hcju/isabelleps huggingface dataset:
abs_steps'_def: fixes c\<^sub>A :: "('a, 'b, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt" shows "abs_steps' c\<^sub>A c\<^sub>C \<equiv> if (\<exists>x m c\<^sub>A_tail. c\<^sub>A = (Skip@[x +=\<^sub>m m]) ;; c\<^sub>A_tail) \<and> (\<exists>x m c\<^sub>C_tail. c\<^sub>C = (Skip@[Eg2_var\<^sub>C_of_Eg1 x +=\<^sub>m m]) ;; c\<^sub>C_tail) then Suc 0 else if \<exists>c\<^sub>A_tail c\<^sub>C_tail. c\<^sub>A = Stop ;; c\<^sub>A_tail \<and> c\<^sub>C = Stop ;; c\<^sub>C_tail then Suc 0 else if \<exists>x then\<^sub>A else\<^sub>A tail\<^sub>A then\<^sub>C else\<^sub>C tail\<^sub>C. c\<^sub>A = Stmt.If (bexp.Eq x 0) then\<^sub>A else\<^sub>A ;; tail\<^sub>A \<and> c\<^sub>C = (reg\<^sub>C \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 x)) ;; Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) then\<^sub>C else\<^sub>C ;; tail\<^sub>C \<or> c\<^sub>C = Stop ;; Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) then\<^sub>C else\<^sub>C ;; tail\<^sub>C then 0 else Suc 0"
RedCastNull: fixes C :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>Cast C null,s\<rangle> \<rightarrow> \<langle>null,s\<rangle>"
greaterThanLessThan_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "ord.greaterThanLessThan less l u \<equiv> ord.greaterThan less l \<inter> ord.lessThan less u"
strict_trans: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y \<and> (y, x) \<notin> relation"   and "y \<succeq>[relation] z \<and> (z, y) \<notin> relation" shows "x \<succeq>[relation] z \<and> (z, x) \<notin> relation"
dg_Dom_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<CC>\<lparr>Dom\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
LU_INCR: shows "l u \<le> Min {l v |v. (u, v) \<in> cf.E}"
from_nat_to_nat: fixes x :: "'c" shows "from_nat (to_nat x) = x"
appliedUnitPropagate_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedUnitPropagate stateA stateB \<equiv> \<exists>uc ul. uc el getF stateA \<and> isUnitClause uc ul (elements (getM stateA)) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA @ [(ul, False)]"
Ana_vars_2: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "Ana t = (K, M)" shows "fv\<^sub>s\<^sub>e\<^sub>t (set M) \<subseteq> fv t"
direct_decomp_unique: fixes A :: "'b set"   and ss :: "'b set list"   and qs :: "'b list"   and qs' :: "'b list" assumes "direct_decomp A ss"   and "qs \<in> listset ss"   and "qs' \<in> listset ss"   and "sum_list qs = sum_list qs'" shows "qs = qs'"
Sigma: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Sum>x\<in>A. sum (g x) (B x)) = (\<Sum>(x, y)\<in>Sigma A B. g x y)"
bexec: shows "steps0 (Suc 0, la, ra) B stpb = (sb, lb, rb)"
meet_connection2: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (y \<sqinter> x = x)"
succ_of_pred_def: fixes s :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "succ_of_pred s \<equiv> \<lambda>a. {b. s a b}"
prod_prime_factors: fixes x :: "'b" assumes "x \<noteq> (0::'b)" shows "(\<Prod>p\<in>prime_factors x. p ^ multiplicity p x) = normalize x"
sup1_eq_sup2: shows "(\<squnion>1) = (\<squnion>2)"
Uint32_signed_def: fixes i :: "integer" shows "Uint32_signed i = (if i < - 2147483648 \<or> 2147483648 \<le> i then undefined Uint32 i else Uint32 i)"
left_closed: fixes a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "(a, b) \<in> {(x, y). (x, y) \<in> (R \<times> S) \<times> R \<times> S \<and> x \<sim> y}" shows "a \<in> R \<times> S"
addSubsetSetI_1: fixes s :: "'b set"   and ts :: "'b set set" shows "s - \<Union> ts \<in> addSubsetSet s ts"
simps_6: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> 'a" shows "rec_list f1 f2 [] = f1"
Zfun_le: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'c" assumes "Zfun g F"   and "\<forall>x. norm (f x) \<le> norm (g x)" shows "Zfun f F"
not_finite_nat_set_iff: fixes N :: "nat set" shows "infinite N = (\<forall>n. \<exists>i\<in>N. n \<le> i)"
ab_semigroup_add_ow_Ball_def: fixes S :: "'c set"   and pls :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "ab_semigroup_add_ow S pls = (semigroup_add_ow S pls \<and> (\<forall>a\<in>S. \<forall>b\<in>S. pls a b = pls b a))"
mult_isol: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set"   and z :: "('a \<times> ('b \<times> 'a) list) set" assumes "x \<subseteq> y" shows "t_prod z x \<subseteq> t_prod z y"
assms_1: shows "typing_cond_prot_stateful \<P>"
good_context_3: fixes s :: "'b sparc_state"   and s'' :: "'b sparc_state"   and v1 :: "word32"   and v2 :: "sparc_operation \<times> inst_operand list" assumes "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = False" shows "False"
extend_def: fixes h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program" shows "extend h F = mk_program (extend_set h (Init F), extend_act h ` Acts F, project_act h -` AllowedActs F)"
Sup_exp_def: fixes S :: "('a \<Rightarrow> real) set" shows "Sup_exp S = (if S = {} then \<lambda>s. 0 else (\<lambda>s. Sup {f s |f. f \<in> S}))"
aux5a: fixes b :: "'b"   and a :: "'b"   and p :: "('b, 'c) Combinators list" assumes "DenyAllFromTo b a \<in> set p" shows "a \<in> set (net_list p)"
add_mset_eq_singleton_iff: fixes x :: "'a"   and M :: "'a multiset"   and y :: "'a" shows "(add_mset x M = {#y#}) = (M = {#} \<and> x = y)"
rep_eq: fixes x :: "'b ratfps" shows "ratfps_nth x = (\<lambda>n. let (a, b) = quot_of_fract (Rep_ratfps x) in \<Sum>i = 0..n. coeff a i * ratfps_nth_aux b (n - i))"
smult_l_null: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<zero> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
prefixElementsAreTrailElements: fixes p :: "('b \<times> bool) list"   and M :: "('b \<times> bool) list" assumes "isPrefix p M" shows "set (elements p) \<subseteq> set (elements M)"
1: shows "\<exists>A. A \<in> \<A> \<and> C \<in> A \<and> D \<in> A"
positive_part_def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "pp f x = (if (0::'b) \<le> f x then f x else (0::'b))"
Vagree_univ: fixes a :: "(real, 'd) vec"   and b :: "(real, 'd) vec"   and c :: "(real, 'd) vec"   and d :: "(real, 'd) vec" assumes "Vagree (a, b) (c, d) UNIV" shows "a = c \<and> b = d"
path_components_of_empty_space: fixes X :: "'a topology" assumes "topspace X = {}" shows "path_components_of X = {}"
listassoc_sym: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list" assumes "Group.monoid G"   and "as [\<sim>]\<^bsub>G\<^esub> bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "bs [\<sim>]\<^bsub>G\<^esub> as"
in_idem: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "lcm a (Lcm\<^sub>f\<^sub>i\<^sub>n A) = Lcm\<^sub>f\<^sub>i\<^sub>n A"
alw_ev_stl: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "alw (ev P) (stl \<omega>) = alw (ev P) \<omega>"
mask_add_aligned: fixes p :: "'b word"   and n :: "nat"   and q :: "'b word" assumes "is_aligned p n" shows "p + q AND mask n = q AND mask n"
assms: shows "P \<equiv> compP\<^sub>2 P\<^sub>1"
identity_map: fixes U :: "'a set"   and x :: "'b" assumes "is_open U"   and "x \<in> \<FF> U" shows "\<rho> U U x = x"
IfTrue: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and ps' :: "char list \<Rightarrow> int option"   and c1 :: "com"   and x :: "nat"   and y :: "nat"   and c2 :: "com" assumes "vars b \<subseteq> dom ps"   and "dom ps' = dom ps"   and "pbval b ps"   and "(c1, ps) \<Rightarrow>\<^sub>A x \<Down> ps'"   and "y = x + 1" shows "(IF b THEN c1 ELSE c2, ps) \<Rightarrow>\<^sub>A y \<Down> ps'"
induced_play_def: fixes \<sigma> :: "'b list \<Rightarrow> 'b" shows "induced_play \<sigma> \<equiv> ltake (2 * N) \<circ> lmap last \<circ> ltl \<circ> strategy_progression \<sigma>"
coupling_def: fixes Q :: "'a \<Rightarrow> nat option"   and V :: "'a \<Rightarrow> nat option"   and D :: "'a \<Rightarrow> enat"   and S :: "'a set" shows "coupling Q V D S \<equiv> D = Dijkstra_Abstract.enat_of_option \<circ> V ++ Q \<and> S = dom V \<and> dom V \<inter> dom Q = {}"
p_def: fixes b :: "'b" shows "p b = (if b \<in> B then if b \<in> f' ` A then the_elem (f ` {x \<in> A. f' x = b}) else p' b else b)"
branch_prob_altdef: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "branch_prob \<Y> \<rho> = \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge V V' (x, \<rho>)) \<delta>)) \<partial>state_measure V \<Gamma>"
TCodomain: fixes T :: "('a, 'b) Axioms"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" shows "TCodomain (TermEquivCl' T A e B) = B"
fmap_ex_cof2: fixes f' :: "'g \<Rightarrow> 'h option"   and f :: "'g \<Rightarrow> 'h option"   and P :: "'i \<Rightarrow> 'i \<Rightarrow> 'h option \<Rightarrow> 'h option \<Rightarrow> 'g \<Rightarrow> bool" assumes "dom f' = dom f"   and "\<forall>l\<in>dom f. \<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) (f' l) l)" shows "\<exists>L. finite L \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) (f' l) l)"
eval_elim_cases_3: fixes G :: "prog"   and s :: "st"   and j :: "jump"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Jmp j)\<succ>\<rightarrow> (x, s')" obtains "x = \<diamondsuit>" and "s' = (Some (Jump j), s)"
set_cases: fixes e :: "'a"   and a :: "'a list" assumes "e \<in> set a" obtains z2 :: "'a list" where "a = e # z2__"   | z1 :: "'a" and z2 :: "'a list" where "a = z1 # z2" and "e \<in> set z2"
linepath_trivial: fixes a :: "'a"   and x :: "real" shows "linepath a a x = a"
lin_independent_append_imp_independent_Spans: fixes ms :: "'b list"   and ns :: "'b list" assumes "set ms \<subseteq> V"   and "R_lin_independent (ms @ ns)" shows "add_independentS [RSpan ms, RSpan ns]"
comp_red_monic_basis_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "dickson_grading d" shows "dgrad_p_set_le d (set (comp_red_monic_basis xs)) (set xs)"
submanifold_atlasE: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set"   and c :: "('c, 'd) chart" assumes "submanifold charts k S"   and "c \<in> c_manifold.atlas (manifold.charts_submanifold charts S) k" shows "c \<in> c_manifold.atlas charts k"
gen_qe_eval: fixes xs :: "real list"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (gen_qe var L F) (xs @ x # \<Gamma>))"
card_arcs_da: shows "card (arcs G) = 2 + card (arcs H)"
whisker_left: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.ide f"   and "C.seq \<nu> \<mu>" shows "f \<star>\<^sub>C \<nu> \<cdot>\<^sub>C \<mu> = (f \<star>\<^sub>C \<nu>) \<cdot>\<^sub>C (f \<star>\<^sub>C \<mu>)"
simps_1: fixes s :: "'s" shows "val_of s [] [] = 1"
leg1_commutes: shows "\<mu>'.cod.leg1 \<cdot> \<mu>'.chine = \<mu>'.dom.leg1"
fpslsts_iff: fixes s :: "'b llist"   and A :: "'b set" shows "(s \<in> A\<^sup>\<clubsuit>) = (s \<in> A\<^sup>\<star> \<and> s \<noteq> LNil)"
admissible_wellformed_tokens: fixes p :: "(symbol \<times> character list) list" assumes "admissible p" shows "wellformed_tokens p"
hyps: fixes f :: "bool stream \<Rightarrow> real" assumes "\<forall>x. 0 \<le> f x"   and "f \<in> borel_measurable (F n)" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = (\<Sum>w\<in>range (pseudo_proj_True n). emeasure M (pseudo_proj_True n -` {w} \<inter> space M) * ennreal (f w))"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "Q x y" shows "\<not> Q y x"
simps_14: fixes v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "'b" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> SKIP {vd}) = False"
inv_union: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "inv t1"   and "inv t2" shows "inv (Set2_Join.union join t1 t2)"
hyps_2: fixes xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and ys :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "xsa__ = xs @ ys" shows "\<exists>l2a. path l2a l2 ys \<and> path l1 l2a xs"
card_subgrp_dvd: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := P\<rparr>)" shows "card H dvd order (G\<lparr>carrier := P\<rparr>)"
\<psi>_mapsto: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "\<psi>D (b, a) \<in> HomD.set (b, a) \<rightarrow> D.hom b a"
hyps_1: fixes m :: "nat"   and n :: "nat" assumes "d = m - n"   and "lset P \<subseteq> S"   and "valid_path P"   and "path_conforms_with_strategy p P well_ordered_strategy"   and "n < m"   and "enat m < llength P" shows "(path_strategies P $ m, path_strategies P $ n) \<in> r"
restrict_endo: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and Y :: "'b set set" assumes "ChamberComplexEndomorphism X f"   and "ChamberComplex.ChamberSubcomplex X Y"   and "order.greater_eq Y (f \<turnstile> Y)" shows "ChamberComplexEndomorphism Y (restrict1 f (\<Union> Y))"
fin: fixes n :: "nat" shows "emeasure M {x \<in> space M. 0 < F n x} < \<infinity>"
beta: fixes x :: "V"   and A :: "V"   and b :: "V \<Rightarrow> V" assumes "x \<in> elts A" shows "app (VLambda A b) x = b x"
deadlocked'D2: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "multithreaded_base.deadlocked' final r s"   and "final_thread.not_final_thread final s t" obtains "t \<in> multithreaded_base.deadlocked final r s"
allocationRightUniqueRangeDomain: fixes a :: "('a set \<times> 'b) set"   and G :: "'a set"   and N :: "'b set" assumes "a \<in> possible_allocations_rel G N" shows "runiq a \<and> runiq (a\<inverse>) \<and> Domain a partitions G \<and> Range a \<subseteq> N"
char_eq_0_iff: fixes a :: "nat" shows "(dc2c \<chi> a = 0) = (a \<notin> carrier G)"
assms_1: shows "A \<TTurnstile> (I \<rightarrow>) P"
trim_enabled: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "enabled (shd (trim rs s)) s"
reads_restr_sem: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "'n"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n" shows "\<forall>v\<in>local.writes n. sem n \<sigma> v = sem n \<sigma>' v"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "\<alpha> \<mu>\<nu>\<tau> \<equiv> \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
prop_equiv_subset: fixes P :: "'b ltln set" shows "{abs_ltln\<^sub>P \<psi> |\<psi>. nested_prop_atoms \<psi> \<subseteq> P} \<subseteq> {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P}"
assms_1: shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
cmp_in_hom_2: fixes a :: "'b"   and b :: "'b" assumes "B.ide a"   and "B.ide b"   and "src\<^sub>B a = trg\<^sub>B b" shows "\<guillemotleft>\<Phi>\<^sub>F (a, b) : F a \<star>\<^sub>C F b \<Rightarrow>\<^sub>C F (a \<star>\<^sub>B b)\<guillemotright>"
negligible_times: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "negligible f"   and "negligible g" shows "negligible (\<lambda>\<eta>. f \<eta> * g \<eta>)"
multiplicity_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p x = (if finite {n. dvd.dvd times (power.power one times p n) x} then Max {n. dvd.dvd times (power.power one times p n) x} else 0)"
properties_for_sort_key: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a" assumes "class.linorder less_eq less"   and "mset ys = mset xs"   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"   and "linorder.sorted less_eq (map f ys)" shows "linorder.sort_key less_eq f xs = ys"
reachable_target: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a"   and r :: "(nat \<times> 'a) list" assumes "q \<in> greachable A w V p"   and "gpath A w V r q" shows "gtarget r q \<in> greachable A w V p"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "Left_a.arr f"   and "Left_a.cod f = b" shows "Left_a.comp b f = f"
Quotient_Tree\<^sub>\<alpha>: shows "Quotient (=\<^sub>\<alpha>) abs_Tree\<^sub>\<alpha> rep_Tree\<^sub>\<alpha> cr_Tree\<^sub>\<alpha>"
of: shows "DP.ord (\<lambda>p\<in>prime_factors n. 1) = n"
choice: fixes Q :: "'q \<Rightarrow> 'r \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. Q x y" shows "\<exists>f. \<forall>x. Q x (f x)"
0: fixes x :: "real" assumes "x \<in> {0<..<1}" shows "isCont f x"
fresh_notin_analz_knows_Spy: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs" shows "Key K \<notin> analz (knows Spy evs)"
Gamma_complex_real: fixes z :: "complex" assumes "z \<in> \<real>" shows "Gamma z \<in> \<real>"
mod_ring_locale_axioms: shows "mod_ring_locale TYPE('a) p"
inca_ubx_compiler_axioms: shows "inca_ubx_compiler Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> Finca_empty Finca_get"
abs_eq: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "local.pmf_of_set S = embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
f1: shows "fst (memory_read asi address s) \<noteq> None"
comp_red_basis_aux_base: fixes ys :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "comp_red_basis_aux [] ys = ys"
neg_neg_linear: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "b < (0::'a)" shows "a < b \<or> a = b \<or> b < a"
key_sets_crypts_snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "key_sets \<lbrace>X, Y\<rbrace> (crypts H) \<subseteq> key_sets Y (crypts H)"
assms_1: shows "\<forall>asset\<in>support_set pf. integrable M (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"
frag_extend_diff: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and a :: "'c \<Rightarrow>\<^sub>0 int"   and b :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_extend f (a - b) = frag_extend f a - frag_extend f b"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) (Y a') (Y (\<psi> (a, a') e))"
PO_a0i_agreement_init: shows "init a0i \<subseteq> a0i_agreement"
oin: fixes T :: "'a set" assumes "T \<in> \<C>" shows "openin (top_of_set U) T"
4: shows "snd ` set ((t, x) # txs) = {x} \<union> snd ` set txs"
maddux_3_21_pp: fixes x :: "'a"   and y :: "'a" shows "- - x \<squnion> - x \<sqinter> y = - - x \<squnion> y"
replay_def: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and P :: "proofterm" shows "replay thy P \<equiv> if \<forall>x\<in>set (hyps P). term_ok thy x \<and> typ_of x = Some propT then replay'' thy [] (fst ` (fv_Proof P \<union> FV (set (hyps P)))) (hyps P) P else None"
fold_cong: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and s :: "'b"   and t :: "'b"   and B :: "'a set" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "s = t"   and "A = B" shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
mono_def: fixes f :: "'s" shows "local.mono f = (arr f \<and> inj_on ((\<cdot>) f) {g. seq f g})"
pred_mono_strong: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a stream"   and Pa :: "'a \<Rightarrow> bool" assumes "pred_stream P x"   and "\<And>z. \<lbrakk>z \<in> sset x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_stream Pa x"
poly_lift_hom_X_var': fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e" assumes "cring S"   and "\<phi> \<in> ring_hom R S" shows "poly_lift_hom R S \<phi> (X_poly R) = X_poly S"
inject_3: fixes x4 :: "nat"   and y4 :: "nat" shows "(REL x4 = REL y4) = (x4 = y4)"
\<gamma>3_def: fixes x :: "real" shows "\<gamma>3 x = \<alpha> (2 * \<Lambda> + x + 2 * \<rho> * (rmax + \<beta>)) + \<Lambda> + 2 * \<rho> * \<beta>"
mono_LeadsTo_o: fixes h :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'b" assumes "mono h" shows "(\<Inter>j. {s. j \<le> g s} \<longmapsto>w {s. j \<le> f s}) \<subseteq> (\<Inter>k. {s. k \<le> h (g s)} \<longmapsto>w {s. k \<le> h (f s)})"
exp_tends_to_zero: fixes c :: "'a" assumes "(0::'a) < c" shows "\<exists>x. b ^ x \<le> c"
interval_integrable_mirror: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b" shows "interval_lebesgue_integrable lborel a b (\<lambda>x. f (- x)) = interval_lebesgue_integrable lborel (- b) (- a) f"
size_neq_size_imp_neq: fixes x :: "'a"   and y :: "'a" assumes "size x \<noteq> size y" shows "x \<noteq> y"
product_def: fixes A :: "'a set"   and B :: "'b set" shows "Product_Type.product A B = A \<times> B"
leD: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<not> y < x"
sup_aci_1: fixes x :: "'a"   and y :: "'a" shows "x + y = y + x"
add_zero_l: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and x :: "'b" assumes "class.join_semilattice_zero plus less_eq less zero" shows "plus zero x = x"
scalar_prod_right_zero: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v \<bullet> 0\<^sub>v n = (0::'b)"
Infty_neq_0_2: shows "0 \<noteq> \<infinity>"
bind_UNION: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'e set" shows "Set.bind A f = \<Union> (f ` A)"
simps_10: fixes r :: "'c rexp" shows "\<guillemotleft>Pr r\<guillemotright> = Pr \<guillemotleft>r\<guillemotright>"
poly_of_vec_vCons: fixes a :: "'b"   and v :: "'b vec" shows "poly_of_vec (vCons a v) = monom a (dim_vec v) + poly_of_vec v"
rel_aext_false: fixes a :: "'c \<Longrightarrow> 'b" shows "false \<oplus>\<^sub>r a = false"