hcju/isabelleps huggingface dataset:
\<F>_drop: fixes \<phi> :: "'b ltl"   and \<G>' :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and S :: "'b ltl set"   and i :: "nat"   and j :: "nat" assumes "\<PP>\<^sub>\<infinity> \<phi> \<G>' w"   and "S \<Turnstile>\<^sub>P \<F> \<phi> w \<G>' (i + j)" shows "S \<Turnstile>\<^sub>P \<F> \<phi> (suffix i w) \<G>' j"
R_def: fixes h :: "'a"   and w :: "nat \<times> nat"   and e :: "nat" shows "local.\<Sigma>_protocols_base.R h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
inc\<^sub>2\<sigma>: shows "l\<^sub>2 \<cdot>\<^sub>l \<sigma> \<in> C\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>"
mat1_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and ze :: "'a"   and on :: "'a" assumes "i < n"   and "j < n" shows "mat1I ze on n ! i ! j = (if i = j then on else ze)"
1: shows "\<guillemotleft>\<psi> : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (\<psi> \<otimes>\<^sub>D \<psi>)"
add_order_gt0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.monoid_add plus zero" shows "order.greater (monoid_add.add_order plus zero a) 0 = (\<exists>n. order.greater n 0 \<and> power.power zero plus a n = zero)"
scaleR2_eq: fixes e :: "real"   and xr :: "real \<times> real"   and P :: "real \<times> real" assumes "0 < e" shows "ccw' 0 xr (e *\<^sub>R P) = ccw' 0 xr P"
Polys_alt: fixes X :: "'a set" shows "P[X] = {p. indets p \<subseteq> X}"
assms_2: fixes R :: "('a, 'b) labeled_graph \<times> ('a, 'b) labeled_graph" assumes "R \<in> Rs" shows "subgraph (fst R) (snd R) \<and> finite_graph (fst R)"
disc2_noC: shows "\<forall>a. \<not> disc2 (Src_Ports a)"
1: shows "from_conjugate_space ` closure X \<subseteq> closure (from_conjugate_space ` X)"
Counter_global_def: fixes n :: "nat" shows "Counter_global n = {.x. \<forall>k. count x k \<le> n.} \<circ> [: x \<leadsto> y . y = count x :]"
take_bit_Suc: fixes n :: "nat"   and a :: "'a" shows "take_bit (Suc n) a = take_bit n (a div (2::'a)) * (2::'a) + a mod (2::'a)"
case1a: shows "sint (SCAST(32 \<rightarrow> 64) w2 + 0x7FFFFFFF) = sint (SCAST(32 \<rightarrow> 64) w2) + sint 0x7FFFFFFF"
fst_set_prappend_signed_list: fixes xs :: "('a \<times> bool) list"   and ys :: "('a \<times> bool) list" shows "order.greater_eq (fst ` (set xs \<union> set ys)) (fst ` set (prappend_signed_list xs ys))"
induct: fixes x1 :: "char list \<Rightarrow> ty"   and x2 :: "aexp"   and x3 :: "ty"   and P :: "(char list \<Rightarrow> ty) \<Rightarrow> aexp \<Rightarrow> ty \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> i. P \<Gamma> (Ic i) Ity"   and "\<And>\<Gamma> r. P \<Gamma> (Rc r) Rty"   and "\<And>\<Gamma> x. P \<Gamma> (V x) (\<Gamma> x)"   and "\<And>\<Gamma> a1 \<tau> a2. \<lbrakk>\<Gamma> \<turnstile> a1 : \<tau>; P \<Gamma> a1 \<tau>; \<Gamma> \<turnstile> a2 : \<tau>; P \<Gamma> a2 \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (Plus a1 a2) \<tau>" shows "P x1 x2 x3"
sub_inf_left_zero: fixes x :: "'a" shows "top \<sqinter> - x = - x"
ssubst_insert_if: fixes V :: "name set"   and t :: "tm"   and i :: "name" assumes "finite V" shows "ssubst t (insert i V) F' = (if i \<in> V then ssubst t V F' else subst i (F' i) (ssubst t V F'))"
strict_mono_leD: fixes r :: "'b \<Rightarrow> 'c"   and m :: "'b"   and n :: "'b" assumes "strict_mono r"   and "m \<le> n" shows "r m \<le> r n"
comp_assoc: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'i"   and h :: "'f \<Rightarrow> 'h" shows "f \<circ> g \<circ> h = (f \<circ>\<circ>\<circ> (\<circ>)) g h"
tendsto_add_smallo: fixes f1 :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and f2 :: "'b \<Rightarrow> 'c" assumes "(f1 \<longlongrightarrow> a) F"   and "f2 \<in> o[F](f1)" shows "((\<lambda>x. f1 x + f2 x) \<longlongrightarrow> a) F"
insert_before_list_distinct: fixes x :: "'a"   and xs :: "'a list"   and ref :: "'a option" assumes "x \<notin> set xs"   and "distinct xs" shows "distinct (insert_before_list x ref xs)"
internal_inv_Leaf: fixes b :: "bool list" shows "\<not> internal b Leaf"
mult_matrix_assoc: fixes fmul1 :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b"   and fadd1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fadd2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'b matrix"   and C :: "'d matrix" assumes "\<forall>a. fmul1 (0::'c) a = (0::'b)"   and "\<forall>a. fmul1 a (0::'b) = (0::'b)"   and "\<forall>a. fmul2 (0::'b) a = (0::'b)"   and "\<forall>a. fmul2 a (0::'d) = (0::'b)"   and "fadd1 (0::'b) (0::'b) = (0::'b)"   and "fadd2 (0::'b) (0::'b) = (0::'b)"   and "\<forall>a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"   and "associative fadd1"   and "associative fadd2"   and "\<forall>a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"   and "\<forall>a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"   and "\<forall>a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)" shows "mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B) C = mult_matrix fmul1 fadd1 A (mult_matrix fmul2 fadd2 B C)"
dis_is_nbr: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set" assumes "discharge_invar c s t u fo lo no f l n" shows "neighbor_invar c s t f l n"
maybe_double_negation: fixes x :: "trilean" shows "\<not>? \<not>? x = x"
poly: fixes X :: "'a set" assumes "X \<in> \<F>" shows "polytope X"
Consts_mkinterp: fixes C :: "char \<Rightarrow> real"   and F :: "char \<Rightarrow> real \<Rightarrow> real"   and P :: "char \<Rightarrow> real \<Rightarrow> bool"   and G :: "char \<Rightarrow> (variable \<Rightarrow> real) set \<Rightarrow> (variable \<Rightarrow> real) set" shows "Consts (mkinterp (C, F, P, G)) = C"
respection: shows "rel_weakly_respects_barbs (indRelLTPO TRel) (STCalWB SWB TWB)"
eq_GSpan_single: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and v :: "'c" assumes "IrrFinGroupRepresentation G smult V"   and "v \<in> V"   and "v \<noteq> (0::'c)" shows "V = R_scalar_mult.RSpan (aezfun_setspan G) smult [v]"
analyzed_if_all_analyzed_in: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set" assumes "\<forall>t\<in>M. analyzed_in t M" shows "analyzed M"
callee_invariant_on_cong: fixes I :: "'a \<Rightarrow> bool"   and I' :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('b, 'd) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and callee' :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'a) spmf" assumes "I = I'"   and "outs_\<I> \<I> = outs_\<I> \<I>'"   and "\<And>s x. \<lbrakk>I' s; x \<in> outs_\<I> \<I>'\<rbrakk> \<Longrightarrow> (set_spmf (callee s x) \<subseteq> responses_\<I> \<I> x \<times> Collect I') = (set_spmf (callee' s x) \<subseteq> responses_\<I> \<I>' x \<times> Collect I')" shows "callee_invariant_on callee I \<I> = callee_invariant_on callee' I' \<I>'"
path_SeqFirst: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and l :: "nat"   and c\<^sub>2 :: "cmd" assumes "While_CFG.path prog n as (_ l _)" shows "While_CFG.path (prog;; c\<^sub>2) n as (_ l _)"
nres_order_simps_5: fixes X :: "'a set"   and Y :: "'a set" shows "(RES X \<le> RES Y) = (X \<subseteq> Y)"
coherence_key_fact: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a \<and> Diag a"   and "Ide b \<and> Diag b"   and "Ide c \<and> Diag c" shows "\<lbrace>(a \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> b) \<^bold>\<Down> c\<rbrace> \<cdot> (\<lbrace>a \<^bold>\<Down> b\<rbrace> \<otimes> \<lbrace>c\<rbrace>) = (\<lbrace>a \<^bold>\<Down> b \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> c\<rbrace> \<cdot> (\<lbrace>a\<rbrace> \<otimes> \<lbrace>b \<^bold>\<Down> c\<rbrace>)) \<cdot> \<a>[\<lbrace>a\<rbrace>, \<lbrace>b\<rbrace>, \<lbrace>c\<rbrace>]"
lsl_bres_closed: fixes x :: "'b"   and y :: "'b" shows "\<nu> (\<nu> x \<rightarrow> \<nu> y) = \<nu> x \<rightarrow> \<nu> y"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> ns_public"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> ns_public; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> ns_public; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) # evs1)"   and "\<And>evs2 NB A' B NA A. \<lbrakk>evs2 \<in> ns_public; P evs2; Nonce NB \<notin> used evs2; Says A' B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) # evs2)"   and "\<And>evs3 A B NA B' NB. \<lbrakk>evs3 \<in> ns_public; P evs3; Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3; Says B' A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) (Nonce NB)) # evs3)" shows "P x"
t_add_closed: fixes x :: "'a"   and y :: "'a" shows "t (!x + !y) = !x + !y"
induct: fixes P :: "'t list \<Rightarrow> bool"   and a0 :: "'t list" assumes "P []"   and "\<And>x ys. P ys \<Longrightarrow> P (x # ys)" shows "P a0"
fresh_subst_ident: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y"   and "fresh xs x Y" shows "Y #[X / x]_xs = Y"
UNIV_4: shows "UNIV = insert 1 (insert 2 (insert 3 (\<eta> 4)))"
conj_refine_left: fixes R :: "'b upred"   and Q :: "'b upred"   and P :: "'b upred" assumes "(Q \<Rightarrow> P) \<sqsubseteq> R" shows "P \<sqsubseteq> (Q \<and> R)"
row_space_is_preserved: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec" assumes "invertible P" shows "row_space (P ** A) = row_space A"
cnj_idem: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and cnj :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.cnj divide inverse times one plus zero minus uminus cnj" shows "cnj (cnj a) = a"
wt_attack_if_tfr_attack_pts: fixes \<S>\<^sub>0 :: "('fun, 'var) strand_step list set"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>s\<^sub>t\<^sub>s \<S>\<^sub>0"   and "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>s\<^sub>t ` \<S>\<^sub>0))"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>s\<^sub>t ` \<S>\<^sub>0))"   and "\<forall>S\<in>\<S>\<^sub>0. list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>\<^sub>0) \<union> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>\<^sub>0))"   and "(\<S>\<^sub>0, []) \<Rightarrow>\<^sup>\<bullet>\<^sup>* (\<S>, \<A>)"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "\<I> \<Turnstile> \<langle>\<A>\<rangle>" shows "\<exists>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> \<I>\<^sub>\<tau> \<Turnstile> \<langle>\<A>\<rangle> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)"
append_eq_append_conv2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and ts :: "'a list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
common_reparam_exists_imp_eq_line_integral_1: fixes basis :: "(real \<times> real) set"   and one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "finite basis"   and "finite one_chain1"   and "finite one_chain2"   and "boundary_chain one_chain1"   and "boundary_chain one_chain2"   and "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"   and "common_reparam_exists one_chain1 one_chain2"   and "\<forall>(k, \<gamma>)\<in>one_chain1. valid_path \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain2. valid_path \<gamma>" shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
abs_fresh_1: fixes b :: "NominalSequents.var"   and a :: "NominalSequents.var"   and x :: "'a" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
Lower_memI: fixes A :: "'c set"   and L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "\<And>y. y \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "x \<in> carrier L" shows "x \<in> Lower L A"
inj_monoid_add_hom_axioms: shows "inj_monoid_add_hom hom"
B'_wf: shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) B'"
simps_2: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_Wbis tval aval indis (c1 ;; c2) = (PL_Indis.SC_WbisT tval aval indis c1 \<and> PL_Indis.SC_Wbis tval aval indis c2 \<or> PL_Indis.SC_Wbis tval aval indis c1 \<and> PL_Indis.SC_discr aval indis c2 \<or> PL_Indis.SC_ZObis tval aval indis (c1 ;; c2) \<or> PL_Indis.SC_WbisT tval aval indis (c1 ;; c2))"
is_arrD_1: fixes \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
eval_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (divide F G) = (\<lambda>x. eval F x / eval G x)"
that: fixes h :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "h \<in> set hs"   and "fst h \<noteq> 0"   and "h \<noteq> p"   and "h \<noteq> q"   and "lt (fst h) adds\<^sub>t term_of_pair (lcs (lp (fst p)) (lp (fst q)), component_of_term (lt (fst p)))"   and "(p, h) \<in>\<^sub>p snd ` set ps"   and "(q, h) \<in>\<^sub>p snd ` set ps" shows "thesis"
iso_iff: shows "(f \<in> Group.iso (reduced_homology_group (int p) (nsphere p)) (reduced_homology_group (1 + int p) (nsphere (Suc p)))) = (carrier (reduced_homology_group (1 + int p) (nsphere (Suc p))) \<subseteq> f ` carrier (reduced_homology_group (int p) (nsphere p)) \<and> (\<forall>x\<in>carrier (reduced_homology_group (int p) (nsphere p)). f x = \<one>\<^bsub>reduced_homology_group (1 + int p) (nsphere (Suc p))\<^esub> \<longrightarrow> x = \<one>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub>))"
Close1: fixes P :: "pi"   and y :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>ly in P''\<rightarrow>a<x> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>y> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> <\<nu>y>(P' \<parallel> Q')"
mono_Sync_D2: fixes P :: "'c process"   and Q :: "'c process"   and A :: "'c set"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P \<lbrakk>A\<rbrakk> S) \<longrightarrow> Ra (P \<lbrakk>A\<rbrakk> S) s = Ra (Q \<lbrakk>A\<rbrakk> S) s"
valid_4: shows "valid_sharing \<S>\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b'"
delete_twist: fixes x :: "'a"   and y :: "'a"   and al :: "('a \<times> 'b) list" shows "delete x (delete y al) = delete y (delete x al)"
chain_imp_trancl: fixes S :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. (S i, S (Suc i)) \<in> r"   and "i < j" shows "(S i, S j) \<in> r\<^sup>+"
rel_FGf_pos_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and tytok_F :: "('ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'aj \<times> 'aj \<times> ('ak, 'aj, 'al, 'v, 'am, 'ac, 'an) G) itself"   and tytok_FGf :: "('ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'ak \<times> 'aj \<times> 'al \<times> 'v \<times> 'am \<times> 'ac \<times> 'an) itself" assumes "rel_F_pos_distr_cond Co1 Co1' Co2 Co2' (=) (=) Contra1 Contra1' Contra2 Contra2' (=) (=) tytok_F" shows "rel_FGf_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_FGf"
ic_union_ci_not_closed: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b"   and "b < c" shows "\<not> closed ({a..<b} \<union> {b<..c})"
intros_4: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "Le a \<prec> Lt b"
Union: shows "(SUP i. emeasure M (\<Union> (Q' ` {..i}))) = emeasure M (\<Union>i. \<Union> (Q' ` {..i}))"
Cod_g: shows "\<AA>\<lparr>Cod\<rparr>\<lparr>g\<rparr> = c"
hoareT_sound2_part: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}" shows "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}"
subnode_dag_cons: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and no :: "ref" assumes "Dag p low high t"   and "no \<in> set_of t" shows "\<exists>not. Dag no low high not"
sum_Ns_eq: shows "(\<Sum>d | d dvd order (mult_of R). card {a \<in> carrier (mult_of R). G.ord a = d}) = order (mult_of R)"
primitive_extractor_correct_8: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "(\<not> has_disc_negated disc False m) = (getNeg as = [])"
emeasure_notin_sets: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<notin> sets M" shows "emeasure M A = 0"
ord_spmf_lossless_spmfD1: fixes R :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and p :: "'h spmf"   and q :: "'h spmf" assumes "ord_spmf R p q"   and "lossless_spmf p" shows "rel_spmf R p q"
nfa_accepting_set_of_bv: fixes A :: "bool list bdd list \<times> bool list"   and q :: "bool list" shows "nfa_accepting A q = (set_of_bv (snd A) \<inter> set_of_bv q \<noteq> {})"
sl_cons_right_invert: fixes ba' :: "'a list"   and t :: "'b"   and bc :: "'b list"   and R :: "('a \<times> 'b) set" assumes "(ba', t # bc) \<in> seq_lift R" obtains xs :: "'a list" and x :: "'a" where "ba' = x # xs" and "(xs, bc) \<in> seq_lift R" and "(x, t) \<in> R"
multiplicative_function_inverse: fixes f :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f" shows "multiplicative_function (\<lambda>n. inverse (f n))"
construct_in_span: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "vs1.independent B" shows "construct B f v \<in> vs2.span (f ` B)"
One_neq_0: shows "One \<noteq> (0::'b)"
m_x_range: fixes m :: "Mode" assumes "x \<in> (conc.restrict_modes (map snd cms\<^sub>C) (range var\<^sub>C_of) ! i) m" shows "x \<in> range var\<^sub>C_of \<and> x \<in> (map snd cms\<^sub>C ! i) m"
finite_mins: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "QI c_k c w" shows "finite (Args_min_on (c_k i j) {i + 1..j})"
lim: shows "(\<lambda>n. real (2 * b) * Re (zeta 3) * real (Lcm {1..n}) ^ 3 / 27 ^ n) \<longlonglongrightarrow> 0"
sorted_wrt_distinct_linorder_nth: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "distinct xs"   and "i < length xs"   and "j < length xs" shows "((xs ! i, xs ! j) \<in> R) = (i \<le> j)"
l2: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and p :: "'b"   and W :: "'b set" assumes "with_w quorum_of"   and "p \<in> with_w.L quorum_of W" shows "\<exists>Q\<subseteq>W. quorum_of p Q"
simps_1: fixes s :: "'c"   and s' :: "'c" shows "lts.path s [] s' = (s' = s)"
simps_2: fixes ys :: "('a \<times> bool) list" shows "prappend_signed_list [] ys = ys"
plus_Quantity_ext_def: fixes x :: "('d, 'c, 'e) Quantity_scheme"   and y :: "('d, 'c, 'e) Quantity_scheme" assumes "Quantity.dim x = Quantity.dim y" shows "x + y = \<lparr>mag = mag x + mag y, dim = Quantity.dim x, \<dots> = Quantity.more x + Quantity.more y\<rparr>"
p': fixes asa :: "'b list"   and bsa :: "'b list" assumes "as @ x # y # bs = asa @ bsa" shows "\<forall>x\<in>set asa. \<forall>y\<in>set bsa. x \<le> y"
assms: shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
assms: shows "n < \<^bold>|w\<^bold>|"
arith_simps_16: shows "neg_numeral_class.sub num.One num.One = (0::'a)"
L1_def: fixes x :: "nat"   and y :: "nat" shows "L1 x y = {(x, y), (x, y + 1), (x + 1, y + 1)}"
S5nD: fixes M :: "('e, 'f, 'g) KripkeStructure"   and a :: "'e" assumes "S5n M" shows "equiv (worlds M) (relations M a)"
elem_of_symcl_1: fixes a :: "'a"   and b :: "'a"   and Rel :: "('a \<times> 'a) set" assumes "(a, b) \<in> Rel" shows "(a, b) \<in> symcl Rel"
minus: fixes x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> X x \<in> UNIV"   and "x \<in> diff_fun_space"   and "y \<in> diff_fun_space" shows "X (x - y) = X x - X y"
comb_seq_def: fixes i :: "nat"   and \<omega> :: "nat \<Rightarrow> 'b"   and \<omega>' :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "comb_seq i \<omega> \<omega>' j = (if j < i then \<omega> j else \<omega>' (j - i))"
aform_vals_empty: fixes e' :: "nat \<Rightarrow> real" shows "aform_vals e' [] = []"
height_btree_last: fixes sub :: "'b btree"   and x :: "'b"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "height (Node ((sub, x) # ts) t) = max (height (Node ts sub)) (Suc (height t))"