hcju/isabelleps huggingface dataset:
converged_cd_same_suc: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<forall>k k'. IFC_def.cs prob \<pi> k = IFC_def.cs prob \<pi>' k' \<and> IFC_def.is_cdi prob l \<pi> k \<longrightarrow> \<pi> (Suc k) = \<pi>' (Suc k')"   and "l < m"   and "IFC_def.cs prob \<pi> m = IFC_def.cs prob \<pi>' m'" shows "\<exists>l'. IFC_def.cs prob \<pi> l = IFC_def.cs prob \<pi>' l'"
three_neq_aux_1: shows "\<^bold>1 \<noteq> \<^bold>2"
distinct_length_filter: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "distinct xs" shows "length (filter P xs) = card ({x. P x} \<inter> set xs)"
rotate1_hd_tl: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "rotate1 xs = tl xs @ [hd xs]"
is_dghm_axioms: shows "\<HH>' : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
octo_power_commutes: fixes x :: "octo"   and y :: "nat" shows "x ^ y * x = x * x ^ y"
dvd_abs_mult_left_int: fixes a :: "int"   and y :: "int"   and x :: "int" shows "(\<bar>a\<bar> * y dvd x) = (a * y dvd x)"
x: shows "x \<in> vec.span {row j A |j. j \<noteq> i}"
refl_lifted: fixes A :: "'a set" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}" shows "(SOME y. y \<in> A, SOME y. y \<in> A) \<in> r"
assms_5: shows "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"
vunion_vempty_right: fixes a :: "V" shows "a \<union>\<^sub>\<circ> 0 = a"
simps: fixes t :: "'b trm" shows "proper_subterms_of t = {s. \<exists>p. p \<noteq> [] \<and> subterm t p s}"
run_one_step_basic_relop_f_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x301 :: "t"   and x302 :: "relop_f"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Relop_f x301 x302) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
vpa_induct: fixes xs :: "'edge list"   and ys :: "'edge list"   and P :: "'edge list \<Rightarrow> 'edge list \<Rightarrow> bool" assumes "valid_path_aux xs ys"   and "\<And>cs. P cs []"   and "\<And>cs a as. \<lbrakk>intra_kind (kind a); valid_path_aux cs as; P cs as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q r p fs. \<lbrakk>kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; valid_path_aux (a # cs) as; P (a # cs) as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q p f. \<lbrakk>kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; cs = []; valid_path_aux [] as; P [] as\<rbrakk> \<Longrightarrow> P cs (a # as)"   and "\<And>cs a as Q p f c' cs'. \<lbrakk>kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; cs = c' # cs'; valid_path_aux cs' as; a \<in> get_return_edges c'; P cs' as\<rbrakk> \<Longrightarrow> P cs (a # as)" shows "P xs ys"
assms: shows "\<forall>x\<in>carrier. convex (at_least_as_good x carrier relation)"
is_non_volatile_Read\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_non_volatile_Read\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = (\<not> volatile)"
dependent_single: fixes x :: "'b" shows "local.dependent {x} = (x = (0::'b))"
CatchSkip: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" shows "\<Gamma>\<turnstile> (Catch Skip c\<^sub>2, s) \<rightarrow> (Skip, s)"
cong: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" shows "comm_monoid_mult.Prod_any times one \<equiv> comm_monoid_mult.Prod_any times one"
D_dest'': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and x :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "x = linordered_ab_monoid_add.D plus less_eq m i j k" shows "x \<in> {linordered_ab_monoid_add.len plus m i j xs |xs. set xs \<subseteq> {0..k}}"
cnj_cnj: fixes q :: "quat" shows "Quaternions.cnj (Quaternions.cnj q) = q"
inf_ref_infiter: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> c\<^sup>\<infinity>"
set_intros_2: fixes y :: "'d"   and x2 :: "'d stream"   and x1 :: "'d" assumes "y \<in> sset x2" shows "y \<in> sset (x1 ## x2)"
Sup_le_iff: fixes A :: "'a set"   and b :: "'a" shows "(Sup A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
ok_finite: fixes e :: "'b environment" assumes "ok e" shows "finite (env_dom e)"
concat_conv_foldr: fixes xss :: "'a list list" shows "concat xss = foldr (@) xss []"
AE_distr_iff: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and P :: "'e \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "almost_everywhere (distr M N f) P = (AE x in M. P (f x))"
s_2: fixes x :: "'a" assumes "x \<in> s" shows "{x} \<in> sets M"
set_iterator_product_it_b_cong: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_a :: "'e set"   and it_b :: "'e \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it_b' :: "'e \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" assumes "set_iterator it_a S_a"   and "\<And>a. a \<in> S_a \<Longrightarrow> it_b a = it_b' a" shows "set_iterator_product it_a it_b = set_iterator_product it_a it_b'"
subset_closure: fixes s :: "'b set"   and T :: "'b set set" shows "s \<subseteq> closure\<^bsub>T\<^esub> s"
tdghm_NTMap_vsubset_Vset: shows "\<NN>\<lparr>NTMap\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
homotopic_points_eq_path_component: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" shows "homotopic_loops S (linepath a a) (linepath b b) = path_component S a b"
rcp_specD6: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "rcp_spec rcp"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ bs) data"   and "set sps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "(p, q) \<in> set sps" shows "(red (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (rcp gs bs ps sps data))))\<^sup>*\<^sup>* (spoly (fst p) (fst q)) 0"
theFTOne_length_finished: fixes n :: "nat"   and cl :: "('g, 'h, 'i) cmd list"   and dl :: "('j, 'k, 'l) cmd list" assumes "n \<in> theFTOne cl dl" shows "n < length cl \<and> finished (cl ! n) \<or> n < length dl \<and> finished (dl ! n)"
exhaust: fixes y :: "exesignature" obtains x1 :: "(String.literal \<times> typ) list" and x2 :: "(String.literal \<times> nat) list" and x3 :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" where "y = ExeSignature x1 x2 x3"
groupParts_same_fw_wi2: fixes V :: "'a wordinterval list"   and c :: "parts_connection"   and rs :: "'a simple_rule list" assumes "V \<in> set (groupWIs c rs)" shows "\<forall>ip1\<in>wordinterval_list_to_set V. \<forall>ip2\<in>wordinterval_list_to_set V. same_fw_behaviour_one ip1 ip2 c rs"
TimesLR_ZeroL: fixes r :: "'a language" shows "TimesLR Zero r = Zero"
not_gr_zero: fixes n :: "'c multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
Prefixes_singleton: fixes xs :: "'a list" shows "Prefixes {xs} = set (prefixes xs)"
split: fixes P :: "'h \<Rightarrow> bool"   and f1 :: "'i \<Rightarrow> 'h"   and f2 :: "'j \<Rightarrow> 'k \<Rightarrow> 'h"   and generat :: "('i, 'j, 'k) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
conseq2: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and Q' :: "'b assn"   and Q :: "'b assn" assumes "G|-{P}. c .{Q'}"   and "\<forall>Z s. Q' Z s \<longrightarrow> Q Z s" shows "G|-{P}. c .{Q}"
body_def: fixes t :: "sterm" shows "body t = (\<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> lc (t\<^bsup>[Fvar s,Fvar p]\<^esup>)))"
unit_vectors_carrier_vec: fixes n :: "nat" shows "unit_vectors n \<subseteq> carrier_vec n"
dims_output_deep_model': fixes rs :: "nat list"   and j :: "nat"   and m :: "real mat convnet" assumes "1 \<le> length rs"   and "\<And>r. r \<in> set rs \<Longrightarrow> 0 < r"   and "j < rs ! 0"   and "remove_weights m = deep_model'_l rs" shows "dims (tensors_from_net m $ j) = replicate (2 ^ (length rs - 1)) (last rs)"
walk_drop: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and n :: "nat" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (drop n xs)"
prodag_aGroup: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "aGroup (a\<Pi>\<^bsub>I\<^esub> A)"
map_respect_eq: shows "rel_fun (=) (rel_fun (equivclp cancel1) (equivclp cancel1)) map map"
elem_min_elems: fixes c :: "'b"   and B :: "'b set" assumes "c \<in> min_elems B" shows "c \<in> B"
mono2mono_max_enat: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> enat"   and t' :: "'a \<Rightarrow> enat" assumes "monotone orda (\<le>) t"   and "monotone orda (\<le>) t'" shows "monotone orda (\<le>) (\<lambda>x. max (t x) (t' x))"
gen_dfs_fe_inv_pres_visited: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a"   and it :: "'a set"   and V' :: "'a set"   and t :: "'a" assumes "gen_dfs_pre E U S V u0"   and "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 it V' False"   and "t \<in> it"   and "it \<subseteq> E `` {u0}"   and "t \<in> V'" shows "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 (it - {t}) V' False"
cases: fixes x :: "('c, 'd) Term.term \<times> ('c, 'd) Term.term" obtains s :: "('c, 'd) Term.term" and t :: "('c, 'd) Term.term" where "x = (s, t)"
add2_None: fixes x :: "'a option" shows "x ++ None = None"
evaluate_determ_1: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'c state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v"   and r1a :: "'c state \<times> (v, v) result"   and r1b :: "'c state \<times> (v, v) result" assumes "evaluate_match ck env s v pes v' r1a"   and "evaluate_match ck env s v pes v' r1b" shows "r1a = r1b"
direct_decomp_perm: fixes A :: "'c set"   and ss1 :: "'c set list"   and ss2 :: "'c set list" assumes "direct_decomp A ss1"   and "ss1 <~~> ss2" shows "direct_decomp A ss2"
that_2: fixes Pa :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Ea :: "char list \<Rightarrow> ty option"   and e :: "expr"   and es :: "expr list"   and T :: "ty"   and Ts :: "ty list" assumes "x = Pa"   and "xa = Ea"   and "xb = e # es"   and "xc = T # Ts"   and "Pa,Ea \<turnstile> e :: T"   and "Pa,Ea \<turnstile> es [::] Ts" shows "thesis"
obj_trg: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.obj (S.trg \<mu>)"
rep_mkarr: fixes f :: "'c local.term" assumes "Arr f" shows "rep (mkarr f) = \<^bold>\<parallel>f\<^bold>\<parallel>"
lookup_product: fixes f :: "'b \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> 'd"   and rbt1 :: "('b, 'e) mapping_rbt"   and rbt2 :: "('c, 'f) mapping_rbt"   and a :: "'b"   and b :: "'c" assumes "ID ccompare \<noteq> None"   and "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.product f rbt1 rbt2) (a, b) = (case RBT_Mapping2.lookup rbt1 a of None \<Rightarrow> None | Some c \<Rightarrow> map_option (f a c b) (RBT_Mapping2.lookup rbt2 b))"
containFalsum: fixes \<Gamma> :: "'b form multiset"   and R :: "('b sequent list \<times> 'b sequent) set"   and \<Delta> :: "'b form multiset" assumes "ff \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable R*"
comp_fun_idem_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.semilattice_inf inf less_eq less" shows "comp_fun_idem inf"
regular_inf_dense: fixes x :: "'a" shows "\<exists>y z. regular y \<and> dense z \<and> x = y \<sqinter> z"
lens_create_def: fixes X :: "'e \<Longrightarrow> 'd"   and v :: "'e" shows "create\<^bsub>X\<^esub> v = put\<^bsub>X\<^esub> src\<^bsub>X\<^esub> v"
interv_sum_list_conv_sum_set_nat: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" assumes "class.monoid_add plus zero" shows "sum_list (map f [m..<n]) = sum f (set [m..<n])"
funpow_shift1_degree: fixes k :: "nat"   and p :: "'b poly" shows "Polynomial_Expression.degree ((shift1 ^^ k) p) = k + Polynomial_Expression.degree p"
continuous_onI: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x e. \<lbrakk>0 < e; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e" shows "continuous_on s f"
terminal_tinfinite: fixes xs :: "('a, 'b) tllist" assumes "\<not> tfinite xs" shows "terminal xs = undefined"
abc_fourtimes_def: shows "abc_fourtimes = (let (aprog, ary, fp) = rec_ci rec_fourtimes in aprog [+] dummy_abc (Suc 0))"
convergent_erf_remainder_integral: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "convergent (\<lambda>m. integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * n)))"
ide_char: fixes a :: "'a \<times> 'a \<times> 'a" shows "B.VVV.ide a = (B.VVV.arr a \<and> B.VxVxV.ide a)"
Ivl_simps_1: fixes a :: "'b"   and b :: "'b" shows "lower (Ivl a b) = min a b"
bbox_bdia_demorgan: fixes R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'f" shows "[R| f x = -  \<langle>R| (\<lambda>y. - f y) x"
unique_circline_type_zero_0': shows "(circline_type circline_point_0 = 0 \<and> 0\<^sub>h \<in> circline_set circline_point_0) \<and> (\<forall>H. circline_type H = 0 \<and> 0\<^sub>h \<in> circline_set H \<longrightarrow> H = circline_point_0)"
cnj_add: fixes x :: "octo"   and y :: "octo" shows "Octonions.cnj (x + y) = Octonions.cnj x + Octonions.cnj y"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P'"   and "a \<noteq> b" shows "<\<nu>b>P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>b> \<prec> P'"
degree_mult_Var: fixes A :: "int mpoly"   and x' :: "nat" shows "MPoly_Type.degree (A * Var x') x' = (if A = 0 then 0 else Suc (MPoly_Type.degree A x'))"
leastPD3: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "leastP P x"   and "y \<in> Collect P" shows "x \<le> y"
cadjoint_eqI: fixes F :: "'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'b" assumes "\<And>x y. \<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>" shows "G\<^sup>\<dagger> = F"
mem_equal_mod_user_reg: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and new_user_reg1 :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and user_reg2 :: "'b word \<Rightarrow> word5 \<Rightarrow> word32" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>user_reg := new_user_reg1\<rparr>) (s2\<lparr>user_reg := user_reg2\<rparr>) pa"
Cidl: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f"
monic_Mp_prod_mset: fixes fs :: "int poly multiset" assumes "\<And>f. f \<in># fs \<Longrightarrow> monic (Mp f)" shows "monic (Mp (\<Prod>\<^sub># fs))"
P3: shows "\<lfloor>\<lambda>w. \<forall>x. (Abstract x \<^bold>\<rightarrow> Dependent x) w\<rfloor>"
flip_at_simps_2: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow> b) \<bullet> b = a"
nonzero_pow_nonzero: fixes a :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier P"   and "a \<noteq> \<zero>\<^bsub>P\<^esub>" shows "a [^]\<^bsub>P\<^esub> n \<noteq> \<zero>\<^bsub>P\<^esub>"
E: shows "Crypt (sesK (d * e)) (Passwd 0) \<in> synth (analz (A \<union> spies evsFC5))"
less_eq_option_Some_None: fixes x :: "'a" shows "(Some x \<le> None) = False"
comm_inv_char: fixes x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C"   and "y \<in> carrier C"   and "x \<otimes>\<^bsub>C\<^esub> y = \<one>\<^bsub>C\<^esub>" shows "inv\<^bsub>C\<^esub> x = y"
nat_of_bool_bound: fixes b :: "bool" shows "of_bool b < 2"
nat_minus_mod_plus_right: fixes n :: "nat"   and x :: "nat"   and m :: "nat" shows "(n + x - n mod m) mod m = x mod m"
indep_in_imp_subset_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X" shows "\<exists>B. basis_in \<E> B \<and> X \<subseteq> B"
is_full_pmdl_subset: fixes B1 :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B2 :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_full_pmdl B1"   and "is_full_pmdl B2"   and "component_of_term ` Keys B1 \<subseteq> component_of_term ` Keys B2" shows "pmdl B1 \<subseteq> pmdl B2"
ev_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "ev \<phi> \<equiv> until tr \<phi>"
projection_sequence: fixes xs :: "'b list"   and X :: "'b set"   and Y :: "'b set" shows "xs \<upharpoonleft> X \<upharpoonleft> Y = xs \<upharpoonleft> (X \<inter> Y)"
width_def: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.width T bag \<equiv> if V\<^bsub>T\<^esub> = {} then 0 else TreeDecomposition.max_bag_card T bag - 1"
simps_1: fixes g1 :: "'d \<Rightarrow> 'c"   and g2 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "shd (unfold_stream g1 g2 a) = g1 a"
degeneralize_nodes_card: fixes A :: "('d, 'c) dgca" shows "card (DCA.nodes (degeneralize A)) \<le> max 1 (length (dgca.rejecting A)) * card (DGCA.nodes A)"
distinct_45: fixes x51 :: "('d, 'f) ODE"   and x52 :: "('d, 'e, 'f) formula"   and x61 :: "('d, 'e, 'f) hp"   and x62 :: "('d, 'e, 'f) hp" shows "EvolveODE x51 x52 \<noteq> (x61 \<union>\<union> x62)"
reduction_step3: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf"   and out :: "unit" shows "\<exists>A1. \<bar>spmf (R1_14_interm2 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar>"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "pre (C\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)"
d2: shows "diagram ars (\<upsilon>, \<tau>', \<tau>'', \<upsilon>')"
normalize_gauss_int_of_nat: fixes n :: "nat" shows "normalize (of_nat n) = of_nat n"
equivalent_objects_def: fixes a :: "'c"   and b :: "'c" shows "D.equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright> \<and> D.equivalence_map f"
lm108: fixes Z :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "Z \<subseteq> X \<union> Y" shows "sum (X <| Y) (Z - X) = 0"
path_continuous_image: fixes g :: "real \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "path g"   and "continuous_on (path_image g) f" shows "path (f \<circ> g)"
Int_stableI: fixes A :: "'d set set" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> a \<inter> b \<in> A" shows "Int_stable A"
case_2: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
order_le_less_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y < z" shows "x < z"
almost_full_on_imp_reflp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "almost_full_on P A" shows "reflp_on P A"
fmdom'_restrict_fset_precise: fixes A :: "'d fset"   and m :: "'d \<rightharpoonup>\<^sub>f 'e" shows "fmdom (fmrestrict_fset A m) = fmdom m |\<inter>| A"
padic_add_inv: fixes p :: "int"   and x :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)" shows "\<exists>y\<in>carrier (padic_int p). y \<oplus>\<^bsub>padic_int p\<^esub> x = \<zero>\<^bsub>padic_int p\<^esub>"
path_to_def: fixes q :: "hf" shows "path_to q \<equiv> SOME u. u \<in> left_lang q"
Try_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (try e catch(C V) e2, xs) (try e' catch(C V) e2, xs')"
dist_z: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat" shows "Graph.dist c v 0 v"
big_d': fixes ys :: "nat list"   and b :: "nat list"   and a :: "nat list"   and i :: "nat" assumes "length ys \<le> length b" shows "set (big_d' a b ys i) = hlde_ops.Di' a b i ys"
member_mset_pos_iff_zcount: fixes x :: "'a"   and M :: "'a zmultiset" shows "(x \<in># mset_pos M) = (0 < zcount M x)"
prim_pyth_tripleE: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "prim_pyth_triple x y z" obtains u :: "nat" and v :: "nat" where "0 < v" and "v < u" and "coprime u v" and "\<not> (odd u \<and> odd v)" and "z = u\<^sup>2 + v\<^sup>2" and "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (x \<preceq> y \<and> y \<preceq> x)"
all_envirE: fixes p :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "all_envir p" obtains "p x"
parts_subset: fixes H :: "msg set" shows "H \<subseteq> parts H"
factorization_to_fract: fixes q :: "'b fract poly"   and p :: "'b poly"   and r :: "'b fract poly" assumes "q \<noteq> 0"   and "map_poly to_fract p = q * r" shows "\<exists>q' r' c. c \<noteq> 0 \<and> q = smult c (map_poly to_fract q') \<and> r = smult (inverse c) (map_poly to_fract r') \<and> content_ff (map_poly to_fract q') =dff 1 \<and> p = q' * r'"
S: shows "S = (\<lambda>s. if 3 \<le> s a then s else s(a := 3))"