hcju/isabelleps huggingface dataset:
elims_3: fixes x :: "'v"   and xa :: "'v list" assumes "\<not> adj_path x xa" obtains v :: "'v" where "x = v__" and "xa = []" and "v__ \<notin> V"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "\<not> (adjacent v u \<and> adj_path u us)"
in_codomains_char: fixes b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(b \<in> codomains f) = ((f \<noteq> Null \<and> Dom f \<in> {A. |A| <o \<AA>} \<and> Cod f \<in> {A. |A| <o \<AA>} \<and> Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)) \<and> b = MkIde (Cod f))"
list_repr_def: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list" shows "list_repr i ba xs = (i \<le> length xs \<and> fst ba = rev (take i xs) \<and> snd ba = drop i xs)"
exr_iff: shows "(\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec (qs1 @ qs2) r) = (\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec qs1 r @ consistent_sign_vec qs2 r)"
2: shows "\<guillemotleft>\<nu> : trg\<^sub>B f \<star>\<^sub>B f \<rightarrow>\<^sub>B trg\<^sub>B f' \<star>\<^sub>B f'\<guillemotright>"
cont_on_left: shows "continuous_on (path_image (\<lambda>y. twoC (0, y))) F"
simps: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.findMin' (BsSkewBinomialHeapStruc.Element e a q) = (e, a)"
independent_is_basis: fixes B :: "(real, 'a) vec set" shows "(independent B \<and> card B = CARD('a)) = is_basis B"
isom_gch_units_transpTr0: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "Gchain n h"   and "i \<le> n"   and "j \<le> n"   and "i < j"   and "isom_Gchains n (transpos i j) g h" shows "{i. i \<le> n \<and> g i \<cong> E} - {i, j} = {i. i \<le> n \<and> h i \<cong> E} - {i, j}"
distinct_10: fixes x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree"   and x2 :: "('a, 'b, 'c, 'd) Tree" shows "tAct x41 x42 x43 \<noteq> tNot x2"
sets_vimage_algebra2: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and M :: "'d measure" assumes "f \<in> X \<rightarrow> space M" shows "sets (vimage_algebra X f M) = {f -` A \<inter> X |A. A \<in> sets M}"
cntbl: shows "uncountable {x. f x \<noteq> 0}"
power_decreasing: fixes n :: "nat"   and N :: "nat"   and a :: "'a" assumes "n \<le> N"   and "(0::'a) \<le> a"   and "a \<le> (1::'a)" shows "a ^ N \<le> a ^ n"
takecptn_is_cptn: fixes c :: "('b com option \<times> 'b) list"   and j :: "nat" assumes "c \<in> cptn" shows "take (Suc j) c \<in> cptn"
K: shows "norm z < K \<and> ereal K < conv_radius f"
sem_def: fixes prob :: "('c, 'a, 'b, 'd) ifc_problem"   and n :: "'c"   and \<sigma> :: "'a \<Rightarrow> 'b" shows "IFC_def.sem prob n \<sigma> = snd (IFC_def.step prob (n, \<sigma>))"
simps_26: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Tee_local i) ts = (if i < length (local \<C>) then type_update ts [TSome (local \<C> ! i)] (Type [local \<C> ! i]) else Bot)"
path_component_of_unique: fixes x :: "'a"   and C :: "'a set"   and X :: "'a topology" assumes "x \<in> C"   and "path_connectedin X C"   and "\<And>C'. \<lbrakk>x \<in> C'; path_connectedin X C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C" shows "path_component_of_set X x = C"
one: fixes f :: "nat \<Rightarrow> 'a" assumes "multiplicative_function f" shows "f 1 = (1::'a)"
free_tv_subst_var: fixes S :: "nat \<Rightarrow> typ"   and v :: "nat" shows "free_tv (S v) \<subseteq> insert v (cod S)"
axioms_1: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" assumes "l_adopt_node_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes known_ptrs adopt_node" shows "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
consistent_sign_vec_copr_def: fixes qs :: "real poly list"   and x :: "real" shows "consistent_sign_vec_copr qs x \<equiv> map (\<lambda>q. if 0 < poly q x then 1 else - 1) qs"
inj: shows "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})"
higher_differentiable_on_snd: fixes S :: "'b set"   and T :: "'c set"   and k :: "nat" shows "higher_differentiable_on (S \<times> T) snd k"
ring_simprules_15: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
iso_assoc': 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"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "category.iso V (bicategory.\<a>' V H \<a> src trg f g h)"
fold_matches_first_step_not_None': fixes xs :: "typ list"   and ys :: "typ list"   and x :: "typ"   and y :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "length xs = length ys"   and "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'" shows "raw_match' x y subs \<noteq> None"
of_nat_add: fixes m :: "nat"   and n :: "nat" shows "\<guillemotleft>m + n\<guillemotright>\<^sub>\<nat> = \<guillemotleft>m\<guillemotright>\<^sub>\<nat> \<oplus> \<guillemotleft>n\<guillemotright>\<^sub>\<nat>"
step: fixes name :: "name"   and rhs :: "pterm"   and rs :: "(name \<times> pterm) fset" assumes "(name, rhs) |\<in>| rs" shows "rs \<turnstile>\<^sub>p Pconst name \<longrightarrow> rhs"
case_3: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case flat_complete_lattice.Top of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f3"
IJ: fixes i :: "'a set" assumes "i \<in> \<I>" shows "i \<lless> J i \<and> finite i"
RSubmodule_RSpan: fixes ms :: "'v list" assumes "set ms \<subseteq> V" shows "Subspace (RSpan ms)"
simps_3: fixes \<phi> :: "('d, 'e, 'f) formula"   and \<sigma> :: "('d, 'e, 'f) subst" shows "Fsubst (! \<phi>) \<sigma> = ! (Fsubst \<phi> \<sigma>)"
pairwise_coprime_dvd': fixes A :: "'c set"   and f :: "'c \<Rightarrow> nat"   and n :: "nat"   and j :: "nat" assumes "finite A"   and "\<And>i j. \<lbrakk>i \<in> A; j \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> coprime (f i) (f j)"   and "n = prod f A"   and "\<forall>a\<in>A. f a dvd j" shows "n dvd j"
assms_1: shows "partn_lst_VWF \<beta> \<alpha> n"
g_size_impl: shows "map_size \<alpha> invar g_size"
sq_norm_pos: fixes j :: "nat" assumes "j < length fs" shows "(0::'a) < \<parallel>gso j\<parallel>\<^sup>2"
eq: shows "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
distinct_15: fixes x91 :: "nat"   and x92 :: "'a fm" shows "TrueF \<noteq> ExN x91 x92"
ntrunc_def: fixes k :: "nat"   and N :: "('f, 'g) Old_Datatype.node set" shows "ntrunc k N \<equiv> {n \<in> N. ndepth n < k}"
comp_in_homE: fixes g :: "'c"   and f :: "'c"   and a :: "'c"   and c :: "'c" assumes "\<guillemotleft>C g 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>"
oDP: shows "order (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)) = n"
par_strict_not_col_3: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ParStrict Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet C D A"
alphaAbs_qAbs_imp_alphaAbs_all_distinct_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_distinct_qAFresh xs x X xs' x' X'"
fully_faithful_functor_axioms: shows "fully_faithful_functor (\<cdot>) (\<cdot>) ((\<otimes>) (cod \<iota>))"
order_smult: fixes c :: "'c"   and x :: "'c"   and p :: "'c poly" assumes "c \<noteq> (0::'c)" shows "order x (smult c p) = order x p"
mult_pos_pos: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less zero (times a b)"
variety_ofD: fixes a :: "'c \<Rightarrow> 'd"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "a \<in> \<V> F"   and "f \<in> F" shows "poly_eval a f = (0::'d)"
distinct_31: fixes x31 :: "POPLmarkRecord.type"   and x32 :: "trm"   and x6 :: "(char list \<times> trm) list" shows "(\<lambda><:x31. x32) \<noteq> Rcd x6"
vle: fixes n :: "nat" assumes "0 < n" shows "v n \<le> real n * v 1"
converse_reachable_cases: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v" obtains "u = v" and "u \<in> pverts (mk_symmetric G)"   | w :: "'a" where "(u, w) \<in> parcs (mk_symmetric G)" and "w \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v"
iwlsFSw_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "wlsFSw MOD" shows "iwlsFSw (fromMOD MOD)"
groupWIs_groupWIs1_equi: fixes c :: "parts_connection"   and rs :: "'b simple_rule list" shows "groupWIs1 c rs = groupWIs c rs"
older_seniors_recursive_card: fixes x :: "nat"   and n :: "nat" assumes "older_seniors x n \<noteq> {}" shows "card (older_seniors x n) = Suc (card (older_seniors (Max (older_seniors x n)) n))"
eq2: shows "partial_state.encode2 st1.dims0 st1.vars1' = partial_state.encode1 st2.dims0 st2.vars1'"
all_edges_finite: fixes V :: "nat set" assumes "finite V" shows "finite (all_edges V)"
sign_simps_32: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "am * bm * ck = am * (bm * ck)"
z_signs_R2: fixes signs :: "rat list"   and I :: "nat list" assumes "list_all (\<lambda>i. i < length signs) I"   and "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs" shows "prod_list (map ((!) signs) I) = 1 \<or> prod_list (map ((!) signs) I) = 0 \<or> prod_list (map ((!) signs) I) = - 1"
steps: shows "\<Gamma>\<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"
float_round_up_le: fixes x :: "real"   and xa :: "float"   and prec :: "nat" assumes "x \<le> real_of_float xa" shows "x \<le> real_of_float (float_round_up prec xa)"
ide_def: fixes a :: "'a" shows "L.ide a \<equiv> L.comp a a \<noteq> L.null \<and> (\<forall>f. (L.comp f a \<noteq> L.null \<longrightarrow> L.comp f a = f) \<and> (L.comp a f \<noteq> L.null \<longrightarrow> L.comp a f = f))"
DistinctNetsDenyAllow: fixes b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and po :: "int" assumes "DenyAllFromTo b c \<in> set p"   and "AllowPortFromTo a d po \<in> set p"   and "allNetsDistinct p"   and "dom (C (DenyAllFromTo b c)) \<inter> dom (C (AllowPortFromTo a d po)) \<noteq> {}" shows "b = a \<and> c = d"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>v. P (Suc v div 10) \<Longrightarrow> P (Suc v)" shows "P a0"
cEnum_char_def: shows "CENUM(char) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
kE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "kE \<xi> \<equiv> \<lambda>x. fromFull (tpOfV x) (\<xi> x)"
simps_2: fixes P :: "'b memref \<Rightarrow> bool"   and r :: "'b memref"   and rs :: "'b memref list" shows "outstanding_refs P (r # rs) = (if P r then address_of r \<union> outstanding_refs P rs else outstanding_refs P rs)"
supp_def: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" assumes "class.pt permute" shows "pt.supp permute x = {a. infinite {b. permute (a \<rightleftharpoons> b) x \<noteq> x}}"
prob_space_lim_stream: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "prob_space (K.lim_stream x)"
dvd: shows "x ^ a - 1 dvd x ^ (a * q) - 1"
norm_PX_0_False: fixes i :: "nat"   and Q :: "pol" shows "isnorm (PX (Pc 0) i Q) = False"
assms: fixes ss :: "('a, 'b) tm list"   and \<zeta> :: "('a, 'b) hd" assumes "\<And>s. s \<in> set ss \<Longrightarrow> P s" shows "P (apps (Hd \<zeta>) ss)"
m_zmax_pdsI_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and I :: "'c set" shows "m_zmax_pdsI K n P I = m_zmax n (m_zmax_pdsI_hom K P I) + 1"
ord_eq_less_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a = b"   and "b \<prec> c" shows "a \<prec> c"
closest_is_closest_finite: fixes s :: "('a, 'b) IEEE.float set"   and v :: "('a, 'b) IEEE.float \<Rightarrow> real"   and x :: "real"   and p :: "('a, 'b) IEEE.float \<Rightarrow> bool" assumes "finite s"   and "s \<noteq> {}" shows "is_closest v s x (closest v p s x)"
integrable_diff: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x - g x)"
int_not_undef: fixes ra :: "32 Word.word" assumes "real_of_int (sint ra) < real_of_int (sint POS_INF)"   and "real_of_int (sint NEG_INF) < real_of_int (sint ra)" shows "ra \<noteq> NEG_INF - 1"
inject_8: fixes x81 :: "nat"   and x82 :: "msg"   and y81 :: "nat"   and y82 :: "msg" shows "(Crypt x81 x82 = Crypt y81 y82) = (x81 = y81 \<and> x82 = y82)"
iso_onto_image: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "Group.group H" shows "(f \<in> Group.iso G (subgroup_generated H (f ` carrier G))) = (f \<in> hom G H \<and> inj_on f (carrier G))"
LPDi_Star: fixes j :: "nat"   and i :: "nat"   and r :: "mregex" assumes "j \<le> i" shows "LPDi j (MStar r) \<subseteq> {MStar r} \<union> MTimesR (\<Union>j\<le>i. LPDi j r) (MStar r)"
summable_sums: assumes "summable f" shows "f sums suminf f"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp Adj'.\<phi>C Adj'.\<phi>D F G' \<phi>' \<psi>' Adj'.\<eta> Adj'.\<epsilon> Adj'.\<Phi> Adj'.\<Psi>"
recvmsg_simps_2: fixes P :: "'a \<Rightarrow> bool"   and ips :: "nat set"   and m :: "'a" shows "recvmsg P (groupcast ips m) = True"
bangTauInduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'d \<Rightarrow> 'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> \<tau> \<prec> P'"   and "\<And>P' C. \<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P' \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (P' \<parallel> !P)"   and "\<And>P' C. \<lbrakk>\<Psi> \<rhd> !P \<longmapsto> \<tau> \<prec> P'; \<And>C. Prop C \<Psi> (!P) P'\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (P \<parallel> P')"   and "\<And>M N P' K xvec P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P''; \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>M N P' K xvec P'' C. \<lbrakk>\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; \<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''; \<Psi> \<turnstile> M \<leftrightarrow> K; xvec \<sharp>* \<Psi>; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* C\<rbrakk> \<Longrightarrow> Prop C \<Psi> (P \<parallel> !P) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"   and "\<And>P' C. \<lbrakk>\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<tau> \<prec> P'; \<And>C. Prop C \<Psi> (P \<parallel> !P) P'; guarded P\<rbrakk> \<Longrightarrow> Prop C \<Psi> (!P) P'" shows "Prop C \<Psi> (!P) P'"
otrans_from_sterms': fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p' :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "((\<sigma>, p'), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "wellformed \<Gamma>"   and "p' \<in> sterms \<Gamma> p" shows "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
unat_mask_eq: fixes n :: "nat" shows "unat (mask n) = mask (min LENGTH('b) n)"
i_aggregate_hd: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "i_aggregate f k ag 0 = ag (f \<Down> k)"
dvd_mult_right: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times b c"
l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_child_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs"
nat_pow_comm: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier G" shows "x [^] n \<otimes> x [^] m = x [^] m \<otimes> x [^] n"
has_codomain_iff_arr: fixes f :: "'a" shows "(A.codomains f \<noteq> {}) = A.arr f"
nprv_clear2_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<psi> :: "'c"   and \<phi>2 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2} \<psi>"
Eq\<^sub>0_prop: fixes v :: "i"   and F :: "\<o>"   and G :: "\<o>"   and \<phi> :: "\<o> \<Rightarrow> \<o>" assumes "[F \<^bold>=\<^sub>0 G in v]" shows "[\<phi> F in v] = [\<phi> G in v]"
dis_rews_5: shows "is_snil\<cdot>[::] = TT"
hyps_1: shows "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<tau> \<prec> C"
filter_def: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b set" shows "Set.filter P A = {a \<in> A. P a}"
len_comp: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and c :: "nat"   and xs :: "nat list"   and b :: "nat"   and ys :: "nat list" shows "linordered_monoid.len m a c (xs @ b # ys) = dbm_add (linordered_monoid.len m a b xs) (linordered_monoid.len m b c ys)"
t: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> t"
HEndPhase0_blocksRead: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p" shows "\<exists>d. blocksRead s p d \<noteq> {}"
sp_fundefs_def: fixes F :: "'k \<Rightarrow> ('l, 'm, 'k, 'd, 'e, 'g, 'n, 'o) Ubx.instr fundef option" shows "sp_fundefs F \<equiv> \<forall>f fd. F f = Some fd \<longrightarrow> sp_fundef F fd (body fd) = Ok [None]"
fst_diff: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "fst (x - y) = fst x - fst y"
DBM_le_subset: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and u :: "'c \<Rightarrow> 'b"   and v :: "'c \<Rightarrow> nat" assumes "\<forall>i j. i \<le> n \<longrightarrow> j \<le> n \<longrightarrow> M i j \<preceq> M' i j"   and "u \<in> [M]\<^bsub>v,n\<^esub>" shows "u \<in> [M']\<^bsub>v,n\<^esub>"
division_points_finite: fixes d :: "'a set set"   and i :: "'a set" assumes "d division_of i" shows "finite (division_points i d)"
QQ1_a_def: shows "QQ1_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . root = nil \<and> p' = nil \<and> t' = nil \<and> mrk' = mrk \<and> left' = left \<and> right' = right \<and> atom' = atom :]"
simps_2: fixes uu :: "real list" shows "eval TrueF uu = True"
f8a: shows "\<not> (\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
cases: fixes a :: "('test, 'atom) com" assumes "siso a" obtains c :: "('test, 'atom) com" where "a = c" and "\<And>s c' s'. (c, s) \<rightarrow>c (c', s') \<Longrightarrow> siso c'" and "\<And>s t c' s'. \<lbrakk>s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')" and "\<And>s t s'. \<lbrakk>s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "\<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
negate_negate_dfa: fixes A :: "nat bdd list \<times> bool list" shows "negate_dfa (negate_dfa A) = A"
that: assumes "border xs ys"   and "length xs < length ys" shows "thesis"
is_nth_power_upto_unit_unit: fixes x :: "'a"   and n :: "nat" assumes "is_unit x" shows "is_nth_power_upto_unit n x"