hcju/isabelleps huggingface dataset:
pdg_dghm_proj_is_dghm: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> 2\<^sub>\<nat>" shows "\<pi>\<^sub>D\<^sub>G (2\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>) i : (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>) \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> (if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)"
develops_at_within_def: fixes e :: "nat \<Rightarrow> real"   and a :: "float interval list"   and I :: "float interval list" shows "develops_at_within e a I = (a all_subset I \<and> (\<forall>i<length I. e i \<in>\<^sub>r I ! i - a ! i))"
fsafe_Geq_simps: fixes t1 :: "('a, 'c) trm"   and t2 :: "('a, 'c) trm" shows "fsafe (Geq t1 t2) = (dsafe t1 \<and> dsafe t2)"
simps_1: fixes s :: "nat"   and lm :: "nat list"   and p :: "abc_inst list" shows "abc_steps_l (s, lm) p 0 = (s, lm)"
a_1: shows "f =m Polynomial.smult c (\<Prod>\<^sub># fs) \<and> (\<forall>f\<in>#fs. irreducible\<^sub>d_m f \<and> monic (Mp f))"
icard_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "icard A \<le> icard B"
rel_fundefs_rewrite: fixes F1 :: "'fenv_std"   and F2 :: "'fenv_inca"   and f :: "'fun"   and fd2 :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and instr :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr"   and F2' :: "'fenv_inca" assumes "rel_fundefs (Fstd_get F1) (Finca_get F2)"   and "Finca_get F2 f = Some fd2"   and "Finca_add F2 f (rewrite_fundef_body fd2 pc instr) = F2'"   and "norm_instr (body fd2 ! pc) = norm_instr instr" shows "rel_fundefs (Fstd_get F1) (Finca_get F2')"
globalsimp: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "(nat \<Rightarrow> 'c) \<times> 'd" shows "global P s = P (fst s)"
iT_Div_0_icard_if: fixes I :: "nat set" shows "icard (I \<oslash> 0) = enat (if I = {} then 0 else Suc 0)"
DC: shows "\<forall>ts'. ts \<^bold>\<Rightarrow> ts' \<longrightarrow> (\<forall>x xa. x \<noteq> xa \<longrightarrow> (\<forall>v'. move ts ts' v = xa > v' \<longrightarrow> (\<nexists>v u. v'=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts' x = ext v \<and> restrict v (res ts') x = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (res ts') xa = lan v \<and> |lan v| = 1) \<and> True)) \<and> True))) \<longrightarrow> (\<forall>ts'a. ts' \<^bold>\<leadsto> ts'a \<longrightarrow> (\<forall>v'. move ts' ts'a (move ts ts' v) = xa > v' \<longrightarrow> (\<nexists>v u. v'=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts'a x = ext v \<and> restrict v (res ts'a) x = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts'a xa = ext v \<and> restrict v (res ts'a) xa = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)))))"
not_empty_eq_Iic_eq_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'b" assumes "class.preorder less_eq less" shows "{} \<noteq> ord.atMost less_eq h"
zeta_partial_sum_le_pos': assumes "0 < s"   and "s \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>x. real x powr - s) x - x powr (1 - s) / (1 - s)\<bar> \<le> c"
pos_le_divide_eq: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "(0::'b) < c" shows "(a \<le> b / c) = (a * c \<le> b)"
smult_smult_assoc: fixes a :: "'b"   and b :: "'b"   and v :: "'b Matrix.vec" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v v) = a * b \<cdot>\<^sub>v v"
summable_Digamma: fixes z :: "'a" assumes "z \<noteq> (0::'a)" shows "summable (\<lambda>n. inverse (of_nat (Suc n)) - inverse (z + of_nat n))"
CCexp_simps_4: fixes e :: "exp"   and x :: "Vars.var"   and a :: "Arity" shows "ccExp_syn a (App e x) = ccExp_syn (inc\<cdot>a) e \<squnion> {x} G\<times> insert x (fv e)"
measurable_T_tail: fixes n :: "nat" shows "T \<in> tail_algebra (Suc n) \<rightarrow>\<^sub>M tail_algebra n"
convolution_transfer: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "Abs_aezfun (convolution f g) = Abs_aezfun f * Abs_aezfun g"
rel_FGco_neg_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'x \<Rightarrow> bool"   and Co3 :: "'r \<Rightarrow> 'y \<Rightarrow> bool"   and Co4 :: "'s \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'u \<Rightarrow> 'ab \<Rightarrow> bool"   and Co1' :: "'w \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2' :: "'x \<Rightarrow> 'ad \<Rightarrow> bool"   and Co3' :: "'y \<Rightarrow> 'ae \<Rightarrow> bool"   and Co4' :: "'z \<Rightarrow> 'af \<Rightarrow> bool"   and Contra1' :: "'aa \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2' :: "'ab \<Rightarrow> 'ah \<Rightarrow> bool"   and Co6 :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Co6' :: "'aj \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra3 :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra3' :: "'am \<Rightarrow> 'an \<Rightarrow> bool"   and Contra4 :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra4' :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and tytok_F :: "('ar \<times> 'as \<times> 'at \<times> 'p \<times> 'w \<times> 'ac \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and tytok_G :: "('p \<times> 'w \<times> 'ac \<times> 'q \<times> 'x \<times> 'ad \<times> 'v) itself"   and Co5 :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Co5' :: "'av \<Rightarrow> 'aw \<Rightarrow> bool"   and tytok_FGco :: "('ar \<times> 'as \<times> 'at \<times> 'v \<times> 'ax) itself" assumes "rel_F_neg_distr_cond (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) (rel_G Co1' Co2' Co3' Co4' Contra1' Contra2') Co3 Co3' Co6 Co6' Contra1 Contra1' Contra3 Contra3' Contra4 Contra4' tytok_F"   and "rel_G_neg_distr_cond Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_FGco_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Co6 Co6' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' tytok_FGco"
af_subsequence_R_GF_advice: fixes i :: "nat"   and n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "i \<le> n"   and "suffix n w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> n])[X]\<^sub>\<nu>"   and "\<And>j. j \<le> i \<Longrightarrow> suffix n w \<Turnstile>\<^sub>n af \<psi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>" shows "suffix (Suc n) w \<Turnstile>\<^sub>n af (\<phi> R\<^sub>n \<psi>) (prefix (Suc n) w)[X]\<^sub>\<nu>"
Vr_prime_maximalTr2: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> vp K v"   and "x \<noteq> \<zero>"   and "Suc 0 < n" shows "x \<notin> Vr K v \<diamondsuit>\<^sub>p (x^\<^bsup>K n\<^esup>) \<and> x^\<^bsup>K (n - Suc 0)\<^esup> \<notin> Vr K v \<diamondsuit>\<^sub>p (x^\<^bsup>K n\<^esup>)"
simps_2: fixes A :: "'c set"   and As :: "'c set list" shows "listset (A # As) = set_Cons A (listset As)"
lowlink_setI: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c"   and p :: "'c list"   and w :: "'c" assumes "DFS_invar G param s"   and "param_DFS_defs.lowlink_path G s v p w"   and "w \<in> dom (discovered s)"   and "(v, w) \<in> (g_E G)\<^sup>*"   and "(w, v) \<in> (g_E G)\<^sup>*" shows "w \<in> param_DFS_defs.lowlink_set G s v"
all_envir_def: fixes p :: "'b \<Rightarrow> bool" shows "all_envir p = (\<forall>x. p x)"
mult_left_sub_dist_sup_right: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<sqinter> z \<le> x \<sqinter> (y \<squnion> z)"
coprime_commute: 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 b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one b a = algebraic_semidom.coprime times one a b"
locally_finite_in_def: fixes X :: "'c topology"   and \<A> :: "'c set set" shows "locally_finite_in X \<A> = (\<Union> \<A> \<subseteq> topspace X \<and> (\<forall>x\<in>topspace X. \<exists>V. openin X V \<and> x \<in> V \<and> finite {U \<in> \<A>. U \<inter> V \<noteq> {}}))"
compatible_modes_old_vars: fixes cms\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list"   and cms\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "abs.compatible_modes (map snd cms\<^sub>A)"   and "length cms\<^sub>A = length cms"   and "length cms\<^sub>C = length cms"   and "\<forall>i<length cms. ((cms\<^sub>A ! i, mem\<^sub>A), cms\<^sub>C ! i, mem\<^sub>C) \<in> \<R>_rel (cms ! i)" shows "conc.compatible_modes (conc.restrict_modes (map snd cms\<^sub>C) (range var\<^sub>C_of))"
bounded_inner_imp_bdd_above: fixes s :: "'b set"   and a :: "'b" assumes "bounded s" shows "bdd_above ((\<lambda>x. x \<bullet> a) ` s)"
incl: fixes A :: "'a set"   and B :: "'a set set"   and b :: "'a set" assumes "Congruence.partition A B"   and "b \<in> B" shows "b \<subseteq> A"
is_lubD1: fixes S :: "'a set"   and x :: "'a" assumes "S <<| x" shows "S <| x"
distinct_1: shows "SizeOp \<noteq> ToUpperCaseOp"
\<sigma>'_def: fixes i :: "nat" shows "\<sigma>' i = (if i \<in> net_tree_ips n then the (fst (netgmap sr s') i) else \<sigma> i)"
foo_mhead_def: shows "foo_mhead = \<lparr>access = Public, static = False, pars = [z], resT = Class Base\<rparr>"
af\<^sub>G_simps_4: fixes \<phi> :: "'a ltl"   and x :: "'a set"   and xs :: "'a set list" shows "af\<^sub>G (X \<phi>) (x # xs) = af\<^sub>G \<phi> xs"
factorize_polys_map_distinct: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs" shows "distinct fss"
delete_min_del_hd: fixes t :: "interval_tree" assumes "t \<noteq> Tip" shows "fst (del_min t) # in_traverse (snd (del_min t)) = in_traverse t"
Ramsey2: fixes Z :: "'b set"   and f :: "'b set \<Rightarrow> nat"   and s :: "nat" assumes "infinite Z"   and "\<forall>x\<in>Z. \<forall>y\<in>Z. x \<noteq> y \<longrightarrow> f {x, y} < s" shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t)"
pathin_const: fixes X :: "'b topology"   and a :: "'b" shows "pathin X (\<lambda>x. a) = (a \<in> topspace X)"
varsAppendClauses: fixes clause1 :: "Literal list"   and clause2 :: "Literal list" shows "vars (clause1 @ clause2) = vars clause1 \<union> vars clause2"
2: shows "Domain a - X \<union> {i} = Domain a \<union> {i} - X"
g0: fixes h :: "int poly" shows "0 \<le> mahler_measure h"
plus_ccsubspace_def: shows "(+) = (\<squnion>)"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright> \<equiv> B.arr \<mu> \<and> src\<^sub>B \<mu> = a \<and> trg\<^sub>B \<mu> = b"
pickBinp_def: fixes binp :: "'c \<Rightarrow> 'd set option" shows "pickBinp binp \<equiv> lift pick binp"
abs_eq: fixes xc :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'c"   and xb :: "'d \<Rightarrow> 'e"   and xa :: "'d \<Rightarrow> 'f"   and x :: "'d \<Rightarrow> 'g" shows "trop xc (mk\<^sub>e xb) (mk\<^sub>e xa) (mk\<^sub>e x) = mk\<^sub>e (\<lambda>b. xc (xb b) (xa b) (x b))"
is_run_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "graph_defs.is_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<equiv> r 0 \<in> g_V0 (b_graph.to_gbg_ext (degeneralize_ext ma) m) \<and> ipath (g_E (b_graph.to_gbg_ext (degeneralize_ext ma) m)) r"
minus_left: fixes a :: "'a"   and b :: "'b" shows "- a ** b = - (a ** b)"
empty_correct: fixes t :: "('a, 'b) FingerTree" shows "(FingerTree.toList t = []) = (t = FingerTree.empty)"
inj_image_mem_iff: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'d"   and A :: "'d set" assumes "inj f" shows "(f a \<in> f ` A) = (a \<in> A)"
all_greater_dim_poly_zero: fixes p :: "linear_poly"   and x :: "nat" assumes "dim_poly p \<le> x" shows "Abstract_Linear_Poly.coeff p x = 0"
ptyping_match_1: fixes p :: "pat"   and T :: "POPLmarkRecord.type"   and \<Delta> :: "binding list"   and t :: "trm" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>"   and "[] \<turnstile> t : T"   and "t \<in> value" shows "\<exists>ts. \<turnstile> p \<rhd> t \<Rightarrow> ts"
eq_\<I>_converterD_WT1: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv1 :: "('h, 'i, 'j, 'k) converter"   and conv2 :: "('h, 'i, 'j, 'k) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
pw_rwof_step': fixes step :: "'c \<Rightarrow> 'c nres"   and s :: "'c"   and m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and s' :: "'c" assumes "nofail (step s)"   and "rwof m0 cond step s"   and "cond s"   and "inres (step s) s'" shows "rwof m0 cond step s'"
hb_irrefl: fixes m1 :: "'a \<times> ('a, 'b) operation"   and m2 :: "'a \<times> ('a, 'b) operation" assumes "hb m1 m2" shows "m1 \<noteq> m2"
induced_automorphism_on_simplices_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and x :: "'b set"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "x \<in> f \<turnstile> X"   and "v \<in> x" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g v = g v"
valid_cons': fixes \<Gamma> :: "binding list"   and x :: "vrs"   and Q :: "ty"   and \<Delta> :: "binding list" assumes "\<turnstile> (\<Gamma> @ VarB x Q # \<Delta>) ok" shows "\<turnstile> (\<Gamma> @ \<Delta>) ok"
ide_cod: fixes f :: "'b" assumes "D.arr f" shows "D.ide (D.cod f)"
eq: fixes T :: "'b set" shows "{x \<in> S. f x \<in> T} = S - {x \<in> S. f x \<in> - T}"
ope: shows "openin (top_of_set (S \<times> T)) U"
Sigma_empty1: fixes B :: "'c \<Rightarrow> 'd set" shows "Sigma {} B = {}"
uses'_code_def: fixes "uses" :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and phis :: "'b \<Rightarrow> 'c \<times> 'a \<Rightarrow> 'a list option"   and g :: "'b"   and "next" :: "'c \<times> 'a"   and n :: "'c" shows "CFG_SSA_Transformed_notriv_base.uses'_code uses phis g next n \<equiv> CFG_SSA_Transformed_notriv_base.substNext_code phis g next ` uses g n"
cong_ex_bigtheta: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f2 :: "'b \<Rightarrow> 'c"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> \<Omega>[F](g1)) = (f2 \<in> \<Omega>[F](g2))"
has_field_derivative_Arctan: fixes z :: "complex" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "(Arctan has_field_derivative inverse (1 + z\<^sup>2)) (at z)"
order_char_1: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (y \<sqinter> - x \<le> x)"
foldri_simp_1: fixes c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldri [] c f \<sigma> = \<sigma>"
foundation16: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> \<bottom> \<and> X \<tau> \<noteq> null)"
pred_sets1: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and N :: "'d measure" assumes "{x \<in> space M. P x} \<in> sets M"   and "f \<in> N \<rightarrow>\<^sub>M M" shows "pred N (\<lambda>x. P (f x))"
a_distinct_lists_def: fixes h :: "(_) heap" shows "local.a_distinct_lists h = distinct (concat (map (\<lambda>ptr. |h \<turnstile> get_child_nodes ptr|\<^sub>r) |h \<turnstile> object_ptr_kinds_M|\<^sub>r @ map (\<lambda>document_ptr. |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) |h \<turnstile> document_ptr_kinds_M|\<^sub>r))"
bisim1NewArray: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and T :: "ty" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,newA T\<lfloor>e\<rceil>,h \<turnstile> (newA T\<lfloor>e'\<rceil>, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
k_invariantAD: fixes ec :: "'rep"   and A :: "('ma, 'mt) AlgState" assumes "k_isNode ec"   and "k_memb ec A"   and "k_invariant A" shows "\<exists>acts. lookup aOps (aActs A) ec = Some acts \<and> set acts = set (simAction a ec)"
fv_ik_subset_fv_st': fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>t S) \<subseteq> fv\<^sub>s\<^sub>t S"
USUP_as_Inf_collect: fixes A :: "'f set"   and f :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Squnion> P \<in> A \<bullet> f P) = \<Squnion>(f ` A)"
timpl_closure'_timpls_trancl_subset': fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b} \<subseteq> timpl_closure' c"
jordan_block_char_poly: fixes n :: "nat"   and a :: "'b" shows "char_poly (jordan_block n a) = [:- a, 1::'b:] ^ n"
isgcd_divides_l: fixes a :: "'a"   and b :: "'a" assumes "a divides b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a gcdof a b"
empty_subsetI: fixes A :: "'b set" shows "order.greater_eq A {}"
snd_eqD: fixes x :: "'h"   and y :: "'g"   and a :: "'g" assumes "snd (x, y) = a" shows "y = a"
order_le_less_subst2: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a \<le> b"   and "f b < c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a < c"
simps_1: fixes q :: "rat" shows "convert_info (Inl q) = Rat_Info q"
inv_inv_next_plane0: fixes p :: "nat" shows "invariant Invariants.inv next_plane0\<^bsub>p\<^esub>"
in_zero_stalk: fixes V :: "'c set" assumes "V \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" shows "(V, zero_str\<^sub>Y V) \<in> stfx.zero_stalk V"
mmult_fa_def: fixes l :: "nat"   and m :: "nat"   and n :: "nat"   and AS :: "floatarith list"   and BS :: "floatarith list" shows "mmult_fa l m n AS BS = concat (map (\<lambda>i. map (\<lambda>k. inner_floatariths (map (\<lambda>j. AS ! (i * m + j)) [0..<m]) (map (\<lambda>j. BS ! (j * n + k)) [0..<m])) [0..<n]) [0..<l])"
cases: fixes a1 :: "char list list"   and a2 :: "char list list"   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"   and C :: "char list" assumes "P,C \<turnstile> a1 \<sqsubset>\<^sup>1 a2" obtains Cs :: "char list list" and Ds :: "char list list" where "a1 = Cs__" and "a2 = Ds" and "Subobjs P C Cs__" and "Subobjs P C Ds" and "Cs__ = butlast Ds"   | Cs :: "char list list" and D :: "char list" where "a1 = Cs" and "a2 = [D]" and "Subobjs P C Cs" and "P \<turnstile> last Cs \<prec>\<^sub>S D"
retraction_def: fixes g :: "'a \<times> 'a" shows "VoV.retraction g \<equiv> \<exists>f. VoV.ide (VoV.comp g f)"
map_le_upd: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and b :: "'b option" assumes "f \<subseteq>\<^sub>m g" shows "f(a := b) \<subseteq>\<^sub>m g(a := b)"
out_eq_p: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s"   and "n \<le> j" shows "a j = p"
LDD1_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and \<beta> :: "'a \<times> ('b \<times> 'a) list"   and \<alpha> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>1 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>2 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>3 :: "'a \<times> ('b \<times> 'a) list" shows "LDD1 ars r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (local_diagram1 ars \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LD_1' r (hd (labels \<beta>)) (hd (labels \<alpha>)) (labels \<sigma>1) (labels \<sigma>2) (labels \<sigma>3))"
cut_greater_not_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<notin> I" shows "x \<notin> I \<down>> t"
fe_def: shows "fe = infiniteExecutionCfg cfg fStepCfg fStepMsg"
msg_distinct_108: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Aenc X Y \<noteq> Enc X' Y'"
frontier_of_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" shows "X frontier_of S \<subseteq> topspace X"
list_upd_conv: fixes h :: "'c \<Rightarrow> 'c"   and p :: "'c"   and y :: "'c"   and q :: "'c" assumes "islist h p"   and "y \<notin> set (list h p)" shows "list (h(y := q)) p = list h p"
inv_5: fixes i :: "nat" assumes "i < m" shows "fs.gs.gso i \<in> Rn"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" shows "\<Gamma>\<turnstile>Throw \<down> Normal s"
disc_6: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "\<not> isSend (Trans x11 x12 x13)"
dom_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VxVxV.arr f" shows "VxVxV.dom f = (local.dom (fst f), VxV.dom (snd f))"
mkarr_memb_1: fixes t :: "'c local.term"   and f :: "'c local.term set" assumes "t \<in> f"   and "arr f" shows "Arr t"
xa_dvd: shows "x ^ a - 1 dvd x ^ b - 1"
order_iff: fixes b :: "'g"   and a :: "'g" shows "(b \<le> a) = (a = sup a b)"
shift_eq_scons: fixes u :: "'b list"   and v :: "'b stream"   and a :: "'b"   and w :: "'b stream" shows "(u @- v = a ## w) = (u = [] \<and> v = a ## w \<or> (\<exists>u'. u = a # u' \<and> u' @- v = w))"
bij_def: fixes f :: "'c \<Rightarrow> 'd" shows "bij f = (inj f \<and> surj f)"
ih1: shows "pi \<bullet> (\<parallel>(A)\<parallel>) = \<parallel>(A)\<parallel>"
p_prime: fixes p :: "nat" assumes "residues_prime p" shows "prime p"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "Ik.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "Ik.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> Ik.satA \<xi> a"
dist_snd_le: fixes x :: "'d \<times> 'c"   and y :: "'d \<times> 'c" shows "dist (snd x) (snd y) \<le> dist x y"
one_fpxs_def: shows "1 \<equiv> Abs_fpxs (\<lambda>r. if r = 0 then 1::'b else (0::'b))"
C1: fixes P' :: "ccs"   and Q' :: "ccs"   and R' :: "ccs"   and T' :: "ccs" assumes "(P', Q') \<in> Rel"   and "(R', T') \<in> Rel'" shows "(P' \<parallel> R', Q' \<parallel> T') \<in> Rel''"
select_edge_code_def: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "select_edge_code node_eq_impl \<equiv> \<lambda>(x, xa, xb, xc). if as_is_empty xc then RETURN (None, x, xa, xb, xc) else let (xd, xe) = as_top xc in if as_get xa (as_length xa - 1) \<le> xd then RETURN (Gen_Set.gen_pick (\<lambda>x. foldli (id x)) xe) \<bind> (\<lambda>xf. let y'a = glist_delete node_eq_impl xf xe; y'b = if is_Nil y'a then as_pop xc else as_set xc (as_length xc - 1) (xd, y'a) in RETURN (Some xf, x, xa, xb, y'b)) else RETURN (None, x, xa, xb, xc)"
distinct_paths: fixes Q :: "'a set"   and R :: "'a set"   and d :: "'a" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "d \<notin> Q"   and "d \<in> R" shows "R \<noteq> Q"