hcju/isabelleps huggingface dataset:
subset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "(A::'b set) \<subseteq> (B::'b set)"   and "(B::'b set) \<subseteq> (C::'b set)" shows "(A::'b set) \<subseteq> (C::'b set)"
take_indices_lem: fixes n :: "nat"   and arb_list :: "'a list list"   and index_list :: "nat list" assumes "n < length (take_indices arb_list index_list)"   and "\<forall>q. List.member index_list q \<longrightarrow> q < length arb_list" shows "\<exists>k<length arb_list. take_indices arb_list index_list ! n = arb_list ! k"
simps_1: fixes x :: "'a" shows "- Fin x = Fin (- x)"
float_of_fp64_inj_iff: fixes r :: "(11, 52) IEEE.float"   and s :: "(11, 52) IEEE.float" shows "(fp64_of_float r = fp64_of_float s) = (r = s)"
size_replicate_mset: fixes n :: "nat"   and M :: "'a" shows "size (replicate_mset n M) = n"
convex_Union_fulldim_cells: fixes \<S> :: "'b set set"   and U :: "'b set" assumes "finite \<S>"   and "\<And>C. C \<in> \<S> \<Longrightarrow> closed C"   and "\<And>C. C \<in> \<S> \<Longrightarrow> convex C"   and "\<Union> \<S> = U"   and "convex U" shows "\<Union> {C \<in> \<S>. aff_dim C = aff_dim U} = U"
sep_empty_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_empty zero \<equiv> \<lambda>h. h = zero"
simps_3: shows "cospan_shape.comp cospan_shape.TT cospan_shape.AT = cospan_shape.AT"
cases: fixes a1 :: "'a"   and a2 :: "'b list"   and a3 :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" assumes "(a1, a2, a3) \<in> trcl t" obtains c :: "'a" where "a1 = c__" and "a2 = []" and "a3 = c__"   | c :: "'a" and a :: "'b" and c' :: "'a" and w :: "'b list" and c'' :: "'a" where "a1 = c" and "a2 = a # w" and "a3 = c''" and "(c, a, c') \<in> t" and "(c', w, c'') \<in> trcl t"
Cons_5: shows "\<forall>e\<in>pred c1. finite (Bexp.vars e)"
H_increasing_less: fixes k :: "nat"   and l :: "nat" assumes "k < l" shows "(case H l of (n, N, j') \<Rightarrow> insert n N) \<subseteq> (case H k of (n, N, j') \<Rightarrow> N)"
default_unique: fixes otherbot :: "'b" assumes "otherbot \<noteq> \<bottom>" shows "\<exists>G nP i F. wf_graph G \<and> \<not> sinvar G nP \<and> F \<in> set_offending_flows G nP \<and> sinvar (delete_edges G F) nP \<and> (\<not> True \<longrightarrow> i \<in> fst ` F \<and> sinvar G (nP(i := otherbot))) \<and> (True \<longrightarrow> i \<in> snd ` F \<and> sinvar G (nP(i := otherbot)))"
alg_dvd_sum: fixes A :: "'b set"   and y :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> y alg_dvd f x" shows "y alg_dvd sum f A"
globs_gupd: fixes r :: "oref"   and obj :: "obj"   and s :: "st" shows "globs (gupd(r\<mapsto>obj) s) = globs s(r \<mapsto> obj)"
filterlim_at_top_mono: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "filterlim f at_top F"   and "\<forall>\<^sub>F x in F. f x \<le> g x" shows "filterlim g at_top F"
sorted_WP2: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "FWNormalisationCore.sorted p l"   and "all_in_list p l"   and "distinct p"   and "allNetsDistinct p"   and "singleCombinators p" shows "wellformed_policy2 p"
funcset: shows "F \<subseteq> arcs G \<rightarrow> arcs G"
tendsto_compose_filtermap: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and T :: "'b"   and F :: "'a filter" shows "((g \<circ> f) \<longlongrightarrow> T) F = (g \<longlongrightarrow> T) (filtermap f F)"
that: fixes ab :: "'a set" assumes "path ab a b"   and "segment a b \<subseteq> ab" shows "thesis"
\<Delta>_def: shows "\<Delta> = delta_price N der matur"
Array_Array_widen: fixes P :: "'c prog"   and T :: "ty"   and U :: "ty" assumes "P \<turnstile> T\<lfloor>\<rceil> \<le> U\<lfloor>\<rceil>" shows "P \<turnstile> T \<le> U"
removeLeaf_heap_is_heap: fixes t :: "'c Tree" assumes "is_heap t"   and "t \<noteq> E" shows "is_heap (snd (removeLeaf t))"
ide_char: fixes a :: "'a \<times> 'a" shows "VV.ide a = (VV.arr a \<and> VxV.ide a)"
False: shows "(p, q) \<notin> snd ` set (apply_ncrit crit data gs bs hs ps)"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "Right_a.isomorphic a a' = (\<exists>f. Right_a.in_hom f a a' \<and> Right_a.iso f)"
fv_iff_occs: fixes x :: "variable"   and \<tau> :: "typ"   and t :: "Core.term" shows "((x, \<tau>) \<in> fv t) = occs (Fv x \<tau>) t"
saturated_def: fixes N :: "'f set" shows "local.no_labels.saturated N = (no_labels.Inf_from N \<subseteq> no_labels.Red_I_\<G> N)"
map_eq_Cons_D: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and y :: "'g"   and ys :: "'g list" assumes "map f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys"
exhaust_sel: fixes hp :: "'b hp" obtains "hp = Hp (un_Hp1 hp) (hps hp)"
remove_cycles_begins_with: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set xs" shows "\<exists>zs. remove_cycles xs x ys = x # zs \<and> x \<notin> set zs"
in_restricted_store_in_unrestricted_store_1: fixes r :: "'d"   and \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and l :: "'e" assumes "r \<in> RID\<^sub>S (\<sigma>(l := None))" shows "r \<in> RID\<^sub>S \<sigma>"
swbl_def: shows "swbl = (\<lambda>A k n. fold (\<lambda>i A. swb A i n) (rev [0..<k]) A)"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "pseudo_ordered_set A less_eq" shows "reflexive A less_eq"
support_wpw0: fixes b :: "bexp"   and c :: "com"   and n :: "nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (wpw b c n Q) \<subseteq> support Q"
mono_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s" shows "Fun.mono_on f A"
LIMSEQ_Suc_n_over_n: shows "(\<lambda>n. of_nat (Suc n) / of_nat n) \<longlonglongrightarrow> (1::'a)"
chain: shows "\<forall>i. (seq i, seq (Suc i)) \<in> S O R"
primes_dvd_imp_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "dvd.dvd times p q" shows "p = q"
abs_o_rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e" assumes "Quotient3 R Abs Rep" shows "Abs \<circ> Rep = id"
Rep_cfun_strict1: fixes x :: "'e" shows "\<bottom>\<cdot>x = \<bottom>"
0: shows "vec_smult \<alpha> (vec_plus (a # as') bs) = \<alpha> * (a + b) # vec_smult \<alpha> (vec_plus as' bs')"
mmulti_join'_correct: fixes A_pos :: "nat set list"   and n :: "nat"   and A_neg :: "nat set list"   and L :: "'b option list set list"   and z :: "'b option list" assumes "A_pos \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L" shows "(z \<in> mmulti_join' A_pos A_neg L) = (wf_tuple n (\<Union>A\<in>set A_pos. A) z \<and> list_all2 (\<lambda>A. (\<in>) (restrict A z)) A_pos (take (length A_pos) L) \<and> list_all2 (\<lambda>A. (\<notin>) (restrict A z)) A_neg (drop (length A_pos) L))"
in_fdom_alt: fixes k :: "'a"   and m :: "('a, 'b) fmap" shows "(k \<in># dom_m m) = (\<not> is_None (fmlookup m k))"
assms: shows "Negligible.negligible (\<lambda>n. discrete_log.advantage n (Schnorr_\<Sigma>_commit.adversary n (\<A> n)))"
snoc: fixes ys :: "'a list"   and y :: "'a" assumes "xs = ys @ [y]" shows "thesis"
reflects_left_adjoint: fixes f :: "'b" assumes "D.ide f"   and "C.is_left_adjoint (F.right_map f)" shows "D.is_left_adjoint f"
simpfm_bound0: fixes p :: "fm" assumes "bound0 p" shows "bound0 (simpfm p)"
size_extend: fixes k :: "'k"   and b :: "bool"   and x :: "'x" shows "size (extend k b x) = SUC k (size x)"
mon_w_unconc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and wa :: "('c, 'f) label list"   and wb :: "('c, 'f) label list" shows "mon_w fg (wa @ wb) = mon_w fg wa \<union> mon_w fg wb"
sporadic_on_e2: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>"
simps_22: fixes vc :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(SKIP {vc} \<le> v ::= va {vb}) = False"
weakStatImpOutputPres: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', M\<langle>N\<rangle>.P, M\<langle>N\<rangle>.Q) \<in> Rel" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<lessapprox><Rel> M\<langle>N\<rangle>.Q"
analz_insert_Hash: fixes X :: "msg"   and H :: "msg set" shows "analz (insert (Hash X) H) = insert (Hash X) (analz H)"
set_bits_aux_def: fixes w :: "'b word"   and n :: "nat" shows "local.set_bits_aux w n = push_bit n w OR take_bit n (set_bits f)"
bounded_linearI': fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" shows "bounded_linear f"
assms_1: shows "\<exists>!x. x \<in> U \<and> P x"
wfP_acyclicP: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfP r" shows "acyclicP r"
LI_completeness: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and S\<^sub>1 :: "('a, 'b) strand_step list"   and \<theta>\<^sub>1 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and public :: "'a \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "intruder_model.constr_sem_c arity public \<I> S\<^sub>1 \<theta>\<^sub>1" shows "\<exists>S\<^sub>2 \<theta>\<^sub>2. ((S\<^sub>1, \<theta>\<^sub>1), S\<^sub>2, \<theta>\<^sub>2) \<in> (intruder_model.LI_rel arity public)\<^sup>* \<and> simple S\<^sub>2 \<and> intruder_model.constr_sem_c arity public \<I> S\<^sub>2 \<theta>\<^sub>2"
eqvt: fixes p :: "perm"   and x :: "'a" shows "p \<bullet> f x = f (p \<bullet> x)"
induct: fixes P :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"   and a0 :: "('a, 'b) rbt"   and a1 :: "'a"   and a2 :: "'b"   and a3 :: "('a, 'b) rbt" assumes "\<And>a k x b s y c. P a k x (Branch R b s y c)"   and "\<And>a k x b s y. P (Branch B a k x b) s y Empty"   and "\<And>a k x b s y va vb vc vd. P (Branch B a k x b) s y (Branch B va vb vc vd)"   and "\<And>a k x b s y c t z. P (Branch R a k x (Branch B b s y c)) t z Empty"   and "\<And>a k x b s y c t z va vb vc vd. P (Branch R a k x (Branch B b s y c)) t z (Branch B va vb vc vd)"   and "\<And>k x. P Empty k x Empty"   and "\<And>va vb vc k x. P (Branch R va vb vc Empty) k x Empty"   and "\<And>va vb vc ve vf vg vh k x. P (Branch R va vb vc (Branch R ve vf vg vh)) k x Empty"   and "\<And>k x va vb vc vd. P Empty k x (Branch B va vb vc vd)"   and "\<And>ve vf vg k x va vb vc vd. P (Branch R ve vf vg Empty) k x (Branch B va vb vc vd)"   and "\<And>ve vf vg vi vj vk vl k x va vb vc vd. P (Branch R ve vf vg (Branch R vi vj vk vl)) k x (Branch B va vb vc vd)" shows "P a0 a1 a2 a3"
left_idem: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (a \<sqinter> (b::'b::semilattice_inf)) = a \<sqinter> b"
that: fixes c :: "real" assumes "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> 2 * (c * f_approx x) \<le> f x"   and "c \<le> c'"   and "0 < c" shows "thesis"
list_all2_iff: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
distinct_89: fixes x61 :: "'a set"   and x62 :: "('a, 'b, 'c) com"   and x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com" shows "While x61 x62 \<noteq> Catch x111 x112"