hcju/isabelleps huggingface dataset:
axioms_3: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e" assumes "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
IH_2: assumes "inv_max_hi r"   and "r \<noteq> \<langle>\<rangle>" shows "\<exists>a\<in>Tree2.set_tree r. high a = max_hi r"
gf_U'_pos: fixes z :: "real"   and K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" assumes "0 < z"   and "z < 1"   and "MC_syntax.U K x y \<noteq> 0" shows "0 < MC_syntax.gf_U' K x y z"
simps_19: fixes v :: "'b formula"   and vb :: "'b formula" shows "safe_formula (Neg (Or (Exists v) vb)) = False"
hyp: shows "span (columns A) = span (columns (Gram_Schmidt_upt_k A k))"
leq_refl: fixes x :: "'d" shows "x \<le> x"
if_then_else_Error_1: fixes a :: "bool"   and b :: "'l"   and c :: "('k, 'l) result"   and d :: "'k" shows "((if a then Ok b else c) = Error d) = (\<not> a \<and> c = Error d)"
compat_tr_compat: fixes NS :: "('b \<times> 'b) set"   and S :: "('b \<times> 'c) set" assumes "NS O S \<subseteq> S" shows "NS\<^sup>* O S \<subseteq> S"
a_type_wf_def: fixes h :: "(_) heap" shows "ShadowRootClass.type_wf h = (DocumentClass.type_wf h \<and> (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). get shadow_root_ptr h \<noteq> None))"
assms_1: shows "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"
agree': shows "Vagree \<nu> \<nu>' (\<Union>i. \<Union> (primify ` FVT (args i)))"
of_nat_tendsto_top_ennreal: shows "of_nat \<longlonglongrightarrow> top"
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> False \<longrightarrow> i \<in> fst ` F \<and> sinvar G (nP(i := otherbot))) \<and> (False \<longrightarrow> i \<in> snd ` F \<and> sinvar G (nP(i := otherbot)))"
distinct3_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points" shows "distinct3 A B C \<equiv> A \<noteq> B \<and> A \<noteq> C \<and> B \<noteq> C"
set_integrable_subset: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "set_integrable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_integrable M B f"
src_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.hseq \<mu> \<nu>" shows "src\<^sub>B (\<mu> \<star>\<^sub>B \<nu>) = src\<^sub>B \<nu>"
lex_append_rightI: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set"   and vs :: "'a list"   and us :: "'a list" assumes "(xs, ys) \<in> lex r"   and "length vs = length us" shows "(xs @ us, ys @ vs) \<in> lex r"
TokensAt_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and I :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.TokensAt \<TT> Lex Doc k I = {uu_. \<exists>t s x l. uu_ = (t, s) \<and> x \<in> bin I k \<and> next_symbol x = Some t \<and> CFG.is_terminal \<TT> t \<and> l \<in> Lex t Doc k \<and> s = take l (drop k Doc)}"
write_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
ordertype_eq_ordertype_iso: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('b \<times> 'b) set"   and B :: "'b set" assumes "wf r"   and "total_on A r"   and "trans r"   and "small A"   and "Field r = A"   and "wf s"   and "total_on B s"   and "trans s"   and "small B"   and "Field s = B" shows "(ordertype A r = ordertype B s) = (\<exists>f. iso r s f)"
card_le_Suc0_iff_eq: fixes A :: "'a set" assumes "finite A" shows "(card A \<le> Suc 0) = (\<forall>a1\<in>A. \<forall>a2\<in>A. a1 = a2)"
test_eq: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c"   and q :: "'c" assumes "drat test not" shows "(p = q) = (test p = test q)"
is_aligned_ucastI: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n" shows "is_aligned (UCAST('b \<rightarrow> 'c) w) n"
fps_binomial_def: fixes a :: "'a" shows "fps_binomial a = Abs_fps ((gchoose) a)"
assms: shows "(P \<U> Q \<U> R) \<sigma>"
exec_ge: shows "\<forall>i\<le>Suc x. 0 < rec_exec f (xs @ [i])"
direct_decompI_alt: fixes ss :: "'b set list"   and A :: "'b set" assumes "\<And>qs. qs \<in> listset ss \<Longrightarrow> sum_list qs \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>!qs. qs \<in> listset ss \<and> a = sum_list qs" shows "direct_decomp A ss"
SELECT_refine_iff: fixes P :: "'c \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and P' :: "'d \<Rightarrow> bool" shows "(SELECT P \<le> \<Down> (\<langle>R\<rangle>option_rel) (SELECT P')) = ((Ex P' \<longrightarrow> Ex P) \<and> (\<forall>x. P x \<longrightarrow> (\<exists>x'. (x, x') \<in> R \<and> P' x')))"
real_of_tighten_bounds_many: fixes i :: "nat"   and x :: "real_alg_3" shows "real_of_3 ((tighten_bounds_3 ^^ i) x) = real_of_3 x"
not_element_restrict: fixes var :: "'a"   and b :: "'a ifex"   and val :: "bool" shows "var \<notin> ifex_var_set (restrict b var val)"
alive_init: fixes T :: "Javatype"   and s :: "Store" shows "alive (init T) s"
arcsin_minus: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "arcsin (- x) = - arcsin x"
F_3: fixes i :: "nat"   and x :: "'a" shows "F i x < top"
finite_lattice_complete_top_greatest: fixes x :: "'a" shows "x \<le> top"
prems: shows "range_vars \<delta> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (s # S) = {}"
l2_obs_secrecy: shows "oreach l2 \<subseteq> l2_secrecy"
l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_attribute get_attribute_locs \<equiv> type_wf = DocumentClass.type_wf \<and> get_attribute = l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute \<and> get_attribute_locs = l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute_locs"
Sup_aform'_def: fixes p :: "nat"   and X :: "'b \<times> 'b pdevs" shows "Sup_aform' p X = eucl_truncate_up p (fst X + Radius' p X)"
sfilter_smap': fixes p :: "'r \<rightarrow> tr"   and f :: "'s \<rightarrow> 'r"   and xs :: "[:'s:]" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p\<cdot>(smap\<cdot>f\<cdot>xs) = smap\<cdot>f\<cdot>(sfilter\<cdot>(p oo f)\<cdot>xs)"
arr_runit: fixes a :: "'a" assumes "ide a" shows "arr \<r>[a]"
index_uminus_mat_2: fixes A :: "'a Matrix.mat" shows "dim_row (- A) = dim_row A"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<psi> f = C'.null"
simps_2: fixes p :: "real \<times> real \<times> real" shows "getPoly (LeqUni p) = p"
IH_1: fixes t :: "'a bro" assumes "t \<in> B h" shows "ins x t \<in> Bp h"
add_def: fixes m :: "'a"   and n :: "'a" shows "add m n = rec n (\<lambda>_. succ) m"
d1: shows "H \<turnstile> All2 z r (All2 z' r (SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x EQ Var x' IMP Var y EQ Var y')))))))"
run_one_step_basic_nop_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Nop) = (s', vs', res)" shows "\<exists>r. res = RSNormal r"
C: shows "set xs' \<subseteq> range p \<union> range q"
iteration_unfoldl': fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z \<cdot> y + z \<cdot> x \<cdot> x\<^sup>\<infinity> \<cdot> y = z \<cdot> x\<^sup>\<infinity> \<cdot> y"
oreachable_local': fixes \<sigma> :: "'a"   and p :: "'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'c"   and \<sigma>' :: "'a"   and p' :: "'b" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "(\<sigma>', p') \<in> oreachable A S U"
quasinorm_sum_subset_1: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 \<subseteq>\<^sub>N N1 +\<^sub>N N2"
assms_2: shows "i \<noteq> 0 \<or> j \<noteq> 0"
get_scdom_component_ptrs_same_scope_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr' \<rightarrow>\<^sub>r sc"
WRP_fresh_iff: fixes a :: "atom"   and x :: "tm"   and y :: "tm" shows "a \<sharp> WRP x y = (a \<sharp> x \<and> a \<sharp> y)"
sba1_bot_unique: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - x) = - (y \<squnion> - y)"
and_protocol_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "and_protocol A B = (let (a1, b1) = A; (a2, b2) = B in coin_spmf \<bind> (\<lambda>\<sigma>. let s0 = \<sigma> \<oplus> (a1 \<oplus> False \<and> b1 \<oplus> False); s1 = \<sigma> \<oplus> (a1 \<oplus> False \<and> b1 \<oplus> True); s2 = \<sigma> \<oplus> (a1 \<oplus> True \<and> b1 \<oplus> False); s3 = \<sigma> \<oplus> (a1 \<oplus> True \<and> b1 \<oplus> True) in protocol_14_OT (s0, s1, s2, s3) (a2, b2) \<bind> (\<lambda>(uu_, s). return_spmf (\<sigma>, s))))"
distinct_1: fixes x1 :: "'e"   and x21 :: "'d"   and x22 :: "('d, 'e) Term.term list" shows "Var x1 \<noteq> Fun x21 x22"
get_owner_document_def: fixes ptr :: "(_) object_ptr" shows "get_owner_document ptr = invoke get_owner_document_tups ptr ()"
fmrel_on_fset_fmmerge1: fixes zm :: "'c \<rightharpoonup>\<^sub>f 'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xm :: "'c \<rightharpoonup>\<^sub>f 'a"   and ym :: "'c \<rightharpoonup>\<^sub>f 'a" assumes "\<And>x y z. \<lbrakk>z \<in> fmran' zm; f x z; f y z\<rbrakk> \<Longrightarrow> f (g x y) z"   and "subtuple f xm zm"   and "subtuple f ym zm" shows "subtuple f (fmmerge g xm ym) zm"
vec_of_list_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "vec_of_list (map f xs) = map_vec f (vec_of_list xs)"
Pabs_2: shows "pre_strong_term_class.wellformed (Pabs cs)"
inline_aux_def: fixes callee :: "'p \<Rightarrow> 'n \<Rightarrow> ('o \<times> 'p, 'q, 'r) gpv" shows "inline_aux callee \<equiv> corec_gpv (\<lambda>state. case state of Inl a \<Rightarrow> map_spmf (\<lambda>uu. case uu of Inl a \<Rightarrow> map_generat id id ((\<circ>) Inl) (Pure (fst a, snd a)) | Inr a \<Rightarrow> IO (id (fst a)) (\<lambda>input. Inr (Inr (snd (snd a), fst (snd a) input)))) (inline1 callee (fst a) (snd a)) | Inr a \<Rightarrow> map_spmf (\<lambda>uu. case uu of Inl (Inl aa) \<Rightarrow> map_generat id id ((\<circ>) Inl) (Pure (fst aa, snd aa)) | Inl (Inr aa) \<Rightarrow> IO (id (fst aa)) (\<lambda>input. Inr (Inr (snd (snd aa), fst (snd aa) input))) | Inr aa \<Rightarrow> IO (id (fst aa)) (\<lambda>input. Inr (Inr (fst a, snd aa input)))) (the_gpv (snd a) \<bind> (\<lambda>generat. case generat of Pure (x, s') \<Rightarrow> map_spmf Inl (inline1 callee (fst a x) s') | IO out c \<Rightarrow> return_spmf (Inr (out, c)))))"
Ghost\<^sub>s\<^sub>b: shows "r = Ghost\<^sub>s\<^sub>b A L R W"
simps_4: fixes a :: "bool"   and ds :: "bool list" shows "delete (a # ds) Leaf = Leaf"
ltl_simps_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (x21 ## x22) = x22"
ars_r_def: fixes x :: "('b \<times> 'b) set" shows "rel_antidomain_kleene_algebra.ads_d x = rel_ad (rel_ad x)"
subset_AboveS_UnderS: fixes B :: "'c set"   and r :: "('c \<times> 'c) set" assumes "B \<subseteq> Field r" shows "B \<subseteq> AboveS r (UnderS r B)"
SD_efficient_def: fixes p :: "'b pmf" shows "SD_efficient R' p = (\<not> (\<exists>q\<in>lotteries_on alts'. p \<prec>[Pareto (SD \<circ> R')] q))"
AE_measure_pmf_iff: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b pmf" shows "almost_everywhere (measure_pmf M) P = (\<forall>y\<in>set_pmf M. P y)"
assoc_coherence: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "F \<a>\<^sub>C[a, b, c] \<cdot>\<^sub>D \<phi> (a \<otimes>\<^sub>C b, c) \<cdot>\<^sub>D (\<phi> (a, b) \<otimes>\<^sub>D F c) = \<phi> (a, b \<otimes>\<^sub>C c) \<cdot>\<^sub>D (F a \<otimes>\<^sub>D \<phi> (b, c)) \<cdot>\<^sub>D \<a>\<^sub>D[F a, F b, F c]"
ProofStepInduct_y_eq_new_last_axioms: shows "ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
runit'_ide_eq: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<r>\<^sup>-\<^sup>1[f] = \<lparr>Chn = \<langle>Chn f \<lbrakk>Leg0 (Dom f), C.cod (Leg0 (Dom f))\<rbrakk> Leg0 (Dom f)\<rangle>, Dom = Cod f, Cod = \<lparr>Leg0 = \<p>\<^sub>0[Leg0 (Dom f), C.cod (Leg0 (Dom f))], Leg1 = Leg1 (Dom f) \<cdot> \<p>\<^sub>1[Leg0 (Dom f), C.cod (Leg0 (Dom f))]\<rparr>\<rparr>"
fmdomE: fixes x :: "'a"   and m :: "('a, 'b) fmap" assumes "x |\<in>| fmdom m" obtains y :: "'b" where "fmlookup m x = Some y"
lebesgue_real_scale: fixes c :: "real" assumes "c \<noteq> 0" shows "lebesgue = density (distr lebesgue lebesgue ((*) c)) (\<lambda>x. ennreal \<bar>c\<bar>)"
main: fixes z :: "char list +\<^sub>\<bottom> 'd" shows "monotone sum_bot.le_fun sum_bot_ord (\<lambda>g. case map_of (map (\<lambda>(k, p). (k, \<lambda>y. p y g)) ps) (Xml.tag x) of None \<Rightarrow> z | Some p \<Rightarrow> p x)"
ord_of_nat_eq_initial: fixes n :: "nat" shows "n\<^sub>\<nat> = set (ord_of_nat ` {..<n})"
fags_cond_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "fags_cond f z i a = (commute_bpp f (aug_pm_set z i {a}) \<and> assoc_bpp (aug_pm_set z i {a}) f \<and> ipp_cond1 {a} i \<and> ipp_cond2 z {a} i f \<and> ipp_cond3 z i \<and> inv_ipp z i f {a} \<and> commute_bpp f (aug_pm_set z i {a}) \<and> zeroA z i f {a} z \<and> free_gen_condition f i a z)"
simps_979: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x251 :: "t"   and x252 :: "unop_f" shows "(case Unop_f x251 x252 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f25 x251 x252"
result_resumption_lub: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<forall>r\<in>R. Resumption.resumption.is_Done r" shows "Resumption.resumption.result (resumption_lub R) = flat_lub None (Resumption.resumption.result ` R)"
expectation_gpv_le_inline: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> plossless_gpv \<I>' (callee s x)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "expectation_gpv 0 \<I> f gpv \<le> expectation_gpv 0 \<I>' (\<lambda>(x, s). f x) (inline callee gpv s)"
stutter_selection_def: fixes s :: "nat set"   and w :: "'b llist" shows "stutter_selection s w \<equiv> 0 \<in> s \<and> (\<forall>k i. enat i < llength w \<longrightarrow> enat (Suc k) < esize s \<longrightarrow> nth_least s k < i \<longrightarrow> i < nth_least s (Suc k) \<longrightarrow> w ?! i = w ?! nth_least s k) \<and> (\<forall>i. enat i < llength w \<longrightarrow> finite s \<longrightarrow> Max s < i \<longrightarrow> w ?! i = w ?! Max s)"
assms_6: shows "\<not> (c \<or> \<not> p \<and> (p \<or> q \<and> \<not> q))"
ltl_G_stabilize_property: fixes \<G> :: "'b ltl set"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltl" assumes "finite \<G>"   and "Only_G \<G>"   and "\<And>\<chi> j. \<chi> \<in> \<G> \<Longrightarrow> suffix i w \<Turnstile> \<chi> = suffix (i + j) w \<Turnstile> \<chi>"   and "G \<psi> \<in> \<G> \<inter> {\<chi>. w \<Turnstile> F \<chi>}" shows "suffix i w \<Turnstile> G \<psi>"
nextl_Reverse_nfa: fixes Q :: "hf set"   and u :: "'a list" shows "NR.nextl Q u = {q' \<in> dfa.states M. nextl q' (rev u) \<in> Q}"
comps: fixes C :: "'a set" assumes "C \<in> components (U - S)"   and "C \<inter> K \<noteq> {}" shows "C \<inter> L \<noteq> {}"
simps_5: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "FOL_Fitting.size_form (Or p q) = FOL_Fitting.size_form p + FOL_Fitting.size_form q + 1"
Compl1_set_less_eq_aux_conv_Compl_set_less_eq_aux: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_eq_aux less_eq (- set xs) (set ys) = proper_intrvl.Compl_set_less_eq_aux less proper_interval None xs ys"
empty_def: shows "AVL_Set_Code.empty = \<langle>\<rangle>"
affine_binop_def: fixes X :: "real \<times> real pdevs"   and Y :: "real \<times> real pdevs"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and k :: "nat" shows "affine_binop X Y a b c d k = (a * fst X + b * fst Y + c, pdev_upd (add_pdevs (scaleR_pdevs a (snd X)) (scaleR_pdevs b (snd Y))) k d)"
discovered_closed_def: fixes s :: "('v, 'a) state_scheme" shows "discovered_closed s \<equiv> E `` dom (finished s) \<subseteq> dom (discovered s) \<and> (E - pending s) `` set (stack s) \<subseteq> dom (discovered s)"
S_transition_cases: fixes P :: "'state"   and \<alpha>\<^sub>S :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<rightarrow>\<^sub>S \<langle>\<alpha>\<^sub>S,P'\<rangle>" obtains \<alpha> :: "'act" where "\<alpha>\<^sub>S = S_action.Act \<alpha>" and "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"   | \<phi> :: "'pred" where "\<alpha>\<^sub>S = S_action.Pred \<phi>" and "P' = P" and "P \<turnstile> \<phi>"
quot_zero_eq: shows "UNIV // \<approx>{} = {UNIV}"
insert_insert_message: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message" shows "insert_messages (M \<union> {Insert m}) = insert_messages M \<union> {m}"
to_poly_mult_simp_2: fixes b :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "b \<in> carrier R"   and "f \<in> carrier (UP R)" shows "f \<otimes>\<^bsub>UP R\<^esub> to_polynomial R b = b \<odot>\<^bsub>UP R\<^esub> f"
apfst_conv: fixes f :: "'f \<Rightarrow> 'd"   and x :: "'f"   and y :: "'e" shows "apfst f (x, y) = (f x, y)"
monoid_add_isom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "monoid_add_isom hom \<equiv> inj_monoid_add_hom hom \<and> bijective hom"
useful_tautologies_8: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> (\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>(\<phi> \<^bold>\<rightarrow> \<psi>)) in v]"