 1. (if j < r1 then 0 else if j < r2 then i else l) < l\n\ngoal (1 subgoal):\n 1. col (f (\\idx.\n if idx < r1 then 0 else if idx < r2 then i else l)) =\n col (f (\\ia.\n if ia \\ r2 - 1 then 0\n else if ia < r1 then 0 else if ia < r2 then i else l))"} {"_id": "501823", "text": "proof (prove)\nusing this:\n a * b = x * c\n c = (0::'a) \\ a = (0::'a) \\ b = (0::'a)\n\ngoal (1 subgoal):\n 1. x dvd a \\ x dvd b"} {"_id": "501824", "text": "proof (prove)\nusing this:\n y \\ (x, t, v)\n val v\n\ngoal (1 subgoal):\n 1. APP LAM [y].[(y, x)] \\ t\n v \\cbv ([(y, x)] \\ t)[y::=v]"} {"_id": "501825", "text": "proof (prove)\ngoal (1 subgoal):\n 1. invertible (fst (Gauss_Jordan_in_ij_PA (P, A) i j))"} {"_id": "501826", "text": "proof (prove)\ngoal (1 subgoal):\n 1. [\\\\<^bold>\\x.\n \\A!,x\\<^sup>P\\ \\<^bold>&\n (\\<^bold>\\F.\n \\x\\<^sup>P,F\\ \\<^bold>\\\n \\ F),G\\ in v]"} {"_id": "501827", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\n.\n \\k = 1..card (uverts H).\n real (card (uverts H) ^ k) *\n (real n ^ (2 * card (uverts H) - k) *\n p n powr\n (real (2 * card (uedges H)) - max_density H * real k)) /\n (1 / real (card (uverts H) ^ card (uverts H)) *\n (real n ^ card (uverts H) * p n ^ card (uedges H)))\\<^sup>2) =\n (\\n.\n (\\k = 1..card (uverts H).\n real (card (uverts H) ^ k) *\n (real n ^ (2 * card (uverts H) - k) *\n p n powr\n (real (2 * card (uedges H)) - max_density H * real k))) /\n (1 / real (card (uverts H) ^ card (uverts H)) *\n (real n ^ card (uverts H) * p n ^ card (uedges H)))\\<^sup>2)"} {"_id": "501828", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. g (f x)) \\a\\ l"} {"_id": "501829", "text": "proof (prove)\ngoal (5 subgoals):\n 1. \\a ec t.\n t \\ SPR.jkbpC \\\n spr_simAbs ec = SPRdet.sim_equiv_class a t \\\n local.spr_simObs a ec = envObs a (tLast t)\n 2. \\a ec t.\n t \\ SPR.jkbpC \\\n spr_simAbs ec = SPRdet.sim_equiv_class a t \\\n set (local.spr_simAction a ec) = set (jAction SPR.MC t a)\n 3. \\a ec t.\n t \\ SPR.jkbpC \\\n spr_simAbs ec = SPRdet.sim_equiv_class a t \\\n spr_simAbs ` set (local.spr_simTrans a ec) =\n rel_ext\n (\\uu_.\n \\t' s.\n uu_ = SPRdet.sim_equiv_class a (t' \\ s) \\\n t' \\ s \\ SPR.jkbpC \\\n spr_jview a t' = spr_jview a t)\n 4. MapOps spr_simAbs (\\a. SPRdet.sim_equiv_class a ` SPR.jkbpC)\n acts_MapOps\n 5. MapOps (\\k. (spr_simAbs (fst k), snd k))\n ((\\a. SPRdet.sim_equiv_class a ` SPR.jkbpC) \\ UNIV)\n trans_MapOps"} {"_id": "501830", "text": "proof (prove)\nusing this:\n (\\\\x\\\\, \\)\n \\ {(\\\\x\\\\, \\),\n (\\, \\\\x\\\\)}\n\ngoal (1 subgoal):\n 1. \\\\x\\\\ \\ \\"} {"_id": "501831", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\is_rendered_commutative_by (\\ J.AA) (\\ J.BB);\n \\p0 p1.\n mkCone p0 p1 \\\n \\j.\n if j = J.AA then p0\n else if j = J.BB then p1\n else if j = J.AT then f0 \\ p0\n else if j = J.BT then f1 \\ p1\n else if j = J.TT then f0 \\ p0 else C.null;\n \\f. \\ J.arr f \\ \\ f = C.null;\n \\f' f.\n f = J.AA \\ (f' = J.AA \\ f' = J.AT) \\\n f = J.BB \\ (f' = J.BB \\ f' = J.BT) \\\n f = J.AT \\ f' = J.TT \\\n f = J.BT \\ f' = J.TT \\\n f = J.TT \\ f' = J.TT \\\n \\ (f' \\\\<^sub>J f) = local.map f' \\ \\ f;\n \\g f.\n J.seq g f =\n (f = J.AA \\ (g = J.AA \\ g = J.AT) \\\n f = J.BB \\ (g = J.BB \\ g = J.BT) \\\n f = J.AT \\ g = J.TT \\\n f = J.BT \\ g = J.TT \\ f = J.TT \\ g = J.TT);\n \\f.\n J.arr f \\\n \\ (J.cod f) \\ a = \\ f\\\n \\ f1 \\ \\ J.BB = \\ J.TT"} {"_id": "501832", "text": "proof (prove)\nusing this:\n A * B =\n upd_rows 0 UNIV\n (\\i. \\j\\UNIV. to_fun A i j *s Square_Matrix.row B j)\n upd_rows 0 UNIV (\\i. Square_Matrix.row B (?f i)) = perm_rows B ?f\n Square_Matrix.det A =\n (\\p | p permutes UNIV.\n of_int (sign p) * (\\i\\UNIV. to_fun A i (p i)))\n\ngoal (1 subgoal):\n 1. Square_Matrix.det (A * B) = Square_Matrix.det A * Square_Matrix.det B"} {"_id": "501833", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length xs \\ n \\ pmf (bv n) xs = 0"} {"_id": "501834", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gcd (a * n div d) n = n div d * gcd a d"} {"_id": "501835", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\, \\' \\\\<^sub>C\n map_converter id id f g id_converter \\ \\\n \\ \\ map_\\ f g \\'"} {"_id": "501836", "text": "proof (state)\nthis:\n ((ps' @ ps, a) # branch) !. v = Some (qs, a') \\\n ((ps' @ ps, a) # branch) !. v = Some (ps' @ qs, a')\n\ngoal (1 subgoal):\n 1. \\v qs a v' p k i xs w w' rs b ps aa branch.\n \\((ps, aa) # branch) !. v = Some (qs, a); qs !. v' = Some p;\n descendants k i ((ps, aa) # branch) xs;\n descendants k i ((ps' @ ps, aa) # branch) xs; (w, w') \\ xs;\n ((ps, aa) # branch) !. w = Some (rs, b); rs !. w' = Some p;\n Nom a at b in (ps, aa) # branch\\\n \\ descendants k i ((ps' @ ps, aa) # branch)\n ({(v, v')} \\ xs)"} {"_id": "501837", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(\\x. f (- inverse x))\n \\ L' at_top (\\x. g (- inverse x));\n (\\x. f (inverse x))\n \\ L' at_top (\\x. g (inverse x))\\\n \\ f \\ L (at 0) g"} {"_id": "501838", "text": "proof (state)\nthis:\n \\M \\ multisets A; M \\ multisets A; M \\ {#}\\\n \\ mulex_on P A M (M + M)\n M \\ multisets A\n add_mset a M \\ multisets A\n add_mset a M \\ {#}\n\ngoal (1 subgoal):\n 1. \\x K.\n \\\\M \\ multisets A; K \\ multisets A;\n K \\ {#}\\\n \\ mulex_on P A M (M + K);\n M \\ multisets A; add_mset x K \\ multisets A;\n add_mset x K \\ {#}\\\n \\ mulex_on P A M (M + add_mset x K)"} {"_id": "501839", "text": "proof (prove)\nusing this:\n x \\ carrier R\n y \\ carrier R\n 1 < n\n\ngoal (1 subgoal):\n 1. x \\\\<^bsub>R\\<^esub> y \\ carrier R"} {"_id": "501840", "text": "proof (prove)\nusing this:\n \\\\\\<^sub>P, eP\\ P\\i \\\\\\<^sub>P @\n \\, eP'\\\n {$$} \\ e, eP, eV : \\\n\ngoal (1 subgoal):\n 1. (\\e' eV'.\n \\{$$} \\ e', eP', eV' : \\;\n \\[], e\\ I\\i \\[], e'\\;\n \\\\, eV\\ V\\i \\[], eV'\\\\\n \\ thesis) \\\n thesis"} {"_id": "501841", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd' Ch"} {"_id": "501842", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (x \\ i.R (y ## s)) = (x = y \\ x \\ i.R s)"} {"_id": "501843", "text": "proof (prove)\ngoal (1 subgoal):\n 1. continuous_map euclidean euclidean (\\f. f x)"} {"_id": "501844", "text": "proof (prove)\ngoal (1 subgoal):\n 1. mustT c1 s"} {"_id": "501845", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf'\\<^sub>s\\<^sub>s\\<^sub>t X (T1 @ T2 @ T3 @ T4 @ T5)"} {"_id": "501846", "text": "proof (prove)\nusing this:\n ES_valid ES1\n ES_valid ES2\n composable ES1 ES2\n isViewOn\n \\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>, N = {},\n C = C\\<^bsub>\\\\<^esub>\\\n E\\<^bsub>ES1 \\ ES2\\<^esub>\n isViewOn\n \\V = V\\<^bsub>\\1\\<^esub> \\ N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\n E\\<^bsub>ES1\\<^esub>\n isViewOn\n \\V = V\\<^bsub>\\2\\<^esub> \\ N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\\n E\\<^bsub>ES2\\<^esub>\n V\\<^bsub>\\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>,\n N = {}, C = C\\<^bsub>\\\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES1\\<^esub> =\n V\\<^bsub>\\V = V\\<^bsub>\\1\\<^esub> \\\n N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\\<^esub>\n V\\<^bsub>\\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>,\n N = {}, C = C\\<^bsub>\\\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES2\\<^esub> =\n V\\<^bsub>\\V = V\\<^bsub>\\2\\<^esub> \\\n N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\\\<^esub>\n C\\<^bsub>\\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>,\n N = {}, C = C\\<^bsub>\\\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES1\\<^esub>\n \\ C\\<^bsub>\\V = V\\<^bsub>\\1\\<^esub> \\\n N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\\<^esub>\n C\\<^bsub>\\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>,\n N = {}, C = C\\<^bsub>\\\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES2\\<^esub>\n \\ C\\<^bsub>\\V = V\\<^bsub>\\2\\<^esub> \\\n N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\\\<^esub>\n N\\<^bsub>\\V = V\\<^bsub>\\1\\<^esub> \\\n N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\\<^esub> \\\n N\\<^bsub>\\V = V\\<^bsub>\\2\\<^esub> \\\n N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\\\<^esub> =\n {}\n N\\<^bsub>\\V = V\\<^bsub>\\1\\<^esub> \\\n N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES1\\<^esub> =\n {} \\\n N\\<^bsub>\\V = V\\<^bsub>\\2\\<^esub> \\\n N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\\\<^esub> \\\n E\\<^bsub>ES2\\<^esub> =\n {}\n\ngoal (1 subgoal):\n 1. Compositionality ES1 ES2\n \\V = V\\<^bsub>\\\\<^esub> \\ N\\<^bsub>\\\\<^esub>,\n N = {}, C = C\\<^bsub>\\\\<^esub>\\\n \\V = V\\<^bsub>\\1\\<^esub> \\ N\\<^bsub>\\1\\<^esub>,\n N = {}, C = C\\<^bsub>\\1\\<^esub>\\\n \\V = V\\<^bsub>\\2\\<^esub> \\ N\\<^bsub>\\2\\<^esub>,\n N = {}, C = C\\<^bsub>\\2\\<^esub>\\"} {"_id": "501847", "text": "proof (prove)\nusing this:\n 0 < real (prod_list ys)\n\ngoal (1 subgoal):\n 1. sum_list (map (log b) (map real (y # ys))) =\n log b (real (prod_list (y # ys)))"} {"_id": "501848", "text": "proof (state)\nthis:\n \\x1 x2.\n x1 \\ unit_disc \\\n x2 \\ unit_disc \\\n \\ poincare_collinear {a, x1, x2} \\\n (\\a1\\unit_disc.\n \\a2\\unit_disc.\n \\ poincare_on_line a a1 a2 \\\n \\ poincare_ray_meets_line a a1 x1 x2 \\\n \\ poincare_ray_meets_line a a2 x1 x2 \\\n (\\a'\\unit_disc.\n poincare_in_angle a' a1 a a2 \\\n poincare_ray_meets_line a a' x1 x2))\n\ngoal (1 subgoal):\n 1. \\a1\\unit_disc.\n \\a2\\unit_disc.\n \\ poincare_on_line a a1 a2 \\\n \\ poincare_ray_meets_line a a1 x1 x2 \\\n \\ poincare_ray_meets_line a a2 x1 x2 \\\n (\\a'\\unit_disc.\n poincare_in_angle a' a1 a a2 \\\n poincare_ray_meets_line a a' x1 x2)"} {"_id": "501849", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\(c, s) \\ t; s \\ S\\\n \\ t \\ post (Collecting.lfp c (step S))"} {"_id": "501850", "text": "proof (prove)\ngoal (1 subgoal):\n 1. VARtoCorrectSYSTEM"} {"_id": "501851", "text": "proof (prove)\ngoal (1 subgoal):\n 1. single_valued (\\n. (f ^^ n) {})"} {"_id": "501852", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\P \\ g \\ R;\n \\,\\\n \\\\<^sub>t\\<^bsub>/F\\<^esub> R c Q,A\\\n \\ \\,\\\n \\\\<^sub>t\\<^bsub>/F\\<^esub> P\n guaranteeStrip f g c Q,A"} {"_id": "501853", "text": "proof (prove)\nusing this:\n \\m\\M. \\n\\M. dist (X m) (X n) < r / sqrt 2\n \\m\\N. \\n\\N. dist (Y m) (Y n) < r / sqrt 2\n\ngoal (1 subgoal):\n 1. \\m\\max M N.\n \\n\\max M N. dist (X m, Y m) (X n, Y n) < r"} {"_id": "501854", "text": "proof (prove)\ngoal (1 subgoal):\n 1. normal_form F (- p) = - normal_form F p"} {"_id": "501855", "text": "proof (prove)\ngoal (1 subgoal):\n 1. e \\ UNIV \\ {- 1..1} \\\n \\pdevs_val e x\\ \\ tdev x"} {"_id": "501856", "text": "proof (prove)\nusing this:\n x =\n (\\i = 0.. (\\G\\Gs. G)\n then \\Gs\\ x ! i else (0::'a))\n \\i (\\G\\Gs. G)\n then \\Gs\\ x ! i else (0::'a)) =\n (0::'a)\n\ngoal (1 subgoal):\n 1. x \\ 0"} {"_id": "501857", "text": "proof (prove)\nusing this:\n closed S\n closed T\n S \\ T = {}\n\ngoal (1 subgoal):\n 1. (\\f.\n \\continuous_on UNIV f;\n \\x. f x \\ closed_segment a b;\n \\x. x \\ S \\ f x = a;\n \\x. x \\ T \\ f x = b\\\n \\ thesis) \\\n thesis"} {"_id": "501858", "text": "proof (prove)\nusing this:\n P \\ \\e_,s\\<^sub>0_\\ \\\n \\Val v_,(h_, l_)\\\n P \\ \\e_,s\\<^sub>0_\\ \\*\n \\Val v_,(h_, l_)\\\n l'_ = l_(V_ \\ v_)\n\ngoal (1 subgoal):\n 1. P \\ \\V_:=e_,s\\<^sub>0_\\ \\*\n \\unit,(h_, l'_)\\"} {"_id": "501859", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\start_heap P a = \\(C, S)\\;\n Subobjs P C Cs\\\n \\ \\fs. (Cs, fs) \\ S"} {"_id": "501860", "text": "proof (prove)\nusing this:\n Re B \\ 0\n\ngoal (1 subgoal):\n 1. (sgn (Re B))\\<^sup>2 = 1"} {"_id": "501861", "text": "proof (prove)\ngoal (1 subgoal):\n 1. LIM x F. - g x :> at_top"} {"_id": "501862", "text": "proof (prove)\ngoal (1 subgoal):\n 1. flat ((x # xs) ## ws) = x ## flat (if xs = [] then ws else xs ## ws)"} {"_id": "501863", "text": "proof (prove)\nusing this:\n CONSTRAINT (IS_PURE single_valued) K\n CONSTRAINT (IS_PURE IS_LEFT_UNIQUE) K\n CONSTRAINT is_pure V\n CONSTRAINT (is_unused_elem dflt) V\n\ngoal (1 subgoal):\n 1. True \\\n comp_PRE\n ((the_pure K \\\\<^sub>r the_pure V) \\\\<^sub>r\n \\the_pure K, the_pure V\\map_rel)\n (\\_. True) (\\x ((a, b), ba). b \\ dflt)\n (\\x.\n nofail (uncurry2 (RETURN \\\\\\ op_map_update) x))\n x_"} {"_id": "501864", "text": "proof (state)\nthis:\n prod_list (list_neq x (mean x)) = 1\n\ngoal (1 subgoal):\n 1. x \\ [] \\ het x = 0 \\\n prod_list x = mean x ^ length x"} {"_id": "501865", "text": "proof (prove)\nusing this:\n (?ps2, ?c2) \\ g3mod1 \\ (?ps2, ?c2) \\ modRules2\n\ngoal (1 subgoal):\n 1. g3mod1 \\ modRules2"} {"_id": "501866", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ Hom A B \\\n (\\f\\Hom A B.\n compose (Hom A A) (set_func (u B))\n (set_func (Hom(A,_) \\<^bsub>\\\\<^esub> f)) (Id A))\n x =\n restrict (set_func (u B)) (Hom A B) x"} {"_id": "501867", "text": "proof (prove)\nusing this:\n 1 < card N\n p ^ a = card N * card (stabilizer m)\n\ngoal (1 subgoal):\n 1. p dvd card N"} {"_id": "501868", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.map f2 =\n A1_B.MkArr (\\f1. F (f1, A2.dom f2))\n (\\f1. F (f1, A2.cod f2)) (\\f1. F (f1, f2))"} {"_id": "501869", "text": "proof (prove)\ngoal (1 subgoal):\n 1. N - (m - n) + m = N + n"} {"_id": "501870", "text": "proof (chain)\npicking this:\n t \\ \\ E\n finite (\\ E)"} {"_id": "501871", "text": "proof (prove)\nusing this:\n f permutes carrier\n\ngoal (1 subgoal):\n 1. inv f -` carrier = carrier"} {"_id": "501872", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x.\n x \\ {e. f e \\ bot \\ f e + bot = fe + bot} \\\n (\\y.\n y \\ {e. f e \\ bot \\\n f e + bot = fe + bot} \\\n enum_lex_less_eq x y)"} {"_id": "501873", "text": "proof (prove)\nusing this:\n is \\ dims A\n\ngoal (1 subgoal):\n 1. lookup A is = a * prod_list (map2 (\\i B. lookup B [i]) is Bs)"} {"_id": "501874", "text": "proof (prove)\ngoal (1 subgoal):\n 1. cinner (a / of_complex m) b = cinner a b / cnj m"} {"_id": "501875", "text": "proof (prove)\nusing this:\n replay' thy vs ns Hs P = Some res'\n beta_eta_norm res' = Some res\n thy,set Hs \\ res'\n\ngoal (1 subgoal):\n 1. thy,set Hs \\ res"} {"_id": "501876", "text": "proof (prove)\nusing this:\n dom\\<^bsub>C\\<^esub> f = X\n LSCategory C\n f maps\\<^bsub>C\\<^esub> X to Y\n\ngoal (1 subgoal):\n 1. Hom\\<^bsub>C\\<^esub>[\\,dom\\<^bsub>C\\<^esub> f] = YFtor C @@ X"} {"_id": "501877", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bits_of k \\ []"} {"_id": "501878", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x\\carrier.\n \\c1\\atlas.\n \\c2\\manifold_eucl.atlas k.\n x \\ domain c1 \\\n f ` domain c1 \\ domain c2 \\\n k-smooth_on (codomain c1)\n (apply_chart c2 \\ f \\ inv_chart c1)"} {"_id": "501879", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n \\lleq_charlist (Rep_fin_string x) (Rep_fin_string y);\n lleq_charlist (Rep_fin_string y) (Rep_fin_string x)\\\n \\ x = y"} {"_id": "501880", "text": "proof (prove)\ngoal (1 subgoal):\n 1. wf_J_prog P \\ wwf_J_prog P"} {"_id": "501881", "text": "proof (prove)\nusing this:\n u' \\ [?D]\\<^bsub>v,n\\<^esub> \\\n \\D'.\n A \\ \\l', ?D\\ \\*\\<^bsub>v,n\\<^esub> \\l'', D'\\ \\\n u'' \\ [D']\\<^bsub>v,n\\<^esub>\n A \\ \\l, D\\ \\\\<^bsub>v,n\\<^esub> \\l', D'\\\n u' \\ [D']\\<^bsub>v,n\\<^esub>\n\ngoal (1 subgoal):\n 1. (\\D''.\n \\A \\ \\l', D'\\ \\*\\<^bsub>v,n\\<^esub> \\l'', D''\\;\n u'' \\ [D'']\\<^bsub>v,n\\<^esub>\\\n \\ thesis) \\\n thesis"} {"_id": "501882", "text": "proof (prove)\ngoal (2 subgoals):\n 1. F \\ E \\\n sym {(x, y) |x y.\n nodes_connected\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = F\\\n x y}\n 2. F \\ E \\\n trans\n {(x, y) |x y.\n nodes_connected\n \\nodes = fst ` E \\ (snd \\ snd) ` E,\n edges = F\\\n x y}"} {"_id": "501883", "text": "proof (prove)\ngoal (1 subgoal):\n 1. F g S = F h T"} {"_id": "501884", "text": "proof (prove)\nusing this:\n a \\ set_pmf M\n set_pmf (return_pmf x) = {x}\n\ngoal (1 subgoal):\n 1. (x, a) \\ set_pmf pq"} {"_id": "501885", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\length qs = length as; length as = n; dist_perm s s\\\n \\ dist_perm (steps' s qs as n) (steps' s qs as n)"} {"_id": "501886", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ I \\ \\[N]_f \\ \\P"} {"_id": "501887", "text": "proof (prove)\ngoal (1 subgoal):\n 1. xs = ys"} {"_id": "501888", "text": "proof (prove)\ngoal (1 subgoal):\n 1. s \\ l\\<^sup>\\ \\ b\\<^sup>\\ \\\n r\\<^sup>\\ \\\n q \\\n (a \\ b\\<^sup>\\ \\ r\\<^sup>\\ \\\n q)\\<^sup>\\\n \\ s \\ l\\<^sup>\\ \\ r\\<^sup>\\ \\\n (a \\ b\\<^sup>\\ \\ q \\\n r\\<^sup>\\)\\<^sup>\\"} {"_id": "501889", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map CAs0f [0..\\cl map \\s0f [0..tp. tp = (Bk \\ i, )} p\n {\\tp. tp = (Bk \\ m, Oc \\ rs @ Bk \\ k)}\n is_final (steps0 (Suc 0, Bk \\ i, ) p n)\n (\\tp.\n tp =\n (Bk \\ m,\n Oc \\ rs @\n Bk \\ k)) holds_for steps0 (Suc 0, Bk \\ i, ) p n\n\ngoal (1 subgoal):\n 1. \\stp.\n steps0 (Suc 0, Bk \\ i, ) p stp =\n (0, Bk \\ m, Oc \\ rs @ Bk \\ k)"} {"_id": "501891", "text": "proof (prove)\ngoal (1 subgoal):\n 1. gFreshGVar MOD \\ igFreshIGVar (fromMOD MOD)"} {"_id": "501892", "text": "proof (prove)\nusing this:\n f ` (A - B)\n \\ {X. X \\ A \\ finite X \\ card X = k}\n infinite (f ` (A - B))\n\ngoal (1 subgoal):\n 1. infinite {X. X \\ A \\ finite X \\ card X = k}"} {"_id": "501893", "text": "proof (state)\nthis:\n rho_bound2 (ICf i) \\ rho_bound2 (ICf j)\n\ngoal (1 subgoal):\n 1. \\IC p i t - IC q j t\\\n \\ \\IC p i s - IC q j s\\ + 2 * \\ * (t - s)"} {"_id": "501894", "text": "proof (prove)\ngoal (1 subgoal):\n 1. bij G.face_cycle_succ"} {"_id": "501895", "text": "proof (prove)\ngoal (1 subgoal):\n 1. finite (short_cycles G k)"} {"_id": "501896", "text": "proof (prove)\nusing this:\n domain.gallery (zx # ws @ [zy])\n x = f ` zx\n y = f ` zy\n domain.gallery (zx # ws @ [zy])\n\ngoal (1 subgoal):\n 1. SimplicialComplex.maxsimpchain (f \\ X)\n (x # f \\ ws @ [y])"} {"_id": "501897", "text": "proof (prove)\ngoal (1 subgoal):\n 1. group.ord (Z (p ^ multiplicity p n))\n ((\\p\\prime_factors n. 1) p) =\n p ^ multiplicity p n"} {"_id": "501898", "text": "proof (prove)\ngoal (1 subgoal):\n 1. distinct (insert_spec xs (oid, ref))"} {"_id": "501899", "text": "proof (prove)\nusing this:\n (remove_rels ts, m,\n \\) \\\\<^sub>v\\<^sup>* (remove_rels ts', m', \\')\n\ngoal (1 subgoal):\n 1. (remove_rels ts, m,\n \\) \\\\<^sub>v\\<^sup>* (remove_rels ts', m', \\')"} {"_id": "501900", "text": "proof (prove)\nusing this:\n SN_on r A\n x \\ A\n\ngoal (1 subgoal):\n 1. x \\ SN_part r"} {"_id": "501901", "text": "proof (prove)\ngoal (1 subgoal):\n 1. nprv {eql (Num n) t, neg (eql t (Num n)), LLq (Num n) t}\n (dsj (LLq t (suc (Num n))) (LLq (suc (Num n)) t))"} {"_id": "501902", "text": "proof (prove)\ngoal (1 subgoal):\n 1. signed_take_bit n k = concat_bit n k (- of_bool (bit k n))"} {"_id": "501903", "text": "proof (state)\nthis:\n timpls_transformable_to_pred A (Fun f T) s\n\ngoal (1 subgoal):\n 1. \\x.\n x \\ {s. timpls_transformable_to_pred A (Fun f T)\n s} \\\n x \\ Fun f `\n {S. length T = length S \\\n (\\s\\set S.\n \\t\\set T.\n timpls_transformable_to_pred A t s)}"} {"_id": "501904", "text": "proof (prove)\nusing this:\n i \\ Basis\n\ngoal (1 subgoal):\n 1. \\z b.\n \\z \\ C; b = z * e / (2 * real DIM('a))\\\n \\ i \\ x \\ b \\\n i \\ y \\ b \\\n b \\ i \\ x \\ b \\ i \\ y"} {"_id": "501905", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Ifm vs (x # bs) (subst0 t p) = Ifm vs (Itm vs (x # bs) t # bs) p"} {"_id": "501906", "text": "proof (prove)\nusing this:\n order.pseudominimal_in (Pow A) x\n order.pseudominimal_in ?P ?x \\ ?x \\ order.bottom ?P\n order.bottom (Pow A) = {}\n\ngoal (1 subgoal):\n 1. (\\a.\n order.greater_eq x {a} \\ thesis) \\\n thesis"} {"_id": "501907", "text": "proof (prove)\nusing this:\n Y = lang_rhs (Arden Y yrhs)\n finite (ES \\ {(Y, yrhs)}) \\\n (\\X rhs.\n (X, rhs) \\ ES \\ {(Y, yrhs)} \\\n finite rhs) \\\n (\\(X, rhs)\\ES \\ {(Y, yrhs)}. X = lang_rhs rhs) \\\n distinctness (ES \\ {(Y, yrhs)}) \\\n ardenable_all (ES \\ {(Y, yrhs)}) \\\n validity (ES \\ {(Y, yrhs)})\n\ngoal (1 subgoal):\n 1. \\(X, rhs)\n \\{(Ya, Subst yrhsa Y (Arden Y yrhs)) |Ya yrhsa.\n (Ya, yrhsa) \\ ES}.\n X = lang_rhs rhs"} {"_id": "501908", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (set xs = set ys) =\n ((\\x\\set xs. x \\ set ys) \\\n (\\y\\set ys. y \\ set xs))"} {"_id": "501909", "text": "proof (prove)\nusing this:\n (\\ (?p \\* ?pa) ?a \\\n ?p (aaa ?p ?pb) \\ (?pb \\* ?pa) ?a) \\\n (\\ ?pb (aaa ?p ?pb) \\\n \\ (?p \\* ?pc) ?aa \\ (?pb \\* ?pc) ?aa)\n\ngoal (1 subgoal):\n 1. \\p. ((\\a. a = (0::'a)) \\* p) aa"} {"_id": "501910", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y.\n (x, y) \\ rel_nabla_bin X Q \\\n \\y.\n y = x \\\n (x, x) \\ rel_nabla_bin X Q \\\n (\\y. (x, y) \\ relfdia (X\\<^sup>*) Q) \\\n (x, y) \\ relfdia (X\\<^sup>*) Q"} {"_id": "501911", "text": "proof (prove)\nusing this:\n Fun f ts \\ s\n\ngoal (1 subgoal):\n 1. (\\t.\n \\t \\ set ts; t \\ s\\\n \\ thesis) \\\n thesis"} {"_id": "501912", "text": "proof (prove)\ngoal (1 subgoal):\n 1. j \\ f i"} {"_id": "501913", "text": "proof (prove)\ngoal (1 subgoal):\n 1. ((\\x. sum f (x \\ A)) \\ limA)\n (finite_subsets_at_top B)"} {"_id": "501914", "text": "proof (prove)\ngoal (1 subgoal):\n 1. simplicial_chain p (standard_simplex (Suc p))\n (chain_boundary (Suc p)\n (frag_of (restrict id (standard_simplex (Suc p)))))"} {"_id": "501915", "text": "proof (prove)\nusing this:\n (case c1 (f y) (f z) of Eq \\ c2 (g y) (g z)\n | Lt \\ Lt | Gt \\ Gt) =\n Lt\n\ngoal (1 subgoal):\n 1. c1 (f y) (f z) = Lt \\ c1 (f y) (f z) = Eq \\ c2 (g y) (g z) = Lt"} {"_id": "501916", "text": "proof (prove)\ngoal (19 subgoals):\n 1. \\G c.\n \\(G, c, HighSec) \\ Deriv; G \\ c : HighSec\\\n \\ G \\ c : Sec\n (\\(s, t, \\). s \\\\<^sub>\\ t)\n 2. \\G.\n G \\ Skip : Sec (\\(s, t, \\).\n s \\\\<^sub>\\ t)\n 3. \\e G x.\n Expr_low e \\\n G \\ Assign x\n e : Sec (\\(s, t, \\).\n \\r.\n s =\n (update (fst r) x (evalE e (fst r)),\n snd r) \\\n r \\\\<^sub>\\ t)\n 4. \\G c1 \\ c2 \\.\n \\(G, c1, Sec \\) \\ Deriv; G \\ c1 : Sec \\;\n (G, c2, Sec \\) \\ Deriv; G \\ c2 : Sec \\\\\n \\ G \\ Comp c1\n c2 : Sec\n (\\(s, t, \\).\n \\r.\n \\ (r, t, \\) \\\n (\\w \\.\n r \\\\<^sub>\\ w \\\n \\ (s, w, \\)))\n 5. \\b G c1 \\ c2 \\.\n \\BExpr_low b; (G, c1, Sec \\) \\ Deriv;\n G \\ c1 : Sec \\; (G, c2, Sec \\) \\ Deriv;\n G \\ c2 : Sec \\\\\n \\ G \\ Iff b c1\n c2 : Sec\n (\\(s, t, \\).\n (evalB b (fst t) \\ \\ (s, t, \\)) \\\n (\\ evalB b (fst t) \\ \\ (s, t, \\)))\n 6. \\x G C.\n CONTEXT x = low \\\n G \\ New x\n C : Sec (\\(s, t, \\).\n \\l r.\n l \\ Dom (snd r) \\\n r \\\\<^sub>\\ t \\\n s =\n (update (fst r) x (RVal (Loc l)),\n (l, C, []) # snd r))\n 7. \\y f G x.\n \\CONTEXT y = low; GAMMA f = low\\\n \\ G \\ Get x y\n f : Sec\n (\\(s, t, \\).\n \\r l C Flds v.\n fst r y = RVal (Loc l) \\\n lookup (snd r) l = Some (C, Flds) \\\n lookup Flds f = Some v \\\n r \\\\<^sub>\\ t \\ s = (update (fst r) x v, snd r))\n 8. \\x f e G.\n \\CONTEXT x = low; GAMMA f = low; Expr_low e\\\n \\ G \\ Put x f\n e : Sec\n (\\(s, t, \\).\n \\r l C F h.\n fst r x = RVal (Loc l) \\\n r \\\\<^sub>\\ t \\\n lookup (snd r) l = Some (C, F) \\\n h = (l, C, (f, evalE e (fst r)) # F) # snd r \\ s = (fst r, h))\n 9. \\b G c \\.\n \\BExpr_low b; (G, c, Sec \\) \\ Deriv;\n G \\ c : Sec \\\\\n \\ G \\ While b c : Sec (PhiWhile b \\)\n 10. \\\\ G.\n \\({Sec (FIX \\)} \\ G, body,\n Sec (\\ (FIX \\)))\n \\ Deriv;\n ({Sec (FIX \\)} \\\n G) \\ body : Sec (\\ (FIX \\));\n Monotone \\\\\n \\ G \\ Call : Sec (FIX \\)\nA total of 19 subgoals..."} {"_id": "501917", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x. f (g x)) -` Y \\ space N \\ sets N"} {"_id": "501918", "text": "proof (prove)\ngoal (1 subgoal):\n 1. igSwapIPresIGWlsSTR MOD \\ igSwapInpIPresIGWlsInpSTR MOD"} {"_id": "501919", "text": "proof (prove)\nusing this:\n x + y + z = 1\n x\\<^sup>2 + y\\<^sup>2 + z\\<^sup>2 = 2\n x ^ 3 + y ^ 3 + z ^ 3 = 3\n\ngoal (1 subgoal):\n 1. power_sum_puzzle_example x y z"} {"_id": "501920", "text": "proof (prove)\ngoal (1 subgoal):\n 1. l_new_character_data_get_disconnected_nodes get_disconnected_nodes_locs"} {"_id": "501921", "text": "proof (prove)\nusing this:\n \\D.arr ?f; D.cod ?f = ?b\\\n \\ ?b \\\\<^sub>D ?f = ?f\n C.ide ?f \\ C.iso \\\\<^sub>C[?f]\n C.ide ?f \\ C.iso \\\\<^sub>C[?f]\n C.iso ?f \\ D.iso (F ?f)\n D.iso ?f \\ ?f \\\\<^sub>D D.inv ?f = D.cod ?f\n C.iso ?f \\ F (C.inv ?f) = D.inv (F ?f)\n\ngoal (1 subgoal):\n 1. \\\\<^sub>D\\<^sup>-\\<^sup>1[F f] \\\\<^sub>D\n ((F \\\\<^sub>C[f] \\\\<^sub>D\n F \\\\<^sub>C\\<^sup>-\\<^sup>1[f]) \\\\<^sub>D\n F \\\\<^sub>C[f] \\\\<^sub>D\n D.inv (F \\\\<^sub>C[f])) \\\\<^sub>D\n \\\\<^sub>D[F f] =\n \\\\<^sub>D\\<^sup>-\\<^sup>1[F f] \\\\<^sub>D \\\\<^sub>D[F f]"} {"_id": "501922", "text": "proof (prove)\ngoal (1 subgoal):\n 1. vectorspace.dim class_ring (vs (subspace_sum W1 W2))\n \\ vectorspace.dim class_ring (vs (local.span (set (cols A)))) +\n vectorspace.dim class_ring (vs (local.span (set (cols B))))"} {"_id": "501923", "text": "proof (prove)\ngoal (1 subgoal):\n 1. space\n (uniform_measure lborel {0..1} \\\\<^sub>M\n Pi\\<^sub>M (set xs)\n (\\x. uniform_measure lborel {0..1})) \\\n {}"} {"_id": "501924", "text": "proof (prove)\nusing this:\n \\\\xa y.\n xa \\ y \\\n xa + \\a * (x - x')\\\n \\ y + \\a * (x - x')\\;\n \\x y. x \\ y \\ d + x \\ d + y\\\n \\ \\f x - b -\n (a * fst (fst X) +\n a * pdevs_val e (snd (fst X)))\\\n \\ d + truncate_up p (\\a\\ * snd X)\n\ngoal (1 subgoal):\n 1. \\f x - b - (a * fst (fst X) + a * pdevs_val e (snd (fst X)))\\\n \\ truncate_up p (\\a\\ * snd X) + d"} {"_id": "501925", "text": "proof (prove)\nusing this:\n t \\ SPR.jkbpC\n\ngoal (1 subgoal):\n 1. set (jAction (spr_simMCt t) (spr_sim t) agent) =\n set (jAction spr_simMC (spr_sim t) agent)"} {"_id": "501926", "text": "proof (prove)\nusing this:\n b \\ set as\n a = hd_coeff1 (zlcms (map hd_coeff as)) b\n\ngoal (1 subgoal):\n 1. hd_coeff b \\ 0 &&& divisor b \\ 0"} {"_id": "501927", "text": "proof (prove)\ngoal (1 subgoal):\n 1. hn_refine (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry\n (REC\\<^sub>T aB ax)"} {"_id": "501928", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\ys.\n length xs = length ys \\\n vec_of_list xs \\ vec_of_list ys =\n sum_list (map2 (*) xs ys);\n Suc (length xs) = length ys\\\n \\ vCons a (vec_of_list xs) \\ vec_of_list ys =\n sum_list (map2 (*) (a # xs) ys)"} {"_id": "501929", "text": "proof (prove)\nusing this:\n x = lincomb a (A \\ U \\ (A \\ W - U))\n\ngoal (1 subgoal):\n 1. x =\n lincomb a (A \\ U) \\\\<^bsub>V\\<^esub>\n lincomb a (A \\ W - U)"} {"_id": "501930", "text": "proof (chain)\npicking this:\n mds\\<^sub>C' = mds\\<^sub>C\n (m := {y \\ mds\\<^sub>C m. y \\ Eg2_var\\<^sub>C_of_Eg1 x})"} {"_id": "501931", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\ A D C CongA A C D"} {"_id": "501932", "text": "proof (prove)\nusing this:\n enat i < llength Ns \\ lhd Ns \\ lnth Ns i\n enat (Suc i) < llength Ns\n (enat (Suc ?m) \\ ?n) = (enat ?m < ?n)\n chain (\\) Ns\n \\chain ?R ?xs; enat (Suc ?j) < llength ?xs\\\n \\ ?R (lnth ?xs ?j) (lnth ?xs (Suc ?j))\n \\?N1.0 \\ ?N2.0; ?N2.0 \\ ?N3.0\\\n \\ ?N1.0 \\ ?N3.0\n (?x < ?y) = (?x \\ ?y \\ ?x \\ ?y)\n\ngoal (1 subgoal):\n 1. lhd Ns \\ lnth Ns (Suc i)"} {"_id": "501933", "text": "proof (prove)\ngoal (2 subgoals):\n 1. rat_poly_inv (rat_poly_times B (rat_poly_times B B)) =\n rat_poly_inv (rat_poly_times A (rat_poly_times B A)) -\n rat_poly_times B\n (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B)\n 2. rat_poly_inv (rat_poly_times A (rat_poly_times B A)) -\n rat_poly_times B\n (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B) =\n rat_poly_inv (rat_poly_times B (rat_poly_times B B))"} {"_id": "501934", "text": "proof (prove)\ngoal (1 subgoal):\n 1. poly (of_nat n) x = of_nat n"} {"_id": "501935", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\x.\n \\(a, b, c)\\set les. a * x\\<^sup>2 + b * x + c < 0) =\n ((\\(a, b, c)\\set les.\n \\x. \\y2 + b * y + c < 0) \\\n (\\(a', b', c')\\set les.\n a' = 0 \\\n b' \\ 0 \\\n (\\(d, e, f)\\set les.\n \\y'>- (c' / b').\n \\x\\{- (c' / b')<..y'}.\n d * x\\<^sup>2 + e * x + f < 0) \\\n a' \\ 0 \\\n 4 * a' * c' \\ b'\\<^sup>2 \\\n ((\\(d, e, f)\\set les.\n \\y'>(sqrt (b'\\<^sup>2 - 4 * a' * c') - b') / (2 * a').\n \\x\\{(sqrt (b'\\<^sup>2 - 4 * a' * c') - b') /\n (2 * a')<..y'}.\n d * x\\<^sup>2 + e * x + f < 0) \\\n (\\(d, e, f)\\set les.\n \\y'>(- b' - sqrt (b'\\<^sup>2 - 4 * a' * c')) /\n (2 * a').\n \\x\\{(- b' - sqrt (b'\\<^sup>2 - 4 * a' * c')) /\n (2 * a')<..y'}.\n d * x\\<^sup>2 + e * x + f < 0))))"} {"_id": "501936", "text": "proof (prove)\ngoal (1 subgoal):\n 1. {k. k \\ n \\ \\ cycle_free_up_to m k n} \\ {}"} {"_id": "501937", "text": "proof (prove)\nusing this:\n 0 \\ k\n 0 \\ 1 - k\n\ngoal (1 subgoal):\n 1. \\ \\ G \\ s +\n \\ G \\ s *\n (wp body\n (\\s. \\ \\ G \\ s * (1 - k))\n s +\n wp body (\\s. k) s)\n \\ \\ \\ G \\ s +\n \\ G \\ s *\n wp body\n (\\s.\n \\ \\ G \\ s * (1 - k) + k)\n s"} {"_id": "501938", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\s. snd d = Fun (Set s) []"} {"_id": "501939", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x xa y.\n \\Wellfounded.accp (step1 r) x;\n \\y.\n step1 r y x \\ listsp (Wellfounded.accp r) y;\n xa \\ set x; r y xa\\\n \\ Wellfounded.accp r y"} {"_id": "501940", "text": "proof (prove)\ngoal (1 subgoal):\n 1. CFG_SSA_Transformed (\\_. gen_wf_\\e gen_cfg_wf)\n (\\_. gen_wf_\\n gen_cfg_wf) (\\_. True)\n (\\_. gen_wf_inEdges' gen_cfg_wf)\n (\\_. gen_wf_Entry gen_cfg_wf)\n (\\_. gen_wf_defs gen_cfg_wf)\n (\\_. gen_wf_uses gen_cfg_wf)\n (\\_. gen_wf_defs' gen_cfg_wf)\n (\\_. gen_wf.uses' gen_cfg_wf)\n (\\_. gen_wf.phis' gen_cfg_wf)\n (\\_. gen_wf_var gen_cfg_wf)"} {"_id": "501941", "text": "proof (prove)\ngoal (1 subgoal):\n 1. deterministic_RP St0 \\ None"} {"_id": "501942", "text": "proof (prove)\ngoal (1 subgoal):\n 1. lmap fst\n (lproject {(x, y). y \\ s} (w \\\\ iterates Suc 0)) =\n lmap fst (lproject (UNIV \\ s) (w \\\\ iterates Suc 0))"} {"_id": "501943", "text": "proof (prove)\ngoal (1 subgoal):\n 1. set (Cubic_Polynomials.croots3 p) = {x. poly p x = 0}"} {"_id": "501944", "text": "proof (prove)\nusing this:\n antisym (set l) \\ exe_antisym l\n antisym (set (a # l))\n\ngoal (1 subgoal):\n 1. exe_antisym (a # l)"} {"_id": "501945", "text": "proof (prove)\nusing this:\n \\ \\ Var x : T\n\ngoal (1 subgoal):\n 1. (x, T) \\ set \\"} {"_id": "501946", "text": "proof (prove)\nusing this:\n v \\ elts y'\n u \\ elts (TC v)\n\ngoal (1 subgoal):\n 1. (u, y') \\ VWO"} {"_id": "501947", "text": "proof (prove)\nusing this:\n \\(?y, x) \\ r; ?y \\ A\\\n \\ ordermap (\\ ` A) s (\\ ?y) = ordermap A r ?y\n x \\ A\n\ngoal (1 subgoal):\n 1. ordermap (\\ ` A) s (\\ x) = ordermap A r x"} {"_id": "501948", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\\\<^sup>+ x. emeasure\n (K.lim_stream x \\\n (\\\\.\n K.lim_stream\n (j, trace_at s (x ## \\) j) \\\n (\\\\'.\n return (stream_space S)\n(merge_at (x ## \\) j \\'))))\n A *\n indicator {j<..} (fst x)\n \\K (t, s) =\n \\\\<^sup>+ x. emeasure (K.lim_stream (j, s)) A *\n indicator {j<..} (fst x)\n \\K (t, s)"} {"_id": "501949", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\S. open S \\ open ((\\x. c * x + b) ` S)"} {"_id": "501950", "text": "proof (prove)\nusing this:\n FW M n 0 (v c) = len M 0 (v c) xs\n set xs \\ {0..n}\n 0 \\ set xs\n v c \\ set xs\n distinct xs\n \\k\\n. 0 < k \\ (\\c. v c = k)\n v c \\ n\n\ngoal (1 subgoal):\n 1. dbm_entry_val u None (Some c) (len M 0 (v c) xs)"} {"_id": "501951", "text": "proof (prove)\ngoal (1 subgoal):\n 1. components g * e\\<^sup>T * top\n \\ forest_components v * e\\<^sup>T * top"} {"_id": "501952", "text": "proof (prove)\nusing this:\n y = BRANCH c rs\n\ngoal (1 subgoal):\n 1. thesis"} {"_id": "501953", "text": "proof (prove)\nusing this:\n S t i \\ t ! i \\ S t (i + 1)\n\ngoal (1 subgoal):\n 1. snd (cs (S t (i + 1)) cid) = Done"} {"_id": "501954", "text": "proof (prove)\ngoal (1 subgoal):\n 1. X \\\\<^sub>L Y"} {"_id": "501955", "text": "proof (prove)\ngoal (1 subgoal):\n 1. insertion (nth_default 0 (init @ xs)) p =\n insertion (nth_default 0 (init @ I @ xs))\n (liftPoly (length init) (length I) p)"} {"_id": "501956", "text": "proof (state)\nthis:\n [i.. i \\\n \\h'. effect (for [i.. I j h'"} {"_id": "501957", "text": "proof (prove)\nusing this:\n inj_on Xd Y\n cop ds = Pds'.cop ds\n Pds'.stable_on ds (Pds'.cop ds)\n d \\ ds'\n x \\ Pds'.cop ds\n\ngoal (1 subgoal):\n 1. x \\ Pdds'.CD_on ds (Pds'.cop ds)"} {"_id": "501958", "text": "proof (prove)\ngoal (2 subgoals):\n 1. \\m::nat.\n (0::nat) < m \\\n \\j>i. f (j - i) \\ f m\n 2. \\ (f ` {k::nat. (0::nat) < k}) \\\n (\\j::nat\\{j::nat. i < j}. f (j - i))\nvariables:\n f :: nat \\ 'a\n i :: nat\ntype variables:\n 'a :: refinement_lattice"} {"_id": "501959", "text": "proof (prove)\nusing this:\n T x \\ space M\n A \\ sets (M \\\\<^sub>M N)\n\ngoal (1 subgoal):\n 1. (\\y. indicator A (T x, y)) \\ borel_measurable N"} {"_id": "501960", "text": "proof (prove)\ngoal (1 subgoal):\n 1. local.gen_llength_cons' n s =\n (case lgenerator g s of Done \\ n\n | Skip s' \\ local.gen_llength_cons' n s'\n | Yield x s' \\ local.gen_llength_cons' (eSuc n) s')"} {"_id": "501961", "text": "proof (state)\ngoal (1 subgoal):\n 1. \\E F \\1 \\2 L'.\n D = E @ F \\\n \\ = \\1 @ \\2 \\\n LeftDerivationLadder \\ E L' \\1 \\\n derivation_ge F (length \\1) \\\n LeftDerivation \\ (derivation_shift F (length \\1) 0)\n \\2 \\\n length L' = length L \\\n ladder_i L' 0 = ladder_i L 0 \\\n ladder_last_j L' = ladder_last_j L"} {"_id": "501962", "text": "proof (prove)\nusing this:\n (\\n. F n x) = (\\n. f x \\ of_nat n)\n\ngoal (1 subgoal):\n 1. (\\n. F n x) = f x"} {"_id": "501963", "text": "proof (prove)\nusing this:\n local.inv C R \\ \\ R \\ {}\n sc C U\n\ngoal (1 subgoal):\n 1. sc C U \\\n (\\C'.\n sc C' U \\ sum w C \\ harm d\\<^sup>* * sum w C')"} {"_id": "501964", "text": "proof (prove)\ngoal (1 subgoal):\n 1. valid_ownership \\ (t # ts) \\ valid_ownership \\ ts"} {"_id": "501965", "text": "proof (prove)\ngoal (1 subgoal):\n 1. list_of_oalist_tc (OAlist_tc_update_by_fun k f xs) =\n update_by_fun_pair_tc k f (list_of_oalist_tc xs)"} {"_id": "501966", "text": "proof (prove)\ngoal (1 subgoal):\n 1. Span.dom (\\.map (f, g) \\ SPN h) =\n Span.dom (\\.map (f, g)) \\ Span.dom (SPN h)"} {"_id": "501967", "text": "proof (state)\nthis:\n \\\\<^bsub>P\\<^esub> b = \\ \\ \\\\<^bsub>P\\<^esub> b\n\ngoal (1 subgoal):\n 1. pderiv (a \\\\<^bsub>P\\<^esub> b) =\n pderiv a \\\\<^bsub>P\\<^esub> pderiv b"} {"_id": "501968", "text": "proof (prove)\nusing this:\n \\f : x \\\\<^sub>A x'\\\n \\g : y' \\\\<^sub>B y\\\n \\h : F y \\\\<^sub>A x\\\n B.arr ?y \\ G (F ?y) = ?y\n A.arr ?x \\ F (G ?x) = ?x\n G \\ F = B.map\n F \\ G = A.map\n A.map ?f = (if A.arr ?f then ?f else A.null)\n B.map ?f = (if B.arr ?f then ?f else B.null)\n\ngoal (1 subgoal):\n 1. G (C f (C h (F g))) = D (G f) (D (G h) g)"} {"_id": "501969", "text": "proof (prove)\nusing this:\n e \\ b\n \\\\ \\c,Normal e\\ \\ Abrupt u\n\ngoal (1 subgoal):\n 1. \\\\ \\While b\n c,Normal\ne\\ \\ Abrupt u"} {"_id": "501970", "text": "proof (prove)\ngoal (1 subgoal):\n 1. map_indets g \\ map_indets f = id"} {"_id": "501971", "text": "proof (prove)\nusing this:\n l = l1 @ x # l2\n l1 = l1' @ y # c\n l2' = c @ x # l2\n\ngoal (1 subgoal):\n 1. l = l1' @ y # c @ x # l2"} {"_id": "501972", "text": "proof (prove)\nusing this:\n disjoint_family F\n emeasure lborel (\\x. F x \\ box a b)\n \\ emeasure lborel (box a b)\n\ngoal (1 subgoal):\n 1. (\\i. content (F i \\ box a b)) sums\n content (\\x. F x \\ box a b)"} {"_id": "501973", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x y z.\n \\x \\ carrier rec_monoid_rng_of_frac;\n y \\ carrier rec_monoid_rng_of_frac;\n z \\ carrier rec_monoid_rng_of_frac\\\n \\ x \\\\<^bsub>rec_monoid_rng_of_frac\\<^esub>\n y \\\\<^bsub>rec_monoid_rng_of_frac\\<^esub>\n z =\n x \\\\<^bsub>rec_monoid_rng_of_frac\\<^esub>\n (y \\\\<^bsub>rec_monoid_rng_of_frac\\<^esub>\n z)"} {"_id": "501974", "text": "proof (prove)\ngoal (3 subgoals):\n 1. \\Mapping.lookup\n (fst (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))) n =\n None;\n finite\n (Mapping.keys\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))));\n \\n v.\n (n, v)\n \\ Mapping.keys\n (snd (foldr (aux_2 g) ns\n (Mapping.empty, Mapping.empty))) \\\n length (predecessors g n) \\ 1;\n \\v.\n Mapping.lookup\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty)))\n (Entry g, v)\n \\ {None, Some []}\\\n \\ lookupDef g n ` uses g n = lookup_multimap u' n\n 2. \\m.\n \\m \\ n;\n Mapping.lookup\n (fst (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))) n =\n None;\n finite\n (Mapping.keys\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))));\n \\n v.\n (n, v)\n \\ Mapping.keys\n (snd (foldr (aux_2 g) ns\n (Mapping.empty, Mapping.empty))) \\\n length (predecessors g n) \\ 1;\n \\v.\n Mapping.lookup\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty)))\n (Entry g, v)\n \\ {None, Some []}\\\n \\ Mapping.lookup u' m =\n Mapping.lookup\n (fst (foldr (aux_2 g) ns\n (Mapping.empty, Mapping.empty)))\n m\n 3. \\m v.\n \\Mapping.lookup\n (fst (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty)))\n n =\n None;\n finite\n (Mapping.keys\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))));\n \\n v.\n (n, v)\n \\ Mapping.keys\n (snd (foldr (aux_2 g) ns\n (Mapping.empty, Mapping.empty))) \\\n length (predecessors g n) \\ 1;\n \\v.\n Mapping.lookup\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty)))\n (Entry g, v)\n \\ {None, Some []}\\\n \\ (if m \\ phiDefNodes_aux g v\n(filter\n (\\n.\n (n, v)\n \\ Mapping.keys\n (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))))\n (\\n g))\nn \\\n v \\ uses g n\n then Some\n (map (\\m. lookupDef g m v)\n (predecessors g m))\n else Mapping.lookup\n (snd (foldr (aux_2 g) ns\n (Mapping.empty, Mapping.empty)))\n (m, v)) =\n Mapping.lookup p' (m, v)"} {"_id": "501975", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\bounded_clinear A; A (1::'a) = (0::'b)\\\n \\ A = (\\_. 0::'b)"} {"_id": "501976", "text": "proof (prove)\nusing this:\n init_config c\n\ngoal (1 subgoal):\n 1. inv_implications_nonneg c"} {"_id": "501977", "text": "proof (prove)\ngoal (1 subgoal):\n 1. open X \\\n open X \\ X = X \\ Collect (Domainp (pcr_nonzero (=)))"} {"_id": "501978", "text": "proof (prove)\nusing this:\n intra_kind (kind a')\n upd_rev_cs cs (a # as') = upd_rev_cs cs as'\n\ngoal (1 subgoal):\n 1. upd_rev_cs cs ((a # as') @ [a']) = upd_rev_cs cs (as' @ [a'])"} {"_id": "501979", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (\\ \\ f = g) = (\\ \\ g = f)"} {"_id": "501980", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\n.\n (u n = - 1 \\ z n) \\\n (u n = 1 \\ \\ z n) \\\n - 1 \\ u n \\ u n \\ 1"} {"_id": "501981", "text": "proof (prove)\nusing this:\n \\ \\g gpv' \\ \\\n expectation_gpv fail \\ (\\_. c) gpv' \\ max c fail\n fail \\ c\n expectation_gpv' gpv' \\ expectation_gpv fail \\ (\\a. c) gpv'\n \\ \\g gpv' \\\n\ngoal (1 subgoal):\n 1. expectation_gpv' gpv' \\ c"} {"_id": "501982", "text": "proof (prove)\ngoal (1 subgoal):\n 1. length (map tpOf (map Var (getVars \\l))) =\n length \\l \\\n (\\il))).\n map tpOf (map Var (getVars \\l)) ! i = \\l ! i)"} {"_id": "501983", "text": "proof (prove)\nusing this:\n \\\\ \\sequence Seq\n ((x # xs) @\n ys),Normal s\\ =n\\ t\n xs = []\n\ngoal (1 subgoal):\n 1. \\s'.\n \\\\ \\sequence Seq\n(x # xs),Normal s\\ =n\\ s' \\\n \\\\ \\sequence Seq\nys,s'\\ =n\\ t"} {"_id": "501984", "text": "proof (prove)\nusing this:\n sub P =\n ksrp\n (\\\\<^bsub>sKs p\\<^esub>t \\sKs\n p \\ t\\\\<^bsub>n\\<^sub>r\\<^esub>)\n \\sKs p\\\\<^bsub>t\n\\sKs p \\ t\\\\<^bsub>n\\<^sub>r\\<^esub>\\<^esub>\n\ngoal (1 subgoal):\n 1. sub P\n \\ bbrp\n (\\\\<^bsub>the_bb\\<^esub>t\n \\sKs p \\ t\\\\<^bsub>n\\<^sub>r\\<^esub>)"} {"_id": "501985", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\x - y\\\\<^sub>Q =\n \\x\\\\<^sub>Q - \\y\\\\<^sub>Q"} {"_id": "501986", "text": "proof (prove)\nusing this:\n finite A\n a \\ A\n \\c \\ carrier R; g \\ A \\ carrier M\\\n \\ c \\\\<^bsub>M\\<^esub> finsum M g A =\n (\\\\<^bsub>M\\<^esub>x\\A. c \\\\<^bsub>M\\<^esub>\n g x)\n c \\ carrier R\n g \\ insert a A \\ carrier M\n\ngoal (1 subgoal):\n 1. c \\\\<^bsub>M\\<^esub> finsum M g (insert a A) =\n (\\\\<^bsub>M\\<^esub>x\\insert a\n A. c \\\\<^bsub>M\\<^esub> g x)"} {"_id": "501987", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\valid_prefixes rtbl; has_default_route rtbl\\\n \\ \\p. rpf_loose rtbl p"} {"_id": "501988", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inj_comm_ring_hom (map_poly hom)"} {"_id": "501989", "text": "proof (prove)\nusing this:\n \\q\\set qs. degree q = 0 \\\n consistent_sign_vectors qs UNIV =\n {map (\\x.\n if poly x 0 < 0 then - 1 else if poly x 0 = 0 then 0 else 1)\n qs}\n\ngoal (1 subgoal):\n 1. set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"} {"_id": "501990", "text": "proof (prove)\nusing this:\n a \\ arcs G\n a \\ {uv, rev_G uv}\n\ngoal (1 subgoal):\n 1. edge_succ HM (rev_G a) = (edge_succ HM \\ rev_H) a"} {"_id": "501991", "text": "proof (prove)\nusing this:\n T \\ Pow S\n ts \\ lists T\n us \\ ssubseqs ts\n S_reduced_for w us\n\ngoal (1 subgoal):\n 1. word_length T w = S_length w"} {"_id": "501992", "text": "proof (prove)\nusing this:\n safe_reach_upto ?n safe_free_flowing (ts, m, \\)\n\ngoal (1 subgoal):\n 1. safe_reach_upto k safe_free_flowing (ts, m, \\)"} {"_id": "501993", "text": "proof (prove)\nusing this:\n (?a, the_NF A ?a) \\ A\\<^sup>!\n (a, b) \\ A\n\ngoal (1 subgoal):\n 1. the_NF A a = the_NF A b"} {"_id": "501994", "text": "proof (prove)\nusing this:\n - real_of ?x = real_of (ma_uminus ?x)\n\ngoal (1 subgoal):\n 1. - real_of_u r = real_of_u (mau_uminus r)"} {"_id": "501995", "text": "proof (prove)\ngoal (1 subgoal):\n 1. inline callee (map_gpv f g gpv) s =\n map_gpv (map_prod f id) id (inline (\\s x. callee s (g x)) gpv s)"} {"_id": "501996", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (DF\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P A \\\\<^sub>F\\<^sub>D\n DF A &&&\n DF A \\\\<^sub>F\\<^sub>D RUN A) &&&\n CHAOS A \\\\<^sub>F\\<^sub>D DF A &&&\n CHAOS\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P A \\\\<^sub>F\\<^sub>D\n CHAOS A &&&\n CHAOS\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P A \\\\<^sub>F\\<^sub>D\n DF\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P A"} {"_id": "501997", "text": "proof (prove)\nusing this:\n M 0 0 \\ Le 0\n i = 0\n dbm_int M n\n Le 0 \\ M 0 0\n\ngoal (1 subgoal):\n 1. \\ \\ Le 0"} {"_id": "501998", "text": "proof (prove)\ngoal (3 subgoals):\n 1. (\\(x, y). Pxy (x, y) * log b (Pxy (x, y) / Py y))\n \\ borel_measurable (S \\\\<^sub>M T)\n 2. (\\x. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)))\n \\ borel_measurable (S \\\\<^sub>M T)\n 3. AE x in S \\\\<^sub>M\n T. (case x of\n (x, y) \\\n Pxy (x, y) * log b (Pxy (x, y) / Py y)) =\n Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x))"} {"_id": "501999", "text": "proof (prove)\nusing this:\n f ((0::'a) - of_nat 1) = f (0::'a)\n\ngoal (1 subgoal):\n 1. f (- (1::'a)) = f (0::'a)"} {"_id": "502000", "text": "proof (prove)\ngoal (1 subgoal):\n 1. (CDPlayer_CTRL_States, CDPlayer_CTRL_Init, CDPlayer_CTRL_Labels,\n CDPlayer_CTRL_Delta)\n \\ seqauto"} {"_id": "502001", "text": "proof (prove)\ngoal (1 subgoal):\n 1. deg_comp cmp (splus t u) (splus t v) = Lt"} {"_id": "502002", "text": "proof (prove)\ngoal (1 subgoal):\n 1. \\l ta.\n \\lock_thread_ok ls ts; ts t = \\xw\\;\n thread_oks ts tas; has_lock (upd_locks (ls $ l) t (las $ l)) ta;\n ta \\ t\\\n \\ \\xw.\n redT_updTs ts tas ta = \\xw\\"} {"_id": "502003", "text": "proof (state)\nthis:\n a \\ domain\n\ngoal (1 subgoal):\n 1. a \\ domain \\\n (SOME aa. \\a\\ = \\aa\\) \\ a"} {"_id": "502004", "text": "proof (chain)\npicking this:\n \\ = \\\\<^bsub>G\\carrier := H\\\\<^esub>"} {"_id": "502005", "text": "proof (prove)\nusing this:\n \\?w \\ W; ?s \\ S\\\n \\ dual_order.poset_simplex_map \\