hcju/isabelleps huggingface dataset:
strip_lemma: 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 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 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 \<leadsto>\<^sup>* s\<^sub>2"   and "(s\<^sub>1, s\<^sub>2') \<in> [\<leadsto>]\<^sup>="   and "reachable s\<^sub>1"   and "infinite UNIV"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> s\<^sub>2 \<leadsto>\<^sup>* s\<^sub>3 \<and> s\<^sub>2' \<leadsto>\<^sup>* s\<^sub>3'"
FF_def: shows "FF \<equiv> \<lambda>\<mu>\<nu>. if S.VV.arr \<mu>\<nu> then (local.map (fst \<mu>\<nu>), local.map (snd \<mu>\<nu>)) else B.VV.null"
face_cycle_eq: fixes b :: "'b"   and a :: "'b" assumes "b \<in> face_cycle_set a" shows "face_cycle_set b = face_cycle_set a"
tt_monomial: fixes c :: "'c"   and v :: "'t" assumes "c \<noteq> (0::'c)" shows "tt (monomial c v) = v"
WHILE_def: shows "cWHILE \<equiv> cWHILEI (\<lambda>_. True)"
set_filter_axioms: shows "set_filter rs.\<alpha> rs.invar rs.\<alpha> rs.invar Benchmark_ICF.rs_filter"
map\<^sub>0_def: fixes a :: "'a" shows "\<tau>'\<tau>.map\<^sub>0 a = map\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a"
shift1_inv_finite_snd: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv (qs, B)" shows "finite B"
mu_repr_def: fixes mu :: "int iarray iarray"   and fs :: "int vec list" shows "mu_repr mu fs = (mu = IArray.of_fun (\<lambda>i. IArray.of_fun (d\<mu> fs i) i) m)"
tree_rank_estimate: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e a r ts)" shows "size (BinomialHeapStruc.tree_to_multiset (BinomialHeapStruc.Node e a r ts)) = 2 ^ r"
kSup_distr: fixes F :: "('a \<Rightarrow> 'c set) set"   and g :: "'c \<Rightarrow> 'b set" shows "\<Squnion> F \<circ>\<^sub>K g = (\<Squnion>f\<in>F. f \<circ>\<^sub>K g)"
associatedI2': fixes a :: "'a"   and b :: "'a"   and u :: "'a" assumes "a = b \<otimes> u"   and "u \<in> Units G"   and "b \<in> carrier G" shows "a \<sim> b"
cases: fixes x :: "'b tag_const" obtains t :: "'b" where "x = \<tau>\<^sub>c\<^sub>s\<^sub>t t"
append_eqvt: fixes p :: "perm"   and xs :: "'b list"   and ys :: "'b list" shows "p \<bullet> (xs @ ys) = p \<bullet> xs @ p \<bullet> ys"
8: shows "(top * e * (w\<^sup>T \<sqinter> - v * - v\<^sup>T \<sqinter> w\<^sup>\<star> * e\<^sup>T * top)\<^sup>\<star>)\<^sup>T \<le> w\<^sup>\<star> * e\<^sup>T * top \<sqinter> - v"
emeasure_count_space_finite: fixes X :: "'c set"   and A :: "'c set" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (count_space A) X = of_nat (card X)"
distinct_10: fixes x3 :: "trm"   and x2 :: "name" shows "suc x3 \<noteq> Var x2"
closed_segment_surface: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "{a--b} = {x \<in> {x. x \<bullet> (b - a) \<in> {a \<bullet> (b - a)..b \<bullet> (b - a)}}. (x - a) \<bullet> rot (b - a) = 0}"
runit_def: fixes f :: "'a" shows "\<r>[f] \<equiv> right_hom_with_unit.runit (\<cdot>) (\<star>) \<a> \<i>[src f] (src f) f"
YM4_analz_knows_Spy: fixes A :: "agent"   and Y :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Gets A \<lbrace>Crypt (shrK A) Y, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "X \<in> analz (knows Spy evs)"
NCut_axioms: shows "NCut c s t (cf.reachableNodes s)"
lens_indep_comm: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'f"   and u :: "'d" assumes "x \<bowtie> y" shows "put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
is_rcoset_replist_Cons: fixes G :: "'a set"   and H :: "'a set"   and g :: "'a"   and gs :: "'a list" assumes "Group G" shows "Group.is_rcoset_replist G H (g # gs) = (g \<in> G \<and> set gs \<subseteq> G \<and> Group.rcoset_rel G H `` {g} \<notin> set (map (\<lambda>x. Group.rcoset_rel G H `` {x}) gs) \<and> distinct (map (\<lambda>x. Group.rcoset_rel G H `` {x}) gs) \<and> G = Group.rcoset_rel G H `` {g} \<union> (\<Union>x\<in>set gs. Group.rcoset_rel G H `` {x}))"
hbase_funC: fixes c :: "nat"   and n :: "C"   and ns :: "C list" assumes "c \<noteq> 0"   and "C (n # ns) \<in> hbase_ext (Suc c)" shows "C (funC n (Suc c) @ ns) \<in> hbase_ext (Suc c)"
T: fixes x :: "'a"   and y :: "'a" shows "eNorm N1 (x + y) + eNorm N2 (x + y) \<le> ennreal (max (defect N1) (defect N2)) * (eNorm N1 x + eNorm N2 x) + ennreal (max (defect N1) (defect N2)) * (eNorm N1 y + eNorm N2 y)"
assms_2: fixes cc :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "terminationPseudo 1 cc Q"
singleton_ne_empty_zmset: fixes x :: "'b" shows "{#x#}\<^sub>z \<noteq> {#}\<^sub>z"
ZObisT_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> ZOretrT (theta \<union> ZObisT)" shows "theta \<subseteq> ZObisT"
atMost_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a" shows "ord.atMost less_eq u \<equiv> {x. less_eq x u}"
rel_fun_image2p: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "rel_fun R (BNF_Greatest_Fixpoint.image2p f g R) f g"
Fun_def: fixes f :: "'s" shows "S.Fun f = restrict (S.some_img \<circ> (\<cdot>) f \<circ> inv_into (S.hom S.unity (S.dom f)) S.some_img) (S.set (S.dom f))"
from_vec_mult: fixes M :: "(('c, 'd) vec, 'd) vec"   and N :: "(('c, 'd) vec, 'd) vec" shows "from_vec (M ** N) = from_vec M * from_vec N"
a_def: shows "a \<equiv> \<lambda>n. \<Sum>k\<le>n. p k * q ^ (fact (Suc n) - fact (Suc k))"
assms: shows "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) ils"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.VV.arr f" shows "G.FoH\<^sub>C.map f = D.null"
L_A_def: fixes x :: "'a"   and y :: "'a" shows "L_A x y = seq [question (Atom x), Atom y, Atom y]"
A_2: shows "\<Sigma> \<in> br_invar \<delta>"
uniformity_complex_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
hash_blindable_def: fixes h :: "'a \<Rightarrow> 'b" shows "hash_blindable h = hash_blindable' \<circ> map_blindable\<^sub>m h id"
B_def: shows "local.B \<equiv> \<chi>i j. Re (A $h i $h j)"
closure_to_conjugate_space: fixes X :: "'d set" shows "closure (to_conjugate_space ` X) = to_conjugate_space ` closure X"
reachable_trans: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
small_insert_V: fixes X :: "V set"   and a :: "V" assumes "small X" shows "small (insert a X)"
ipurge_ref_aux_empty: fixes u :: "'b"   and I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and X :: "'c set" assumes "u \<in> sinks_aux I D U xs"   and "\<And>x. x \<in> X \<Longrightarrow> (u, D x) \<in> I" shows "ipurge_ref_aux I D U xs X = {}"
imp_cnj_compat_eqvR: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (eqv \<chi>1 \<chi>2) (eqv (cnj \<phi> \<chi>1) (cnj \<phi> \<chi>2)))"
g3cp_def: shows "g3cp \<equiv> {(xa, x). g3cpp xa x}"
assms_1: shows "valid ((Mn n f, xs, []) # rest)"
intros_3: fixes \<phi> :: "'b" shows "Formula.hereditarily_fs (Formula.Pred\<^sub>\<alpha> \<phi>)"
transversal_segment_reverse: fixes x :: "'a"   and y :: "'a" assumes "rev.transversal_segment x y" shows "rev.transversal_segment y x"
le_finE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2" obtains v\<^sub>1 :: "'a list" where "(traces.eq_swap ind)\<^sup>*\<^sup>* (w\<^sub>1 @ v\<^sub>1) w\<^sub>2"
simps: fixes M :: "('a, 'b, 'c) FSM"   and p :: "'c"   and xs :: "'a list"   and q :: "'c" shows "d_reaches M p xs q = (\<exists>tr ys. d_reached_by M p xs q tr ys)"
one_dim_iso_minus: fixes a :: "'c"   and b :: "'c" shows "one_dim_iso (a - b) = one_dim_iso a - one_dim_iso b"
finsum_all0: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = \<zero>\<^bsub>M\<^esub>" shows "finsum M f A = \<zero>\<^bsub>M\<^esub>"
assms: shows "run_one_step d i (s, vs, ves, $Testop x281 x282) = (s', vs', res)"
r: shows "s to x' in ([(x, x')] \<bullet> t) \<mapsto> s to x' in ([(x, x')] \<bullet> t')"
if_reg_stop_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and c\<^sub>A_then :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_else :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_then :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_else :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Stmt.If (bexp.Eq x 0) c\<^sub>A_then c\<^sub>A_else ;; c\<^sub>A_tail"   and "c\<^sub>C = Stop ;; Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<in> mds\<^sub>A AsmNoWrite \<and> x \<notin> mds\<^sub>A AsmNoReadOrWrite"   and "ev\<^sub>B\<^sub>C mem\<^sub>C (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A (bexp.Eq x 0)"   and "\<forall>mem\<^sub>A' mem\<^sub>C' mem\<^sub>A'' mem\<^sub>C''. mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> mem\<^sub>A'' = mem\<^sub>A' \<and> ev\<^sub>B\<^sub>C mem\<^sub>C' (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A' (bexp.Eq x 0) \<and> ev\<^sub>B\<^sub>C mem\<^sub>C'' (bexp\<^sub>C.Eq reg\<^sub>C 0) = ev\<^sub>B mem\<^sub>A'' (bexp.Eq x 0) \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A, \<langle>Stmt.If (bexp\<^sub>C.Eq reg\<^sub>C 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
modulo_tree_def: fixes x :: "'b tree"   and y :: "'b tree" shows "x mod y = pure (mod) \<diamondop> x \<diamondop> y"
VP_G_def: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" shows "VP_G G = ((\<forall>C m l A B I. G\<down>(C, m, l) = Some (A, B, I) \<longrightarrow> G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I) \<and> (\<forall>C m par code l0 T MI Anno. mbody_is C m (par, code, l0) \<longrightarrow> MST\<down>(C, m) = Some (T, MI, Anno) \<longrightarrow> G \<rhd> \<lbrace> (\<lambda>s0 s. s = mkState s0) \<rbrace> C,m,l0 \<lbrace> mkPost T \<rbrace> mkInv MI))"
abs_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "abs_nat 0 = zero"
infA: shows "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<in> fps L f"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "\<tau>.B.inverse_arrows f g" shows "B f g = \<tau>.B.dom g"
focus_sum: fixes X :: "'c set"   and f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and I :: "'e set" shows "focus X (sum f I) = (\<Sum>i\<in>I. focus X (f i))"
frefD: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set"   and y :: "'c"   and x :: "'a" assumes "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"   and "P y"   and "(x, y) \<in> R" shows "(f x, g y) \<in> S"
lift_pref_profile_permute_alts: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and alts :: "'a set"   and alts' :: "'a set"   and agents :: "'b set"   and agents' :: "'b set"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "alts \<subseteq> alts'" shows "lift_pref_profile agents alts agents' alts' (permute_profile \<sigma> R) = permute_profile \<sigma> (lift_pref_profile agents alts agents' alts' R)"
strict_implies_order: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p b a" shows "ord_p b a"
res_units_eq: shows "Units R = {x. 0 < x \<and> x < int p \<and> coprime x (int p)}"
swapEnvIm_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option" shows "swapEnvIm xs x y rho \<equiv> \<lambda>zs. lift (swap xs x y) (rho zs)"
r_witness: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "\<exists>j. 0 < x $h j \<and> r x = (A *v x) $h j / x $h j"
h2: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) (B\<^sup>2 * C - A\<^sup>2) = Bv\<^sup>2 * Cv - Av\<^sup>2"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2))"
x2: shows "lincomb_list (lc(j := lc j - lc i * c)) (us[i := us ! i + c \<cdot>\<^sub>v us ! j]) = (lc j - lc i * c) \<cdot>\<^sub>v us ! j + (lc i \<cdot>\<^sub>v us ! i + lc i \<cdot>\<^sub>v (c \<cdot>\<^sub>v us ! j) + M1)"
circline_eq_clmat_refl: fixes H :: "circline_mat" shows "circline_eq_clmat H H"
map_snd_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
"2_2": shows "Bex (greachable A w V u) (gaccepting A)"
ilt_interleave1: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "w \<in> w1 \<otimes> w2"   and "w1 \<noteq> []" shows "w2 \<prec> w"
\<psi>v_1: shows "\<psi> v1 = w1 +o \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {s1}\<rangle>"
HAStates_SA_mem: fixes SA :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a" assumes "SA \<in> SAs A"   and "S \<in> States SA" shows "S \<in> HAStates A"
V_arrow_elim_strong: fixes u :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and c :: "'a" assumes "u \<in> V (T\<^sub>1\<rightarrow>T\<^sub>2)" obtains a :: "name" and t :: "trm" where "a \<sharp> c" and "u = Lam [a].t" and "\<forall>v\<in>V T\<^sub>1. \<exists>v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
project_coord_inner_Basis: fixes i :: "'a"   and x :: "'a"   and b :: "'a"   and n :: "real" assumes "i \<in> Basis" shows "project_coord x b n \<bullet> i = (if i = b then n else if i = - b then - n else x \<bullet> i)"
f_shrink_append_mod: fixes xs :: "'b message_af list"   and k :: "nat"   and ys :: "'b message_af list" assumes "length xs mod k = 0" shows "(xs @ ys) \<div> k = xs \<div> k @ ys \<div> k"
token_run_step': fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "\<UU>.token_run x (Suc n) = \<up>af\<^sub>G\<^sub>\<UU> (\<UU>.token_run x n) (w n)"
length_remdups_concat: fixes xss :: "'g list list" shows "length (remdups (concat xss)) = card (\<Union> (set ` set xss))"
dual_undual: fixes x' :: "'c dual" shows "dual (undual x') = x'"
orbit_eqI_1: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y = f x" shows "y \<in> orbit f x"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_all_front_tickfree_traces1: fixes A :: "'a set" shows "T (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {s. front_tickFree s \<and> set s \<subseteq> ev ` A \<union> {tick}}"
proots_within_proots: fixes p :: "'a poly"   and s :: "'a set" shows "proots_within p s \<subseteq> proots p"
sign_simps_1: fixes x :: "'a" assumes "(0::'a) < x" shows "sign x = 1"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'a"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation"   and a4 :: "nat \<times> nat \<Rightarrow> 'a"   and a5 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4, a5) \<in> pr_algo_lts" obtains f :: "nat \<times> nat \<Rightarrow> 'a" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fa__" and "a2 = la__" and "a3 = PUSH" and "a4 = push_effect fa__ e" and "a5 = la__" and "push_precond fa__ la__ e"   | f :: "nat \<times> nat \<Rightarrow> 'a" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL" and "a4 = f" and "a5 = relabel_effect f l u" and "relabel_precond f l u"
simps_11: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and y :: "nat"   and \<omega> :: "Formula.agg_type \<times> event_data"   and b :: "nat"   and f :: "Formula.trm"   and \<phi> :: "Formula.formula" shows "Formula.sat \<sigma> V v i (formula.Agg y \<omega> b f \<phi>) = (let M = {(x, ecard Zs) |x Zs. Zs = {zs. length zs = b \<and> Formula.sat \<sigma> V (zs @ v) i \<phi> \<and> Formula.eval_trm (zs @ v) f = x} \<and> Zs \<noteq> {}} in (M = {} \<longrightarrow> fv \<phi> \<subseteq> {0..<b}) \<and> v ! y = eval_agg_op \<omega> M)"
Pi_I: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
cont': shows "continuous_on {real_of_int a..real_of_int b} f'"
mbisim_thrNone_eq: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2" shows "(thr s1 t = None) = (thr s2 t = None)"
split_req_3: fixes xs :: "('a btree \<times> 'a) list"   and p :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list" assumes "split xs p = (ls, (sub, sep) # rs)"   and "sorted_less (separators xs)" shows "p \<le> sep"
prems_1: shows "(v \<lhd> \<sigma>, u \<lhd> \<sigma>) \<in> trm_ord"
complement_blocks_wf: fixes bl :: "'a set" assumes "bl \<in># \<B>\<^sup>C" shows "bl \<subseteq> \<V>"
Methd_triple_valid2_SucI: fixes G :: "prog"   and n :: "nat"   and P :: "'b assn"   and C :: "qtname"   and sig :: "sig"   and Q :: "'b assn" assumes "G\<Turnstile>n\<Colon>{Normal P} body G C sig-\<succ> {Q}" shows "G\<Turnstile>Suc n\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
weakStatImpReflexive: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "{(\<Psi>, P, P) |\<Psi> P. True} \<subseteq> Rel" shows "\<Psi> \<rhd> P \<lessapprox><Rel> P"
permute_atom_def: fixes p :: "'a perm"   and a :: "'a" shows "permute_atom p a = Rep_perm p a"
nat_pow_one: fixes n :: "nat" shows "[n] \<cdot> \<zero> = \<zero>"
Un_empty_left: fixes B :: "'c set" shows "{} \<union> B = B"
lookup_empty_trie: fixes ks :: "'c list" shows "lookup_trie empty_trie ks = None"
compact_cball: fixes x :: "'a"   and e :: "real" shows "compact (cball x e)"
f_eq': fixes x :: "int" shows "f x = m * x + f 0"
le2: fixes n :: "nat" shows "\<integral>\<^sup>+ x. ennreal (norm (F n x - f x)) \<partial>M \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) + ennreal (norm (F n x)) \<partial>M"
chain: shows "Complete_Partial_Order.chain (rel_prod (ord_spmf (=)) (ord_spmf (=))) Y"
absorb2: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le>s a" shows "signed.min a b = b"
simps_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
d_delta_arctan_upper_14_neg: fixes x :: "real" shows "diff_delta_arctan_upper_14 x < 0"
M\<^sub>R_3: shows "\<forall>i\<le>n. M\<^sub>R i i = Le 0"
strict_implies_not_eq: fixes b :: "'c list"   and a :: "'c list" assumes "strict_prefix b a" shows "a \<noteq> b"
cofactor_def: shows "cofactor \<equiv> map_fun id of_fun (\<lambda>A i j. Square_Matrix.det (minor A i j))"
cl_least: fixes c :: "'b set"   and L :: "'b set set"   and r :: "'b set" assumes "c \<in> L"   and "r \<subseteq> c" shows "cl L r \<subseteq> c"
le_max3I2: fixes n3 :: "nat"   and n1 :: "nat"   and n2 :: "nat" shows "n3 \<le> max n1 (max n2 n3)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s2))"
prems_4: shows "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
pairs_top_closed: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "(top, Filters.filter.Abs_filter {top}) \<in> triple.pairs phi"
Nonce_EphK: fixes X :: "nonce_t"   and X' :: "ephkey" shows "Nonce X \<noteq> EphK X'"
norm_cauchy_schwarz_abs_eq: fixes x :: "'a"   and y :: "'a" shows "(\<bar>x \<bullet> y\<bar> = norm x * norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x)"