phanerozoic/HOL4 · Datasets at Hugging Face

fact stringlengths 9 20.3k	type stringclasses 2 values	library stringclasses 116 values	imports listlengths 0 239	filename stringlengths 21 103	symbolic_name stringlengths 1 92	docstring null
FDOM_update : ∀xs root n q. FDOM (update xs root n k q) = FDOM q ∪ (set (MAP FST xs) DIFF FDOM k)	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FDOM_update	null
has_path_exists_known : ∀edges root l n. has_path edges root l n ∧ root ∈ FDOM known ∧ n ∉ FDOM (known: num \|-> res_info) ⇒ ∃m1 m2 l1 l2 l3. has_path edges root l1 m1 ∧ m1 ∈ FDOM known ∧ has_edge edges m1 l2 m2 ∧ m2 ∉ FDOM known ∧ l1 + l2 ≤ l	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	has_path_exists_known	null
FLOOKUP_update_neq : ∀i ns n d k q j v. i ≠ j ⇒ FLOOKUP (update ns n d k (q \|+ (j,v))) i = FLOOKUP (update ns n d k q) i	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_neq	null
FLOOKUP_update_cons_neq : ∀i ns n d k q j v m. i ≠ j ⇒ FLOOKUP (update ((j,m)::ns) n d k q) i = FLOOKUP (update ns n d k q) i	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_cons_neq	null
pull_if : (f (if b then x else y) = if b then f x else f y) ∧ ((if b then g1 else g2) z = if b then g1 z else g2 z)	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	pull_if	null
FLOOKUP_update_pass : ∀i k ns h n d h1 q. (∀l. MEM (i,l) ns ⇒ h ≤ l) ⇒ FLOOKUP (update ns n d k (q \|+ (i,<\|prev_node := n; dist_from_root := d + h\|>))) i = SOME <\|prev_node := n; dist_from_root := d + h\|>	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_pass	null
FLOOKUP_update_skip : ∀i k ns h n d h1 q l. FLOOKUP k i = NONE ∧ MEM (i,l) ns ∧ l < h ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + h < x.dist_from_root) ⇒ FLOOKUP (update ns n d k (q \|+ (i,<\|prev_node := n; dist_from_root := d + h\|>))) i = FLOOKUP (update ns n d k q) i	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_skip	null
FLOOKUP_update_MEM : ∀i k ns h n d h1 q. FLOOKUP k i = NONE ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + h < x.dist_from_root) ⇒ FLOOKUP (update ns n d k (q \|+ (i,<\|prev_node := n; dist_from_root := d + h\|>))) i = if ∃l. MEM (i,l) ns ∧ l < h then FLOOKUP (update ns n d k q) i else SOME <\|prev_node := n; dist_from_root := d + h\|>	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_MEM	null
update_thm : ∀i ns n d k q v. FLOOKUP (update ns n d k q) i = SOME v ∧ i ∉ FDOM k ⇒ (* the original value in q survived ) (FLOOKUP q i = SOME v ∧ ( all matching pairs in ns must have been ignored ) ∀len. MEM (i,len) ns ∧ FLOOKUP k i = NONE ⇒ v.dist_from_root ≤ d + len) ∨ ( mem in ns has been picked ) ∃len. MEM (i,len) ns ∧ v.dist_from_root = d + len ∧ v.prev_node = n ∧ (∀len2. MEM (i,len2) ns ⇒ len ≤ len2) ∧ ( any mapping in q must have been larger *) ∀w. FLOOKUP q i = SOME w ⇒ d + len ≤ w.dist_from_root	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	update_thm	null
FLOOKUP_update_k : ∀ns q. i ∈ FDOM k ⇒ FLOOKUP (update ns n d k q) i = FLOOKUP q i	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	FLOOKUP_update_k	null
update_not_eq : ∀ns n d k q i v. FLOOKUP (update ns n d k q) i = SOME v ∧ v.prev_node ≠ n ⇒ FLOOKUP q i = SOME v ∧ ∀len1. MEM (i,len1) ns ∧ i ∉ FDOM k ⇒ v.dist_from_root ≤ len1 + d	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	update_not_eq	null
update_eq : ∀ns d k q i v. FLOOKUP (update ns v.prev_node d k q) i = SOME v ∧ (∀w. FLOOKUP q i = SOME w ⇒ w ≠ v) ⇒ ∃len. i ∉ FDOM k ∧ MEM (i,len) ns ∧ v.dist_from_root = len + d ∧ (∀len1. MEM (i,len1) ns ⇒ len ≤ len1) ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + len ≤ x.dist_from_root)	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	update_eq	null
dijkstra_thm : ∀root known queue edges res. calc known queue edges res ⇒ FLOOKUP known root = SOME (root_node root) ⇒ DISJOINT (FDOM known) (FDOM queue) ∧ (* content in known is shortest ) correct_result edges root known ∧ ( all neighbours of known are in known or queue ) (∀n len i. n ∈ FDOM known ∧ has_edge edges n len i ⇒ i ∈ FDOM known ∨ i ∈ FDOM queue) ∧ ( each queue points at shortest node in known *) (∀n v. FLOOKUP queue n = SOME v ⇒ ∃x len. FLOOKUP known v.prev_node = SOME x ∧ has_edge edges v.prev_node len n ∧ v.dist_from_root = x.dist_from_root + len ∧ ∀other y len1. FLOOKUP known other = SOME y ∧ has_edge edges other len1 n ⇒ v.dist_from_root ≤ y.dist_from_root + len1) ⇒ correct_result edges root res ∧ complete_result edges root res	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	dijkstra_thm	null
dijkstra_imp_correct_result : dijkstra root edges result ⇒ correct_result edges root result ∧ complete_result edges root result	theorem	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	dijkstra_imp_correct_result	null
min_elem_def = min_elem (n:num) v queue ⇔ FLOOKUP queue n = SOME v ∧ ∀m w. FLOOKUP queue m = SOME w ⇒ v.dist_from_root ≤ w.dist_from_root	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	min_elem_def	null
update_def = update [] n d k q = q ∧ update ((i,len)::es) n d k q = case FLOOKUP k i of \| SOME _ => update es n d k q \| NONE => case FLOOKUP q i of \| NONE => update es n d k (q \|+ (i,<\| prev_node := n ; dist_from_root := d + len \|>)) \| SOME j => if j.dist_from_root ≤ d + len then update es n d k q else update es n d k (q \|+ (i,<\| prev_node := n ; dist_from_root := d + len \|>))	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	update_def	null
root_node_def = root_node r = <\| prev_node := r; dist_from_root := 0 \|>	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	root_node_def	null
dijkstra_def = dijkstra root edges result ⇔ calc FEMPTY (FEMPTY \|+ (root, root_node root)) edges result	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	dijkstra_def	null
correct_result_def = correct_result edges root known ⇔ (∀n v. FLOOKUP known n = SOME v ⇒ has_path edges root v.dist_from_root n ∧ (∀l. l < v.dist_from_root ⇒ ~ has_path edges root l n) ∧ (n = root ⇒ v.prev_node = root) ∧ (n ≠ root ⇒ ∃w l. FLOOKUP known v.prev_node = SOME w ∧ has_edge edges v.prev_node l n ∧ v.dist_from_root = l + w.dist_from_root))	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	correct_result_def	null
complete_result_def = complete_result edges root known ⇔ (∀n len. has_path edges root len n ⇒ n ∈ FDOM known)	definition	examples/algorithms	[]	examples/algorithms/dijkstraScript.sml	complete_result_def	null
domain_spt_fold_union : ! tree y. domain (spt_fold union y tree) = domain y UNION {n \| ?k aSet. lookup k tree = SOME aSet /\ n IN domain aSet}	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	domain_spt_fold_union	null
num_set_tree_union_empty : ! t1 t2 . isEmpty (num_set_tree_union t1 t2) <=> isEmpty t1 /\ isEmpty t2	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	num_set_tree_union_empty	null
wf_num_set_tree_union : ! t1 t2 result . wf t1 /\ wf t2 ==> wf (num_set_tree_union t1 t2)	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	wf_num_set_tree_union	null
domain_num_set_tree_union : ! t1 t2 . domain (num_set_tree_union t1 t2) = domain t1 UNION domain t2	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	domain_num_set_tree_union	null
num_set_tree_union_sym : ! t1 t2 . num_set_tree_union t1 t2 = num_set_tree_union t2 t1	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	num_set_tree_union_sym	null
lookup_num_set_tree_union : !t1 t2 n. lookup n (num_set_tree_union t1 t2) = case lookup n t1 of SOME x => ( case lookup n t2 of SOME y => SOME (union x y) \| NONE => SOME x) \| NONE => lookup n t2	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	lookup_num_set_tree_union	null
wf_spt_fold_union : ! tree y. (! n x . (lookup n tree = SOME x) ==> wf x) /\ wf y ==> wf (spt_fold union y tree)	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	wf_spt_fold_union	null
wf_closure_spt : ! reachable tree. wf reachable /\ (! n x . lookup n tree = SOME x ==> wf x) ==> wf (closure_spt reachable tree)	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	wf_closure_spt	null
adjacent_domain : ! tree x y . is_adjacent tree x y ==> x IN domain tree	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	adjacent_domain	null
reachable_domain : ! tree x y . is_reachable tree x y ==> (x = y \/ x IN domain tree)	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	reachable_domain	null
reachable_LHS_NOTIN : !tree n x. is_reachable tree n x /\ n NOTIN domain tree ==> n = x	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	reachable_LHS_NOTIN	null
rtc_is_adjacent : (! k . k IN t ==> ! n . (is_adjacent fullTree k n ==> n IN t)) ==> ! x y . RTC (is_adjacent fullTree) x y ==> x IN t ==> y IN t	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	rtc_is_adjacent	null
is_adjacent_num_set_tree_union : ! t1 t2 n m . is_adjacent t1 n m ==> is_adjacent (num_set_tree_union t1 t2) n m	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	is_adjacent_num_set_tree_union	null
is_reachable_num_set_tree_union : ! t1 t2 n m . is_reachable t1 n m ==> is_reachable (num_set_tree_union t1 t2) n m	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	is_reachable_num_set_tree_union	null
closure_spt_lemma : ! reachable tree closure (roots : num set). closure_spt reachable tree = closure /\ roots ⊆ domain reachable /\ (!k. k IN domain reachable ==> ?n. n IN roots /\ is_reachable tree n k) ==> domain closure = {a \| ?n. n IN roots /\ is_reachable tree n a}	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	closure_spt_lemma	null
closure_spt_thm : ! tree start. domain (closure_spt start tree) = {a \| ?n. n IN domain start /\ is_reachable tree n a}	theorem	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	closure_spt_thm	null
num_set_tree_union_def = (num_set_tree_union LN t2 = t2 : num_set num_map) /\ (num_set_tree_union (LS a) t = case t of \| LN => LS a \| LS b => LS (union a b) \| BN t1 t2 => BS t1 a t2 \| BS t1 b t2 => BS t1 (union a b) t2) /\ (num_set_tree_union (BN t1 t2) t = case t of \| LN => BN t1 t2 \| LS a => BS t1 a t2 \| BN t1' t2' => BN (num_set_tree_union t1 t1') (num_set_tree_union t2 t2') \| BS t1' a t2' => BS (num_set_tree_union t1 t1') a (num_set_tree_union t2 t2')) /\ (num_set_tree_union (BS t1 a t2) t = case t of \| LN => BS t1 a t2 \| LS b => BS t1 (union a b) t2 \| BN t1' t2' => BS (num_set_tree_union t1 t1') a (num_set_tree_union t2 t2') \| BS t1' b t2' => BS (num_set_tree_union t1 t1') (union a b) (num_set_tree_union t2 t2'))	definition	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	num_set_tree_union_def	null
closure_spt_def = closure_spt (reachable: num_set) tree = let sets = inter tree reachable in let nodes = spt_fold union LN sets in if subspt nodes reachable then reachable else closure_spt (union reachable nodes) tree Termination WF_REL_TAC `measure (\ (r,t). size (difference (spt_fold union LN t) r))` >> rw[] >> gvs[subspt_domain, SUBSET_DEF] >> irule size_diff_less >> gvs[domain_union, domain_spt_fold_union, lookup_inter] >> EVERY_CASE_TAC >> gvs[] >> simp[PULL_EXISTS, GSYM CONJ_ASSOC] >> goal_assum drule >> goal_assum drule >> simp[] >> goal_assum (drule_at Any) >> qexists_tac `k` >> simp[]	definition	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	closure_spt_def	null
is_adjacent_def = is_adjacent tree x y = ? aSetx. ( lookup x tree = SOME aSetx ) /\ ( lookup y aSetx = SOME () )	definition	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	is_adjacent_def	null
is_reachable_def = is_reachable tree = RTC (is_adjacent tree)	definition	examples/algorithms	[]	examples/algorithms/spt_closureScript.sml	is_reachable_def	null
partition_LENGTH : !ks n reach xs ys zs xs1 ys1 zs1. partition n ks reach (xs,ys,zs) = (xs1,ys1,zs1) ==> LENGTH xs1 <= LENGTH xs + LENGTH ks /\ LENGTH ys1 <= LENGTH ys + LENGTH ks /\ LENGTH zs1 <= LENGTH zs + LENGTH ks	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	partition_LENGTH	null
top_sort_example : top_sort [(0,fromAList[]); (* 0 has no deps ) (1,fromAList[(2,());(0,())]); ( 1 depens on 2 and 0 ) (2,fromAList[(1,())]); ( 2 depends on 1 ) (3,fromAList[(1,())])] ( 3 depends on 1 ) = [[0]; [1; 2]; [3]] ( 0 defined first, 1 and 2 are mutual, 3 is last *)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_example	null
top_sort_any_example : top_sort_any [("A",[]); (* A has no deps ) ("B",["C";"A"]); ( B depens on C and A ) ("C",["B";"X"]); ( C depends on B and X -- X is intentionally not defined here ) ("D",["B"])] ( D depends on B ) = [["A"]; ["B"; "C"]; ["D"]] ( A defined first, B and C are mutual, D is last *)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_example	null
needs_eq_is_adjacent : !x y tree. needs x y tree <=> is_adjacent tree x y	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	needs_eq_is_adjacent	null
partition_lemma : !n ks reach acc xas ybs zcs as bs cs. partition n ks reach acc = (xas,ybs,zcs) /\ acc = (as,bs,cs) ==> ?xs ys zs. xas = xs ++ as /\ ybs = ys ++ bs /\ zcs = zs ++ cs /\ (set (xs ++ ys ++ zs) = set ks) /\ (!x. MEM x ks ==> (MEM x xs <=> ¬ needs x n reach)) /\ (!y. MEM y ks ==> (MEM y ys <=> needs y n reach /\ needs n y reach)) /\ (!z. MEM z ks ==> (MEM z zs <=> needs z n reach /\ ¬ needs n z reach))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	partition_lemma	null
partition_thm : !n ks reach as bs cs res. partition n ks reach (as, bs, cs) = res ==> ?xs ys zs. res = (xs ++ as, ys ++ bs, zs ++ cs) /\ (set (xs ++ ys ++ zs) = set ks) /\ (!x. MEM x ks ==> (MEM x xs <=> ¬ needs x n reach)) /\ (!y. MEM y ks ==> (MEM y ys <=> needs y n reach /\ needs n y reach)) /\ (!z. MEM z ks ==> (MEM z zs <=> needs z n reach /\ ¬ needs n z reach))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	partition_thm	null
ALL_DISTINCT_APPEND_SWAP : !l1 l2. ALL_DISTINCT (l1 ++ l2) <=> ALL_DISTINCT (l2 ++ l1)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALL_DISTINCT_APPEND_SWAP	null
partition_ALL_DISTINCT : !n ks reach acc xs ys zs as bs cs. partition n ks reach acc = (xs,ys,zs) /\ acc = (as,bs,cs) /\ ALL_DISTINCT (ks ++ as ++ bs ++ cs) ==> ALL_DISTINCT (xs ++ ys ++ zs)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	partition_ALL_DISTINCT	null
top_sort_aux_correct : !ns reach acc resacc. top_sort_aux ns reach acc = resacc /\ (!n m. TC (\a b. needs a b reach) n m ==> needs n m reach) /\ ALL_DISTINCT ns ==> ?res. resacc = res ++ acc /\ ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set ns /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: ) res = xss ++ [ys] ++ zss /\ MEM y ys ==> ( for any defined dependencies of y ) !dep. needs y dep reach /\ MEM dep ns ==> ( mutual dependencies must by in ys, all others in xss *) (¬ needs dep y reach <=> MEM dep (FLAT xss)) /\ ( needs dep y reach <=> MEM dep ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_aux_correct	null
domain_lookup_num_set : !t k. k IN domain t <=> lookup k t = SOME ()	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	domain_lookup_num_set	null
trans_clos_correct : !nexts n m. needs n m (trans_clos nexts) \/ n = m <=> is_reachable nexts n m	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	trans_clos_correct	null
trans_clos_correct_imp : !nexts n m. (TC (\x y. needs x y nexts)) n m ==> needs n m (trans_clos nexts)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	trans_clos_correct_imp	null
trans_clos_TC_closed : !t n m. TC (\a b. needs a b (trans_clos t)) n m ==> needs n m (trans_clos t)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	trans_clos_TC_closed	null
top_sort_correct_weak : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: ) res = xss ++ [ys] ++ zss /\ MEM y ys ==> ( all dependencies of y must be defined in ys or earlier, i.e. xss *) ?deps. ALOOKUP lets y = SOME deps /\ !d. d IN domain deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_correct_weak	null
top_sort_correct : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: ) res = xss ++ [ys] ++ zss /\ MEM y ys ==> ( for any defined dependencies of y *) !dep. is_reachable (fromAList lets) y dep /\ MEM dep (MAP FST lets) ==> (¬ is_reachable (fromAList lets) dep y <=> MEM dep (FLAT xss)) /\ ( is_reachable (fromAList lets) dep y <=> MEM dep ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_correct	null
top_sort_aux_same_partition : ∀ns reach acc xs. MEM xs (top_sort_aux ns reach acc) ⇒ ∀x y nexts. (∀n m. TC (λa b. needs a b reach) n m ⇒ needs n m reach) ∧ (∀as x y. MEM as acc ∧ x ≠ y ∧ MEM x as ∧ MEM y as ⇒ needs x y reach) ∧ x ≠ y ∧ MEM x xs ∧ MEM y xs ⇒ needs x y reach	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_aux_same_partition	null
top_sort_same_partition : MEM xs (top_sort graph) ∧ x ≠ y ∧ MEM x xs ∧ MEM y xs ⇒ needs x y (trans_clos $ fromAList graph)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_same_partition	null
SCC_REFL : SCC E x x	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	SCC_REFL	null
SCC_SYM : SCC E x y ⇔ SCC E y x	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	SCC_SYM	null
SCC_TRANS : SCC E x y ∧ SCC E y z ⇒ SCC E x z	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	SCC_TRANS	null
SCC_EQUIV : EQUIV (SCC E)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	SCC_EQUIV	null
is_reachable_IMP_MEM : ALL_DISTINCT (MAP FST graph) ⇒ x ≠ y ⇒ is_reachable (fromAList graph) x y ⇒ MEM x (MAP FST graph)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	is_reachable_IMP_MEM	null
top_sort_SCC : ALL_DISTINCT (MAP FST graph) ∧ MEM x l ∧ MEM l (top_sort graph) ⇒ SCC (is_adjacent (fromAList graph)) x = set l	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_SCC	null
to_nums_correct : !xs b res. to_nums b xs = res /\ ALL_DISTINCT (MAP FST b) ==> domain res = {c \| ?d. MEM d xs /\ ALOOKUP b d = SOME c}	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	to_nums_correct	null
ALL_DISTINCT_MAPi_ID : !l. ALL_DISTINCT (MAPi (\i _. i) l)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALL_DISTINCT_MAPi_ID	null
ALL_DISTINCT_FLAT : ∀l. ALL_DISTINCT (FLAT l) ⇔ (∀l0. MEM l0 l ⇒ ALL_DISTINCT l0) ∧ (∀i j. i < j ∧ j < LENGTH l ⇒ ∀e. MEM e (EL i l) ⇒ ¬MEM e (EL j l))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALL_DISTINCT_FLAT	null
ALOOKUP_MAPi_ID : !l k. ALOOKUP (MAPi (\i n. (i,n)) l) k = if k < LENGTH l then SOME (EL k l) else NONE	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALOOKUP_MAPi_ID	null
ALOOKUP_MAPi_ID_f : !l k. ALOOKUP (MAPi (\i n. (i,f n)) l) k = if k < LENGTH l then SOME (f (EL k l)) else NONE	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALOOKUP_MAPi_ID_f	null
MAPi_MAP : !l. MAPi (\i n. f n) l = MAP f l	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	MAPi_MAP	null
MEM_ALOOKUP : !l k v. ALL_DISTINCT (MAP FST l) ==> (MEM (k,v) l <=> ALOOKUP l k = SOME v)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	MEM_ALOOKUP	null
top_sort_any_correct_weak : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort_any lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: ) res = xss ++ [ys] ++ zss /\ MEM y ys ==> ( all dependencies of y must be defined in ys or earlier, i.e. xss *) ?deps. ALOOKUP lets y = SOME deps /\ !d. MEM d deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_correct_weak	null
depends_on : depends_on alist = RTC (depends_on1 alist)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	depends_on	null
top_sort_any_depends_on_weak : !lets res. (!xss ys zss y. res = xss ++ [ys] ++ zss /\ MEM y ys ==> ?deps. ALOOKUP lets y = SOME deps /\ !d. MEM d deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)) ==> !a b. depends_on lets a b ==> !xss ys zss y. res = xss ++ [ys] ++ zss /\ MEM a ys ==> MEM b (FLAT xss ++ ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_depends_on_weak	null
top_sort_any_correct : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort_any lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: ) res = xss ++ [ys] ++ zss /\ MEM y ys ==> ( all dependencies of y must be defined in ys or earlier, i.e. xss *) !dep. depends_on lets y dep ==> MEM dep (FLAT xss ++ ys) /\ (depends_on lets dep y ==> MEM dep ys)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_correct	null
MEM_MEM_top_sort : ALL_DISTINCT (MAP FST l) ∧ MEM xs $ top_sort l ∧ MEM n xs ⇒ MEM n (MAP FST l)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	MEM_MEM_top_sort	null
ALL_DISTINCT_enc_graph : ALL_DISTINCT (MAPi ($o FST o (λi (n,ns). (i,to_nums (MAPi (λi n. (n,i)) (MAP FST graph)) ns))) graph)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ALL_DISTINCT_enc_graph	null
RTC_ALOOKUP_enc_graph_IMP_depends_on : ALL_DISTINCT (MAP FST graph) ⇒ ∀n n'. RTC (λx y. ∃aSetx. ALOOKUP (MAPi (λi (n,ns). (i,to_nums (MAPi (λi n. (n,i)) (MAP FST graph)) ns)) graph) x = SOME aSetx ∧ lookup y aSetx = SOME ()) n n' ⇒ depends_on graph (EL n (MAP FST graph)) (EL n' (MAP FST graph))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	RTC_ALOOKUP_enc_graph_IMP_depends_on	null
top_sort_any_same_partition : ALL_DISTINCT (MAP FST graph) ∧ MEM xs (top_sort_any graph) ∧ MEM x xs ∧ MEM y xs ⇒ depends_on graph x y	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_same_partition	null
top_sort_any_SCC : ALL_DISTINCT (MAP FST graph) ∧ MEM x l ∧ MEM l (top_sort_any graph) ⇒ SCC (depends_on1 graph) x = set l	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_SCC	null
cycle_CASES_lem : ∀x z. TC R x z ⇒ x = z ⇒ (R x z ∨ (∃y. x ≠ y ∧ TC R x y ∧ TC R y z))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	cycle_CASES_lem	null
cycle_CASES : TC R x x ⇔ (R x x ∨ (∃y. x ≠ y ∧ TC R x y ∧ TC R y x))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	cycle_CASES	null
depends_on_IMP_MEM : ∀x y. depends_on graph x y ⇒ x ≠ y ⇒ MEM y (MAP FST graph)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	depends_on_IMP_MEM	null
TC_depends_on_SCC : x ≠ y ⇒ SCC (depends_on1 graph) x y = (TC_depends_on graph x y ∧ TC_depends_on graph y x)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	TC_depends_on_SCC	null
ONE_LT_LENGTH_ALL_DISTINCT_IMP : 1 < LENGTH l ∧ ALL_DISTINCT l ⇒ ∃x y. MEM x l ∧ MEM y l ∧ x ≠ y	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	ONE_LT_LENGTH_ALL_DISTINCT_IMP	null
DISTINCT_IMP_ONE_LT_LENGTH : MEM x l ∧ MEM y l ∧ x ≠ y ⇒ 1 < LENGTH l	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	DISTINCT_IMP_ONE_LT_LENGTH	null
has_cycle_correct : ALL_DISTINCT (MAP FST graph) ⇒ (has_cycle graph ⇔ ∃x. TC_depends_on graph x x)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	has_cycle_correct	null
TC_depends_on_weak_IMP_MEM : ∀x y. TC_depends_on_weak graph x y ⇒ MEM x (MAP FST graph)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	TC_depends_on_weak_IMP_MEM	null
TC_depends_on_weak_IMP_TC_depends_on : ∀x y. TC_depends_on_weak graph x y ⇒ MEM y (MAP FST graph) ⇒ TC_depends_on graph x y	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	TC_depends_on_weak_IMP_TC_depends_on	null
TC_depends_on_TC_depends_on_weak_thm : ∀x y. TC_depends_on graph x y = (TC_depends_on_weak graph x y ∧ MEM y (MAP FST graph))	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	TC_depends_on_TC_depends_on_weak_thm	null
TC_depends_on_weak_cycle_thm : TC_depends_on_weak graph x x = TC_depends_on graph x x	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	TC_depends_on_weak_cycle_thm	null
has_cycle_correct2 : ALL_DISTINCT (MAP FST graph) ⇒ (has_cycle graph ⇔ ∃x. TC_depends_on_weak graph x x)	theorem	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	has_cycle_correct2	null
needs_def = needs n m (reach:num_set num_map) = case lookup n reach of \| NONE => F (* cannot happen *) \| SOME s => IS_SOME (lookup m s)	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	needs_def	null
partition_def = partition n [] reach acc = acc /\ partition n (k::ks) reach (xs,ys,zs) = if needs k n reach then if needs n k reach then partition n ks reach (xs,k::ys,zs) else partition n ks reach (xs,ys,k::zs) else partition n ks reach (k::xs,ys,zs)	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	partition_def	null
top_sort_aux_def = top_sort_aux [] reach acc = acc /\ top_sort_aux (n::ns) reach acc = let (xs,ys,zs) = partition n ns reach ([],[],[]) in top_sort_aux xs reach ((n::ys) :: top_sort_aux zs reach acc) Termination WF_REL_TAC `measure (LENGTH o FST)` \\ rw [] \\ pop_assum (assume_tac o GSYM) \\ imp_res_tac partition_LENGTH \\ fs []	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_aux_def	null
trans_clos_def = (* computes the transitive closure for each node given nexts *) trans_clos nexts = map (\x. closure_spt x nexts) nexts	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	trans_clos_def	null
top_sort_def = top_sort (let_bindings : (num (* name ) # num_set ( free vars *)) list) = let roots = MAP FST let_bindings in let nexts = fromAList let_bindings in let reach = trans_clos nexts in top_sort_aux roots reach []	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_def	null
to_nums_def = to_nums b [] = LN /\ to_nums b (x::xs) = case ALOOKUP b x of \| NONE => to_nums b xs \| SOME n => insert n () (to_nums b xs)	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	to_nums_def	null
top_sort_any_def = top_sort_any (lets : ('a # 'a list) list) = if NULL lets (* so that HD names, below, is well defined *) then [] else let names = MAP FST lets in let to_num = MAPi (\i n. (n,i)) names in let from_num = fromAList (MAPi (\i n. (i,n)) names) in let nesting = top_sort (MAPi (\i (n,ns). (i,to_nums to_num ns)) lets) in MAP (MAP (\n. case lookup n from_num of SOME m => m \| _ => HD (names))) nesting	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	top_sort_any_def	null
has_cycle_def = has_cycle graph = ((EXISTS (λl. 1 < LENGTH l) $ top_sort_any graph) ∨ EXISTS (λ(v,l). MEM v l) graph)	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	has_cycle_def	null
SCC_def = SCC E x y ⇔ RTC E x y ∧ RTC E y x	definition	examples/algorithms	[]	examples/algorithms/topological_sortScript.sml	SCC_def	null

FDOM_update : ∀xs root n q. FDOM (update xs root n k q) = FDOM q ∪ (set (MAP FST xs) DIFF FDOM k)

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FDOM_update

null

has_path_exists_known : ∀edges root l n. has_path edges root l n ∧ root ∈ FDOM known ∧ n ∉ FDOM (known: num |-> res_info) ⇒ ∃m1 m2 l1 l2 l3. has_path edges root l1 m1 ∧ m1 ∈ FDOM known ∧ has_edge edges m1 l2 m2 ∧ m2 ∉ FDOM known ∧ l1 + l2 ≤ l

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

has_path_exists_known

null

FLOOKUP_update_neq : ∀i ns n d k q j v. i ≠ j ⇒ FLOOKUP (update ns n d k (q |+ (j,v))) i = FLOOKUP (update ns n d k q) i

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_neq

null

FLOOKUP_update_cons_neq : ∀i ns n d k q j v m. i ≠ j ⇒ FLOOKUP (update ((j,m)::ns) n d k q) i = FLOOKUP (update ns n d k q) i

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_cons_neq

null

pull_if : (f (if b then x else y) = if b then f x else f y) ∧ ((if b then g1 else g2) z = if b then g1 z else g2 z)

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

pull_if

null

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_pass

null

FLOOKUP_update_skip : ∀i k ns h n d h1 q l. FLOOKUP k i = NONE ∧ MEM (i,l) ns ∧ l < h ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + h < x.dist_from_root) ⇒ FLOOKUP (update ns n d k (q |+ (i,<|prev_node := n; dist_from_root := d + h|>))) i = FLOOKUP (update ns n d k q) i

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_skip

null

FLOOKUP_update_MEM : ∀i k ns h n d h1 q. FLOOKUP k i = NONE ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + h < x.dist_from_root) ⇒ FLOOKUP (update ns n d k (q |+ (i,<|prev_node := n; dist_from_root := d + h|>))) i = if ∃l. MEM (i,l) ns ∧ l < h then FLOOKUP (update ns n d k q) i else SOME <|prev_node := n; dist_from_root := d + h|>

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_MEM

null

update_thm : ∀i ns n d k q v. FLOOKUP (update ns n d k q) i = SOME v ∧ i ∉ FDOM k ⇒ (* the original value in q survived *) (FLOOKUP q i = SOME v ∧ (* all matching pairs in ns must have been ignored *) ∀len. MEM (i,len) ns ∧ FLOOKUP k i = NONE ⇒ v.dist_from_root ≤ d + len) ∨ (* mem in ns has been picked *) ∃len. MEM (i,len) ns ∧ v.dist_from_root = d + len ∧ v.prev_node = n ∧ (∀len2. MEM (i,len2) ns ⇒ len ≤ len2) ∧ (* any mapping in q must have been larger *) ∀w. FLOOKUP q i = SOME w ⇒ d + len ≤ w.dist_from_root

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

update_thm

null

FLOOKUP_update_k : ∀ns q. i ∈ FDOM k ⇒ FLOOKUP (update ns n d k q) i = FLOOKUP q i

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

FLOOKUP_update_k

null

update_not_eq : ∀ns n d k q i v. FLOOKUP (update ns n d k q) i = SOME v ∧ v.prev_node ≠ n ⇒ FLOOKUP q i = SOME v ∧ ∀len1. MEM (i,len1) ns ∧ i ∉ FDOM k ⇒ v.dist_from_root ≤ len1 + d

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

update_not_eq

null

update_eq : ∀ns d k q i v. FLOOKUP (update ns v.prev_node d k q) i = SOME v ∧ (∀w. FLOOKUP q i = SOME w ⇒ w ≠ v) ⇒ ∃len. i ∉ FDOM k ∧ MEM (i,len) ns ∧ v.dist_from_root = len + d ∧ (∀len1. MEM (i,len1) ns ⇒ len ≤ len1) ∧ (∀x. FLOOKUP q i = SOME x ⇒ d + len ≤ x.dist_from_root)

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

update_eq

null

dijkstra_thm : ∀root known queue edges res. calc known queue edges res ⇒ FLOOKUP known root = SOME (root_node root) ⇒ DISJOINT (FDOM known) (FDOM queue) ∧ (* content in known is shortest *) correct_result edges root known ∧ (* all neighbours of known are in known or queue *) (∀n len i. n ∈ FDOM known ∧ has_edge edges n len i ⇒ i ∈ FDOM known ∨ i ∈ FDOM queue) ∧ (* each queue points at shortest node in known *) (∀n v. FLOOKUP queue n = SOME v ⇒ ∃x len. FLOOKUP known v.prev_node = SOME x ∧ has_edge edges v.prev_node len n ∧ v.dist_from_root = x.dist_from_root + len ∧ ∀other y len1. FLOOKUP known other = SOME y ∧ has_edge edges other len1 n ⇒ v.dist_from_root ≤ y.dist_from_root + len1) ⇒ correct_result edges root res ∧ complete_result edges root res

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

dijkstra_thm

null

dijkstra_imp_correct_result : dijkstra root edges result ⇒ correct_result edges root result ∧ complete_result edges root result

theorem

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

dijkstra_imp_correct_result

null

min_elem_def = min_elem (n:num) v queue ⇔ FLOOKUP queue n = SOME v ∧ ∀m w. FLOOKUP queue m = SOME w ⇒ v.dist_from_root ≤ w.dist_from_root

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

min_elem_def

null

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

update_def

null

root_node_def = root_node r = <| prev_node := r; dist_from_root := 0 |>

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

root_node_def

null

dijkstra_def = dijkstra root edges result ⇔ calc FEMPTY (FEMPTY |+ (root, root_node root)) edges result

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

dijkstra_def

null

correct_result_def = correct_result edges root known ⇔ (∀n v. FLOOKUP known n = SOME v ⇒ has_path edges root v.dist_from_root n ∧ (∀l. l < v.dist_from_root ⇒ ~ has_path edges root l n) ∧ (n = root ⇒ v.prev_node = root) ∧ (n ≠ root ⇒ ∃w l. FLOOKUP known v.prev_node = SOME w ∧ has_edge edges v.prev_node l n ∧ v.dist_from_root = l + w.dist_from_root))

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

correct_result_def

null

complete_result_def = complete_result edges root known ⇔ (∀n len. has_path edges root len n ⇒ n ∈ FDOM known)

definition

examples/algorithms

[]

examples/algorithms/dijkstraScript.sml

complete_result_def

null

domain_spt_fold_union : ! tree y. domain (spt_fold union y tree) = domain y UNION {n | ?k aSet. lookup k tree = SOME aSet /\ n IN domain aSet}

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

domain_spt_fold_union

null

num_set_tree_union_empty : ! t1 t2 . isEmpty (num_set_tree_union t1 t2) <=> isEmpty t1 /\ isEmpty t2

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

num_set_tree_union_empty

null

wf_num_set_tree_union : ! t1 t2 result . wf t1 /\ wf t2 ==> wf (num_set_tree_union t1 t2)

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

wf_num_set_tree_union

null

domain_num_set_tree_union : ! t1 t2 . domain (num_set_tree_union t1 t2) = domain t1 UNION domain t2

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

domain_num_set_tree_union

null

num_set_tree_union_sym : ! t1 t2 . num_set_tree_union t1 t2 = num_set_tree_union t2 t1

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

num_set_tree_union_sym

null

lookup_num_set_tree_union : !t1 t2 n. lookup n (num_set_tree_union t1 t2) = case lookup n t1 of SOME x => ( case lookup n t2 of SOME y => SOME (union x y) | NONE => SOME x) | NONE => lookup n t2

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

lookup_num_set_tree_union

null

wf_spt_fold_union : ! tree y. (! n x . (lookup n tree = SOME x) ==> wf x) /\ wf y ==> wf (spt_fold union y tree)

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

wf_spt_fold_union

null

wf_closure_spt : ! reachable tree. wf reachable /\ (! n x . lookup n tree = SOME x ==> wf x) ==> wf (closure_spt reachable tree)

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

wf_closure_spt

null

adjacent_domain : ! tree x y . is_adjacent tree x y ==> x IN domain tree

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

adjacent_domain

null

reachable_domain : ! tree x y . is_reachable tree x y ==> (x = y \/ x IN domain tree)

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

reachable_domain

null

reachable_LHS_NOTIN : !tree n x. is_reachable tree n x /\ n NOTIN domain tree ==> n = x

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

reachable_LHS_NOTIN

null

rtc_is_adjacent : (! k . k IN t ==> ! n . (is_adjacent fullTree k n ==> n IN t)) ==> ! x y . RTC (is_adjacent fullTree) x y ==> x IN t ==> y IN t

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

rtc_is_adjacent

null

is_adjacent_num_set_tree_union : ! t1 t2 n m . is_adjacent t1 n m ==> is_adjacent (num_set_tree_union t1 t2) n m

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

is_adjacent_num_set_tree_union

null

is_reachable_num_set_tree_union : ! t1 t2 n m . is_reachable t1 n m ==> is_reachable (num_set_tree_union t1 t2) n m

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

is_reachable_num_set_tree_union

null

closure_spt_lemma : ! reachable tree closure (roots : num set). closure_spt reachable tree = closure /\ roots ⊆ domain reachable /\ (!k. k IN domain reachable ==> ?n. n IN roots /\ is_reachable tree n k) ==> domain closure = {a | ?n. n IN roots /\ is_reachable tree n a}

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

closure_spt_lemma

null

closure_spt_thm : ! tree start. domain (closure_spt start tree) = {a | ?n. n IN domain start /\ is_reachable tree n a}

theorem

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

closure_spt_thm

null

num_set_tree_union_def = (num_set_tree_union LN t2 = t2 : num_set num_map) /\ (num_set_tree_union (LS a) t = case t of | LN => LS a | LS b => LS (union a b) | BN t1 t2 => BS t1 a t2 | BS t1 b t2 => BS t1 (union a b) t2) /\ (num_set_tree_union (BN t1 t2) t = case t of | LN => BN t1 t2 | LS a => BS t1 a t2 | BN t1' t2' => BN (num_set_tree_union t1 t1') (num_set_tree_union t2 t2') | BS t1' a t2' => BS (num_set_tree_union t1 t1') a (num_set_tree_union t2 t2')) /\ (num_set_tree_union (BS t1 a t2) t = case t of | LN => BS t1 a t2 | LS b => BS t1 (union a b) t2 | BN t1' t2' => BS (num_set_tree_union t1 t1') a (num_set_tree_union t2 t2') | BS t1' b t2' => BS (num_set_tree_union t1 t1') (union a b) (num_set_tree_union t2 t2'))

definition

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

num_set_tree_union_def

null

closure_spt_def = closure_spt (reachable: num_set) tree = let sets = inter tree reachable in let nodes = spt_fold union LN sets in if subspt nodes reachable then reachable else closure_spt (union reachable nodes) tree Termination WF_REL_TAC `measure (\ (r,t). size (difference (spt_fold union LN t) r))` >> rw[] >> gvs[subspt_domain, SUBSET_DEF] >> irule size_diff_less >> gvs[domain_union, domain_spt_fold_union, lookup_inter] >> EVERY_CASE_TAC >> gvs[] >> simp[PULL_EXISTS, GSYM CONJ_ASSOC] >> goal_assum drule >> goal_assum drule >> simp[] >> goal_assum (drule_at Any) >> qexists_tac `k` >> simp[]

definition

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

closure_spt_def

null

is_adjacent_def = is_adjacent tree x y = ? aSetx. ( lookup x tree = SOME aSetx ) /\ ( lookup y aSetx = SOME () )

definition

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

is_adjacent_def

null

is_reachable_def = is_reachable tree = RTC (is_adjacent tree)

definition

examples/algorithms

[]

examples/algorithms/spt_closureScript.sml

is_reachable_def

null

partition_LENGTH : !ks n reach xs ys zs xs1 ys1 zs1. partition n ks reach (xs,ys,zs) = (xs1,ys1,zs1) ==> LENGTH xs1 <= LENGTH xs + LENGTH ks /\ LENGTH ys1 <= LENGTH ys + LENGTH ks /\ LENGTH zs1 <= LENGTH zs + LENGTH ks

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

partition_LENGTH

null

top_sort_example : top_sort [(0,fromAList[]); (* 0 has no deps *) (1,fromAList[(2,());(0,())]); (* 1 depens on 2 and 0 *) (2,fromAList[(1,())]); (* 2 depends on 1 *) (3,fromAList[(1,())])] (* 3 depends on 1 *) = [[0]; [1; 2]; [3]] (* 0 defined first, 1 and 2 are mutual, 3 is last *)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_example

null

top_sort_any_example : top_sort_any [("A",[]); (* A has no deps *) ("B",["C";"A"]); (* B depens on C and A *) ("C",["B";"X"]); (* C depends on B and X -- X is intentionally not defined here *) ("D",["B"])] (* D depends on B *) = [["A"]; ["B"; "C"]; ["D"]] (* A defined first, B and C are mutual, D is last *)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_example

null

needs_eq_is_adjacent : !x y tree. needs x y tree <=> is_adjacent tree x y

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

needs_eq_is_adjacent

null

partition_lemma : !n ks reach acc xas ybs zcs as bs cs. partition n ks reach acc = (xas,ybs,zcs) /\ acc = (as,bs,cs) ==> ?xs ys zs. xas = xs ++ as /\ ybs = ys ++ bs /\ zcs = zs ++ cs /\ (set (xs ++ ys ++ zs) = set ks) /\ (!x. MEM x ks ==> (MEM x xs <=> ¬ needs x n reach)) /\ (!y. MEM y ks ==> (MEM y ys <=> needs y n reach /\ needs n y reach)) /\ (!z. MEM z ks ==> (MEM z zs <=> needs z n reach /\ ¬ needs n z reach))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

partition_lemma

null

partition_thm : !n ks reach as bs cs res. partition n ks reach (as, bs, cs) = res ==> ?xs ys zs. res = (xs ++ as, ys ++ bs, zs ++ cs) /\ (set (xs ++ ys ++ zs) = set ks) /\ (!x. MEM x ks ==> (MEM x xs <=> ¬ needs x n reach)) /\ (!y. MEM y ks ==> (MEM y ys <=> needs y n reach /\ needs n y reach)) /\ (!z. MEM z ks ==> (MEM z zs <=> needs z n reach /\ ¬ needs n z reach))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

partition_thm

null

ALL_DISTINCT_APPEND_SWAP : !l1 l2. ALL_DISTINCT (l1 ++ l2) <=> ALL_DISTINCT (l2 ++ l1)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALL_DISTINCT_APPEND_SWAP

null

partition_ALL_DISTINCT : !n ks reach acc xs ys zs as bs cs. partition n ks reach acc = (xs,ys,zs) /\ acc = (as,bs,cs) /\ ALL_DISTINCT (ks ++ as ++ bs ++ cs) ==> ALL_DISTINCT (xs ++ ys ++ zs)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

partition_ALL_DISTINCT

null

top_sort_aux_correct : !ns reach acc resacc. top_sort_aux ns reach acc = resacc /\ (!n m. TC (\a b. needs a b reach) n m ==> needs n m reach) /\ ALL_DISTINCT ns ==> ?res. resacc = res ++ acc /\ ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set ns /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: *) res = xss ++ [ys] ++ zss /\ MEM y ys ==> (* for any defined dependencies of y *) !dep. needs y dep reach /\ MEM dep ns ==> (* mutual dependencies must by in ys, all others in xss *) (¬ needs dep y reach <=> MEM dep (FLAT xss)) /\ ( needs dep y reach <=> MEM dep ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_aux_correct

null

domain_lookup_num_set : !t k. k IN domain t <=> lookup k t = SOME ()

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

domain_lookup_num_set

null

trans_clos_correct : !nexts n m. needs n m (trans_clos nexts) \/ n = m <=> is_reachable nexts n m

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

trans_clos_correct

null

trans_clos_correct_imp : !nexts n m. (TC (\x y. needs x y nexts)) n m ==> needs n m (trans_clos nexts)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

trans_clos_correct_imp

null

trans_clos_TC_closed : !t n m. TC (\a b. needs a b (trans_clos t)) n m ==> needs n m (trans_clos t)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

trans_clos_TC_closed

null

top_sort_correct_weak : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: *) res = xss ++ [ys] ++ zss /\ MEM y ys ==> (* all dependencies of y must be defined in ys or earlier, i.e. xss *) ?deps. ALOOKUP lets y = SOME deps /\ !d. d IN domain deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_correct_weak

null

top_sort_correct : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: *) res = xss ++ [ys] ++ zss /\ MEM y ys ==> (* for any defined dependencies of y *) !dep. is_reachable (fromAList lets) y dep /\ MEM dep (MAP FST lets) ==> (¬ is_reachable (fromAList lets) dep y <=> MEM dep (FLAT xss)) /\ ( is_reachable (fromAList lets) dep y <=> MEM dep ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_correct

null

top_sort_aux_same_partition : ∀ns reach acc xs. MEM xs (top_sort_aux ns reach acc) ⇒ ∀x y nexts. (∀n m. TC (λa b. needs a b reach) n m ⇒ needs n m reach) ∧ (∀as x y. MEM as acc ∧ x ≠ y ∧ MEM x as ∧ MEM y as ⇒ needs x y reach) ∧ x ≠ y ∧ MEM x xs ∧ MEM y xs ⇒ needs x y reach

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_aux_same_partition

null

top_sort_same_partition : MEM xs (top_sort graph) ∧ x ≠ y ∧ MEM x xs ∧ MEM y xs ⇒ needs x y (trans_clos $ fromAList graph)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_same_partition

null

SCC_REFL : SCC E x x

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

SCC_REFL

null

SCC_SYM : SCC E x y ⇔ SCC E y x

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

SCC_SYM

null

SCC_TRANS : SCC E x y ∧ SCC E y z ⇒ SCC E x z

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

SCC_TRANS

null

SCC_EQUIV : EQUIV (SCC E)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

SCC_EQUIV

null

is_reachable_IMP_MEM : ALL_DISTINCT (MAP FST graph) ⇒ x ≠ y ⇒ is_reachable (fromAList graph) x y ⇒ MEM x (MAP FST graph)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

is_reachable_IMP_MEM

null

top_sort_SCC : ALL_DISTINCT (MAP FST graph) ∧ MEM x l ∧ MEM l (top_sort graph) ⇒ SCC (is_adjacent (fromAList graph)) x = set l

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_SCC

null

to_nums_correct : !xs b res. to_nums b xs = res /\ ALL_DISTINCT (MAP FST b) ==> domain res = {c | ?d. MEM d xs /\ ALOOKUP b d = SOME c}

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

to_nums_correct

null

ALL_DISTINCT_MAPi_ID : !l. ALL_DISTINCT (MAPi (\i _. i) l)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALL_DISTINCT_MAPi_ID

null

ALL_DISTINCT_FLAT : ∀l. ALL_DISTINCT (FLAT l) ⇔ (∀l0. MEM l0 l ⇒ ALL_DISTINCT l0) ∧ (∀i j. i < j ∧ j < LENGTH l ⇒ ∀e. MEM e (EL i l) ⇒ ¬MEM e (EL j l))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALL_DISTINCT_FLAT

null

ALOOKUP_MAPi_ID : !l k. ALOOKUP (MAPi (\i n. (i,n)) l) k = if k < LENGTH l then SOME (EL k l) else NONE

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALOOKUP_MAPi_ID

null

ALOOKUP_MAPi_ID_f : !l k. ALOOKUP (MAPi (\i n. (i,f n)) l) k = if k < LENGTH l then SOME (f (EL k l)) else NONE

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALOOKUP_MAPi_ID_f

null

MAPi_MAP : !l. MAPi (\i n. f n) l = MAP f l

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

MAPi_MAP

null

MEM_ALOOKUP : !l k v. ALL_DISTINCT (MAP FST l) ==> (MEM (k,v) l <=> ALOOKUP l k = SOME v)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

MEM_ALOOKUP

null

top_sort_any_correct_weak : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort_any lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: *) res = xss ++ [ys] ++ zss /\ MEM y ys ==> (* all dependencies of y must be defined in ys or earlier, i.e. xss *) ?deps. ALOOKUP lets y = SOME deps /\ !d. MEM d deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_correct_weak

null

depends_on : depends_on alist = RTC (depends_on1 alist)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

depends_on

null

top_sort_any_depends_on_weak : !lets res. (!xss ys zss y. res = xss ++ [ys] ++ zss /\ MEM y ys ==> ?deps. ALOOKUP lets y = SOME deps /\ !d. MEM d deps /\ MEM d (MAP FST lets) ==> MEM d (FLAT xss ++ ys)) ==> !a b. depends_on lets a b ==> !xss ys zss y. res = xss ++ [ys] ++ zss /\ MEM a ys ==> MEM b (FLAT xss ++ ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_depends_on_weak

null

top_sort_any_correct : !lets res. ALL_DISTINCT (MAP FST lets) /\ res = top_sort_any lets ==> ALL_DISTINCT (FLAT res) /\ set (FLAT res) = set (MAP FST lets) /\ !xss ys zss y. (* for any element ys in the result res, for any y in that ys: *) res = xss ++ [ys] ++ zss /\ MEM y ys ==> (* all dependencies of y must be defined in ys or earlier, i.e. xss *) !dep. depends_on lets y dep ==> MEM dep (FLAT xss ++ ys) /\ (depends_on lets dep y ==> MEM dep ys)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_correct

null

MEM_MEM_top_sort : ALL_DISTINCT (MAP FST l) ∧ MEM xs $ top_sort l ∧ MEM n xs ⇒ MEM n (MAP FST l)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

MEM_MEM_top_sort

null

ALL_DISTINCT_enc_graph : ALL_DISTINCT (MAPi ($o FST o (λi (n,ns). (i,to_nums (MAPi (λi n. (n,i)) (MAP FST graph)) ns))) graph)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ALL_DISTINCT_enc_graph

null

RTC_ALOOKUP_enc_graph_IMP_depends_on : ALL_DISTINCT (MAP FST graph) ⇒ ∀n n'. RTC (λx y. ∃aSetx. ALOOKUP (MAPi (λi (n,ns). (i,to_nums (MAPi (λi n. (n,i)) (MAP FST graph)) ns)) graph) x = SOME aSetx ∧ lookup y aSetx = SOME ()) n n' ⇒ depends_on graph (EL n (MAP FST graph)) (EL n' (MAP FST graph))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

RTC_ALOOKUP_enc_graph_IMP_depends_on

null

top_sort_any_same_partition : ALL_DISTINCT (MAP FST graph) ∧ MEM xs (top_sort_any graph) ∧ MEM x xs ∧ MEM y xs ⇒ depends_on graph x y

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_same_partition

null

top_sort_any_SCC : ALL_DISTINCT (MAP FST graph) ∧ MEM x l ∧ MEM l (top_sort_any graph) ⇒ SCC (depends_on1 graph) x = set l

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_SCC

null

cycle_CASES_lem : ∀x z. TC R x z ⇒ x = z ⇒ (R x z ∨ (∃y. x ≠ y ∧ TC R x y ∧ TC R y z))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

cycle_CASES_lem

null

cycle_CASES : TC R x x ⇔ (R x x ∨ (∃y. x ≠ y ∧ TC R x y ∧ TC R y x))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

cycle_CASES

null

depends_on_IMP_MEM : ∀x y. depends_on graph x y ⇒ x ≠ y ⇒ MEM y (MAP FST graph)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

depends_on_IMP_MEM

null

TC_depends_on_SCC : x ≠ y ⇒ SCC (depends_on1 graph) x y = (TC_depends_on graph x y ∧ TC_depends_on graph y x)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

TC_depends_on_SCC

null

ONE_LT_LENGTH_ALL_DISTINCT_IMP : 1 < LENGTH l ∧ ALL_DISTINCT l ⇒ ∃x y. MEM x l ∧ MEM y l ∧ x ≠ y

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

ONE_LT_LENGTH_ALL_DISTINCT_IMP

null

DISTINCT_IMP_ONE_LT_LENGTH : MEM x l ∧ MEM y l ∧ x ≠ y ⇒ 1 < LENGTH l

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

DISTINCT_IMP_ONE_LT_LENGTH

null

has_cycle_correct : ALL_DISTINCT (MAP FST graph) ⇒ (has_cycle graph ⇔ ∃x. TC_depends_on graph x x)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

has_cycle_correct

null

TC_depends_on_weak_IMP_MEM : ∀x y. TC_depends_on_weak graph x y ⇒ MEM x (MAP FST graph)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

TC_depends_on_weak_IMP_MEM

null

TC_depends_on_weak_IMP_TC_depends_on : ∀x y. TC_depends_on_weak graph x y ⇒ MEM y (MAP FST graph) ⇒ TC_depends_on graph x y

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

TC_depends_on_weak_IMP_TC_depends_on

null

TC_depends_on_TC_depends_on_weak_thm : ∀x y. TC_depends_on graph x y = (TC_depends_on_weak graph x y ∧ MEM y (MAP FST graph))

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

TC_depends_on_TC_depends_on_weak_thm

null

TC_depends_on_weak_cycle_thm : TC_depends_on_weak graph x x = TC_depends_on graph x x

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

TC_depends_on_weak_cycle_thm

null

has_cycle_correct2 : ALL_DISTINCT (MAP FST graph) ⇒ (has_cycle graph ⇔ ∃x. TC_depends_on_weak graph x x)

theorem

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

has_cycle_correct2

null

needs_def = needs n m (reach:num_set num_map) = case lookup n reach of | NONE => F (* cannot happen *) | SOME s => IS_SOME (lookup m s)

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

needs_def

null

partition_def = partition n [] reach acc = acc /\ partition n (k::ks) reach (xs,ys,zs) = if needs k n reach then if needs n k reach then partition n ks reach (xs,k::ys,zs) else partition n ks reach (xs,ys,k::zs) else partition n ks reach (k::xs,ys,zs)

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

partition_def

null

top_sort_aux_def = top_sort_aux [] reach acc = acc /\ top_sort_aux (n::ns) reach acc = let (xs,ys,zs) = partition n ns reach ([],[],[]) in top_sort_aux xs reach ((n::ys) :: top_sort_aux zs reach acc) Termination WF_REL_TAC `measure (LENGTH o FST)` \\ rw [] \\ pop_assum (assume_tac o GSYM) \\ imp_res_tac partition_LENGTH \\ fs []

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_aux_def

null

trans_clos_def = (* computes the transitive closure for each node given nexts *) trans_clos nexts = map (\x. closure_spt x nexts) nexts

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

trans_clos_def

null

top_sort_def = top_sort (let_bindings : (num (* name *) # num_set (* free vars *)) list) = let roots = MAP FST let_bindings in let nexts = fromAList let_bindings in let reach = trans_clos nexts in top_sort_aux roots reach []

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_def

null

to_nums_def = to_nums b [] = LN /\ to_nums b (x::xs) = case ALOOKUP b x of | NONE => to_nums b xs | SOME n => insert n () (to_nums b xs)

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

to_nums_def

null

top_sort_any_def = top_sort_any (lets : ('a # 'a list) list) = if NULL lets (* so that HD names, below, is well defined *) then [] else let names = MAP FST lets in let to_num = MAPi (\i n. (n,i)) names in let from_num = fromAList (MAPi (\i n. (i,n)) names) in let nesting = top_sort (MAPi (\i (n,ns). (i,to_nums to_num ns)) lets) in MAP (MAP (\n. case lookup n from_num of SOME m => m | _ => HD (names))) nesting

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

top_sort_any_def

null

has_cycle_def = has_cycle graph = ((EXISTS (λl. 1 < LENGTH l) $ top_sort_any graph) ∨ EXISTS (λ(v,l). MEM v l) graph)

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

has_cycle_def

null

SCC_def = SCC E x y ⇔ RTC E x y ∧ RTC E y x

definition

examples/algorithms

[]

examples/algorithms/topological_sortScript.sml

SCC_def

null