hcju/isabelleps huggingface dataset:
applicableExplainCharacterization: fixes stateA :: "State" shows "applicableExplain stateA = (\<exists>l reason. getConflictFlag stateA = True \<and> l el getC stateA \<and> formulaEntailsClause (getF stateA) reason \<and> isReason reason (opposite l) (elements (getM stateA)))"
simps_1: fixes n :: "nat" shows "selecthelp [] n = (if n \<le> 0 then [\<zero>] else \<zero> # selecthelp [] (n - 1))"
predtrans_UNIV: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "mono \<tau>" shows "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subseteq> (\<tau> ^^ i) UNIV"
flowsto_def: fixes X0 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and T :: "real set"   and CX :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and X1 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set" shows "flowsto X0 T CX X1 = (\<forall>(x0, d0)\<in>X0. \<exists>h\<in>T. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0<--<h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX))"
pullback_closed: fixes f :: "'a \<Rightarrow> 'b"   and T :: "('a, 'c) partial_object_scheme"   and S :: "('b, 'd) partial_object_scheme"   and g :: "'b \<Rightarrow> 'e"   and U :: "('e, 'f) partial_object_scheme" assumes "f \<in> struct_maps T S"   and "g \<in> struct_maps S U" shows "pullback T f g \<in> struct_maps T U"
tinst_imp_ex_raw_match_term: fixes t1 :: "Core.term"   and t2 :: "Core.term" assumes "tinst t1 t2" shows "\<exists>subs. raw_match_term t2 t1 Map.empty = Some subs"
rel_witness_generat1: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) generat"   and y :: "('e, 'g, 'i) generat" assumes "rel_generat A C R x y" shows "rel_generat (\<lambda>a (a', b). a = a' \<and> A a' b) (\<lambda>c (c', d). c = c' \<and> C c' d) (\<lambda>r (r', s). r = r' \<and> R r' s) x (rel_witness_generat (x, y))"
approx_HFinite_mult_cancel: fixes a :: "'a star"   and w :: "'a star"   and z :: "'a star" assumes "a \<in> HFinite - Infinitesimal"   and "a * w \<approx> a * z" shows "w \<approx> z"
initQ_def: shows "initQ \<equiv> [s \<mapsto> 0::'a]"
nf_unlift: fixes n :: "'a itrm"   and x :: "'a itrm" assumes "n \<simeq> x"   and "n \<in> CF" shows "CF_pure n \<leftrightarrow> unlift x"
infsetsum'_mono_set: fixes B :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> B - A \<Longrightarrow> (0::'d) \<le> f x"   and "A \<subseteq> B"   and "infsetsum'_converges f A"   and "infsetsum'_converges f B" shows "infsetsum' f A \<le> infsetsum' f B"
assms_2: shows "\<pi> m = ipd (\<pi> k)"
natural_isomorphism_axioms: shows "natural_isomorphism Adj.DopxC.comp Adj.DopxD.comp Adj'.DopxG.map Adj'.DopxG.map Adj'.DopxG.map"
frechet_derivative_at': fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "(f has_derivative s) (at x)" shows "frechet_derivative f (at x) = s"
dilating_fun_image_strict: fixes f :: "nat \<Rightarrow> nat"   and r :: "'b run"   and m :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating_fun f r" shows "{k. f m < k \<and> k < f n \<and> hamlet (Rep_run r k c)} = f ` {k. m < k \<and> k < n \<and> hamlet (Rep_run r (f k) c)}"
in_strand_fv_subset_rcv: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "fv t \<subseteq> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S))"
fds_nth_zeta: shows "($) fds_zeta = (\<lambda>n. if n = 0 then 0::'b else (1::'b))"
aux0_4: fixes a :: "'a"   and x :: "'a list"   and y :: "'a list" shows "(a \<in> set x \<or> a \<in> set y) = (a \<in> set (x @ y))"
finite_total_preorder_on_def: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "finite_total_preorder_on carrier le \<equiv> total_preorder_on carrier le \<and> finite_total_preorder_on_axioms carrier"
power_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_power [x, y] = x ^ y"
arrE: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" obtains "f \<noteq> null" and "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom f) (Cod f) (Map f)"
prems_1: shows "\<forall>t\<in>set (a # ts). rank t = 0 \<and> children t = []"
star_right_induct_mult: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> y" shows "y * x\<^sup>\<star> \<le> y"
cong_cong_mod_int: fixes a :: "int"   and b :: "int"   and m :: "int" shows "[a = b] (mod m) = [a mod m = b mod m] (mod m)"
fresh_atom_at_base: fixes a :: "atom"   and b :: "'a" shows "a \<sharp> atom b = a \<sharp> b"
in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "C.arr f"   and "C.dom f = a"   and "C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
ccINF_lower2: 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 :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
prems_3: shows "of_weak_ranking (A # As) = restrict_relation (\<Union> (set (A # As))) le"
abs_neg_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (uminus one) = one"
ctr: fixes out :: "'b"   and c :: "('a, 'b, 'c) rpv" shows "Generative_Probabilistic_Value.Pause out c = GPV (return_spmf (IO out c))"
finite_ReZ_segments_part_circlepath: fixes z0 :: "complex"   and r :: "real"   and st :: "real"   and tt :: "real"   and z :: "complex" shows "finite_ReZ_segments (part_circlepath z0 r st tt) z"
relevant_vars_subts: fixes ts :: "fterm list"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" assumes "\<forall>x\<in>vars\<^sub>t\<^sub>s ts. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "ts \<cdot>\<^sub>t\<^sub>s \<sigma>\<^sub>1 = ts \<cdot>\<^sub>t\<^sub>s \<sigma>\<^sub>2"
cond_seq_left_distr: fixes b :: "('c, 'b) urel"   and P :: "('c, 'b) urel"   and Q :: "('c, 'b) urel"   and R :: "'b hrel" assumes "out\<alpha> \<sharp> b" shows "(P \<triangleleft> b \<triangleright> Q) ;; R = P ;; R \<triangleleft> b \<triangleright> Q ;; R"
axioms_2: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NFlow c s t f" shows "Flow c s t f"
iMOD_div_mod_gr0_not_ex: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "k < m"   and "0 < m mod k" shows "\<nexists>r' m'. [ r', mod m' ] = [ r, mod m ] \<oslash> k"
subgroup_mult_equality: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and h1 :: "'c"   and h2 :: "'c" assumes "Group.group G"   and "subgroup H G"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> h2 = h1 \<otimes>\<^bsub>G\<^esub> h2"
collapse: fixes msllist :: "'a msllist" assumes "msllist \<noteq> MSLNil" shows "MSLCons (un_MSLCons1 msllist) (un_MSLCons2 msllist) = msllist"
lfp_unfold: fixes f :: "'c \<Rightarrow> 'c" assumes "mono f" shows "lfp f = f (lfp f)"
nn_integral_eP: fixes \<omega> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "\<omega> \<in> space (Pi\<^sub>M {0..<n} M)"   and "f \<in> borel_measurable (Pi\<^sub>M {0..<Suc n} M)" shows "integral\<^sup>N (eP n \<omega>) f = \<integral>\<^sup>+ x. f (\<omega>(n := x)) \<partial>P n \<omega>"
InitProcessingReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
andalso_cong: fixes P :: "tr"   and P' :: "tr"   and Q :: "tr"   and Q' :: "tr" assumes "P = P'"   and "P' = TT \<Longrightarrow> Q = Q'" shows "(P andalso Q) = (P' andalso Q')"
lcp_lenI: fixes i :: "nat"   and u :: "'b list"   and v :: "'b list" assumes "i < min \<^bold>|u\<^bold>| \<^bold>|v\<^bold>|"   and "take i u = take i v"   and "u ! i \<noteq> v ! i" shows "i = \<^bold>|u \<and>\<^sub>p v\<^bold>|"
Gamma_fresh: shows "\<forall>(x, y)\<in>set pi. x \<sharp> \<Gamma> \<and> y \<sharp> \<Gamma>"
tau_star: fixes x :: "'a" shows "\<tau> x\<^sup>\<star> = 1\<^sub>\<sigma> + \<tau> x"
eq1: shows "{Suc 0..Suc d} = Suc ` {..d}"
empty_dilated_prefix: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "n < f 0" shows "\<not> hamlet (Rep_run r n c)"
SComplex_Infinitesimal_zero: fixes x :: "complex star" assumes "x \<in> SComplex"   and "x \<in> Infinitesimal" shows "x = 0"
assms_2: assumes "v = None"   and "v' = None" shows "n \<le> \<Down> Rb n'"
outer_prod_dim: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec"   and m :: "nat" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec m" shows "outer_prod v w \<in> carrier_mat n m"
pmod_const_2: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list"   and q :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "degree p < degree q" shows "L.pmod p q = p"
argmax_3: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "snd (argmax f xs) = Max (f ` set xs)"
children_eq2_h3: fixes ptr' :: "(_) object_ptr" shows "|h3 \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
ts_tp_le_def: fixes ts :: "nat"   and tp :: "nat"   and tstp :: "nat + nat" shows "ts_tp_le ts tp tstp = (case tstp of Inl ts' \<Rightarrow> ts \<le> ts' | Inr tp' \<Rightarrow> tp \<le> tp')"
inter_finite: shows "\<forall>bl\<in>#remove1_mset blv \<B>. finite (bl \<inter> blv)"
orecvmsg_simps_1: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and m :: "'e" shows "orecvmsg P \<sigma> (broadcast m) = True"
b: shows "weak_unit b \<and> ide b \<and> S.ide b"
simps_2: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalQuadraticUni a b c d (LessUni p) = map_atomUni (quadraticSubUni a b c d) (convertDerivativeUni p)"
filtered_lb: fixes X :: "'a set" assumes "filtered X" shows "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y"
blocks_nempty: fixes bl :: "'a set" assumes "bl \<in># \<B>\<^sup>+" shows "bl \<noteq> {}"
in_components_subset: fixes c :: "'a set"   and s :: "'a set" assumes "c \<in> components s" shows "c \<subseteq> s"
interleave_simps: fixes a3 :: "'b list"   and a1 :: "'b list"   and a2 :: "'b list" shows "(a3 \<in> a1 \<otimes> a2) = (a1 = [] \<and> a2 = [] \<and> a3 = [] \<or> (\<exists>xs ys zs x. a1 = x # xs \<and> a2 = ys \<and> a3 = x # zs \<and> zs \<in> xs \<otimes> ys) \<or> (\<exists>xs ys zs x. a1 = xs \<and> a2 = x # ys \<and> a3 = x # zs \<and> zs \<in> xs \<otimes> ys))"
assms_2: shows "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>"
inv_op_closed1: fixes x :: "'a"   and h :: "'a" assumes "x \<in> carrier G"   and "h \<in> H" shows "inv x \<otimes> h \<otimes> x \<in> H"
MakeFtorMor: fixes f :: "'d"   and F :: "('e, 'g, 'd, 'h, 'f, 'i, 'j) Functor_scheme" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "MakeFtor F ## f = F ## f"
elims: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and x :: "'c"   and xa :: "nat"   and y :: "nat set" assumes "semi_mojmir_def.configuration \<delta> q\<^sub>0 w x xa = y" obtains q :: "'c" and n :: "nat" where "x = q" and "xa = n" and "y = {x. x \<le> n \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = q}"
SDG_cdep_edge: fixes n :: "'node SDG_node"   and m :: "'node"   and n' :: "'node SDG_node"   and m' :: "'node" assumes "n = CFG_node m"   and "n' = CFG_node m'"   and "m controls m'" shows "n \<longrightarrow>\<^bsub>cd\<^esub> n'"
assms: shows "dep_var_set PROB (\<Union> (set lvs)) vs"
ex_iter_fSucn: fixes flat :: "(nat \<times> form) list list list \<Rightarrow> (nat \<times> form) list list"   and l :: "(nat \<times> form) list list" shows "(\<exists>m. iter (\<lambda>x. flat (map subs x)) l m = []) = (if l = [] then True else \<exists>n. iter (\<lambda>x. flat (map subs x)) (flat (map subs l)) n = [])"
elims_2: fixes x :: "ereal"   and xa :: "ereal" assumes "x < xa" obtains xb :: "real" and y :: "real" where "x = ereal xb__" and "xa = ereal y" and "xb__ < y"   | xb :: "real" where "x = ereal xb" and "xa = \<infinity>"   | r :: "real" where "x = - \<infinity>" and "xa = ereal r"   | "x = - \<infinity>" and "xa = \<infinity>"
SD_singleton_left: fixes x :: "'alt"   and q :: "'alt pmf" assumes "x \<in> alts"   and "q \<in> lotteries_on alts" shows "SD (Pareto R) (return_pmf x) q = (\<forall>y\<in>set_pmf q. Pareto R x y)"
set_sum_eq: shows "(\<Sum>s\<in>set signs \<inter> consistent_sign_vec qs ` set (characterize_root_list_p p). z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})) = (\<Sum>s\<in>set (remdups (map (consistent_sign_vec qs) (characterize_root_list_p p))). z_R I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))"
mangoldt_primepow': fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "mangoldt (p ^ k) = of_real (ln (real p))"
c4_def: shows "c4 = Max {c2 / min 1 (min ((b / 2) powr (p + 1)) ((b * 3 / 2) powr (p + 1))) |b. b \<in> set bs}"
dvd_add_triv_left_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus a b) = dvd.dvd times a b"
m_children: shows "queue_to_multiset (children (getMinTree q)) = tree_to_multiset (getMinTree q) - {#(val (getMinTree q), prio (getMinTree q))#}"
fadd_def: fixes m :: "roundmode"   and a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "fadd m a b = (if is_nan a \<or> is_nan b \<or> is_infinity a \<and> is_infinity b \<and> sign a \<noteq> sign b then some_nan else if is_infinity a then a else if is_infinity b then b else zerosign (if is_zero a \<and> is_zero b \<and> sign a = sign b then sign a else if m = To_ninfinity then 1 else 0) (IEEE.round m (valof a + valof b)))"
unit_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>C \<i>\<^sub>C[a] = a"
indis_trans: fixes indis :: "('a \<times> 'a) set"   and s :: "'a"   and s' :: "'a"   and s'' :: "'a" assumes "PL_Indis indis"   and "(s, s') \<in> indis"   and "(s', s'') \<in> indis" shows "(s, s'') \<in> indis"
isodistinct_GIrrRep_repset: fixes V :: "('b, 'g) aezfun set"   and W :: "('b, 'g) aezfun set" assumes "of_nat (card G) \<noteq> (0::'b)"   and "V \<in> GIrrRep_repset"   and "W \<in> GIrrRep_repset"   and "V \<noteq> W" shows "\<nexists>T. FGModuleIso G (*) V (*) T W"
weak_hb_def: fixes m1 :: "'msgid \<times> 'op"   and m2 :: "'msgid \<times> 'op" shows "weak_hb m1 m2 \<equiv> hb m1 m2 \<or> m1 = m2"
execute_bind_2: fixes f :: "'a Heap"   and h :: "heap"   and g :: "'a \<Rightarrow> 'b Heap" assumes "execute f h = None" shows "execute (f \<bind> g) h = None"
Ord_linear: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" shows "k \<^bold>\<in> l \<or> k = l \<or> l \<^bold>\<in> k"
fps_integral0_deriv: fixes a :: "'a fps" shows "fps_integral0 (fps_deriv a) = a - fps_const (fps_nth a 0)"
entries_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "j < length \<GG>" shows "\<GG> ! i \<subseteq> \<GG> ! j"
simps: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme"   and gens :: "'c set" shows "(a \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) = (a = \<one>\<^bsub>G\<^esub> \<or> (\<exists>x. a = x \<and> x \<in> gens) \<or> (\<exists>x. a = inv\<^bsub>G\<^esub> x \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) \<or> (\<exists>x y. a = x \<otimes>\<^bsub>G\<^esub> y \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub> \<and> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>))"
h2: shows "MPoly (higherPoly\<^sub>0 0 z (mapping_of a - mapping_of b)) = MPoly (higherPoly\<^sub>0 0 z (mapping_of a)) - MPoly (higherPoly\<^sub>0 0 z (mapping_of b))"
body': shows "unwrap oo body oo wrap = body' oo unwrap oo wrap"
fun_eq_iff: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "(f = g) = (\<forall>x. f x = g x)"
code: fixes xs :: "'a msstream" shows "msllist_of_msstream xs = MSLCons (case xs of MSSCons x xs \<Rightarrow> x) (case xs of MSSCons x xs \<Rightarrow> msllist_of_msstream xs)"
valid_maximal_conforming_path_0: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "\<not> lnull P"   and "Digraph.valid_path G P"   and "Digraph.maximal_path G P"   and "ParityGame.path_conforms_with_strategy G p P \<sigma>" shows "vmc_path G P (P $ 0) p \<sigma>"
executable_edgeSet': fixes prog :: "program"   and g :: "gState\<^sub>I" assumes "gState_inv prog g"   and "program_inv prog" shows "executable (states prog) g \<le> SPEC (\<lambda>cs. \<forall>(e, p)\<in>set cs. e \<in> edgeSet (states prog !! pState.idx p) \<and> pState_inv prog p \<and> cl_inv (g, p))"
linear_add: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and b1 :: "('b, 'c) vec"   and b2 :: "('b, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (b1 + b2) = f b1 + f b2"
l4_16R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.FSC Bet Cong A B C D A' B' C' D'"   and "A \<noteq> B"   and "Bet A B C" shows "Cong C D C' D'"
sgn2_def: fixes x :: "nat" shows "sgn2 x \<equiv> case x of 0 \<Rightarrow> 1 | Suc y \<Rightarrow> 0"
bin_nth_bl: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n < m" shows "bit w n = rev (bin_to_bl m w) ! n"
exhaust: fixes y :: "'c point" obtains "y = Infinity"   | x21 :: "'c" and x22 :: "'c" where "y = Point x21 x22"
dom_hgf: shows "\<D>\<^sub>\<circ> hgf = 3\<^sub>\<nat>"
Basic_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and t :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>Basic f \<down> t"
intra_SDG_path_Append: fixes n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and ns :: "'node SDG_node list" assumes "n'' i-ns'\<rightarrow>\<^sub>d* n'"   and "n i-ns\<rightarrow>\<^sub>d* n''" shows "n i-ns @ ns'\<rightarrow>\<^sub>d* n'"
one_eq_numeral_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(one = numeral.numeral one plus n) = (num.One = n)"
test_s_prod_aux1: fixes P :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" assumes "P \<subseteq> 1\<^sub>\<sigma>"   and "(a, A) \<in> P \<cdot> R" shows "(a, {a}) \<in> P \<and> (a, A) \<in> R"
jordan_nf_exists: fixes A :: "'a mat"   and n :: "nat"   and as :: "'a list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])" shows "\<exists>n_as. jordan_nf A n_as"
dom_glb_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(ad (ad x) \<le> ad (ad y) \<cdot> ad (ad z)) = (ad (ad x) \<le> ad (ad y) \<and> ad (ad x) \<le> ad (ad z))"
has_extend_witness': fixes n :: "nat"   and S :: "'a set"   and xs :: "'a list" assumes "has n S"   and "hasw xs S"   and "length xs < n" shows "\<exists>x. hasw (x # xs) S"
min_prio_def: shows "min_prio = Min (\<omega> ` V)"
fun_upds_def: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b set"   and y :: "'a"   and z :: "'b" shows "(f(xs [:=] y)) z = (if z \<in> xs then y else f z)"
P: shows "P = (\<lambda>((s, l, r), ss, x). s = start_of ly e)"
Interp_def: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset" shows "Interp N C = interp N C \<union> production N C"
not_free_1: shows "\<forall>i\<in>frees s. n \<le> i"
next_InvIncomingInfoUpright: fixes s :: "('c, 'd) configuration stream" assumes "holds InvIncomingInfoUpright s"   and "next s" shows "nxt (holds InvIncomingInfoUpright) s"
bypass: fixes z :: "'v" assumes "z \<in> set p" shows "z \<notin> fst ` \<E> S"
all_final_except_def: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and Ts :: "'f set" shows "all_final_except s Ts \<equiv> \<forall>t. not_final_thread s t \<longrightarrow> t \<in> Ts"
strict_order_iff: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
alphaAbs_qAbs_imp_alphaAbs_all_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_qAFresh xs x X xs' x' X'"
A1b: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<^bold>\<not>\<P> x \<^bold>\<rightarrow> \<P> (\<rightharpoondown>x)) w)\<rfloor>"
E: shows "\<forall>X\<in>A. X = (\<Union>n\<le>length xs. \<Union>W\<in>R X n. W) \<and> (\<forall>W\<in>R X 0. xs \<notin> sentences P \<and> None \<notin> set xs \<and> (xs, W) \<in> failures P \<or> xs \<in> sentences P \<and> (\<exists>U V. (xs, U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length xs}. \<forall>W\<in>R X n. take (length xs - n) xs \<in> sentences P \<and> (drop (length xs - n) xs, W) \<in> failures Q)"
plus_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x + y = (fst x + fst y, snd x + snd y)"
finite_finfun_default: fixes f :: "('a, 'b) finfun" shows "finite {a. finfun_apply f a \<noteq> finfun_default f}"
apply_cltn2_incident: fixes p :: "proj2"   and l :: "proj2_line"   and A :: "cltn2" shows "proj2_incident p (apply_cltn2_line l A) = proj2_incident (apply_cltn2 p (cltn2_inverse A)) l"