hcju/isabelleps huggingface dataset:
Par_is_action: fixes A1 :: "('c, 'd) action"   and A2 :: "('c, 'd) action"   and cs :: "'c set"   and ns1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and ns2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "R (\<lambda>(S, S'). (\<nexists>tr1 tr2. (action.relation_of A1\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr1 = tr S)) (S, S') \<and> (spec False (wait S) (action.relation_of A2) ;; (\<lambda>(S, S'). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<and> (\<nexists>tr1 tr2. (spec False (wait S) (action.relation_of A1) ;; (\<lambda>(S, uu_). tr1 = tr S)) (S, S') \<and> (action.relation_of A2\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<turnstile> \<lambda>(S, S'). \<exists>s1 s2. ((\<lambda>(A, A'). action.relation_of A1\<^sup>t\<^sub>f (A, s1) \<and> action.relation_of A2\<^sup>t\<^sub>f (A, s2)) ;; M_par s1 ns1 s2 ns2 cs) (S, S')) \<in> {p. is_CSP_process p}"
limit_def: fixes x :: "nat \<Rightarrow> 'd" shows "limit x \<equiv> {a. \<exists>\<^sub>\<infinity>n. x n = a}"
exists_path_subpath_to_frontier: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "pathstart g \<in> closure S"   and "pathfinish g \<notin> S" obtains h :: "real \<Rightarrow> 'a" where "path h" and "pathstart h = pathstart g" and "path_image h \<subseteq> path_image g" and "path_image h - {pathfinish h} \<subseteq> interior S" and "pathfinish h \<in> frontier S"
scf_list_subset: fixes scf :: "nat \<Rightarrow> nat"   and xs :: "'a list" shows "set (scf_list scf xs) \<subseteq> set xs"
summable_minus: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "summable (\<lambda>n. - f n)"
SLC_ifTrue: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e1 :: "('r, 'l, 'v) expr"   and e2 :: "('r, 'l, 'v) expr"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1]))"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
Exit_path_Low_path: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>* (_Exit_)"   and "inner_node n" obtains a' :: "'edge" and as' :: "'edge list" where "as = as' @ [a']" and "n -as'\<rightarrow>* (_Low_)" and "kind a' = (\<lambda>s. True)\<^sub>\<surd>"
4: shows "card (atoms (tseytin_tran F)) \<le> card (atoms F) + card (\<Union>f\<in>set (subformulae F). atoms (tseytin_toatom F f))"
lconjseq_snoc: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and xs :: "'b list"   and x :: "'b" assumes "class.group_add minus plus zero uminus" shows "group_add.lconjseq minus plus (xs @ [x]) = group_add.lconjseq minus plus xs @ [minus (plus (monoid_add.sum_list plus zero xs) x) (monoid_add.sum_list plus zero xs)]"
real_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and real_of :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_embedding plus zero minus uminus less_eq less sgn abs times one real_of" shows "real_of (plus x y) = real_of x + real_of y"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs)" shows "fs.gs.d k = (\<Prod>j<k. \<parallel>fs.gs.gso j\<parallel>\<^sup>2)"
Suc: shows "card {a \<in> Collect even. a < 2 * n} = n"
Ex_E_with_renaming: fixes i :: "name"   and i' :: "name"   and A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert ((i \<leftrightarrow> i') \<bullet> A) H \<turnstile> B"   and "atom i' \<sharp> (A, i, B)"   and "\<forall>C\<in>H. atom i' \<sharp> C" shows "insert (SyntaxN.Ex i A) H \<turnstile> B"
Suc_eq_numeral: fixes n :: "nat"   and k :: "num" shows "(Suc n = numeral k) = (n = pred_numeral k)"
finite_proj_diag: fixes d :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and P :: "'d \<Rightarrow>\<^sub>F 'e"   and Q :: "'d \<Rightarrow>\<^sub>F 'f"   and S :: "'d set" shows "finite ((\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)"
map_simp_ide: fixes a :: "'c local.term set" assumes "\<F>\<^sub>SC.ide a" shows "\<mu>.map a = a"
ss_cons: fixes CT :: "nat \<Rightarrow> classDef option"   and C0 :: "nat"   and D0 :: "nat"   and Cs :: "nat list"   and Ds :: "nat list" assumes "CT \<turnstile> C0 <: D0"   and "CT \<turnstile>+ Cs <: Ds" shows "CT \<turnstile>+ (C0 # Cs) <: (D0 # Ds)"
pivotal_def: fixes F :: "(indi \<Rightarrow> alt \<Rightarrow> real) \<Rightarrow> alt \<Rightarrow> real"   and i :: "indi"   and b :: "alt" assumes "arrow F" shows "arrow.pivotal F i b \<equiv> \<exists>P. Extreme P b \<and> b \<cdot>< P i \<and> b \<cdot>< F P \<and> F (P(i := mktop (P i) b)) <\<cdot> b"
\<G>\<F>_af: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "\<G>\<F> (af \<phi> (prefix i w)) (suffix i w) = \<G>\<F> \<phi> (suffix i w)"
Abs_segment_inverse: fixes y :: "'b \<times> 'b" assumes "y \<in> {x. fst x \<le> snd x}" shows "Rep_segment (Abs_segment y) = y"
flowD_source_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "flow (residual_network f) f" shows "d_IN f (source (residual_network f)) = 0"
foo2: fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and project_'a :: "'d \<Rightarrow> 'e"   and inject_'a :: "'e \<Rightarrow> 'd"   and project_Nat_nat_Nat_nat_fun :: "'d \<Rightarrow> nat \<Rightarrow> nat"   and inject_Nat_nat_Nat_nat_fun :: "(nat \<Rightarrow> nat) \<Rightarrow> 'd"   and project_Int_int :: "'d \<Rightarrow> int"   and inject_Int_int :: "int \<Rightarrow> 'd"   and project_Nat_nat :: "'d \<Rightarrow> nat"   and inject_Nat_nat :: "nat \<Rightarrow> 'd"   and i :: "int"   and s :: "'c \<Rightarrow> 'd" assumes "foo f a b c project_'a inject_'a project_Nat_nat_Nat_nat_fun inject_Nat_nat_Nat_nat_fun project_Int_int inject_Int_int project_Nat_nat inject_Nat_nat" shows "lookup project_Int_int a (update project_Int_int inject_Int_int a (K_statefun i) s) = i"
oconf_lconf: fixes G :: "prog"   and s :: "st"   and obj :: "obj"   and r :: "oref" assumes "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r" shows "G,s\<turnstile>values obj[\<Colon>\<preceq>]var_tys G (tag obj) r"
compute_float_round_up: fixes prec :: "nat"   and x :: "float" shows "float_round_up prec x = - float_round_down prec (- x)"
Red_preserves_sconf: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
co_path_append: fixes p1 :: "('b \<times> 'b) list"   and p2 :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "[last p1, hd p2] \<notin> {[(fst e, w), (w, snd e)], [(snd e, w), (w, fst e)]}" shows "co_path e w (p1 @ p2) = co_path e w p1 @ co_path e w p2"
abelian_subgroup_a_kernel: shows "abelian_subgroup (a_kernel R S h) R"
reinterpret: fixes t1 :: "t"   and t2 :: "t"   and \<C> :: "t_context" assumes "t1 \<noteq> t2"   and "t_length t1 = t_length t2" shows "\<C> \<turnstile> [Cvtop t1 Reinterpret t2 None] : [t2] _> [t1]"
ifte_thms_3: fixes e1 :: "'b"   and e2 :: "'b" shows "If TT then e1 else e2 = e1"
finlsts_induct: fixes x :: "'b llist"   and A :: "'b set"   and P :: "'b llist \<Rightarrow> bool" assumes "x \<in> A\<^sup>\<star>"   and "\<And>l. l = LNil \<Longrightarrow> P l"   and "\<And>a l. \<lbrakk>l \<in> A\<^sup>\<star>; P l; a \<in> A\<rbrakk> \<Longrightarrow> P (a ## l)" shows "P x"
integral_le: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f integrable_on S"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "integral S f \<le> integral S g"
simps_3: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula" shows "(\<A>::'b \<Rightarrow> bool) \<Turnstile> \<^bold>\<not> (F::'b formula) = (\<not> \<A> \<Turnstile> F)"
that: fixes y :: "'b" assumes "fmlookup m x = Some y" shows "thesis"
5: shows "f t x \<inter> snd ` set txs = {}"
perm_simps_4: fixes p :: "perm"   and var :: "Vars.var"   and exp :: "exp" shows "p \<bullet> Lam [var]. exp = Lam [(p \<bullet> var)]. (p \<bullet> exp)"
is_lasso_def: fixes m :: "'a"   and L :: "'Q lasso" shows "gb_graph.is_lasso (to_gbg_ext m) L \<equiv> graph.is_lasso_pre (to_gbg_ext m) L \<and> (\<forall>A\<in>gbg_F (to_gbg_ext m). set (lasso_cycle L) \<inter> A \<noteq> {})"
valuation_some_in_prime_divisor: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "valuation K (SOME w. w \<in> P\<^bsub> K v\<^esub>)"
intros: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
match_4: fixes g :: "'a1"   and f :: "'a1"   and h :: "'a1" assumes "A1.seq g f"   and "A1.seq h g" shows "A1.seq h (g \<cdot>\<^sub>A\<^sub>1 f)"
mult_divides: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R) - {\<zero>\<^bsub>poly_ring R\<^esub>}"   and "c \<otimes>\<^bsub>poly_ring R\<^esub> a divides\<^bsub>poly_ring R\<^esub> c \<otimes>\<^bsub>poly_ring R\<^esub> b" shows "a divides\<^bsub>poly_ring R\<^esub> b"
paths_def: shows "paths = {(loc1, xs). impWitnessPath c loc1 loc2 xs t}"
step_seq_tau: shows "\<tau> \<equiv> \<tau>\<^sub>s"
Confidentiality_Tgs: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Key servK \<notin> analz (knows Spy evs)"
mset_left_cancel_elem: fixes a :: "'c"   and b :: "'c"   and A :: "'c multiset" assumes "a \<in># {#b#} + A"   and "a \<noteq> b" shows "a \<in># A"
ExistsE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and n :: "'a"   and b :: "('a, 'b) form" assumes "G \<turnstile> Exists a"   and "a[App n []/0] # G \<turnstile> b"   and "news n G"   and "new n a"   and "new n b" shows "G \<turnstile> b"
project_constrains_mono: fixes D :: "'c set"   and C :: "'c set"   and F :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "D \<subseteq> C"   and "project h C F \<in> A co B" shows "project h D F \<in> A co B"
path_append: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a list"   and p :: "'b"   and s :: "'a list" assumes "transition_system.path execute enabled r p"   and "transition_system.path execute enabled s (fold execute r p)" shows "transition_system.path execute enabled (r @ s) p"
l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"
dvd_times_left_cancel_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times a b) (times a c) = dvd.dvd times b c"
onorm_Sup_on_ball': fixes r :: "real"   and \<tau> :: "real"   and x :: "'c"   and f :: "'c \<Rightarrow>\<^sub>L 'd" assumes "0 < r"   and "\<tau> < 1" shows "\<exists>\<xi>\<in>ball x r. \<tau> * r * norm f \<le> norm (f *\<^sub>v \<xi>)"
map_le_conv: fixes h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option" shows "(h\<^sub>0' \<subseteq>\<^sub>m h\<^sub>0 \<and> h\<^sub>0' \<noteq> h\<^sub>0) = (\<exists>h\<^sub>1. h\<^sub>0 = h\<^sub>0' ++ h\<^sub>1 \<and> h\<^sub>0' \<bottom> h\<^sub>1 \<and> h\<^sub>0' \<noteq> h\<^sub>0)"
simps_4: fixes m :: "'node"   and x :: "nat" shows "valid_SDG_node (Actual_in (m, x)) = (\<exists>a Q r p fs ins outs. valid_edge a \<and> kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<and> sourcenode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length ins)"
assms_2: shows "theta\<inverse> \<subseteq> Sretr (theta\<inverse>)"
finite_union: shows "finite (vars1 \<union> vars2 \<union> vars3)"
mset_concat: fixes xs :: "'a list list" shows "mset (concat xs) = fold (+) (map mset xs) {#}"
in_orbitsI: fixes a :: "'b"   and f :: "'b perm" assumes "a \<in> affected f" shows "orbit f a \<in> orbits f"
NA: shows "A (Q' Q \<pi> u) (\<pi>' Q \<pi> u) = {(u, v)} \<union> A Q \<pi>"
alphaAbs_all_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
ordered_keys_infinite: fixes m :: "('b, 'c) mapping" assumes "infinite (keys m)" shows "ordered_keys m = []"
apex_def: shows "Cod'.apex \<equiv> Maps.dom Cod'.leg0"
in_mono: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
lossless_sub_gpvsI: fixes gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "lossless_spmf (the_gpv gpv)"   and "\<And>gpv'. gpv' \<in> sub_gpvs \<I> gpv \<Longrightarrow> lossless_gpv \<I> gpv'" shows "lossless_gpv \<I> gpv"
vars\<^sub>s\<^sub>t_def: fixes S :: "('e, 'd) strand_step list" shows "vars\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map vars\<^sub>s\<^sub>t\<^sub>p S))"
proj2_Col_abs: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec"   and r :: "(real, 3) vec"   and i :: "real"   and j :: "real"   and k :: "real" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "r \<noteq> 0"   and "i \<noteq> 0 \<or> j \<noteq> 0 \<or> k \<noteq> 0"   and "i *\<^sub>R p + j *\<^sub>R q + k *\<^sub>R r = 0" shows "proj2_Col (proj2_abs p) (proj2_abs q) (proj2_abs r)"
left_idem: fixes a :: "'b upred"   and b :: "'b upred" shows "(a \<and> a \<and> b) = (a \<and> b)"
Cauchy_integral_formula_weak: fixes S :: "complex set"   and k :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "convex S"   and "finite k"   and "continuous_on S f"   and "\<And>x. x \<in> interior S - k \<Longrightarrow> f field_differentiable at x"   and "z \<in> interior S - k"   and "valid_path \<gamma>"   and "path_image \<gamma> \<subseteq> S - {z}"   and "pathfinish \<gamma> = pathstart \<gamma>" shows "((\<lambda>w. f w / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
longest_common_prefix_max_prefix: fixes ps :: "'c list"   and xs :: "'c list"   and ys :: "'c list" assumes "prefix ps xs"   and "prefix ps ys" shows "prefix ps (longest_common_prefix xs ys)"
neq_Nil_conv2: fixes xs :: "'d list" shows "(xs \<noteq> []) = (\<exists>x xs'. xs = xs' @ [x])"
sign_simps_8: fixes ab :: "'b"   and bb :: "'b"   and cb :: "'b" shows "ab + (bb - cb) = ab + bb - cb"
H: shows "\<exists>hs. rev v @ x # h = [x, x] @ hs"
assms: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
inject: fixes x :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and ya :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "(Simplex_State x = Simplex_State ya) = (x = ya)"
atLeastLessThan_empty_iff: fixes a :: "'a \<times> ('a, 'b) operation"   and b :: "'a \<times> ('a, 'b) operation" shows "(hb.atLeastLessThan a b = {}) = (\<not> hb a b)"
assms_2: shows "kKS key \<in> specKeysSecrets P"
disc_1: fixes x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval" shows "is_INS (INS x11 x12 x13)"
cball_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_total A"   and "(A ===> A ===> (=)) dist dist" shows "(A ===> (=) ===> rel_set A) cball cball"
is_OK_returns_result_I: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and y :: "'ar" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y" shows "h \<turnstile> ok f"
subst_noop_2: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var" shows "\<Gamma>[y::h=y] = \<Gamma>"
measurable_predpow: fixes T :: "'a \<Rightarrow> bool"   and M :: "'a measure"   and R :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "Measurable.pred M T"   and "\<And>Q. Measurable.pred M Q \<Longrightarrow> Measurable.pred M (R Q)" shows "Measurable.pred M ((R ^^ n) T)"
rcos_module_rev: fixes x :: "nat"   and x' :: "nat" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "x' \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H" shows "x' \<in> H #>\<^bsub>G\<^esub> x"
inR2: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, G, H, \<Delta>" shows "\<Gamma> \<Rightarrow> G, H, F, \<Delta>"
dvd_mult_unit_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (times c b) = dvd.dvd times a c"
lipschitz_onI: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and L :: "real" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> dist (f x) (f y) \<le> L * dist x y"   and "0 \<le> L" shows "L-lipschitz_on X f"
some_basis_def: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "vector_space scale" shows "vector_space.some_basis scale V = Eps (vector_space.is_basis scale V)"
interfree_aux_def: shows "interfree_aux \<equiv> \<lambda>(co, q, co'). co' = None \<or> (\<forall>(r, a)\<in>atomics (the co'). \<parallel>= (q \<inter> r)  a q \<and> (co = None \<or> (\<forall>p\<in>assertions (the co). \<parallel>= (p \<inter> r)  a p)))"
mut_m'_def: fixes m :: "'a"   and m' :: "'a" shows "mut_m' m m' \<equiv> m \<noteq> m'"
simps_1: fixes d :: "'a" shows "norm_instr (Inca.instr.IPush d) = Std.instr.IPush d"
\<I>_ideal_def: shows "\<I>_ideal \<equiv> \<lambda>\<eta>. \<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>))"
encode_operator_effect_valid_2: fixes t :: "nat"   and op :: "(nat \<times> nat) strips_operator"   and sv :: "sat_plan_variable"   and vs :: "'a list"   and asses :: "'a list" assumes "t < h"   and "op \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)"   and "sv \<in> atoms \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) op))) \<^bold>\<or> \<^bold>\<not> (Atom (State (Suc t) (index vs v)))) asses" shows "valid_state_var sv"
asym: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># b"   and "b \<subset># a" shows "False"
distinct_23: fixes x31 :: "acom"   and x32 :: "acom"   and x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom" shows "x31;; x32 \<noteq> {x61} WHILE x62 DO x63"
AL_update5: fixes L :: "('b \<times> 'a) list"   and b :: "'b"   and M :: "'a option"   and a :: "'b"   and X :: "'a" assumes "L\<down>b = M"   and "a \<noteq> b" shows "L[a\<mapsto>X]\<down>b = M"
less: shows "less_multiset_ext\<^sub>D\<^sub>M R M N"
Red1CallExternal: fixes s :: "'heap \<times> 'addr val list"   and a :: "'addr"   and T :: "htype"   and P :: "'addr J1_prog"   and M :: "String.literal"   and Ts :: "ty list"   and Tr :: "ty"   and D :: "String.literal"   and t :: "'thread_id"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and uf :: "bool" assumes "typeof_addr (hp s) a = \<lfloor>T\<rfloor>"   and "P \<turnstile> class_type_of T sees M: Ts\<rightarrow>Tr = Native in D"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),hp s\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "e' = extRet2J (addr a\<bullet>M(map Val vs)) va"   and "s' = (h', lcl s)" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>M(map Val vs),s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>"
height_node: fixes m :: "nat"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" assumes "HBT m" shows "height (node l a r) = max (height l) (height r) + 1"
red_preserves_lconf: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> e : T"   and "P,h \<turnstile> l (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
Chains_inits_DiffI: fixes R :: "('b \<times> 'b) set set"   and s :: "('b \<times> 'b) set" assumes "R \<in> Chains init_seg_of" shows "{r - s |r. r \<in> R} \<in> Chains init_seg_of"
S10: shows "nat_to_sch (c_pair 3 0) = Base_snd"
complex_not_root_unity: fixes n :: "nat" assumes "1 \<le> n" shows "\<exists>u. cmod u = 1 \<and> u ^ n \<noteq> 1"