hcju/isabelleps huggingface dataset:
analz_image_keys_lemma: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> analz (G \<union> H) \<longrightarrow> X \<in> analz H" shows "(X \<in> analz (G \<union> H)) = (X \<in> analz H)"
meet_above_iff: fixes z :: "'c"   and x :: "'c"   and y :: "'c" shows "(z \<sqsubseteq> x \<sqinter> y) = (z \<sqsubseteq> x \<and> z \<sqsubseteq> y)"
inter_add_right2: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" assumes "x \<in># N" shows "N \<inter># (x, M) = x, (N - (x, {#})) \<inter># M"
monom_one_Suc: fixes n :: "nat" shows "monom (UP Zp) \<one> (Suc n) = monom (UP Zp) \<one> n \<otimes>\<^bsub>UP Zp\<^esub> monom (UP Zp) \<one> 1"
simps_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Abrupt x2 = Abrupt y2) = (x2 = y2)"
is_class_def: fixes P :: "'b prog"   and C :: "String.literal" shows "is_class P C \<equiv> class P C \<noteq> None"
forest_components_star: fixes x :: "'a" assumes "x \<sqinter> id x \<le> top" shows "top = top \<sqinter> top"
mono_D21: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "D (S `;` Q) \<subseteq> D (S `;` P)"
iT_iff_5: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "(x \<in> [ r, mod m, c ]) = (x mod m = r mod m \<and> r \<le> x \<and> x \<le> r + m * c)"
ed_ou_def: fixes X :: "'a set" shows "ed_ou X = (I \<circ> K = K \<circ> I)"
Domain_Id_on: fixes A :: "'a set" shows "r2s (s2r A) = A"
diff_less_mono: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a < b"   and "c \<le> a" shows "a - c < b - c"
independent_split_2: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "R.independent K (Us @ Vs)" shows "R.independent K Us"
rtrancl_beta_AppL: fixes s :: "Core.term"   and s' :: "Core.term"   and t :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" shows "s $ t \<rightarrow>\<^sub>\<beta>\<^sup>* s' $ t"
assms: shows "other.s \<delta> \<le> u \<delta>"
last_index_size_conv: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "length xs = n" shows "(last_index xs x = n) = (x \<notin> set xs)"
taut2emptyrem: fixes Cn :: "'b set \<Rightarrow> 'b set"   and \<phi> :: "'b"   and A :: "'b set" assumes "Tarskian_logic Cn"   and "\<phi> \<in> Cn {}" shows "Tarskian_logic.remainder Cn A \<phi> = {}"
componentsE: fixes S :: "'a set"   and U :: "'a set" assumes "S \<in> components U" obtains x :: "'a" where "x \<in> U" and "S = connected_component_set U x"
\<phi>_in_hom_2: shows "\<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0\<guillemotright>"
rep_eq: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "'a mpoly" shows "insertion x xa = insertion_aux x (mapping_of xa)"
zeros: fixes z :: "complex"   and m :: "nat" assumes "z \<in> A m"   and "exp z = 1" shows "\<exists>k\<in>{- int m..int m}. z = complex_of_real (2 * pi * real_of_int k) * \<i>"
collect_locksI: fixes las :: "'c lock_actions"   and l :: "'c" assumes "Lock \<in> set (las $ l)" shows "l \<in> collect_locks las"
rep_eq: fixes x :: "(nat, rat) fmap" shows "Rep_linear_poly (LinearPoly x) = get_var_coeff x"
assms_1: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
scale_left_distrib: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
AssertionStatImpClosed: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and p :: "(name \<times> name) list" assumes "\<Psi> \<hookrightarrow> \<Psi>'" shows "p \<bullet> \<Psi> \<hookrightarrow> p \<bullet> \<Psi>'"
dependent_def: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and P :: "'e set" assumes "vector_space scale" shows "module.dependent scale P = (\<exists>a\<in>P. a \<in> module.span scale (P - {a}))"
64260: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - - (x \<sqinter> y)) = - x"
rtrancl_list_impl: fixes r :: "('b \<times> 'b) list"   and as :: "'b list" shows "set (rtrancl_list_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>*}"
quad_part_1_eq: fixes var :: "nat"   and L :: "real list"   and p :: "real mpoly"   and deg :: "nat"   and D :: "real"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and d :: "real mpoly"   and C :: "real" assumes "var < length L"   and "MPoly_Type.degree p var = deg"   and "D \<noteq> 0"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = A"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = B"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = D" shows "aEval (Eq p) (L[var := (A + B * C) / D]) = aEval (Eq (quadratic_part_1 var a b d (Eq p))) (L[var := C])"
induct: fixes x :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "x \<in> Induction_ax"   and "\<And>v x x0 w xw xevw allw allvw ax. \<lbrakk>SubstForm v 0 x x0; SubstForm v w x xw; SubstForm v (q_Eats v w) x xevw; AbstForm w 0 (q_Imp x (q_Imp xw xevw)) allw; AbstForm v 0 (q_All allw) allvw; AbstForm v 0 x ax; v \<noteq> w; VarNonOccForm w x\<rbrakk> \<Longrightarrow> P (q_Imp x0 (q_Imp (q_All allvw) (q_All ax)))" shows "P x"
map_le_map_add: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "f \<subseteq>\<^sub>m f \<Oplus> g"
ipurge_tr_rev_aux_nil: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list" shows "(ipurge_tr_rev_aux I D U xs = []) = (\<forall>u\<in>U. \<not> (\<exists>v\<in>D ` set xs. (v, u) \<in> I))"
R39_4: shows "pmf (sds R39) d = 1 / 2"
ex1: shows "\<exists>!B. chamber B \<and> B \<notin> f \<turnstile> X \<and> f ` B = f ` C"
tanh_real_neg_iff: fixes x :: "real" shows "(tanh x < 0) = (x < 0)"
oexp_mono_le: fixes \<gamma> :: "V"   and \<beta> :: "V"   and \<alpha> :: "V" assumes "\<gamma> \<le> \<beta>"   and "\<alpha> \<noteq> 0"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<alpha> \<up> \<gamma> \<le> \<alpha> \<up> \<beta>"
invariant_pairI: fixes A :: "('e \<times> 'f, 'g) automaton"   and P :: "'e \<times> 'f \<Rightarrow> bool"   and I :: "'g \<Rightarrow> bool" assumes "\<And>\<xi> p. (\<xi>, p) \<in> init A \<Longrightarrow> P (\<xi>, p)"   and "\<And>\<xi> p \<xi>' p' a. \<lbrakk>(\<xi>, p) \<in> reachable A I; P (\<xi>, p); ((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P (\<xi>', p')" shows "A \<TTurnstile> (I \<rightarrow>) P"
ord_conv_Least: fixes x :: "'a" assumes "x \<in> carrier G"   and "\<exists>n>0. x [^] n = \<one>" shows "ord x = (LEAST n. 0 < n \<and> x [^] n = \<one>)"
Delay_msg1: fixes x :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list" assumes "msg (Suc 0) x"   and "Delay x i1 d y i2" shows "msg (Suc 0) y"
inj_dual_iff: fixes dual :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.dual dual" shows "(dual x = dual y) = (x = y)"
bisim2: fixes cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a"   and s :: "bool list \<Rightarrow> bool list option" shows "rel_spmf (\<lambda>(x, uu_, s) (y, s'). x = y \<and> s = s') (exec_gpv (\<lambda>(s', s) y. map_spmf (\<lambda>((x, s'), s). (x, (), s)) (exec_gpv prf.random_oracle (prf_encrypt_oracle () y) s)) (\<A>2 cipher \<sigma>) ((), s)) (exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) s)"
prj_chine_assoc_3: shows "trg_trg_f.Prj\<^sub>0\<^sub>0 \<cdot> trg_trg_f.chine_assoc = trg_trg_f.Prj\<^sub>0"
FOO: shows "Result.those (map (compile_env_entry \<A> \<O>) (Finca_to_list e)) = Ok xs"
corec_transfer: fixes S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "((S ===> (=)) ===> (S ===> R) ===> (S ===> (=)) ===> (S ===> llist_all2 R) ===> (S ===> S) ===> S ===> llist_all2 R) corec_llist corec_llist"
Rep_apply_matrix: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c matrix"   and j :: "nat"   and i :: "nat" assumes "f (0::'c) = (0::'b)" shows "Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
a1: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>'. p = ( \<Phi>' \<Rightarrow>* \<Psi>' \<oplus> Modal M Ms)"
merge_rt_F\<^sub>m_cong: fixes x :: "('c \<times> 'd list, 'e \<times> 'f list) blindable\<^sub>m"   and y :: "('c \<times> 'd list, 'e \<times> 'f list) blindable\<^sub>m"   and ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and ma' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and mm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option"   and mm' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option"   and ha :: "'c \<Rightarrow> 'e"   and hm :: "'d \<Rightarrow> 'f" assumes "\<And>a b. \<lbrakk>a \<in> set1_blindable\<^sub>m x \<bind> fsts; b \<in> set1_blindable\<^sub>m y \<bind> fsts\<rbrakk> \<Longrightarrow> ma a b = ma' a b"   and "\<And>a b. \<lbrakk>a \<in> set1_blindable\<^sub>m x \<bind> snds \<bind> set; b \<in> set1_blindable\<^sub>m y \<bind> snds \<bind> set\<rbrakk> \<Longrightarrow> mm a b = mm' a b" shows "merge_rt_F\<^sub>m ha ma hm mm x y = merge_rt_F\<^sub>m ha ma' hm mm' x y"
eqvtI: fixes P :: "pi"   and Q :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<approx>\<^sup>s Q" shows "perm \<bullet> P \<approx>\<^sup>s perm \<bullet> Q"
iso_is_arr: fixes f :: "'c local.term set" assumes "\<F>C.iso f" shows "\<F>C.arr f"
p_def: shows "p \<equiv> Poly (map (\<lambda>k. c (N - k)) [0..<Suc N])"
ins_list_Cons: fixes x :: "'b"   and xs :: "'b list" assumes "sorted_less (x # xs)" shows "ins_list x xs = x # xs"
test_bit_numeral: fixes w :: "num"   and n :: "nat" shows "numeral w !! n = (n < LENGTH('b) \<and> bit (numeral w) n)"
tree_type: fixes t :: "'a bro"   and h :: "nat" assumes "t \<in> T (h + 1)" shows "tree t \<in> B (h + 1) \<union> B h"
inner_loop2_correct: fixes succl :: "'a list"   and succ :: "'a set"   and dsti :: "'a"   and dst :: "'a"   and ui :: "'b"   and u :: "'b"   and PREDi :: "'a \<Rightarrow> 'b option"   and PRED :: "'a \<Rightarrow> 'b option"   and Ni :: "'a set"   and N :: "'a set" assumes "(succl, succ) \<in> \<langle>Id\<rangle>list_set_rel"   and "(dsti, dst) \<in> Id"   and "(ui, u) \<in> Id"   and "(PREDi, PRED) \<in> Id"   and "(Ni, N) \<in> Id" shows "inner_loop2 dsti succl ui PREDi Ni \<le> \<Down> Id (add_succ_spec dst succ u PRED N)"
multiple_mask_trivia: fixes m :: "nat"   and n :: "nat"   and x :: "'b word" assumes "m \<le> n" shows "(x AND NOT (mask n)) + (x AND mask n AND NOT (mask m)) = x AND NOT (mask m)"
isGlbD1a: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isGlb R S x" shows "x \<in> R"
catch_fail: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b" assumes "monad_catch return bind fail catch" shows "catch fail m = m"
interval_integral_cong_AE: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and a :: "ereal"   and b :: "ereal" assumes "f \<in> borel_measurable borel"   and "g \<in> borel_measurable borel"   and "AE x\<in>einterval (min a b) (max a b) in lborel. f x = g x" shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
covering_lemma: fixes S :: "'c set"   and a :: "'c"   and b :: "'c"   and g :: "'c \<Rightarrow> 'c set" assumes "S \<subseteq> cbox a b"   and "box a b \<noteq> {}"   and "gauge g" obtains \<D> :: "'c set set" where "countable \<D>" and "\<Union> \<D> \<subseteq> cbox a b" and "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" and "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>" and "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x\<in>S \<inter> K. K \<subseteq> g x" and "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n" and "S \<subseteq> \<Union> \<D>"
rc1: fixes n :: "nat"   and w :: "'a word" assumes "source_size revcast + n = target_size revcast" shows "ucast w = word_reverse (ucast (word_reverse w) << n)"
path_cons_elim: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and a :: "('d \<times> 'e) \<times> 'f"   and r :: "(('d \<times> 'e) \<times> 'f) list"   and p :: "'f" assumes "path A (a # r) p" obtains "snd a \<in> succ A (fst a) p" and "path A r (snd a)"
l2_inv3D: fixes A :: "agent"   and B :: "agent"   and gnx :: "msg"   and Rb :: "fid_t"   and x :: "l2_state" assumes "Auth A B \<langle>Number (Suc 0), gnx, Exp Gen (NonceF (Rb $ ny))\<rangle> \<in> chan x"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv3" shows "\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xgnx = Some gnx \<and> guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))"
igAbsRen_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igAbsRen MOD \<equiv> \<forall>xs y x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
supremum_pair: fixes a :: "'a"   and b :: "'a" shows "supremum {a, b} = {a \<squnion> b}"
case_2: fixes f1 :: "'d option \<Rightarrow> 'e \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case Delete x2 of Insert x xa \<Rightarrow> f1 x xa | Delete x \<Rightarrow> f2 x) = f2 x2"
assms_2: shows "\<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1"
simps_10: fixes t1 :: "Syntax.term"   and t2 :: "Syntax.term"   and f :: "(Syntax.var, Syntax.term) fmap" shows "psubst_term (App t1 t2) f = App (psubst_term t1 f) (psubst_term t2 f)"
Int: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<inter> b \<in> sets M"
wls_swapAbs_preserves_freshAbs: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and xs :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and x :: "'var" assumes "wlsAbs (us, s) A" shows "freshAbs xs (x @xs[y1 \<and> y2]_ys) (A $[y1 \<and> y2]_ys) = freshAbs xs x A"
tgt_V: shows "io_targets PM (initial PM) vs = {target (vs || trV) (initial PM)}"
finprod_Union_disjoint: fixes C :: "'a set set"   and f :: "'a \<Rightarrow> int" assumes "finite C"   and "\<And>A. A \<in> C \<Longrightarrow> finite A \<and> (\<forall>x\<in>A. f x \<in> carrier R)"   and "pairwise disjnt C" shows "finsum R f (\<Union> C) = finsum R (finsum R f) C"
diffconst_sound_lemma: shows "sound (proof_result DiffConstProof)"
cop_F_RH_mono: fixes ds :: "'b set"   and fp :: "'a set"   and x :: "'a" assumes "cop_F_closed_inv ds fp"   and "cop_F_range_inv ds fp"   and "Xd x \<notin> Xd ` Ch (Xh x) fp"   and "x \<in> Rf CH fp" shows "x \<in> Rf CH (cop_F ds fp)"
Nmlize_Vcomp_Cod_Arr: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t \<^bold>\<cdot> t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
axioms_1: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" assumes "FunctorM F" shows "PreFunctor F"
map_simp: fixes f :: "'b" assumes "B.arr f" shows "local.map f = f"
red_not_final_thread: fixes s :: "('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 t :: "'t"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "('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" assumes "s -1-t\<triangleright>ta\<rightarrow> s'" shows "r1.not_final_thread s t"
get_def: shows "State_Monad.get = State (\<lambda>s. (s, s))"
AssertionStatEqEqvt: fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "p \<bullet> \<Psi> \<simeq> \<Psi>' = p \<bullet> \<Psi> \<simeq> p \<bullet> \<Psi>'"
rGamma_nonzero: fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "rGamma z \<noteq> (0::'a)"
rewriteR_comp_comp: fixes g :: "'e \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'e"   and r :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'f" assumes "g \<circ> h = r" shows "f \<circ> g \<circ> h = f \<circ> r"
dioid_sum_prod: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "\<Sum> {f x |x. x \<in> A} \<cdot> \<Sum> {g x |x. x \<in> A} = \<Sum> {f x \<cdot> g y |x y. x \<in> A \<and> y \<in> A}"
triple_valid2_def2: fixes G :: "prog"   and n :: "nat"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" shows "G\<Turnstile>n\<Colon>{P} t\<succ> {Q} = (\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y', s') \<longrightarrow> (\<forall>L. s\<Colon>\<preceq>(G, L) \<longrightarrow> (\<forall>T C A. (normal s \<longrightarrow> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T \<and> \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>t\<guillemotright> A) \<longrightarrow> Q Y' s' Z \<and> s'\<Colon>\<preceq>(G, L)))))"
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xa :: "'e set['g]"   and xb :: "'f set['g]" shows "rel_bset x xa xb = rel_set x (set_bset xa) (set_bset xb)"
openin_Union: fixes K :: "'c set set"   and U :: "'c topology" assumes "\<And>S. S \<in> K \<Longrightarrow> openin U S" shows "openin U (\<Union> K)"
\<rho>_def: shows "\<rho> = hd (renamings_apart (DA0 # CAs0))"
z_neutral: fixes x :: "'a"   and y :: "'a" shows "x \<^bold>* y \<^bold>* \<^bold>1 = x \<^bold>* y"
set_mkIde: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set" assumes "A \<subseteq> Univ"   and "|A| <o natLeq" shows "local.set (mkIde A) = A"
terminatess_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f" shows "\<Gamma>\<turnstile>cs,css \<Down> Fault f"
assms_2: shows "coincide_on \<sigma> \<eta> (vars_of_lit L1)"
loop_def: shows "loop \<equiv> while (dwa_cond (brec_det_algo (build_rqrm \<delta>) Qi \<delta>)) (dwa_step (brec_det_algo (build_rqrm \<delta>) Qi \<delta>)) (dwa_initial (brec_det_algo (build_rqrm \<delta>) Qi \<delta>))"
list_length_hint1a: fixes l :: "'d list" assumes "l \<noteq> []" shows "0 < length l"
pred_inject_2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "pred_arctic_delta P (Num_arc_delta a) = P a"
fair'_2: shows "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}"
discarded_subset: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and D' :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and D :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and hs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and sps :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ap :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data' :: "nat \<times> 'h" assumes "ab_spec ab"   and "D' = D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs (ps -- sps) hs data'))"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "D \<subseteq> (set gs \<union> set bs) \<times> (set gs \<union> set bs)" shows "D' \<subseteq> (set gs \<union> set (ab gs bs hs data')) \<times> (set gs \<union> set (ab gs bs hs data'))"
matchC_TMC_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_TMC theta c d \<equiv> \<forall>s t c' s'. mustT c s \<and> mustT d t \<and> s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> (\<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta)"
UNlift01_shift: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and dl :: "('c, 'd, 'a) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis"   and "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<PL.brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "n \<in> theNFTBoth cl dl"   and "I \<in> P n" shows "PL_Indis.UNlift01 cl dl P F (PL_Indis.shift cl n I) = PL_Indis.shift dl n (F n I)"
arith_simps_8: fixes m :: "num"   and n :: "num" shows "num.Bit1 m + num.Bit0 n = num.Bit1 (m + n)"
elims_2: fixes x :: "'c local.term" assumes "Diag x" obtains "x = \<^bold>\<I>"   | f :: "'c" where "x = \<^bold>\<langle>f__\<^bold>\<rangle>" and "C.arr f__"   | f :: "'c" and u :: "'c local.term" where "x = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u" and "C.arr f \<and> Diag u \<and> u \<noteq> \<^bold>\<I>"
uminus_local_ring_at_def: shows "uminus_local_ring_at \<equiv> cq.uminus_rel"
sym_Un_converse: fixes r :: "('c \<times> 'c) set" shows "sym (r \<union> r\<inverse>)"
path_connected_open_segment: fixes a :: "'b"   and b :: "'b" shows "path_connected {a<--<b}"
defl_app_def: shows "defl \<equiv> \<lambda>x. TC('d)\<cdot>DEFL('c)"
simps_2: fixes \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm" shows "qfree (And \<phi>\<^sub>1 \<phi>\<^sub>2) = (qfree \<phi>\<^sub>1 \<and> qfree \<phi>\<^sub>2)"
Healthy_range: fixes H :: "'d upred \<Rightarrow> 'd upred" assumes "Idempotent H" shows "range H = \<lbrakk>H\<rbrakk>\<^sub>H"
directional_derivative_eq_frechet_derivative: fixes k :: "enat"   and f :: "'c \<Rightarrow> real"   and a :: "'c"   and v :: "'c" assumes "k-smooth_on UNIV f" shows "directional_derivative k a v f = frechet_derivative f (at a) v"
H_eq_one: fixes x :: "'c" assumes "x \<in> cball (0::'c) 1" shows "H x = 1"
one_pos_def: shows "1 \<equiv> Abs_pos (1::'a)"
dfa_Accessible: fixes M :: "'b dfa" assumes "dfa M" shows "dfa (dfa.Accessible_dfa M)"
analz_Un_implSecureSet_aux: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
cycle_free_diag_intro: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs" shows "cycle_free M n"
f_def: shows "f \<equiv> \<lambda>x. ln (ln (x * exp (x * exp x) + 1)) - exp (exp (ln (ln x) + 1 / x))"