hcju/isabelleps huggingface dataset:
proves_beta_norm: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "beta_norm t = Some u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
hom_unique_on_span: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and gens :: "'d set"   and h :: "'d \<Rightarrow> 'f"   and h' :: "'d \<Rightarrow> 'f" assumes "Group.group G"   and "Group.group H"   and "gens \<subseteq> carrier G"   and "h \<in> hom G H"   and "h' \<in> hom G H"   and "\<forall>g\<in>gens. h g = h' g" shows "\<forall>x\<in>\<langle>gens\<rangle>\<^bsub>G\<^esub>. h x = h' x"
trunc_ell2_empty: fixes x :: "'c ell2" shows "trunc_ell2 {} x = 0"
lookup_zero: fixes k :: "'d" shows "lookup 0 k = (0::'c)"
seq_if_composable: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "src \<nu> = trg \<mu>"
eq2: shows "snd (splitAt ram2 vs) = fst (splitAt (hd vs) btw) \<and> tl (fst (splitAt ram1 vs)) = snd (splitAt (hd vs) btw)"
proj2_def: shows "\<pi>\<^sub>2 \<equiv> \<lambda>(dsn, uu_, uu_, uu_, uu_, uu_). dsn"
SUP_cong_simp: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "\<Or> (C ` A) = \<Or> (D ` B)"
subgroupE_1: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "H \<le> G" shows "H \<subseteq> carrier G"
ladder_early_stage_5: fixes index :: "nat" assumes "index + 1 < length L" shows "E ! ladder_n L' index = D ! ladder_n L index"
inc_state_all_le: fixes k :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and a :: "action"   and b :: "nat" assumes "k < length ap"   and "ap ! k = Inc n"   and "(a, b) \<in> set (shift (shift tinc_b (2 * n)) (start_of (layout_of ap) k - Suc 0))" shows "b \<le> start_of (layout_of ap) (length ap)"
FWdelay_bisimulation_lift_aux_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_lift_aux final2 r2 final1 r1 \<tau>move2 \<tau>move1"
prems_4: shows "range_vars (\<sigma>1(v := t)) \<inter> subst_domain \<theta>2 = {}"
ipset_from_cidr_wordlength: fixes ip :: "'a word" shows "ipset_from_cidr ip LENGTH('a) = {ip}"
minus_add_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (plus a b) = plus (uminus a) (uminus b)"
tMap_tLast: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace" shows "tLast (tMap f t) = f (tLast t)"
sel_2: fixes x21 :: "'d"   and x22 :: "'e \<Rightarrow> ('d, 'e, 'f) resumption" shows "output (Pause x21 x22) = x21"
simps_2: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_siso (c1 ;; c2) = (SC_siso c1 \<and> SC_siso c2)"
abs_eq: fixes xa :: "(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 x :: "char list" shows "init_class_fieldmap' xa x = mapping.Mapping (init_class_fieldmap xa x)"
simps_1: shows "ExcessNotAtRecList [] = (\<lambda>g. [])"
MkIde_Dom': fixes a :: "(bool, unit) arr" assumes "ide a" shows "MkIde (Dom a) = a"
compI_1: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e"   and g :: "'e" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "g \<in> Ar\<^bsub>CC\<^esub>"   and "Cod\<^bsub>CC\<^esub> f = Dom\<^bsub>CC\<^esub> g" shows "g \<bullet>\<^bsub>CC\<^esub> f \<in> Ar\<^bsub>CC\<^esub>"
linear_coord: fixes X :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)" shows "linear (*s) (*s) (coord X)"
DERIV_fun_exp: fixes g :: "'a \<Rightarrow> 'a"   and m :: "'a"   and x :: "'a" assumes "(g has_field_derivative m) (at x)" shows "((\<lambda>x. exp (g x)) has_field_derivative exp (g x) * m) (at x)"
inv_inv_eq: fixes f :: "'c \<Rightarrow> 'd" assumes "bij f" shows "inv (inv f) = f"
reachable_in_verts_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "v \<in> verts G"
first: fixes Mkt :: "('b, 'c) discrete_market"   and v :: "real"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and asset :: "'c" shows "remaining_qty Mkt v pf asset (Suc 0) = (\<lambda>w. (v - val_process Mkt pf 0 w) / prices Mkt asset 0 w)"
divisor_mod_0_imp_iMOD_in_i_set_mult: fixes m :: "nat"   and k :: "nat"   and r :: "nat" assumes "m mod k = 0" shows "[ r, mod m ] \<in> i_set_mult k"
lookup_override_on_eq: fixes m1 :: "'e \<Rightarrow> 'd"   and m2 :: "'e \<Rightarrow> 'd"   and S :: "'e set"   and x :: "'e" shows "(m1 ++\<^bsub>S\<^esub> m2) x = (if x \<in> S then m2 x else m1 x)"
assms_4: fixes x :: "'c"   and y :: "'a" shows "snd R x y \<Longrightarrow>\<^sub>t snd R' x y"
big_clocked_unclocked_equiv_timeout: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" shows "(\<forall>r. \<not> evaluate False env s e r) = (\<forall>c. \<exists>s'. evaluate True env (update_clock (\<lambda>_. c) s) e (s', Rerr (Rabort Rtimeout_error)) \<and> clock s' = 0)"
pp_isotone_inf: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) \<le> - - x"
term_subst_eq_rev: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<forall>x\<in>fv t. \<sigma> x = \<tau> x"
assms_3: shows "\<forall>v. enat d \<le> Q v"
backlinks_correct: fixes E :: "('b \<times> 'b) list" shows "backflows (set E) = set (backlinks E)"
simps_3: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (Eq (CN 0 c e)) = c"
x1: shows "x = lc j \<cdot>\<^sub>v us ! j + (lc i \<cdot>\<^sub>v us ! i + lc i \<cdot>\<^sub>v (c \<cdot>\<^sub>v us ! j) + M1)"
starset_Un: fixes A :: "'b set"   and B :: "'b set" shows "*s* (A \<union> B) = *s* A \<union> *s* B"
correctCompositionDiffLevelsA8: shows "correctCompositionDiffLevels sA8"
tvs_occs: fixes v :: "Core.term"   and t :: "Core.term" assumes "occs v t" shows "tvs v \<subseteq> tvs t"
ZFUnionRangeExplode: fixes A :: "ZF set"   and f :: "ZF \<Rightarrow> ZF set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> range explode"   and "A \<in> range explode" shows "\<Union> (f ` A) \<in> range explode"
non_head_prefixes: fixes n :: "nat"   and p :: "'b list" assumes "n < length p" shows "prefixes p ! Suc n \<noteq> []"
unique_connecting_path: fixes v :: "'c"   and w :: "'c" assumes "v \<in> V\<^bsub>T\<^esub>"   and "w \<in> V\<^bsub>T\<^esub>" shows "\<exists>!xs. v \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> w"
Inf_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and x :: "'c" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq (Inf A) x = (\<forall>y. less x y \<longrightarrow> (\<exists>a\<in>A. less a y))"
CFG_wf_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set" shows "CFG_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses"
lideal: fixes r :: "'a"   and a :: "'a" assumes "r \<in> B"   and "a \<in> I" shows "multB r a \<in> I"
assms_4: fixes \<eta> :: "nat" shows "bound_outer \<eta> * max (bound_sim \<eta>) 1 \<le> bound_inner \<eta>"
steps'_set: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list" assumes "length qs = length as"   and "n \<le> length as" shows "set (steps' s qs as n) = set s"
free_vars_expr_rf_to_cexpr: fixes e :: "expr" assumes "randomfree e" shows "free_vars (expr_rf_to_cexpr e) \<subseteq> free_vars e"
lossless_restrict_gpvI: fixes gpv :: "('b, 'out, 'in) gpv" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_gpv \<I> (local.restrict_gpv gpv)"
deg_p1: fixes t :: "'x \<Rightarrow>\<^sub>0 'a" assumes "t \<in> keys p1" shows "deg_pm t = poly_deg p"
conjugate_vec_sprod_comm: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod w v = conjugate w \<bullet> v"
shift_bounds_cl_nat_ivl: fixes g :: "nat \<Rightarrow> 'g"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "prod g {m + k..n + k} = (\<Prod>i = m..n. g (i + k))"
polar_identity: fixes x :: "'b"   and y :: "'b" shows "(norm (x + y))\<^sup>2 = (norm x)\<^sup>2 + (norm y)\<^sup>2 + 2 * Re \<langle>x, y\<rangle>"
degenerate_lots_subset_all: shows "local.degenerate_lotteries \<subseteq> local.\<P>"
manifold_eucl_diff_fun_space_iff: fixes g :: "'c \<Rightarrow> real"   and k :: "enat" shows "(g \<in> manifold_eucl.diff_fun_space k) = k-smooth_on UNIV g"
finite_imageD: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
P1_security: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x < q n"   and "y < q n" shows "local.sim_non_det_def.perfect_sec_P1 n x y"
subseqs_foldr: fixes base :: "int \<times> int poly list"   and elements :: "int poly list"   and n :: "nat"   and out :: "(int \<times> int poly list) list"   and state :: "'a" assumes "subseqs_foldr sl_impl base elements n = (out, state)" shows "sli base elements n state \<and> set out = S base elements n"
mem_ball: fixes y :: "'a"   and x :: "'a"   and e :: "real" shows "(y \<in> ball x e) = (dist x y < e)"
c_swap: shows "c_swap \<le> 8 * (m - Suc i) + 4"
conformsI: fixes s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and x :: "abopt" assumes "\<forall>r. ! obj:globs s r: G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"   and "G,s\<turnstile>locals s[\<sim>\<Colon>\<preceq>]L"   and "\<forall>a. x = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)"   and "x = Some (Jump Ret) \<longrightarrow> locals s Result \<noteq> None" shows "(x, s)\<Colon>\<preceq>(G, L)"
chain_Seq1: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "chain (\<lambda>i. Y i `;` S)"
simps: fixes a1 :: "nat"   and a2 :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" shows "((a1, a2) \<in> Clos rel n) = ((\<exists>x y. a1 = x \<and> a2 = y \<and> (x, y) \<in> rel) \<or> (\<exists>x z y. a1 = x \<and> a2 = y \<and> (x, z) \<in> Clos rel n \<and> (z, y) \<in> Clos rel n \<and> z < n))"
X: shows "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to \<Union> \<B>)) = X"
swapVal_compose: fixes zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b" shows "val ^[x \<and> y]_zs ^[x' \<and> y']_zs' = val ^[x' @zs'[x \<and> y]_zs \<and> y' @zs'[x \<and> y]_zs]_zs' ^[x \<and> y]_zs"
power_eq_if: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a"   and m :: "nat" shows "power.power one times p m = (if m = 0 then one else times p (power.power one times p (m - 1)))"
poly_rel_times: shows "(poly_rel ===> poly_rel ===> poly_rel) (times_poly_i ff_ops) (*)"
echelon_form_of_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "echelon_form_of_upt_k A k bezout = fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
le_append_swap: fixes y :: "nat list"   and v :: "nat list"   and x :: "nat list"   and w :: "nat list" assumes "length y = length v"   and "x @ y \<le>\<^sub>v w @ v" shows "y @ x \<le>\<^sub>v v @ w"
is_natural_2: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd"   and \<tau> :: "'b \<Rightarrow> 'd"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
set_minimize_check_generate: fixes a :: "nat list"   and b :: "nat list" shows "set (minimize (check_generate a b)) = {(x, y). (x, y) \<in> set (check_generate a b) \<and> \<not> (\<exists>(u, v)\<in>set (check_generate a b). u @ v <\<^sub>v x @ y)}"
bits_1_div_2: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide one (numeral.numeral one plus (num.Bit0 num.One)) = zero"
arr_Par_ArrDom_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T" shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
allUses_dominated: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> allUses g n"   and "n \<in> set (\<alpha>n g)" shows "old.dominates g (defNode g v) n"
nDAeqSet: fixes p :: "('b, 'c) Combinators list"   and s :: "('b, 'c) Combinators list" assumes "set p = set s" shows "noDenyAll p = noDenyAll s"
conc_fun_chain: fixes R :: "('b \<times> 'c) set"   and S :: "('c \<times> 'd) set"   and M :: "'d nres" shows "\<Down> R (\<Down> S M) = \<Down> (R O S) M"
basic_trans_rules_31: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
comp_null_2: fixes f :: "'a" shows "hom'.comp f hom'.null = hom'.null"
simps_6: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc ({(I, S, E)} WHILE b DO C) Q = ((\<forall>l s. (I l s \<and> bval b s \<longrightarrow> pre C I l s \<and> 1 + preT C E s + time C s \<le> E s \<and> S s = S (postQ C s)) \<and> (I l s \<and> \<not> bval b s \<longrightarrow> Q l s \<and> 1 \<le> E s \<and> S s = s)) \<and> vc C I)"
oreachable_other: fixes s :: "'a \<times> '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 \<sigma>' :: "'a" assumes "s \<in> oreachable A S U"   and "U (fst s) \<sigma>'" shows "(\<sigma>', snd s) \<in> oreachable A S U"
FAss: fixes A :: "triple set"   and P :: "assn"   and e1 :: "expr"   and Q :: "loc \<Rightarrow> assn"   and e2 :: "expr"   and R :: "assn"   and f :: "fname" assumes "A \<turnstile>\<^sub>e {P} e1 {\<lambda>v s. \<forall>a. v = Addr a \<longrightarrow> Q a s}"   and "\<forall>a. A \<turnstile>\<^sub>e {Q a} e2 {\<lambda>v s. R (upd_obj a f v s)}" shows "A \<turnstile> {P} e1..f:==e2 {R}"
cod_def: fixes f :: "'a1" shows "C1.cod f = (if C1.codomains f \<noteq> {} then SOME b. b \<in> C1.codomains f else C1.null)"
dickson_grading_def: fixes d :: "'a \<Rightarrow> nat" shows "dickson_grading d = ((\<forall>s t. d (s + t) = max (d s) (d t)) \<and> (\<forall>n. almost_full_on (adds) {x. d x \<le> n}))"
clop_idem: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "f \<circ> f = f"
fls_deriv_subdegree0: fixes f :: "'a fls" assumes "fls_subdegree f = 0" shows "0 \<le> fls_subdegree (fls_deriv f)"
chooseNext': fixes g :: "'g" assumes "redundant g" shows "chooseNext' g \<in> dom (phis g) \<and> trivial g (chooseNext g)"
BC_closed_subseteq: fixes U :: "(nat \<Rightarrow> nat option) set"   and V :: "(nat \<Rightarrow> nat option) set" assumes "U \<in> BC"   and "V \<subseteq> U" shows "V \<in> BC"
countable_UN': fixes X :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "countable X"   and "A ` X \<subseteq> sigma_sets UNIV X" shows "\<Union> (A ` X) \<in> sigma_sets UNIV X"
assms: shows "w =\<^sub>I w\<^sub>1 @- w\<^sub>2"
sorted_const: fixes x :: "'b"   and k :: "'c list" shows "sorted (map (\<lambda>y. x) k)"
\<tau>_prefix_conv: fixes p :: "'a Trace.prefix"   and s :: "'a Trace.trace"   and s' :: "'a Trace.trace"   and i :: "nat" assumes "prefix_of p s"   and "prefix_of p s'"   and "i < plen p" shows "\<tau> s i = \<tau> s' i"
sl_def: shows "sl = map fst (stake i0 \<pi>)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs" shows "l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs"
open_inter: fixes U :: "'c set"   and V :: "'c set" assumes "is_open\<^sub>Y U"   and "is_open\<^sub>Y V" shows "is_open\<^sub>Y (U \<inter> V)"
mono_gamma_f: fixes f :: "'av Abs_State.st"   and g :: "'av Abs_State.st" assumes "f \<sqsubseteq> g" shows "\<gamma>\<^sub>f f \<subseteq> \<gamma>\<^sub>f g"
None_eqvt: fixes p :: "perm" shows "p \<bullet> None = None"
non_empty_cycle_root_non_empty: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x \<noteq> 0"
graph_symmetric: fixes G :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "sym_digraph G"   and "u \<rightarrow>\<^bsub>G\<^esub> v" shows "v \<rightarrow>\<^bsub>G\<^esub> u"
univalent_comp_left_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "coreflexive (x[[x]])" shows "x * (y \<sqinter> z) = x * y \<sqinter> x * z"
s2p_def: fixes S :: "'b set" shows "s2p S = (\<lambda>x. x \<in> S)"
fst_gb_schema_direct: fixes sel :: "(('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> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('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) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('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) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('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 bs0 :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list"   and data0 :: "'g" shows "fst ` set (gb_schema_direct sel ap ab compl bs0 data0) = (let data = (length bs0, data0); bs1 = fst (add_indices (bs0, data0) (0, data0)); bs = ab [] [] bs1 data in fst ` set (gb_schema_aux sel ap ab compl [] (count_rem_components bs, data) bs (ap [] [] [] bs1 data)))"
atMost_def: fixes u :: "'a" shows "ord.atMost (\<lambda>x y. y \<le> x) u \<equiv> {x. u \<le> x}"
CM2_S2TTP_parts_knows_Spy: fixes R :: "agent"   and A :: "agent"   and B :: "agent"   and em :: "msg"   and AO :: "nat"   and cleartext :: "nat"   and q :: "nat"   and c :: "msg"   and evs :: "event list" assumes "Gets R \<lbrace>Agent A, Agent B, em, Number AO, Number cleartext, Nonce q, c\<rbrace> \<in> set evs"   and "evs \<in> certified_mail" shows "c \<in> parts (knows Spy evs)"
nth_prime_eqI'': fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "primes_pi (real p) = real n + 1" shows "nth_prime n = p"
MP_subset_SMP_3: fixes M :: "('fun, 'var) Term.term set" shows "M \<subseteq> SMP M"
map_cong0: fixes x :: "('l, 'k, 'm, 'n) G"   and f1 :: "'k \<Rightarrow> 'o"   and g1 :: "'k \<Rightarrow> 'o"   and f2 :: "'m \<Rightarrow> 'p"   and g2 :: "'m \<Rightarrow> 'p"   and f3 :: "'n \<Rightarrow> 'q"   and g3 :: "'n \<Rightarrow> 'q" assumes "\<And>z1. z1 \<in> Gset1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> Gset2 x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> Gset3 x \<Longrightarrow> f3 z3 = g3 z3" shows "Gmap f1 f2 f3 x = Gmap g1 g2 g3 x"
image_is_empty: fixes f :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "(f ` A = {}) = (A = {})"
DBM_set_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "[M]\<^bsub>v,n\<^esub> = [(\<lambda>i j. if i = j then Le (0::'c) else M i j)]\<^bsub>v,n\<^esub>"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and a :: "('fsym, 'psym) atm" shows "satL \<xi> (Pos a) = satA \<xi> a"