hcju/isabelleps huggingface dataset:
Segment_def: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and l1 :: "int"   and code :: "(int \<times> Instr) list" shows "Segment C m l l1 code = (\<exists>Mspec Minv Anno. MST\<down>(C, m) = Some (Mspec, Minv, Anno) \<and> (\<forall>ll\<ge>l. ll < l1 \<longrightarrow> Anno\<down>ll = None \<and> (\<exists>ins. ins_is C m ll ins \<and> code\<down>ll = Some ins)))"
unfold: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f"   and a :: "'e"   and s' :: "'f" assumes "g s = Yield a s'"   and "s' \<in> terminates_on g" shows "s \<in> terminates_on g"
extended_predist_weak_triangle: fixes x :: "'a Gromov_completion"   and z :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_predist x z \<le> sqrt 2 * max (extended_predist x y) (extended_predist y z)"
BlockThrow: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(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" shows "P,E \<turnstile> \<langle>{V:T; Throw r},s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
ZG2_msg_in_parts_spies: fixes B :: "agent"   and F :: "msg"   and B' :: "msg"   and L :: "msg"   and C :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>F, B', L, C, X\<rbrace> \<in> set evs"   and "evs \<in> zg" shows "C \<in> parts (knows Spy evs)"
prems: shows "l \<in> set (subseqs (x # l'))"
distinct_mset_rempdups_union_mset: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "distinct_mset A"   and "distinct_mset B" shows "A \<union># B = remdups_mset (A + B)"
dvd_div_eq_0_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "dvd.dvd times b a" shows "(divide a b = zero) = (a = zero)"
proj2_subset_Col: fixes T :: "proj2 set"   and S :: "proj2 set" assumes "T \<subseteq> S"   and "proj2_set_Col S" shows "proj2_set_Col T"
plus_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "f + Mp g =m f + g"
is_ladder: shows "is_ladder (D @ (i, r) # E) (L @ [mk_deriv_intro ix (Suc (length D + length E)) j])"
monic_map_poly_hom: fixes p :: "'q mod_ring poly" assumes "monic p" shows "monic (map_poly rebase p)"
le_funD: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h"   and x :: "'g" assumes "f \<le> g" shows "f x \<le> g x"
length_remdups_eq: fixes xs :: "'g list" shows "(length (remdups xs) = length xs) = (remdups xs = xs)"
Fields_i_i_oE: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "((char list \<times> char list) \<times> ty) list" assumes "pred.eval (Fields_i_i_o x xa) xb" obtains "x \<turnstile> xa has_fields xb"
dim_Times: fixes S :: "'b set"   and T :: "'c set" assumes "vs1.subspace S"   and "vs2.subspace T" shows "p.dim (S \<times> T) = vs1.dim S + vs2.dim T"
wls_psubst_cong: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wls s X"   and "wlsEnv rho"   and "wlsEnv rho'"   and "\<And>ys y. fresh ys y X \<or> rho ys y = rho' ys y" shows "X #[rho] = X #[rho']"
e'_eq_e: fixes k :: "nat" shows "e' k = e k"
Schnitt: fixes \<Gamma> :: "'b formula list"   and \<DD> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<Delta> :: "'b formula list"   and \<Lambda> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<DD> # \<Theta>"   and "\<DD> # \<Delta> \<Rightarrow> \<Lambda>" shows "\<Gamma> @ \<Delta> \<Rightarrow> \<Theta> @ \<Lambda>"
state_succ_valid_act_disjoint: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "a \<in> PROB"   and "vs \<inter> prob_dom PROB = {}" shows "fmrestrict_set vs (state_succ s a) = fmrestrict_set vs s"
emeasure_distr_restrict: fixes I :: "'b set"   and K :: "'b set"   and Q :: "('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure"   and A :: "('b \<Rightarrow> 'c) set" assumes "I \<subseteq> K"   and "sets Q = sets (Pi\<^sub>M K M)"   and "A \<in> sets (Pi\<^sub>M I M)" shows "emeasure (distr Q (Pi\<^sub>M I M) (\<lambda>\<omega>. restrict \<omega> I)) A = emeasure Q (prod_emb K M I A)"
step_ge_x0: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and t :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "akra_bazzi_recursion x\<^sub>0 x\<^sub>1 k as bs ts"   and "t \<in> set ts"   and "x\<^sub>1 \<le> x" shows "x\<^sub>0 \<le> t x"
RDR_axioms: shows "RDR (\<bullet>) \<gamma> \<bottom>"
lesssub_lengthD: fixes xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" assumes "xs [\<sqsubset>\<^bsub>r\<^esub>] ys" shows "length ys = length xs"
sigma_sets_empty_eq: fixes A :: "'d set" shows "sigma_sets A {} = {{}, A}"
cl_rank_of: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and x :: "'b"   and X :: "'b set" assumes "matroid carrier indep"   and "x \<in> matroid.cl carrier indep X" shows "indep_system.lower_rank_of indep (insert x X) = indep_system.lower_rank_of indep X"
subst_type_1: fixes \<Delta> :: "binding list"   and U :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and u :: "trm" assumes "\<Delta> @ VarB U \<Colon> \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<down>\<^sub>e 1 0 \<Delta> @ \<Gamma> \<turnstile> t[\<parallel>\<Delta>\<parallel> \<mapsto> u] : \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> T"
assms_1: shows "run (Array.freeze a) \<sigma> \<sigma>' r"
ide_cod: fixes f :: "'a1" assumes "C1.arr f" shows "C1.ide (C1.cod f)"
poly_rel_mod: shows "(poly_rel ===> poly_rel ===> poly_rel) (mod_field_poly_i ff_ops) (mod)"
coherent_Prim: fixes f :: "'c" assumes "Arr \<^bold>\<langle>f\<^bold>\<rangle>" shows "coherent \<^bold>\<langle>f\<^bold>\<rangle>"
\<phi>_in_hom: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<guillemotleft>\<phi> y f : y \<rightarrow>\<^sub>D G x\<guillemotright>"
TS: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length v - 2"
Int_Ici_Ico: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "{a..} \<inter> {b..<c} = {max a b..<c}"
Sup_multiset_unbounded: fixes A :: "'k multiset set" assumes "\<not> subset_mset.bdd_above A" shows "Sup A = {#}"
simps_2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and ts :: "'b FOL_Fitting.term list" shows "psubstt f (App x ts) = App (f x) (psubstts f ts)"
simps: fixes a :: "trm" shows "(a \<in> value) = ((\<exists>T t. a = (\<lambda>:T. t)) \<or> (\<exists>T t. a = (\<lambda><:T. t)))"
simps_1: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at [] dir.Left (App t s) = t"
true_lit_def: fixes I :: "'a set"   and L :: "'a literal" shows "(I \<Turnstile>l L) = (if is_pos L then \<lambda>P. P else Not) (atm_of L \<in> I)"
induced_function_support: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and y :: "'a" shows "induced_function A f y = induced_function A f y * indicator (return_time_function A -` {1..}) y"
limit_def: fixes x :: "nat \<Rightarrow> 'c" shows "limit x \<equiv> {a. \<exists>\<^sub>\<infinity>n. x n = a}"
terminates_Normal_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>DynCom c \<down> Normal s" obtains "\<Gamma>\<turnstile>c s \<down> Normal s"
list_all2_op_map_iff: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 (\<lambda>a b. b = f a) xs ys = (map f xs = ys)"
ord_inj: fixes a :: "'a" assumes "a \<in> carrier G" shows "inj_on (([^]) a) {0..ord a - 1}"
seq_assoc: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) ; (b::'a::refinement_lattice) ; (c::'a::refinement_lattice) = a ; (b ; c)"
wf_set_def: fixes n :: "nat"   and V :: "nat set" shows "wf_set n V = (\<forall>x\<in>V. x < n)"
ovalidNF_def: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "ovalidNF P f Q \<equiv> \<forall>s. P s \<longrightarrow> f s \<noteq> None \<and> (\<forall>r. f s = Some r \<longrightarrow> Q r s)"
assoc'_simps_5: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "cod \<a>\<^sup>-\<^sup>1[f, g, h] = (cod f \<star> cod g) \<star> cod h"
preservation': fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and M' :: "'a trm" assumes "\<Gamma> \<turnstile> M : \<tau>"   and "M \<rightarrow>\<beta> M'" shows "\<Gamma> \<turnstile> M' : \<tau>"
errMOD_igWlsAllDisj: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igWlsAllDisj MOD" shows "igWlsAllDisj (errMOD MOD)"
arr_single: fixes x :: "'arr" assumes "x \<in> Arr" shows "arr (MkArr False True [x])"
neq: fixes w :: "real" assumes "w \<in> {0..1}" shows "(1 - x * y) * w \<noteq> 1"
map_F_id: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't, 'u, 'v, 'w) F" shows "map_F id id id id id id id id id x = x"
subcocycle_lim_meas_Inv_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'b \<Rightarrow> real" assumes "fmpt M T" shows "fmpt.subcocycle_lim M T u \<in> borel_measurable (qmpt.Invariants M T)"
hom: fixes g :: "'m"   and g' :: "'m" assumes "g \<in> M"   and "g' \<in> M" shows "T (g + g') = T g + T g'"
labeled_list_insert_eq_cases_1: fixes d :: "'c"   and D :: "('d \<times> 'c) list"   and i :: "'d" assumes "d \<notin> set (unlabel D)" shows "List.insert d (unlabel D) = unlabel (List.insert (i, d) D)"
mat_of_rows_carrier_3: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "dim_col (mat_of_rows n vs) = n"
has_visible_field: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "P \<turnstile> C has F,b:T in D"
measure_countable_Union_approachable: fixes \<D> :: "'a set set"   and e :: "real"   and B :: "real" assumes "countable \<D>"   and "0 < e"   and "\<And>d. d \<in> \<D> \<Longrightarrow> d \<in> lmeasurable"   and "\<And>D'. \<lbrakk>D' \<subseteq> \<D>; finite D'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure lebesgue (\<Union> D') \<le> B" obtains D' :: "'a set set" where "D' \<subseteq> \<D>" and "finite D'" and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) - e < Sigma_Algebra.measure lebesgue (\<Union> D')"
refl_onI: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
counterexample_trace_set: shows "trace_set T\<^sub>c"
int_simps_10: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (#True \<longrightarrow> P) = P"
iter_def: fixes nrandom :: "natural"   and sz :: "natural"   and seed :: "natural \<times> natural" shows "Random_Pred.iter nrandom sz seed = Random_Pred.iter' TYPE('a) nrandom sz seed"
lfilter_lfinite: fixes P :: "'a \<Rightarrow> bool"   and t :: "'a llist" assumes "lfinite (lfilter P t)"   and "\<not> lfinite t" shows "\<exists>n. \<forall>n'>n. \<not> P (lnth t n')"
headn_nz: fixes p :: "'b poly"   and n0 :: "nat"   and m :: "nat" assumes "isnpolyh p n0" shows "(headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
add_node_def: fixes v :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "add_node v g \<equiv> \<lparr>nodes = insert v (nodes g), edges = edges g\<rparr>"
split_scaleR_neg_le: fixes a :: "real"   and x :: "'a" assumes "0 \<le> a \<and> x \<le> (0::'a) \<or> a \<le> 0 \<and> (0::'a) \<le> x" shows "a *\<^sub>R x \<le> (0::'a)"
Pd_shuffle_to_top_linear: fixes Y :: "'a set"   and ds' :: "'b set"   and d :: "'b" assumes "inj_on Xd Y" shows "Linear_order (Pd_shuffle_to_top ds' Y d)"
dims2_def: shows "stm1.dims2 = nths dims {0..<Suc m}"
dom_map_default: fixes k :: "'c"   and v :: "'d"   and f :: "'d \<Rightarrow> 'd"   and xs :: "('c \<times> 'd) list" shows "fst ` set (map_default k v f xs) = insert k (fst ` set xs)"
MakeCat_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "MakeCat C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = restrict (Dom C) (mor\<^bsub>C\<^esub>), Cod = restrict (Cod C) (mor\<^bsub>C\<^esub>), Id = restrict (Category.Id C) (obj\<^bsub>C\<^esub>), Comp = \<lambda>f g. (\<lambda>(x, y)\<in>{(f, g) |f g. f \<approx>>\<^bsub>C\<^esub> g}. x ;;\<^bsub>C\<^esub> y) (f, g), \<dots> = more C\<rparr>"
scaleC_scaleC: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> '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 a :: "complex"   and b :: "complex"   and x :: "'b" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (scaleC b x) = scaleC (a * b) x"
correctCompositionOut_def: fixes x :: "specID" shows "correctCompositionOut x \<equiv> out x = \<Union> (out ` subcomponents x) - loc x \<and> out x \<inter> \<Union> (ins ` subcomponents x) = {}"
to_abs_list_insert: fixes n :: "nat \<times> bool"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and D :: "((('a, 'b, 'c) prot_fun, 'd) Term.term \<times> (('f, 'g, 'e) prot_fun, 'h) Term.term) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "Fun (Val n) [] \<noteq> t" shows "\<alpha>\<^sub>0 D n = \<alpha>\<^sub>0 (List.insert (t, s) D) n"
sep_substate_tran: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq> y \<and> y \<preceq> z" shows "x \<preceq> z"
morph: fixes f :: "'b list \<Rightarrow> 'c list"   and u :: "'b list"   and v :: "'b list" assumes "morphism f" shows "f (u \<cdot> v) = f u \<cdot> f v"
continuous_sqrt: fixes f :: "complex \<Rightarrow> complex" assumes "continuous_on S f"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"   and "\<And>f. \<lbrakk>continuous_on S f; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = exp (g z))" shows "\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
circ_left_unfold: fixes x :: "'a" shows "Rf x (x\<^sup>\<circ>) = x\<^sup>\<circ>"
h5: shows "st_in t = init_state \<and> req t \<noteq> [init] \<longrightarrow> ack t = [init_state] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = init_state"
assms_2: shows "\<forall>x y. x \<le> y \<longrightarrow> g x \<le> g y"
iIN_cut_le: fixes n :: "nat"   and d :: "nat"   and t :: "nat" shows "[n\<dots>,d] \<down>\<le> t = (if t < n then {} else if t \<le> n + d then [n\<dots>,t - n] else [n\<dots>,d])"
mod_partition_iT_Div_Int: fixes k :: "nat"   and d :: "nat"   and A :: "nat set"   and n :: "nat" assumes "0 < k"   and "0 < d" shows "A \<inter> [n * k\<dots>,d * k - Suc 0] \<oslash> k = (A \<oslash> k) \<inter> ([n * k\<dots>,d * k - Suc 0] \<oslash> k)"
bij_image_Collect_eq: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "bij f" shows "f ` Collect P = {y. P (inv f y)}"
Pi_def: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
rank_invar_cons_up: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank_invar (t # bq)"   and "SkewBinomialHeapStruc.rank t' < SkewBinomialHeapStruc.rank t" shows "SkewBinomialHeapStruc.rank_invar (t' # t # bq)"
cases: fixes a1 :: "nat"   and a2 :: "nat"   and xs :: "'a list" assumes "(a1, a2) \<in> inversions xs" obtains i :: "nat" and j :: "nat" where "a1 = i" and "a2 = j" and "i < j" and "j < length xs" and "xs ! j < xs ! i"
decr0_qf: fixes p :: "fm" assumes "bound0 p" shows "qfree (decr0 p)"
simps: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and B :: "('d, 'd cset Monomorphic_Monad.id) nondetT" shows "cr_prob_ndi A p B = rel_set A (set_pmf p) (rcset (extract (run_nondet B)))"
univalent_char: fixes R :: "('d \<times> 'e) set" shows "univalent R = (R\<inverse> O R \<subseteq> Id)"
finite_imp_finite_ReZ_segments: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "finite {t. Re (g t - z) = 0 \<and> 0 \<le> t \<and> t \<le> 1}" shows "finite_ReZ_segments g z"
vdomain_atE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f" and "f\<lparr>a\<rparr> = b"
assms: shows "s \<in> set (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)))"
invariant_reaches1: fixes a :: "'b"   and b :: "'b" assumes "B.reaches1 a b"   and "PB a" shows "QB b"
in_set_impl_in_set_zip1: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c" assumes "length xs = length ys"   and "x \<in> set xs" obtains y :: "'d" where "(x, y) \<in> set (zip xs ys)"
v1: shows "toList (deepR pr m sf) = digitToList pr @ toList m @ nlistToList sf"
b: shows "0 \<le> liminf (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x))"
assms_2: shows "PP \<longrightarrow> P' \<and> Q"
Fun_corestr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "Fun (corestr f) = Fun f"
sat_Since_pointD: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and t :: "nat"   and \<psi> :: "'a MFOTL.formula"   and I :: "\<I>" assumes "MFOTL.sat \<sigma> v i (formula.Since \<phi> (point t) \<psi>)"   and "mem t I" shows "MFOTL.sat \<sigma> v i (formula.Since \<phi> I \<psi>)"
w1case1a': shows "sint (SCAST(32 \<rightarrow> 64) w1 + 0xFFFFFFFF80000001) = sint (SCAST(32 \<rightarrow> 64) w1) + sint (- 0x7FFFFFFF)"
minus_of_int: fixes x :: "complex"   and n :: "int" shows "h (x - of_int n) = h x"
is_prefix_bd'_def: fixes xs :: "'b list"   and ys :: "'b list" shows "is_prefix_bd' xs ys \<equiv> check_prefix_bd xs ys \<bind> (\<lambda>x. (case x of Inl e \<Rightarrow> RETURN (Inl e) | Inr x \<Rightarrow> RETURN (Inr True)) \<bind> case_sum (\<lambda>e. RETURN (Inr False)) (\<lambda>r. RETURN (Inr r)))"
left_idem: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "semilattice f" shows "f a (f a b) = f a b"
concat_vpath_def: fixes p :: "'a list"   and q :: "'a list" shows "concat_vpath p q \<equiv> vwalk_to_vpath (p \<oplus> q)"
vimage_small: fixes A :: "V"   and r :: "V" shows "small {b. \<exists>a\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
HVZK_unfold2_alt: assumes "\<Sigma>1.\<Sigma>_protocol" shows "\<forall>h a e z. e \<in> carrier L \<longrightarrow> h \<in> valid_pub1 \<longrightarrow> (a, z) \<in> set_spmf (S1_raw h e) \<longrightarrow> check1 h a e z"
elts_of_set: fixes X :: "V set" shows "elts (ZFC_in_HOL.set X) = (if small X then X else {})"
closure_of_UNIV: fixes X :: "'a topology" shows "X closure_of UNIV = topspace X"
ocircline_set_def: fixes H :: "ocircline" shows "ocircline_set H = {z. on_ocircline H z}"
times_of_nat_minus_right: fixes l :: "hmultiset"   and m :: "nat"   and n :: "nat" shows "l * (of_nat m - of_nat n) = l * of_nat m - l * of_nat n"
empty_def: shows "PQ.empty = pq.Abs_pq []"
lagrange: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" assumes "Group.group G"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = order G"
simps_978: 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 x241 :: "t"   and x242 :: "unop_i" shows "(case Unop_i x241 x242 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) = f24 x241 x242"
lock_ok_lasE: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" assumes "lock_ok_las ls t las" obtains "\<And>l. lock_actions_ok (ls $ l) t (las $ l)"
set_update_with_aux: fixes xs :: "('a \<times> 'b) list"   and v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "distinct (map fst xs)" shows "list.set (update_with_aux v k f xs) = list.set xs - {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v | Some v \<Rightarrow> v))}"
absorb_iff2: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a \<preceq> b) = (linordered_monoid.max a b = b)"
valid7: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> (X \<triangleq> Y) = true"
dlex_pm_def: shows "dlex_pm \<equiv> dord_pm lex_pm"
eq4: shows "x \<bullet> y1 = x \<bullet> y2"