instruct string | input string | output string |
|---|---|---|
Given a lemma, please prove it. | lemma fconverseE[elim!]:
assumes "x \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>"
obtains a b where "x = [b, a]\<^sub>\<circ>" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" | using assms unfolding fconverse_def
by auto |
Given a lemma, please prove it. | lemma fpairsD[dest]:
assumes "x \<in>\<^sub>\<circ> fpairs r"
shows "x \<in>\<^sub>\<circ> r" and "\<exists>a b. x = [a, b]\<^sub>\<circ>" | using assms unfolding fpairs_def
by auto |
Given a lemma, please prove it. | lemma fun_dual5: "(f = \<partial> \<circ> g \<circ> \<partial>) = (\<partial> \<circ> f \<circ> \<partial> = g)" | by (metis comp_assoc fun_dual1 fun_dual2) |
Given a lemma, please prove it. | lemma Ord_vsubset_Vset_succ[V_cs_intros]:
assumes "Ord \<alpha>" and "B \<subseteq>\<^sub>\<circ> Vset \<alpha>"
shows "B \<subseteq>\<^sub>\<circ> Vset (succ \<alpha>)" | by (intro vsubsetI) (auto simp: assms Vset_trans Ord_vsubset_in_Vset_succI) |
Given a lemma, please prove it. | lemma vsv_vimageI1:
assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "a \<in>\<^sub>\<circ> A"
shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> r `\<^sub>\<circ> A" | using assms
by (simp add: vsv_vimage_eqI) |
Given a lemma, please prove it. | lemma vsv_vlrestriction_vinsert:
assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
shows "r \<restriction>\<^sup>l\<^sub>\<circ> vinsert a A = vinsert \<langle>a, r\<lparr>a\<rparr>\<rangle> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" | using assms
by (auto intro!: vsubset_antisym) |
Given a lemma, please prove it. | lemma dg_prod_vdiff_vunion_Obj_in_Obj:
assumes "J \<subseteq>\<^sub>\<circ> I"
and "b \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gk\<in>\<^sub>\<circ>I -\<^sub>\<circ> J. \<AA> k)\<lparr>Obj\<rparr>"
and "c \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>J. \<AA> j)\<lparr>Obj\<rparr>"
show... | by ( vdiff_of_vunion rule: dg_prod_vunion_Obj_in_Obj assms: assms(2,3) subset: assms(1) ) |
Given a lemma, please prove it. | lemma vimage_eq_imp_vcomp:
assumes "r `\<^sub>\<circ> A = s `\<^sub>\<circ> B"
shows "(t \<circ>\<^sub>\<circ> r) `\<^sub>\<circ> A = (t \<circ>\<^sub>\<circ> s) `\<^sub>\<circ> B" | using assms
by (metis vcomp_vimage) |
Given a lemma, please prove it. | lemma vifintersectionE2[elim]:
assumes "a \<in>\<^sub>\<circ> (\<Inter>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. f i)"
obtains i where "i \<in>\<^sub>\<circ> I" and "a \<in>\<^sub>\<circ> f i" | using assms
by (elim vifintersectionE3) (meson assms VInterE2 app_vimageE) |
Given a lemma, please prove it. | lemma fconst_onE[elim!]:
assumes "x \<in>\<^sub>\<circ> fconst_on A c"
obtains a where "a \<in>\<^sub>\<circ> A" and "x = [a, c]\<^sub>\<circ>" | using assms unfolding fconst_on_def
by auto |
Given a lemma, please prove it. | lemma frestrictionI[intro!]:
assumes "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> A" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A" | using assms unfolding frestriction_def
by simp |
Given a lemma, please prove it. | lemma vsv_vimage_eqI[intro]:
assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "r\<lparr>a\<rparr> = b" and "a \<in>\<^sub>\<circ> A"
shows "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A" | using assms(2)[unfolded vsv_ex1_app2[OF assms(1)]] assms(3)
by auto |
Given a lemma, please prove it. | lemma fimageI1:
assumes "x \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
shows "x \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A" | using assms unfolding fimage_def
by simp |
Given a lemma, please prove it. | lemma vrat_mult_closed:
assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" and "y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
shows "x *\<^sub>\<rat> y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" | proof- have "(x'::rat) * y' \<in> UNIV" for x' y'
by simp from this[untransferred, OF assms] show ?thesis .
qed |
Given a lemma, please prove it. | lemma vrange_VLambda: "\<R>\<^sub>\<circ> (\<lambda>a\<in>\<^sub>\<circ>A. f a) = set (f ` elts A)" | by (intro vsubset_antisym vsubsetI) auto |
Given a lemma, please prove it. | lemma vrat_assoc_law_multiplication:
assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" and "y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" and "z \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
shows "(x *\<^sub>\<rat> y) *\<^sub>\<rat> z = x *\<^sub>\<rat> (y *\<^sub>\<rat> z)" | proof- have "(x' * y') * z' = x' * (y' * z')" for x' y' z' :: rat
by simp from this[untransferred, OF assms] show ?thesis .
qed |
Given a lemma, please prove it. | lemma vimage_VLambda_vrange: "(\<lambda>a\<in>\<^sub>\<circ>A. f a) `\<^sub>\<circ> B = \<R>\<^sub>\<circ> (\<lambda>a\<in>\<^sub>\<circ>A \<inter>\<^sub>\<circ> B. f a)" | unfolding vimage_def
by (simp add: vlrestriction_VLambda) |
Given a lemma, please prove it. | lemma vrestriction_atE2[elim]:
assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A"
obtains a b where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> A" and "r\<lparr>a\<rparr> = b" | using assms unfolding vrestriction_def
by clarsimp |
Given a lemma, please prove it. | lemma vint_mult_closed:
assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
shows "x *\<^sub>\<int> y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" | proof- have "(x'::int) * y' \<in> UNIV" for x' y'
by simp from this[untransferred, OF assms] show ?thesis .
qed |
Given a lemma, please prove it. | lemma app_vimageE[elim]:
assumes "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
obtains a where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" and "a \<in>\<^sub>\<circ> A" | using assms unfolding vimage_def
by auto |
Given a lemma, please prove it. | lemma v11_vconverse_app_in_vdomain:
assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
shows "r\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" | using assms v11_vconverse unfolding vrange_vconverse[symmetric]
by (auto simp: v11_def) |
Given a lemma, please prove it. | lemma drop_suffix_hd_css_step'':
assumes step: "\<Gamma>\<turnstile> (p#ps@cs,css,s) \<rightarrow> (cs',(pnorm@cs,pabr@cs)#css,t)"
shows "\<Gamma>\<turnstile> (p#ps,css,s) \<rightarrow> (cs',(pnorm,pabr)#css,t)" | using drop_suffix_hd_css_step' [OF step]
by auto |
Given a lemma, please prove it. | lemma mset_subst_cls_list_subst_cls_mset: "mset (Cs \<cdot>cl \<sigma>) = (mset Cs) \<cdot>cm \<sigma>" | unfolding subst_cls_mset_def subst_cls_list_def
by auto |
Given a lemma, please prove it. | lemma cs_length_g_one: assumes \<open>length (cs\<^bsup>\<pi>\<^esup> i) \<noteq> 1\<close> obtains k where \<open>cs\<^bsup>\<pi>\<^esup> i = (cs\<^bsup>\<pi>\<^esup> k)@[\<pi> i]\<close> and \<open>i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k\<close> | apply (cases \<open>i\<close> \<open>\<pi>\<close> rule: cs_cases)
using assms cs_not_nil
by auto |
Given a lemma, please prove it. | lemma preprocess'_Tableau_Poly_Mapping_Some': "(Poly_Mapping (preprocess' cs start)) p = Some v
\<Longrightarrow> \<exists> h. poly h = p \<and> \<not> is_monom (poly h) \<and> qdelta_constraint_to_atom h v \<in> flat (set (Atoms (preprocess' cs start)))" | by (induct cs start rule: preprocess'.induct, auto simp: Let_def split: option.splits if_splits) |
Given a lemma, please prove it. | lemma length_locss:
"i < length cs
\<Longrightarrow> length (locss P cs loc ! (length cs - Suc i)) =
locLength P (fst(cs ! (length cs - Suc i)))
(fst(snd(cs ! (length cs - Suc i))))
(snd(snd(cs ! (length cs - Suc i))))" | apply (induct cs, auto)
apply (case_tac "i = length cs")
by (auto simp: nth_Cons') |
Given a lemma, please prove it. | lemma size_jump2: "size (jump l cs) < size cs \<or> jump l cs = cs" | apply(induct cs)
apply simp
apply(case_tac a)
apply auto done |
Given a lemma, please prove it. | lemma fps_XDp_foldr_nth [simp]: "foldr (\<lambda>c r. fps_XDp c \<circ> r) cs (\<lambda>c. fps_XDp c a) c0 $ n =
foldr (\<lambda>c r. (c + of_nat n) * r) cs (c0 + of_nat n) * a$n" | by (induct cs arbitrary: c0) (simp_all add: algebra_simps) |
Given a lemma, please prove it. | lemma weakBisimCasePushRes:
fixes x :: name
and \<Psi> :: 'b
and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"
assumes "x \<sharp> \<Psi>"
and "x \<sharp> (map fst Cs)"
shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(Cases Cs) \<approx> Cases(map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P... | using assms
by(metis bisimCasePushRes strongBisimWeakBisim) |
Given a lemma, please prove it. | lemma non_strict_constr_no_LTPP:
assumes "nonstrict_constrs cs"
shows "\<forall>x \<in> set cs. \<not>(\<exists>a b. LTPP a b = x)" | using assms nonstrict_constr.simps(9)
by blast |
Given a lemma, please prove it. | lemma SubobjsR_subclassRep:
"Subobjs\<^sub>R P C Cs \<Longrightarrow> (C,last Cs) \<in> (subclsR P)\<^sup>*" | apply(erule Subobjs\<^sub>R.induct)
apply simp
apply(simp add: SubobjsR_nonempty) done |
Given a lemma, please prove it. | lemma same_level_path_aux_Append:
"\<lbrakk>same_level_path_aux cs as; same_level_path_aux (upd_cs cs as) as'\<rbrakk>
\<Longrightarrow> same_level_path_aux cs (as@as')" | by(induct rule:slpa_induct,auto simp:intra_kind_def) |
Given a lemma, please prove it. | lemma class_leq_refl[iff]: "class_ex cs c \<Longrightarrow> class_leq cs c c" | using wf
by (simp add: class_leq_def class_ex_def refl_on_def) |
Given a lemma, please prove it. | lemma min_gallery_least_length:
assumes "chamber C" "chamber D" "C\<noteq>D"
defines "Cs \<equiv> ARG_MIN length Cs. gallery (C#Cs@[D])"
shows "min_gallery (C#Cs@[D])" | unfolding Cs_def
using assms gallery_least_length
by (blast intro: min_galleryI_betw arg_min_nat_le) |
Given a lemma, please prove it. | lemma remdups_clss_Nil_iff: "remdups_clss Cs = [] \<longleftrightarrow> Cs = []" | by (cases Cs, simp, hypsubst, subst remdups_clss.simps(2), simp add: Let_def) |
Given a lemma, please prove it. | lemma sort_eqv_trans: "sort_eqv cs x y \<Longrightarrow> sort_eqv cs y z \<Longrightarrow> sort_eqv cs x z" | using sort_eqv_def sort_leq_trans
by blast |
Given a lemma, please prove it. | lemma valid_path_aux_intra_path:
"\<forall>a \<in> set as. intra_kind(kind a) \<Longrightarrow> valid_path_aux cs as" | by(induct as,auto simp:intra_kind_def) |
Given a lemma, please prove it. | lemma presSwap_fromIMor[simp]:
"ipresCons h hA (fromMOD MOD) \<Longrightarrow> ipresSwap h (fromMOD MOD)
\<Longrightarrow> presSwap (fromIMor h) MOD" | unfolding ipresSwap_def presSwap_def
by simp |
Given a lemma, please prove it. | lemma ipresIGFreshAll_termMOD[simp]:
"ipresIGFreshAll h hA termMOD MOD = ipresFreshAll h hA MOD" | unfolding ipresIGFreshAll_def ipresFreshAll_def
by simp |
Given a lemma, please prove it. | lemma fromIMor_termFSwSbMorph[simp]:
assumes "termFSwSbImorph h hA (fromMOD MOD)"
shows "termFSwSbMorph (fromIMor h) (fromIMorAbs hA) MOD" | using assms unfolding termFSwSbImorph_defs1
using assms unfolding termFSwSbImorph_def termFSwSbMorph_def
by simp |
Given a lemma, please prove it. | lemma errMOD_igWlsAbsDisj:
assumes "igWlsAbsDisj MOD"
shows "igWlsAbsDisj (errMOD MOD)" | using assms unfolding errMOD_def igWlsAbsDisj_def
apply clarify subgoal for _ _ _ _ A
by(cases A) fastforce+ . |
Given a lemma, please prove it. | lemma imp_igSubstInpIPresIGWlsInpSTR:
"igSubstIPresIGWlsSTR MOD \<Longrightarrow> igSubstInpIPresIGWlsInpSTR MOD" | by(simp add: igSubstInpIPresIGWlsInpSTR_def igSubstIPresIGWlsSTR_def igSubstInp_def igWlsInp_def liftAll2_def lift_def sameDom_def split: option.splits) (smt option.distinct(1) option.exhaust) |
Given a lemma, please prove it. | lemma comp_ipresIGWlsAll:
assumes "ipresIGWlsAll h hA MOD MOD'" and "ipresIGWlsAll h' hA' MOD' MOD''"
shows "ipresIGWlsAll (h' o h) (hA' o hA) MOD MOD''" | using assms unfolding ipresIGWlsAll_def
using comp_ipresIGWls comp_ipresIGWlsAbs
by auto |
Given a lemma, please prove it. | lemma igFreshIGVar_fromMOD[simp]:
"gFreshGVar MOD \<Longrightarrow> igFreshIGVar (fromMOD MOD)" | unfolding igFreshIGVar_def gFreshGVar_def
by simp |
Given a lemma, please prove it. | lemma ipresCons_imp_ipresSubstAll:
assumes *: "ipresCons h hA MOD" and **: "igSubstCls MOD"
and "igConsIPresIGWls MOD" and "igFreshCls MOD"
shows "ipresSubstAll h hA MOD" | unfolding ipresSubstAll_def
using assms ipresCons_imp_ipresSubst ipresCons_imp_ipresSubstAbs
by auto |
Given a lemma, please prove it. | lemma igSubstAllIPresIGWlsAllSTR_imp_igSubstAllIPresIGWlsAll:
"igSubstAllIPresIGWlsAllSTR MOD \<Longrightarrow> igSubstAllIPresIGWlsAll MOD" | unfolding igSubstAllIPresIGWlsAllSTR_def igSubstAllIPresIGWlsAll_def
using igSubstIPresIGWlsSTR_imp_igSubstIPresIGWls igSubstAbsIPresIGWlsAbsSTR_imp_igSubstAbsIPresIGWlsAbs
by auto |
Given a lemma, please prove it. | theorem wlsFSwSb_recAll_unique_presCons:
assumes "wlsFSwSb MOD" and "presCons h hA MOD"
shows "(wls s X \<longrightarrow> h X = rec MOD X) \<and>
(wlsAbs (us,s') A \<longrightarrow> hA A = recAbs MOD A)" | using assms wlsFSw_recAll_unique_presCons unfolding wlsFSwSb_def
by blast |
Given a lemma, please prove it. | lemma
"(0::nat) mod 0 = 0"
"(x::nat) mod 0 = x"
"(0::nat) mod 1 = 0"
"(1::nat) mod 1 = 0"
"(3::nat) mod 1 = 0"
"(x::nat) mod 1 = 0"
"(0::nat) mod 3 = 0"
"(1::nat) mod 3 = 1"
"(3::nat) mod 3 = 0"
"x mod 3 < 3"
"(x mod 3 = x) = (x < 3)" | using [[z3_extensions]]
by smt+ |
Given a lemma, please prove it. | lemma errMOD_igWlsAllDisj:
assumes "igWlsAllDisj MOD"
shows "igWlsAllDisj (errMOD MOD)" | using assms unfolding igWlsAllDisj_def
using errMOD_igWlsDisj errMOD_igWlsAbsDisj
by auto |
Given a lemma, please prove it. | lemma iwlsFSbSw_fromMOD[simp]:
"wlsFSbSw MOD \<Longrightarrow> iwlsFSbSw (fromMOD MOD)" | unfolding iwlsFSbSw_def wlsFSbSw_def
by simp |
Given a lemma, please prove it. | lemma sub_diff_mod_eq': "
r \<le> t \<Longrightarrow> (k * m + t - (t - r) mod m) mod (m::nat) = r mod m" | apply (simp only: diff_mod_le[of t r m, THEN add_diff_assoc, symmetric])
apply (simp add: sub_diff_mod_eq) done |
Given a lemma, please prove it. | lemma not_SN_onI[intro]: "f 0 \<in> X \<Longrightarrow> chain R f \<Longrightarrow> \<not> SN_on R X" | by (unfold SN_on_def not_not, intro exI conjI) |
Given a lemma, please prove it. | lemma non_empty_cycle_root_loop_converse:
assumes "non_empty_cycle_root r x"
shows "r \<le> x\<^sup>+;r" | using assms less_eq_def non_empty_cycle_root_rtc_tc
by fastforce |
Given a lemma, please prove it. | lemma "(0::real) < 1 + x\<^sup>2" | by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))") |
Given a lemma, please prove it. | lemma oprod_Well_order:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "Well_order (r *o r')" | proof- have "Total r \<and> Total r'"
using WELL WELL'
by (auto simp add: order_on_defs) thus ?thesis
using assms unfolding well_order_on_def
using oprod_Linear_order oprod_wf_Id
by blast
qed |
Given a lemma, please prove it. | lemma subset_AboveS_UnderS: "B \<le> Field r \<Longrightarrow> B \<le> AboveS r (UnderS r B)" | by(auto simp add: AboveS_def UnderS_def) |
Given a lemma, please prove it. | lemma per_add_exp: assumes "u \<le>p r\<^sup>\<omega>" and "m \<noteq> 0" shows "u \<le>p (r\<^sup>@m)\<^sup>\<omega>" | using per_exp_pref[OF per_rootD, OF \<open>u \<le>p r\<^sup>\<omega>\<close>, of m] per_rootD'[OF \<open>u \<le>p r\<^sup>\<omega>\<close>, folded nonzero_pow_emp[OF \<open>m \<noteq> 0\<close>, of r]] unfolding period_root_def .. |
Given a lemma, please prove it. | lemma coprimeI:
assumes "\<And>r. r dvd p \<Longrightarrow> r dvd q \<Longrightarrow> r dvd 1"
shows "coprime p q" | using assms
by (auto simp: coprime_def') |
Given a lemma, please prove it. | lemma li_minus:
assumes "locally_irrefl R A"
shows "locally_irrefl R (A - B)" | using assms unfolding locally_irrefl_def
by (meson in_diffD) |
Given a lemma, please prove it. | lemma SN_on_Image_rtrancl_conv:
"SN_on r A \<longleftrightarrow> SN_on r (r\<^sup>* `` A)" (is "?L \<longleftrightarrow> ?R") | proof assume ?L then show ?R
by (auto simp: SN_on_Image_rtrancl) next assume ?R then show ?L
by (auto simp: SN_on_def)
qed |
Given a lemma, please prove it. | lemma oprod_Preorder: "\<lbrakk>Preorder r; Preorder r'; antisym r; antisym r'\<rbrakk> \<Longrightarrow> Preorder (r *o r')" | unfolding preorder_on_def
using oprod_Refl oprod_trans
by blast |
Given a lemma, please prove it. | lemma pj_invim_mono1:"\<lbrakk>Ring R; ideal R I; ideal (qring R I) J1;
ideal (qring R I) J2; J1 \<subseteq> J2 \<rbrakk> \<Longrightarrow>
(rInvim R (qring R I) (pj R I) J1) \<subseteq> (rInvim R (qring R I) (pj R I) J2)" | apply (rule subsetI)
apply (simp add:rInvim_def)
apply (simp add:subsetD) done |
Given a lemma, please prove it. | lemma rrsmult_supp : "supp (r \<currency> f) \<subseteq> R" | using rightreg_scalar_multD2 suppD_contra
by force |
Given a lemma, please prove it. | lemma False_step_conc[iff]:
"\<And>L R. (False#p,q) : step (conc L R) a =
(\<exists>r. q = False#r \<and> (p,r) : step R a)" | apply (simp add:conc_def step_def)
apply blast done |
Given a lemma, please prove it. | lemma r2f_ad_rel_hom: "\<F> (ad_rel R) = kad (\<F> R)" | by (force simp add: kad_def ad_rel_def r2f_def fun_eq_iff) |
Given a lemma, please prove it. | lemma abs_trans[trans]:
assumes A: "\<Up>R C \<le> B" and B: "\<Up>R' B \<le> A"
shows "\<Up>R' (\<Up>R C) \<le> A" | using assms
by (fastforce simp: pw_le_iff refine_pw_simps) |
Given a lemma, please prove it. | lemma (in ring) carrier_is_subalgebra:
assumes "K \<subseteq> carrier R" shows "subalgebra K (carrier R) R" | using assms subalgebra.intro[OF add.group_incl_imp_subgroup[of "carrier R"], of K] add.group_axioms unfolding subalgebra_axioms_def
by auto |
Given a lemma, please prove it. | lemma (in domain) Units_mult_eq_Units [simp]: "Units (mult_of R) = Units R" | unfolding Units_def
using insert_Diff integral_iff
by auto |
Given a lemma, please prove it. | lemma toCard_pred_toCard:
"\<lbrakk>|A| \<le>o r; Card_order r\<rbrakk> \<Longrightarrow> toCard_pred A r (toCard A r)" | unfolding toCard_def
using someI_ex[OF ex_toCard_pred] . |
Given a lemma, please prove it. | lemma order_asym: "trans R \<Longrightarrow> asym R = irrefl R" | unfolding asym.simps irrefl_def trans_def
by meson |
Given a lemma, please prove it. | lemma runiq_wrt_ex1:
"runiq R \<longleftrightarrow> (\<forall> a \<in> Domain R . \<exists>! b . (a, b) \<in> R)" | using runiq_basic
by (metis Domain.DomainI Domain.cases) |
Given a lemma, please prove it. | lemma per_eq: "x \<le>p r\<^sup>\<omega> \<longleftrightarrow> (\<exists> k z. r\<^sup>@k\<cdot>z = x \<and> z \<le>p r) \<and> r \<noteq> \<epsilon>" | using per_pref[unfolded pref_pow_conv] . |
Given a lemma, please prove it. | lemma rawpsubstT_atrm[simp,intro]:
assumes "r \<in> atrm" and "snd ` (set txs) \<subseteq> var" and "fst ` (set txs) \<subseteq> atrm"
shows "rawpsubstT r txs \<in> atrm" | using assms
by (induct txs arbitrary: r) auto |
Given a lemma, please prove it. | lemma one_add_square_eq_0: "1 + (x)\<^sup>2 \<noteq> (0::real)" | by (sos "((R<1 + (([~1] * A=0) + (R<1 * (R<1 * [x]^2)))))") |
Given a lemma, please prove it. | lemma good_ruleset_alt: "good_ruleset rs = (\<forall>r\<in>set rs. get_action r = Accept \<or> get_action r = Drop \<or>
get_action r = Reject \<or> get_action r = Log \<or> get_action r = Empty)" | unfolding good_ruleset_def
apply(rule Set.ball_cong)
apply(simp_all)
apply(rename_tac r)
by(case_tac "get_action r")(simp_all) |
Given a lemma, please prove it. | lemma divideR_right:
fixes x y :: "'a::real_normed_vector"
shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x" | using scaleR_cancel_left[of r y "x /\<^sub>R r"]
by simp |
Given a lemma, please prove it. | lemma sup_refine[refine]:
assumes "ai \<le>\<Down>R a"
assumes "bi \<le>\<Down>R b"
shows "sup ai bi \<le>\<Down>R (sup a b)" | using assms
by (auto simp: pw_le_iff refine_pw_simps) |
Given a lemma, please prove it. | lemma order_less_trans: "\<lbrakk> order r; x \<sqsubset>\<^sub>r y; y \<sqsubset>\<^sub>r z \<rbrakk> \<Longrightarrow> x \<sqsubset>\<^sub>r z" | by (unfold order_def lesssub_def) blast |
Given a lemma, please prove it. | lemma characteriseAx:
shows "r \<in> Ax \<Longrightarrow> r = ([],\<LM> ff \<RM> \<Rightarrow>* \<Empt>) \<or> (\<exists> i. r = ([], \<LM> At i \<RM> \<Rightarrow>* \<LM> At i \<RM>))" | apply (cases r)
by (rule Ax.cases) auto |
Given a lemma, please prove it. | lemma is_top_sorted_antimono:
assumes "R\<subseteq>R'"
assumes "is_top_sorted R' l"
shows "is_top_sorted R l" | using assms unfolding is_top_sorted_alt
by (auto dest: rtrancl_mono_mp) |
Given a lemma, please prove it. | lemma D_inv: assumes "trans r" and "irrefl r" and "(r|\<tau>'| -s dl r \<sigma>,r|\<tau>| ) \<in> mul_eq r"
and "(r|\<sigma>'| -s dl r \<tau>,r|\<sigma>| ) \<in> mul_eq r"
shows "D r \<tau> \<sigma> \<sigma>' \<tau>'" | using assms unfolding D_def lemma3_2_2
using lemma2_6_6_a[OF assms(1)] union_commute
by metis |
Given a lemma, please prove it. | lemma connected_root_iff4:
assumes "point r"
shows "connected_root r x \<longleftrightarrow> 1;x = r\<^sup>T;x\<^sup>+" | by (metis assms conv_contrav conv_invol conv_one star_conv star_slide_var connected_root_iff3) |
Given a lemma, please prove it. | theorem per_pref: "x \<le>p r\<^sup>\<omega> \<longleftrightarrow> (\<exists> k. x \<le>p r\<^sup>@k) \<and> r \<noteq> \<epsilon>" | using per_pref_ex period_root_def pref_pow_ext' pref_prod_pref
by metis |
Given a lemma, please prove it. | lemma alpha_d_more_eqI:
assumes "tr r = tr r'" "wait r = wait r'" "ref r = ref r'" "more r = more r'"
shows "alpha_d.more r = alpha_d.more r'" | using assms
by (cases r, cases r') auto |
Given a lemma, please prove it. | lemma pred_resumption_antimono:
assumes r: "pred_resumption A C r'"
and le: "resumption_ord r r'"
shows "pred_resumption A C r" | using r monotoneD[OF monotone_results le] monotoneD[OF monotone_outputs le]
by(auto simp add: pred_resumption_def) |
Given a lemma, please prove it. | lemma isLimOrd_succ:
assumes isLimOrd and "i \<in> Field r"
shows "succ i \<in> Field r" | using assms unfolding isLimOrd_def isSuccOrd_def
by (metis REFL in_notinI refl_on_domain succ_smallest) |
Given a lemma, please prove it. | lemma eps_match: "eps test i r \<longleftrightarrow> match test r i i" | by (induction r) (auto dest: antisym[OF match_le match_le]) |
Given a lemma, please prove it. | lemma ltlr_expand_Release:
"\<xi> \<Turnstile>\<^sub>r \<phi> R\<^sub>r \<psi> \<longleftrightarrow> (\<xi> \<Turnstile>\<^sub>r \<psi> and\<^sub>r (\<phi> or\<^sub>r (X\<^sub>r (\<phi> R\<^sub>r \<psi>))))" | by (metis ltln_expand_Release ltlr_to_ltln.simps(5-7,9) ltlr_to_ltln_semantics) |
Given a lemma, please prove it. | lemma trans_llexord:
"transp r \<Longrightarrow> transp (llexord r)" | by(auto intro!: transpI elim: llexord_trans dest: transpD) |
Given a lemma, please prove it. | lemma pure_hn_refineI_no_asm:
assumes "(c,a)\<in>R"
shows "hn_refine emp (return c) emp (pure R) (RETURN a)" | unfolding hn_refine_def
using assms
by (sep_auto simp: pure_def) |
Given a lemma, please prove it. | lemma inverse_pow_pow:
assumes "a \<in> carrier G"
shows "inv (a [^] (r::nat)) = (inv a) [^] r" | proof - have "a [^] r \<in> carrier G"
using assms
by blast then show ?thesis
by (simp add: assms nat_pow_inv)
qed |
Given a lemma, please prove it. | theorem ACI_norm_final[simp]:
"final \<guillemotleft>r\<guillemotright> = final r" | proof (induct r) case (Plus r1 r2) thus ?case
using toplevel_summands_final
by (auto simp: final_PLUS)
qed auto |
Given a lemma, please prove it. | lemma lang_rexp_subst: "lang (rexp_subst f r) = subst_word f ` lang r" | by (induction r) (simp_all add: image_Un) |
Given a lemma, please prove it. | lemma hom_init: "hom_ab (init_a r) = init_b r" | unfolding init_a_def init_b_def hom_ab.simps
by (simp add: nonfinal_empty_mrexp) |
Given a lemma, please prove it. | lemma extended_Ax_prems_empty:
assumes "r \<in> Ax"
shows "fst (extendRule S r) = []" | using assms
apply (cases r)
by (rule Ax.cases) (auto simp add:extendRule_def) |
Given a lemma, please prove it. | lemma "(r 0 0 \<and> r 0 1 \<and> r 0 2 \<and> r 0 3) \<or>
(r 1 0 \<and> r 1 1 \<and> r 1 2 \<and> r 1 3) \<or>
(r 2 0 \<and> r 2 1 \<and> r 2 2 \<and> r 2 3) \<or>
(r 3 0 \<and> r 3 1 \<and> r 3 2 \<and> r 3 3)" | by (metis (full_types) rax) |
Given a lemma, please prove it. | lemma non_empty_cycle_root_msc_plus:
assumes "non_empty_cycle_root r x"
shows "x\<^sup>+;r = x\<^sup>T\<^sup>+;r" | using assms many_strongly_connected_iff_7 non_empty_cycle_root_msc
by fastforce |
Given a lemma, please prove it. | lemma d_delta_lnexp_cf2_nonpos: "diff_delta_lnexp_cf2 x \<le> 0" | unfolding diff_delta_lnexp_cf2_def
by (sos "(((R<1 + ((R<1 * ((R<5/4 * [~3/40*x^2 + 1]^2) + (R<11/1280 * [x^2]^2))) + ((A<1 * R<1) * (R<1/64 * [1]^2))))) & ((R<1 + ((R<1 * ((R<5/4 * [~3/40*x^2 + 1]^2) + (R<11/1280 * [x^2]^2))) + ((A<1 * R<1) * (R<1/64 * [1]^2))))))") |
Given a lemma, please prove it. | lemma lang_final: "final r = ([] \<in> lang n r)" | using concI[of "[]" _ "[]"]
by (induct r arbitrary: n) auto |
Given a lemma, please prove it. | lemma in_rtrancl_insert: "x\<in>R\<^sup>* \<Longrightarrow> x\<in>(insert r R)\<^sup>*" | by (metis in_mono rtrancl_mono subset_insertI) |
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