hcju/isabelleps huggingface dataset:
le_cases: fixes y :: "'a"   and x :: "'a" obtains "y \<le> x"   | "x \<le> y"
finite_field_factorization_i_def: fixes p :: "int"   and ff_ops :: "'b arith_ops_record"   and f :: "'b list" shows "finite_field_factorization_i p ff_ops f = (if degree_i f = 0 then (lead_coeff_i ff_ops f, []) else let a = lead_coeff_i ff_ops f; u = smult_i ff_ops (arith_ops_record.inverse ff_ops a) f; gs = if use_distinct_degree_factorization then distinct_degree_factorization_i p ff_ops u else [(1, u)]; (irr, hs) = List.partition (\<lambda>(i, f). degree_i f = i) gs in (a, map snd irr @ concat (map (\<lambda>(i, g). berlekamp_monic_factorization_i p ff_ops i g) hs)))"
has_contour_integral_lmul: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "(f has_contour_integral i) g" shows "((\<lambda>x. c * f x) has_contour_integral c * i) g"
distinct_1: fixes x21 :: "'c"   and x22 :: "'c" shows "Infinity \<noteq> Point x21 x22"
basis_row_space_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "basis_row_space A = {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \<noteq> 0}"
assoc'_in_hom: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[a, b, c] : a \<otimes> b \<otimes> c \<rightarrow> (a \<otimes> b) \<otimes> c\<guillemotright>"
tp_diff: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "trigpoly f"   and "trigpoly g" shows "trigpoly (\<lambda>x. f x - g x)"
list_of_vec_row_nth: fixes x :: "nat"   and A :: "'c mat"   and i :: "nat" assumes "x < dim_col A" shows "list_of_vec (row A i) ! x = A $$ (i, x)"
map\<^sub>0_def: fixes a :: "'a" shows "\<tau>\<tau>'.map\<^sub>0 a = \<tau>\<^sub>0 a \<star>\<^sub>D map\<^sub>0 a"
subset_singletonD: fixes A :: "'e set"   and x :: "'e" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
iso_class_def: fixes f :: "'a" shows "\<lbrakk>f\<rbrakk> \<equiv> {f'. f \<cong> f'}"
rank_of_insert_le: fixes X :: "'a set"   and x :: "'a" assumes "X \<subseteq> carrier"   and "x \<in> carrier" shows "rank_of (insert x X) \<le> Suc (rank_of X)"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "local.msort_key x = y" obtains xs :: "'a list" where "x = xs" and "y = local.merge_all (local.sequences xs)"
override_empty: fixes p :: "'c \<Rightarrow> 'd option" shows "p \<Oplus> \<emptyset> = p"
tab_simps_3: shows "trg (\<rho> \<cdot> \<phi>) = trg r"
reciprocal_inf_iff: fixes z :: "complex_homo" shows "(reciprocal z = \<infinity>\<^sub>h) = (z = 0\<^sub>h)"
intro\<^sub>1: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<R>\<^sub>1p \<Gamma> x xa xaa xaaa xaaaa xaaaaa) xa) lc'" shows "\<R>p \<Gamma> lc lc'"
runit_simps_5: 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 f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.cod V (bicategory.runit V H \<a> \<i> src f) = f"
floorlog_bounds: fixes x :: "nat"   and b :: "nat" assumes "0 < x"   and "1 < b" shows "b ^ (floorlog b x - 1) \<le> x \<and> x < b ^ floorlog b x"
almost_full_on_mono: fixes A :: "'c set"   and B :: "'c set"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x y. Q x y \<Longrightarrow> P x y"   and "almost_full_on Q B" shows "almost_full_on P A"
reduced_GB_unique_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m"   and "is_reduced_GB G"   and "pmdl G = pmdl F" shows "reduced_GB F = G"
swap: fixes a :: "atom"   and b :: "atom" assumes "sort_of a = sort_of b"   and "a \<noteq> b"   and "a \<in> S"   and "b \<in> S" shows "P (a \<rightleftharpoons> b)"
\<pi>_nth_prime: fixes n :: "nat" shows "primes_pi (real (nth_prime n)) = real n + 1"
assms: shows "n < card {a. a < i \<and> a \<in> S}"
psimps: fixes ts :: "char list" assumes "parse_nodes_dom ts" shows "parse_nodes ts = (if ts = [] \<or> take 2 ts = ''</'' then Parser_Monad.return [] ts else if hd ts \<noteq> CHR ''<'' then (parse_text \<bind> (\<lambda>t. parse_nodes \<bind> (\<lambda>ns. Parser_Monad.return (XML_text (the t) # ns)))) ts else (exactly ''<'' \<bind> (\<lambda>_. parse_name \<bind> (\<lambda>n. parse_attributes \<bind> (\<lambda>atts. oneof [''/>'', ''>''] \<bind> (\<lambda>e ts'. if e = ''/>'' then (parse_nodes \<bind> (\<lambda>cs. Parser_Monad.return (XML n atts [] # cs))) ts' else (parse_nodes \<bind> (\<lambda>cs. exactly ''</'' \<bind> (\<lambda>_. exactly n \<bind> (\<lambda>_. exactly ''>'' \<bind> (\<lambda>_. parse_nodes \<bind> (\<lambda>ns. Parser_Monad.return (XML n atts cs # ns))))))) ts'))))) ts)"
graeffe_poly_impl_hom: fixes f :: "int poly"   and m :: "nat" shows "of_int_poly (graeffe_poly_impl f m) = graeffe_poly_impl (of_int_poly f) m"
arc_implies_awalk: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs (subdivide G e w)" shows "S.awalk (fst e) [e] (snd e)"
expectation_gpv_try_gpv: fixes fail :: "ennreal"   and \<I> :: "('h, 'i) \<I>"   and f :: "'j \<Rightarrow> ennreal"   and gpv :: "('j, 'h, 'i) gpv"   and gpv' :: "('j, 'h, 'i) gpv" shows "expectation_gpv fail \<I> f (TRY gpv ELSE gpv') = expectation_gpv (expectation_gpv fail \<I> f gpv') \<I> f gpv"
polyadd_norm: fixes p :: "'b poly"   and q :: "'b poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p +\<^sub>p q)"
ts: shows "deg_fun t < deg_fun u \<or> deg_fun t = deg_fun u \<and> ord t u"
INT_E: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and a :: "'b" assumes "b \<in> \<Inter> (B ` A)" obtains "b \<in> B a"   | "a \<notin> A"
ch2ch_Pair: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'c" assumes "chain X"   and "chain Y" shows "chain (\<lambda>i. (X i, Y i))"
psi_scale_1: fixes r :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real" shows "psi (r *\<^sub>R a) (r *\<^sub>R b) 0 = (if 0 < r then psi a b 0 else if r < 0 then psi 0 b a else True)"
proves_beta_step_pre: 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 vs :: "(variable \<times> typ) list"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "\<forall>(x, \<tau>)\<in>set vs. (x, \<tau>) \<notin> fv t \<union> FV \<Gamma>"   and "term_ok \<Theta> (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t)"   and "t \<rightarrow>\<^sub>\<beta> 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 (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t) (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) u)"
simps_17: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'i \<Rightarrow> 'f"   and f3 :: "'j \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> 'f"   and f4 :: "('g, 'h, 'j, 'k) resumption \<Rightarrow> 'f"   and x1 :: "'g" shows "(case Terminal_corec x1 of Terminal_corec x \<Rightarrow> f1 x | Linear_corec x xa \<Rightarrow> f2 x xa | Branch_corec x xa \<Rightarrow> f3 x xa | Resumption_corec x \<Rightarrow> f4 x) = f1 x1"
comp_in_homE: fixes g :: "'c"   and f :: "'c"   and a :: "'c"   and c :: "'c" assumes "\<guillemotleft>g \<cdot>\<^sub>C f : a \<rightarrow>\<^sub>C c\<guillemotright>" obtains b :: "'c" where "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>"
rep_equator_reflect_def: shows "rep_equator_reflect \<equiv> vector [vector [1, 0, 0], vector [0, - 1, 0], vector [0, 0, 1]]"
symmetric_axioms: shows "symmetric A (\<lambda>x y. \<not> x \<bowtie> y)"
simps_1: fixes t :: "'d llist" shows "Co_Snapshot.ltake 0 t = []"
ap_set_def: fixes F :: "('d \<Rightarrow> 'c) set"   and X :: "'d set" shows "F \<diamondop> X = {f x |f x. f \<in> F \<and> x \<in> X}"
isRefV_simps_3: fixes s :: "int" shows "isRefV (shortV s) = False"
comp_min_basis_subset': fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (comp_min_basis xs) \<subseteq> {x \<in> set xs. x \<noteq> 0}"
bounded_iff: fixes a :: "'a upred"   and b :: "'a upred"   and c :: "'a upred" shows "((b \<and> c) \<sqsubseteq> a) = (b \<sqsubseteq> a \<and> c \<sqsubseteq> a)"
fv_subset_if_in_strand_ik': fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "fv t \<subseteq> fv\<^sub>s\<^sub>t S"
two_comparisons_into_case_order_10: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and P :: "'c"   and Q :: "'c"   and R :: "'c" assumes "comparator comp" shows "(if x = y then P else if lt_of_comp comp x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
distinct_254: fixes x13 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Roll x13 \<noteq> Prj1 x11"
mv: shows "max_Var_floatarith fa \<le> length vs"
homotopic_through_contractible: fixes S :: "'b set"   and f1 :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g1 :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'c \<Rightarrow> 'd" assumes "continuous_on S f1"   and "f1 ` S \<subseteq> T"   and "continuous_on T g1"   and "g1 ` T \<subseteq> U"   and "continuous_on S f2"   and "f2 ` S \<subseteq> T"   and "continuous_on T g2"   and "g2 ` T \<subseteq> U"   and "contractible T"   and "path_connected U" shows "homotopic_with_canon (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
induct: fixes x :: "'d"   and F :: "'d program"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> reachable F"   and "\<And>s. s \<in> Init F \<Longrightarrow> P s"   and "\<And>act s s'. \<lbrakk>act \<in> Acts F; s \<in> reachable F; P s; (s, s') \<in> act\<rbrakk> \<Longrightarrow> P s'" shows "P x"
span_pad_2: fixes U :: "'a vec set"   and n :: "nat"   and m :: "nat" assumes "U \<subseteq> carrier_vec n" shows "(@\<^sub>v) (0\<^sub>v m) ` LinearCombinations.module.span class_ring (M n) U = LinearCombinations.module.span class_ring (M (n + m)) ((@\<^sub>v) (0\<^sub>v m) ` U)"
embed_Field: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "embed r r' f" shows "f ` Field r \<subseteq> Field r'"
homh: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_cring R S h" shows "h \<in> ring_hom R S"
iMODb_subset_conv: fixes m' :: "nat"   and c' :: "nat"   and r' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "0 < m'"   and "0 < c'" shows "([ r', mod m', c' ] \<subseteq> [ r, mod m, c ]) = (r' mod m = r mod m \<and> r \<le> r' \<and> m' mod m = 0 \<and> r' + m' * c' \<le> r + m * c)"
cases: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ra :: "nat"   and wa :: "nat" assumes "P,E \<turnstile> ra \<leadsto>mrw wa" obtains adal :: "'c \<times> addr_loc" where "ra \<in> read_actions E" and "adal \<in> action_loc P E ra" and "E \<turnstile> wa \<le>a ra" and "wa \<in> write_actions E" and "adal \<in> action_loc P E wa" and "\<And>wa'. \<lbrakk>wa' \<in> write_actions E; adal \<in> action_loc P E wa'\<rbrakk> \<Longrightarrow> E \<turnstile> wa' \<le>a wa \<or> E \<turnstile> ra \<le>a wa'"
Always_ltlp_def: fixes \<phi> :: "'b pltl" shows "G\<^sub>p \<phi> \<equiv> not\<^sub>p F\<^sub>p (not\<^sub>p \<phi>)"
oconf_new: fixes P :: "'b prog"   and h :: "heap"   and obj :: "obj"   and a :: "nat"   and x :: "obj" assumes "P,h \<turnstile> obj \<surd>"   and "h a = None" shows "P,h(a \<mapsto> x) \<turnstile> obj \<surd>"
filterlim_inverse_at_right_top: shows "filterlim inverse (at_right 0) at_top"
igba_impl_rel_eext_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "\<langle>Rm, Rv, Rl\<rangle>igba_impl_rel_eext \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_igba_impl_rel_eext"
linear_independent_injective_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and s :: "('a, 'b) vec set" assumes "linear (*s) (*s) f"   and "vec.independent s"   and "inj_on f (vec.span s)" shows "vec.independent (f ` s)"
p2lNmt: fixes p :: "('a, 'b) Combinators" shows "policy2list p \<noteq> []"
the_g: shows "(THE fa. \<forall>ia. f (X $ i) $ ia = (\<Sum>x\<in>UNIV. fa x * Y $ x $ ia)) = g"
perfect_imp_continuous_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "perfect_map X Y f" shows "continuous_map X Y f"
insert_inc2: fixes a :: "'h"   and A :: "'h set" shows "a \<in> insert a A"
disjoint_second_summands: fixes v :: "i"   and x :: "i"   and y :: "i"   and z :: "i" assumes "P v (x \<oplus> y) \<and> P v (x \<oplus> z)"   and "\<not> O y z" shows "P v x"
param_distinct: fixes A :: "('a \<times> 'b) set" assumes "IS_LEFT_UNIQUE A"   and "single_valued A" shows "(distinct, distinct) \<in> \<langle>A\<rangle>list_rel \<rightarrow> bool_rel"
less_eq_bot_def: fixes x :: "'b bot"   and y :: "'b bot" shows "(x \<le> y) = (case x of bot.Value x \<Rightarrow> case y of bot.Value y \<Rightarrow> x \<le> y | bot.Bot \<Rightarrow> False | bot.Bot \<Rightarrow> True)"
rep_eq: fixes x :: "('d \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e) \<Rightarrow> 'f"   and xa :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and xb :: "'h \<Rightarrow> 'i \<Rightarrow> 'c"   and xc :: "'i dual"   and xd :: "'g dual" shows "un_B (ap_dual x xa xb xc xd) = xb (xa (x (\<lambda>x f. f x)) (un_B xd)) (un_B xc)"
eval_monom_pos: fixes basis :: "(real \<Rightarrow> real) list"   and monom :: "real \<times> real list" assumes "basis_wf basis"   and "0 < fst monom" shows "\<forall>\<^sub>F x in at_top. 0 < eval_monom monom basis x"
Vinv3D: fixes Quorum :: "process set set"   and x :: "v_state" assumes "quorum_process Quorum"   and "x \<in> quorum_process.Vinv3 Quorum" shows "ran (decisions x) \<subseteq> quorum_process.locked Quorum x"
inner_prod_with_itself_real: fixes u :: "complex vec" shows "\<langle>u|u\<rangle> \<in> \<real>"
function_factors_left: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd" shows "(\<forall>x y. g x = g y \<longrightarrow> f x = f y) = (\<exists>h. f = h \<circ> g)"
elims_3: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and xb :: "'a" assumes "valid_graph G"   and "\<not> valid_graph.is_path G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = va__" and "xa = []" and "xb = v'__" and "\<not> (va__ = v'__ \<and> v'__ \<in> nodes G)"   | v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and p :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "\<not> (v = v1 \<and> (v1, w, v2) \<in> edges G \<and> valid_graph.is_path G v2 p v')"
gnz: fixes x :: "'a" assumes "norm (x - \<xi>) \<le> s"   and "norm (x - \<xi>) \<le> r / 2" shows "g x \<noteq> (0::'a)"
complex_mod_rcis: fixes r :: "real"   and a :: "real" shows "cmod (rcis r a) = \<bar>r\<bar>"
BisT_matchT_TMT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>T d" shows "matchT_TMT c d"
prob_space_measure_of_st_vec: fixes x :: "'a st_vec" shows "prob_space (measure_of_st_vec x)"
bind_eq_None_conv: fixes a :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "(Option.bind a f = None) = (a = None \<or> f (the a) = None)"
must_wait_mono': fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and lt :: "'a + 'b + 'b"   and A :: "'b set"   and B :: "'b set" assumes "must_wait s t lt A"   and "A \<subseteq> B" shows "must_wait s t lt B"
hom_induced2: shows "\<exists>hom_relmap. (\<forall>p X S Y T f. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<longrightarrow> hom_relmap p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)) \<and> (\<forall>p X S Y T f c. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> singular_relcycle p X S c \<longrightarrow> hom_relmap (int p) X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)) \<and> (\<forall>p<0. hom_relmap p = (\<lambda>X S Y T f c. undefined))"
dzero: fixes i :: "nat" assumes "i \<le> m" shows "d fs i \<noteq> 0"
sup_bot_left: fixes x :: "'b" shows "\<top> \<sqinter> x = x"
isomorphic_implies_hpar_1: fixes f :: "'b"   and f' :: "'b" assumes "D.isomorphic f f'" shows "D.ide f"
envsOf_preserves_good: fixes P :: "('b, 'c, 'varSort, 'var, 'd) param"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" assumes "goodPar P"   and "rho \<in> envsOf P" shows "goodEnv rho"
gen_dfs_fe_imp_post_brk: 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" 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' True"   and "it \<subseteq> E `` {u0}" shows "gen_dfs_post E U S V u0 V' True"
genPrefix_trans: fixes x :: "'a list"   and y :: "'a list"   and r :: "('a \<times> 'a) set"   and z :: "'a list" assumes "(x, y) \<in> genPrefix r"   and "(y, z) \<in> genPrefix r"   and "trans r" shows "(x, z) \<in> genPrefix r"
covector_mult_closed: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_bounded_idempotent_semiring.covector x" shows "matrix_bounded_idempotent_semiring.covector (x \<odot> y)"
get_child_nodes_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "ptr |\<in>| object_ptr_kinds h"
l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"
X''_closed: fixes X'' :: "'a set"   and ds :: "'b set" assumes "X'' \<subseteq> CD_on ds (cop ds \<union> X'')" shows "X'' \<subseteq> fp_cop_F ds"
ocomplete_no_arrive: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'f"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and T :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" shows "((\<sigma>, s), ii\<not>ni:arrive(m), \<sigma>', s') \<notin> ocnet_sos T"
left_neutral: fixes a :: "'c" shows "- (1::'c) AND a = a"
in_lset_ltlD: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset (ltl xs)" shows "x \<in> lset xs"
residualFresh: fixes x :: "name"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" shows "x \<sharp> \<alpha> \<prec> P = (x \<sharp> subject \<alpha> \<and> (x mem bn \<alpha> \<or> x \<sharp> object \<alpha> \<and> x \<sharp> P))"
strict_coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c < b" shows "c < sup a b"
inext_image2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and I :: "nat set" assumes "strict_mono f" shows "inext (f n) (f ` I) = f (inext n I)"
weak_convex1I: assumes "\<forall>x. convex (at_least_as_good x carrier relation)" shows "weak_convex_pref relation"
crename_NotR_aux: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and N :: "trm"   and c :: "coname" assumes "R[a\<turnstile>c>b] = NotR x.N c" shows "a = c \<and> a = b \<or> a \<noteq> c"
future_divergences_def: fixes P :: "'e process"   and xs :: "'e list" shows "future_divergences P xs \<equiv> {ys. xs @ ys \<in> divergences P}"
T: shows "xps \<in> traces (c_process step out s\<^sub>0)"
F_def: shows "F \<equiv> \<lambda>f x g y. f ; x\<^sup>\<smile> \<cdot> g ; y\<^sup>\<smile>"
trueFormulaWithSingleLiteralClause: fixes literal :: "Literal"   and formula :: "Literal list list"   and valuation :: "Literal list" assumes "formulaTrue (removeAll [literal] formula) (valuation @ [literal])" shows "formulaTrue formula (valuation @ [literal])"
diff_add_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.comm_monoid_diff plus minus zero" shows "minus a (plus a b) = zero"
R1_a_def: shows "R1_a \<equiv> {: stk, mrk \<leadsto> X, mrk' . mrk' = mrk :}"
ccNeighbors_cc_restr: fixes x :: "Vars.var"   and S :: "Vars.var set"   and G :: "CoCalls" shows "ccNeighbors x (G G|` S) = (if x \<in> S then ccNeighbors x G \<inter> S else {})"
diff_cancel: fixes a :: "'g" shows "a - a = (0::'g)"
finite_tree: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and v :: "'e"   and p :: "('b, 'a) antecedent"   and pth :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) list" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst"   and "Vertex_Graph.valid_in_port (Abstract_Task.inPorts anyP antecedent) vertices nodeOf (v, p)"   and "Vertex_Graph.terminal_vertex (Abstract_Task.outPorts anyP antecedent consequent) vertices nodeOf v" shows "tfinite (Tasked_Proof_Graph.tree freshenLC renameLCs subst anyP antecedent vertices nodeOf edges vidx inst v p pth)"
det_mul_finsum_alt: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "Determinant.det (A * B) = Determinant.det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('b) n (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v col A k) {0..<m}))"
norm_eq_L2: fixes x :: "octo" shows "norm x = L2_set (octo_proj x) {..7}"
continuous_on_exp: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. exp (f x))"
of_int_sint: fixes a :: "'a word" shows "word_of_int (sint a) = a"