hcju/isabelleps huggingface dataset:
iwf: fixes n :: "('form, 'rule) graph_node"   and p :: "'form"   and fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and s :: "'subst"   and i :: "nat"   and \<Gamma> :: "'form fset"   and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list"   and c :: "'form" assumes "n \<in> i.R nodes"   and "Reg p |\<in>| outPorts n"   and "list_all2 (\<lambda>ip t. iwf fc t ((\<lambda>h. subst s (freshen i (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma> \<turnstile> subst s (freshen i (labelsIn n ip)))) (inPorts' n) ants"   and "fc n i s (\<Gamma> \<turnstile> c)"   and "c = subst s (freshen i p)" shows "iwf fc (INode n p i s ants) (\<Gamma> \<turnstile> c)"
aboveS_pred: fixes i :: "'a" assumes "isSucc i" shows "local.aboveS (pred i) \<noteq> {}"
lcosI: fixes G :: "('c, 'd) monoid_scheme"   and h :: "'c"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> h \<in> x <#\<^bsub>G\<^esub> H"
strip_all_vars_subst_bv1_Fv: fixes B :: "Core.term"   and lev :: "nat"   and x :: "variable"   and \<tau> :: "typ" shows "strip_all_vars (subst_bv1 B lev (Fv x \<tau>)) = strip_all_vars B"
distinct_1: fixes x1 :: "'b" shows "ev x1 \<noteq> tick"
constn_lemma: fixes n :: "nat"   and x :: "nat" shows "rec_exec (constn n) [x] = n"
e_bounded: fixes t :: "real" assumes "t \<in> T" shows "dist t t0 \<le> b / B"
syzygy_moduleD: fixes s :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "s \<in> pmdl.syzygy_module B" shows "pmdl.represents B s 0"
equality: fixes r :: "('c, 'd, 'e) automaton_scheme"   and r' :: "('c, 'd, 'e) automaton_scheme" assumes "init r = init r'"   and "automaton.trans r = automaton.trans r'"   and "more r = more r'" shows "r = r'"
principal_ideal: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" assumes "Ring R"   and "a \<in> carrier R" shows "ideal R (R \<diamondsuit>\<^sub>p a)"
abc_unhalt: shows "\<forall>stp. case abc_steps_l (0, lm) ap stp of (as, am) \<Rightarrow> as < length ap"
semilattice_axioms: shows "semilattice (\<^bold>\<squnion>)"
add_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b"   and b' :: "'b" shows "a *\<^sub>V (b + b') = a *\<^sub>V b + a *\<^sub>V b'"
prune_decreasing: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "prune A w V \<subseteq> V"
left_lang_UN: fixes qs1 :: "hf set"   and qs2 :: "hf set" assumes "\<Union> (MN.left_lang ` qs1) = \<Union> (MN.left_lang ` qs2)"   and "qs1 \<union> qs2 \<subseteq> MN.accessible" shows "qs1 = qs2"
not_is_empty_alist_of: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "alist_of q \<noteq> []"
ENF_refl_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "SecurityInvariant_withOffendingFlows.ENF_refl sinvar P \<equiv> SecurityInvariant_withOffendingFlows.sinvar_all_edges_normal_form sinvar P \<and> (\<forall>p1. P p1 p1)"
T_measure1_def: fixes A :: "('v \<times> 'v) set" shows "T_measure1 A = card (nodes rg - S A)"
that: fixes h :: "nat \<Rightarrow> int" assumes "h \<in> carrier Zp \<and> f = g \<oplus> [p] \<cdot> \<one> [^] k \<otimes> h" shows "thesis"
simps_18: fixes h :: "hash"   and e :: "Syntax.term" shows "\<lparr>Hashed h e\<rparr> = Hash h"
gmw_base: fixes n :: "nat" shows "gmw_base (S1_14_OT n) (R1_14_OT n) (S2_14_OT n) (R2_14_OT n) (protocol_14_OT n) (adv_14_OT n)"
limit: assumes "\<And>ys. lstrict_prefix ys xs \<Longrightarrow> P ys" shows "P xs"
C_def: shows "C = r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> B \<inter>\<^sub>\<circ> A"
inf_sup_aci_4: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "inf x (inf x y) = inf x y"
rQuots_rev_lQuots: fixes x :: "'b list"   and A :: "'b list set" shows "rQuots x A = rev ` lQuots x (rev ` A)"
nths_append: fixes l :: "'g list"   and l' :: "'g list"   and A :: "nat set" shows "nths (l @ l') A = nths l A @ nths l' {j. j + length l \<in> A}"
lookup_def: fixes destr :: "'f \<Rightarrow> 'e"   and n :: "'g"   and s :: "'g \<Rightarrow> 'f" shows "lookup destr n s = destr (s n)"
1: shows "\<forall>f. snd (Rep_lifted_pair (Rep_regular pf) f) = Rep_phi f (- fst (Rep_lifted_pair (Rep_regular pf) f))"
val2formFormulaTrue: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" shows "formulaTrue (val2form valuation1) valuation2 = (\<forall>literal. literal el valuation1 \<longrightarrow> literal el valuation2)"
abs_eq: fixes xa :: "conc_interface"   and x :: "conc_interface" shows "(abs_interface xa \<otimes> abs_interface x) = abs_interface (xa \<otimes>\<^sub>c x)"
upper_closure: shows "\<pi>\<^sub>*\<^bsub>F\<^esub> \<in> carrier \<Y>\<^bsub>F\<^esub> \<rightarrow> carrier \<X>\<^bsub>F\<^esub>"
preprocess'_unsat: fixes I :: "'a set"   and v :: "nat \<Rightarrow> QDelta"   and s :: "('a \<times> QDelta ns_constraint) list"   and V :: "nat set"   and start :: "nat" assumes "(I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set s"   and "vars_constraints (map snd s) \<subseteq> V"   and "\<forall>var\<in>V. var < start" shows "\<exists>v'. (\<forall>var\<in>V. v var = v' var) \<and> v' \<Turnstile>\<^sub>a\<^sub>s (snd ` (set (Atoms (preprocess' s start)) \<inter> I \<times> UNIV)) \<and> v' \<Turnstile>\<^sub>t Tableau (preprocess' s start)"
impl_of_empty: shows "impl_of Trie2.empty = empty_trie"
decisive_on_def: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "decisive_on A f = (\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> f B \<noteq> {})"
le_Inf_iff: fixes b :: "'d \<times> 'e"   and A :: "('d \<times> 'e) set" shows "less_eq_prod b (Inf_prod A) = (\<forall>a\<in>A. less_eq_prod b a)"
not_square_less_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (times a a) zero"
lset_eq_empty: fixes xs :: "'c llist" shows "(lset xs = {}) = lnull xs"
permutes_insert: fixes a :: "'b"   and S :: "'b set" shows "{p. p permutes insert a S} = (\<lambda>(b, p). Fun.swap a b id \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
\<beta>\<^sub>f: shows "\<guillemotleft>\<beta>\<^sub>f : tab\<^sub>1 f \<star> w\<^sub>f \<rightarrow> tab\<^sub>1 f \<star> w\<^sub>f'\<guillemotright>"
simps_10: fixes v :: "'b"   and uu :: "'b ltln set" shows "prop\<^sub>n(v)[uu]\<^sub>\<nu> = prop\<^sub>n(v)"
mem_read_delayed_write_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> get_delayed_pool s1 = [] \<and> get_delayed_pool s2 = []" shows "fst (memory_read 8 (cpu_reg_val PC (delayed_pool_write s1)) (delayed_pool_write s1)) = fst (memory_read 8 (cpu_reg_val PC (delayed_pool_write s2)) (delayed_pool_write s2))"
ipresCons_imp_ipresSwapAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igSwapCls Delta MOD"   and "FixSyn.igConsIPresIGWls Delta MOD" shows "FixSyn.ipresSwapAll Delta h hA MOD"
simple_rule_dtor_def: fixes r :: "'a simple_rule" shows "simple_rule_dtor r \<equiv> case r of SimpleRule m a \<Rightarrow> (m, a)"
fmdom_notI: fixes m :: "('b, 'a) fmap"   and x :: "'b" assumes "fmlookup m x = None" shows "x |\<notin>| fmdom m"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "src\<^sub>C f = C.null"
eq: fixes t :: "'a \<times> nat" assumes "t \<in> set (dense_repr p)" shows "h [fst t] \<otimes>\<^bsub>A\<^esub> h X [^]\<^bsub>A\<^esub> snd t = (h \<circ> (\<lambda>t. [fst t] \<otimes>\<^bsub>K [X]\<^esub> X [^]\<^bsub>K [X]\<^esub> snd t)) t"
extract_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "List.extract P xs = (case dropWhile (Not \<circ> P) xs of [] \<Rightarrow> None | y # ys \<Rightarrow> Some (takeWhile (Not \<circ> P) xs, y, ys))"
EqOnUOut_def: fixes p :: "pvar"   and p' :: "pvar"   and u :: "'U"   and ou :: "'Out" shows "EqOnUOut p p' u ou \<equiv> Eq (Atom (Obs u ou) p) (Atom (Obs u ou) p')"
intros_1: fixes t :: "('e, 'f, 'g) Tree"   and tset :: "('e, 'f, 'g) Tree set['e]" assumes "t \<in> set_bset tset" shows "(t, tConj tset) \<in> Tree_wf"
simps: fixes a1 :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and a2 :: "('c, 'e) flow_entry_match list"   and a3 :: "'d"   and a4 :: "'e option" shows "guha_table_semantics a1 a2 a3 a4 = ((\<exists>\<gamma> fe p ft1 ft2. a1 = \<gamma> \<and> a2 = ft1 @ fe # ft2 \<and> a3 = p \<and> a4 = Some (ofe_action fe) \<and> \<gamma> (ofe_fields fe) p = True \<and> (\<forall>fe'\<in>set (ft1 @ ft2). ofe_prio fe < ofe_prio fe' \<longrightarrow> \<gamma> (ofe_fields fe') p = False)) \<or> (\<exists>ft \<gamma> p. a1 = \<gamma> \<and> a2 = ft \<and> a3 = p \<and> a4 = None \<and> (\<forall>fe\<in>set ft. \<gamma> (ofe_fields fe) p = False)))"
abs_lattice: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add_abs abs plus zero minus uminus less_eq less inf sup" shows "abs a = sup a (uminus a)"
c_min: fixes D :: "'a literal multiset" assumes "D \<in> N"   and "\<not> INTERP N \<Turnstile> D" shows "C \<le> D"
dom_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.dom (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
ivalid: shows "index_valid (set as) (assert_all_state t as)"
absolutely_integrable_restrict_UNIV: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) absolutely_integrable_on UNIV) = (f absolutely_integrable_on S)"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)"   and "a \<in> H" shows "inv\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> a \<in> H"
int_zone_dbm: fixes v :: "'d \<Rightarrow> nat"   and n :: "nat"   and cc :: "('d, 'e) cconstraint" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>(uu_, d)\<in>collect_clock_pairs cc. d \<in> \<int>"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" obtains M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" where "{u. u \<turnstile> cc} = [M]\<^bsub>v,n\<^esub>" and "dbm_int M n"
on_triple_ID_restrict: fixes G :: "('b, 'c) labeled_graph" shows "on_triple (Id_on (vertices G)) `` edges G = edges (restrict G)"
minus_closed: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<ominus>\<^bsub>G\<^esub> y \<in> carrier G"
rel: shows "SourceTerm SP \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> SourceTerm SQ"
inf_set_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B = Collect ((\<lambda>x. x \<in> A) \<sqinter> (\<lambda>x. x \<in> B))"
vsv_vimageI1: fixes a :: "V"   and A :: "V" 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"
complete_ht_update_normal: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and k :: "'a"   and v :: "'b" assumes "\<not> length l * load_factor \<le> the_size ht * 100" shows "<is_hashtable l ht> hm_update k v ht <is_hashtable (abs_update k v l)>"
persistent_clauses_subsume_redundant_clauses: fixes C :: "'at Literal set"   and R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set"   and N :: "nat" assumes "redundant C (inferred_clause_sets R S N)"   and "all_fulfill finite S"   and "derived_clauses_are_finite R"   and "finite C" shows "redundant C (saturation R S)"
verticesFrom_in': fixes a :: "nat"   and f :: "face"   and v :: "nat" assumes "a \<in> set (verticesFrom f v)"   and "a \<noteq> v" shows "a \<in> \<V> f"
asymD: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "asym R"   and "(x, y) \<in> R" shows "(y, x) \<notin> R"
CDMode_CTRL_Delta_def: shows "CDMode_CTRL_Delta = {(''Playing'', (En ''Next'', defaultguard, defaultaction), ''SelectingNextTrack''), (''SelectingNextTrack'', (En ''Ready'', defaultguard, CDMode_CTRL_Action1), ''Playing''), (''Playing'', (En ''Back'', defaultguard, defaultaction), ''SelectingPreviousTrack''), (''SelectingPreviousTrack'', (En ''Ready'', defaultguard, CDMode_CTRL_Action2), ''Playing''), (''Playing'', (En ''EndOfTitle'', \<lambda>d. Select0 d < Select1 d, defaultaction), ''SelectingNextTrack'')}"
zmset_empty_image_zmset_empty: fixes M :: "'a zmultiset"   and f :: "'a \<Rightarrow> 'b"   and t :: "'b" assumes "\<And>t. zcount M t = 0" shows "zcount (Auxiliary.image_zmset f M) t = 0"
leg1_simps_3: shows "src (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) = src ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)"
Re_sum: fixes f :: "'a \<Rightarrow> complex"   and s :: "'a set" shows "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))"
suc_n_union_of_union: fixes F :: "'a set"   and n :: "nat" shows "{xs. set xs \<subseteq> F \<and> length xs = Suc n} = (\<Union>x\<in>F. \<Union>xs\<in>{xs. set xs \<subseteq> F \<and> length xs = n}. {x # xs})"
isEmpty_def: fixes q :: "('a, 'b) BinomialHeap" shows "BinomialHeap_loc.isEmpty q \<equiv> Rep_BinomialHeap q = []"
pfun_le_plus: fixes x :: "('a, 'b) pfun"   and y :: "('a, 'b) pfun" assumes "pdom x \<inter> pdom y = {}" shows "x \<subseteq>\<^sub>p x + y"
mpoly_base_conv_2: fixes c :: "'a"   and x :: "'a" shows "c = poly [:c:] x"
group_def: fixes G :: "('e, 'f) monoid_scheme" shows "Group.group G \<equiv> Group.monoid G \<and> Group.group_axioms G"
C1'_p: shows "ground\<^sub>l\<^sub>s C\<^sub>1' \<and> falsifies\<^sub>g (B @ [d]) C\<^sub>1'"
sorted_map: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "sorted (map f xs) = sorted_wrt (\<lambda>x y. f x \<le> f y) xs"
VSGeneral_3_times_def: shows "VSGeneral_3_times = VSGeneral \<circ> VSGeneral \<circ> VSGeneral"
a_Inj2_2: fixes b :: "Syntax.term" assumes "\<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], b\<ggreater>" shows "\<exists>eP' eV' \<pi>. {$$} \<turnstile> b, eP', eV' : \<tau>\<^sub>2 \<and> (\<forall>\<pi>\<^sub>P. \<lless>\<pi>\<^sub>P, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>P @ \<pi>, eP'\<ggreater>) \<and> (\<forall>\<pi>'. \<lless>\<pi> @ \<pi>', eV\<ggreater> V\<rightarrow> \<lless>\<pi>', eV'\<ggreater>)"
fls_deriv_one: shows "fls_deriv 1 = 0"
prob_dom_def: fixes prob :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" shows "prob_dom prob \<equiv> \<Union>(s1, s2)\<in>prob. action_dom s1 s2"
incl_subgroup: fixes J :: "'a set"   and I :: "'a set" assumes "subgroup J (G\<lparr>carrier := H\<rparr>)"   and "subgroup I (G\<lparr>carrier := H, carrier := J\<rparr>)" shows "subgroup I (G\<lparr>carrier := H\<rparr>)"
recv_marker_changes_head_only_at_i: fixes ev :: "('p, 's, 'm) event"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and c :: "('p, 's, 'm) configuration"   and c' :: "('p, 's, 'm) configuration"   and i' :: "nat" assumes "ev = RecvMarker i p q"   and "c \<turnstile> ev \<mapsto> c'"   and "i' \<noteq> i" shows "msgs c i' = [] \<or> hd (msgs c i') = hd (msgs c' i')"
record_simps_7: fixes x :: "'b state" shows "update_clock (\<lambda>_. clock x) x = x"
simps: fixes upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and c :: "nat" shows "basis_reduction_main_cost upw i state c = (if i < m then let ((upw', i', state'), c_step) = basis_reduction_step_cost upw i state in basis_reduction_main_cost upw' i' state' (c + c_step) else (state, c))"
unit_vecs_distinct: fixes n :: "nat" shows "distinct (unit_vecs n)"
Lcm_least: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times b a" shows "dvd.dvd times (Lcm A) a"
sets_into_space: fixes x :: "'a set" assumes "x \<in> M" shows "x \<subseteq> \<Omega>"
part_antisymmetry: fixes x :: "i"   and y :: "i" assumes "M"   and "P x y"   and "P y x" shows "x = y"
gallery_overlap_join: fixes xs :: "'w set set list"   and w :: "'w set set"   and ys :: "'w set set list" assumes "gallery (xs @ [w])"   and "gallery (w # ys)" shows "gallery (xs @ w # ys)"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>cones_map f \<chi> f : \<chi>'.A.map a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] D b\<guillemotright>"
cases: fixes a1 :: "sterm"   and a2 :: "sterm" assumes "a1 => a2" obtains x :: "char list" where "a1 = Fvar x" and "a2 = Fvar x"   | f' :: "Label \<Rightarrow> sterm option" and f :: "Label \<Rightarrow> sterm option" and L :: "char list set" and T :: "Sigma.type" where "a1 = Obj f__ T__" and "a2 = Obj f'__ T__" and "dom f'__ = dom f__" and "finite L__" and "\<forall>l\<in>dom f__. \<forall>s p. s \<notin> L__ \<and> p \<notin> L__ \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f__ l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (f'__ l) = \<sigma>[s,p] t)" and "\<forall>l\<in>dom f__. body (the (f__ l))"   | t :: "sterm" and t' :: "sterm" and L :: "char list set" and u :: "sterm" and u' :: "sterm" and l :: "Label" where "a1 = Upd t__ l__ u__" and "a2 = Upd t'__ l__ u'__" and "t__ => t'__" and "lc t__" and "finite La__" and "\<forall>s p. s \<notin> La__ \<and> p \<notin> La__ \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u__\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u'__ = \<sigma>[s,p] t'')" and "body u__"   | f :: "Label \<Rightarrow> sterm option" and T :: "Sigma.type" and f' :: "Label \<Rightarrow> sterm option" and L :: "char list set" and t :: "sterm" and t' :: "sterm" and l :: "Label" where "a1 = Upd (Obj fa__ Ta__) la__ ta__" and "a2 = Obj (f'a__(la__ \<mapsto> t'a__)) Ta__" and "Obj fa__ Ta__ => Obj f'a__ Ta__" and "finite L" and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. ta__\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t'a__ = \<sigma>[s,p] t'')" and "la__ \<in> dom fa__" and "lc (Obj fa__ Ta__)" and "body ta__"   | t :: "sterm" and t' :: "sterm" and u :: "sterm" and u' :: "sterm" and l :: "Label" where "a1 = Call t lb__ u" and "a2 = Call t' lb__ u'" and "t => t'" and "u => u'" and "lc t" and "lc u"   | f :: "Label \<Rightarrow> sterm option" and T :: "Sigma.type" and f' :: "Label \<Rightarrow> sterm option" and l :: "Label" and p :: "sterm" and p' :: "sterm" where "a1 = Call (Obj f T) l p" and "a2 = (the (f' l)\<^bsup>[Obj f' T,p']\<^esup>)" and "Obj f T => Obj f' T" and "l \<in> dom f" and "p => p'" and "lc (Obj f T)" and "lc p"
comp: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'f"   and h :: "'g \<Rightarrow> 'c"   and i :: "'d \<Rightarrow> 'h" shows "map_fun f g \<circ> map_fun h i = map_fun (h \<circ> f) (g \<circ> i)"
mat1_mult_left: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nr :: "nat"   and nc :: "nat"   and m :: "'a list list" assumes "class.semiring_1 one times plus zero"   and "mat nr nc m" shows "mat_multI zero plus times nr (mat1I zero one nr) m = m"
that: fixes b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> A" shows "thesis"
not_empty_eq_Iic_eq_empty: fixes h :: "'a" shows "{} \<noteq> {..h}"
walk_comp: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G xs"   and "Graph.walk G ys"   and "xs = [] \<or> ys = [] \<or> last xs \<rightarrow>\<^bsub>G\<^esub> hd ys" shows "Graph.walk G (xs @ ys)"
valid_write_sops: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>write_sops sb. valid_sop sop"
A: fixes n :: "nat" assumes "enat n < llength ys" shows "(n \<in> A) = (n \<in> B)"
inference_computation: fixes P :: "('a literal multiset \<times> nat) multiset"   and C :: "'a literal multiset"   and i :: "nat"   and N :: "('a literal multiset \<times> nat) multiset"   and n :: "nat"   and Q :: "('a literal multiset \<times> nat) multiset" assumes "\<forall>(D, j)\<in>#P. weight (C, i) \<le> weight (D, j)"   and "N = mset_set ((\<lambda>D. (D, n)) ` concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (set_mset (image_mset fst Q)) C))" shows "({#}, P + {#(C, i)#}, Q, n) \<leadsto>\<^sub>w (N, {#(D, j) \<in># P. D \<noteq> C#}, Q + {#(C, i)#}, Suc n)"
that: fixes X :: "'a set" assumes "x \<in> X"   and "X \<in> P" shows "thesis"
someI2: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and Q :: "'d \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
suffix_take: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>s ys" shows "ys = take (\<^bold>|ys\<^bold>| - \<^bold>|xs\<^bold>|) ys \<cdot> xs"
h1: shows "\<turnstile> I \<longrightarrow> P"
varrep_prop: shows "\<forall>no\<in>Nodes (Suc nb) levellista. var (repc no) \<le> var no \<and> repc (repc no) = repc no"
dlabels_def: shows "dlabels = fold_tree (\<lambda>x. Some [x]) (\<lambda>l r. ap_option (Some (\<lambda>(x, y). x @ y)) (assert (\<lambda>(xs, ys). set xs \<inter> set ys = {}) (ap_option (ap_option (Some Pair) l) r)))"
ind_in_set_mono2: fixes a :: "nat"   and A :: "nat set"   and b :: "nat" assumes "a \<in> A"   and "b \<in> A"   and "ind_in_set A a < ind_in_set A b" shows "a < b"
simps_5: fixes h :: "'heap"   and va :: "32 word" shows "typeof\<^bsub>h\<^esub> (Intg va) = typeof (Intg va)"
ts_b': fixes x :: "nat"   and y :: "nat"   and v :: "nat list"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])"   and "(\<exists>hs. h = [x, x] @ hs) \<or> h = [x] \<or> h = []" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) v = length v - 2 \<and> (\<exists>hs. rev v @ h = [y, y] @ hs) \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) v = ([y, x], rev v @ h)"
cases: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b" assumes "Domainp r a" obtains aa :: "'b" and b :: "'d" where "a = aa" and "r aa b"