hcju/isabelleps huggingface dataset:
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n"   and "n \<le> p" shows "prod g {m..<n} * prod g {n..<p} = prod g {m..<p}"
r_code_const1_aux: fixes i :: "nat"   and r :: "nat"   and c :: "nat" shows "eval r_code_const1_aux [i, r, c] \<down>= quad_encode 3 1 1 (singleton_encode r)"
size_fset_of_list: fixes l :: "'a list" shows "size (fset_of_list l) = length (remdups l)"
FrameStatEq_def: fixes F :: "'b frame"   and G :: "'b frame" shows "F \<simeq>\<^sub>F G \<equiv> F \<hookrightarrow>\<^sub>F G \<and> G \<hookrightarrow>\<^sub>F F"
1: shows "eval (Cn 1 r_pdec1 [Mn1f]) [b] \<down>= pdec1 (the (eval (Mn 1 f) [b]))"
msum_pdevs_raw_def: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "msum_pdevs_raw n x y i = (if i < n then x i else y (i - n))"
mat_diag_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "mat_diag n f = Matrix.mat n n (\<lambda>(i, j). if i = j then f j else (0::'a))"
path_split_snoc: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" obtains a' :: "'edge" and as' :: "'edge list" where "as = as' @ [a']" and "n -as'\<rightarrow>* sourcenode a'" and "valid_edge a'" and "n' = targetnode a'"
bad_sticky1: fixes s2 :: "'s2" assumes "bad2 s2"   and "I2 s2" shows "callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"
ih: fixes M :: "nat multiset" assumes "local.well_base M"   and "\<forall>m\<in>#M. m < n" shows "\<Sum>\<^sub># (image_mset ((^) base) M) < base ^ n"
lless_eq: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> x \<noteq> y)"
fds_newman_def: shows "fds_newman = fds (\<lambda>x. complex_of_real (\<MM> (real x)))"
other_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a" shows "other P I \<sigma> \<sigma>' \<equiv> \<forall>i. if i \<in> I then \<sigma>' i = \<sigma> i else P (\<sigma> i) (\<sigma>' i)"
assms: shows "f \<in> S \<rightarrow> mat_kernel A"
minus_set_def: fixes A :: "'d set"   and B :: "'d set" shows "A - B = Collect ((\<lambda>x. x \<in> A) - (\<lambda>x. x \<in> B))"
profile_bin\<^sub>S\<^sub>t\<^sub>r\<^sub>o\<^sub>n\<^sub>g\<^sub>E\<^sub>q_\<^sub>v_\<^sub>v_axioms: shows "profile_bin\<^sub>S\<^sub>t\<^sub>r\<^sub>o\<^sub>n\<^sub>g\<^sub>E\<^sub>q_\<^sub>v_\<^sub>v f"
contractible_imp_path_connected_space: fixes X :: "'b topology" assumes "contractible_space X" shows "path_connected_space X"
spectral_radius_ev: shows "\<exists>ev v. eigen_vector cA v ev \<and> cmod ev = HMA_Connect.spectral_radius cA"
elims: fixes x :: "'b heap"   and y :: "'b" assumes "get_min x = y" obtains xa :: "'b" and uu_ :: "'b heap list" where "x = Hp xa uu_" and "y = xa"   | "x = heap.Empty" and "y = undefined"
a_type_wf_def: fixes h :: "(_) heap" shows "l_type_wf_def\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_type_wf h = (DocumentClass.type_wf h \<and> (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h \<noteq> None))"
lambda_system_empty: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f" shows "{} \<in> lambda_system \<Omega> M f"
rep_eq: fixes x :: "('b, unit) RBT.rbt" shows "r_min_opt x = rbt_min_opt (rbt.impl_of x)"
reduction_def: fixes L1 :: "'a literal"   and C :: "'a eclause"   and \<sigma>' :: "('a \<times> 'a trm) list"   and t :: "'a trm"   and s :: "'a trm"   and polarity :: "sign"   and L2 :: "'a literal"   and u :: "'a trm"   and u' :: "'a trm"   and p :: "indices list"   and v :: "'a trm"   and D :: "'a eclause"   and I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "reduction L1 C \<sigma>' t s polarity L2 u u' p v D I S \<sigma> = (D \<in> S \<and> eligible_literal L1 C \<sigma>' \<and> eligible_literal L2 D \<sigma>' \<and> ground_clause (subst_cl (cl_ecl D) \<sigma>') \<and> minimal_redex p (t \<lhd> \<sigma>) C S t \<and> coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C)) \<and> allowed_redex u' C \<sigma> \<and> \<not> is_a_variable u' \<and> L1 \<in> cl_ecl C \<and> L2 \<in> cl_ecl D \<and> orient_lit_inst L1 t s polarity \<sigma>' \<and> orient_lit_inst L2 u v pos \<sigma>' \<and> u \<lhd> \<sigma>' \<noteq> v \<lhd> \<sigma>' \<and> u' \<lhd> \<sigma>' = u \<lhd> \<sigma>' \<and> \<not> validate_ground_clause I (subst_cl (cl_ecl D - {L2}) \<sigma>') \<and> (subst_lit L2 \<sigma>', subst_lit L1 \<sigma>') \<in> lit_ord \<and> (\<forall>x\<in>cl_ecl D - {L2}. (subst_lit x \<sigma>', subst_lit L2 \<sigma>') \<in> lit_ord) \<and> all_trms_irreducible (subst_set (trms_ecl D) \<sigma>') (\<lambda>t. trm_rep t S) \<and> I (u \<lhd> \<sigma>') (v \<lhd> \<sigma>') \<and> subterm t p u')"
rbt_lookup_keys: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "dom (rbt_lookup t) = set (RBT_Impl.keys t)"
\<tau>move1CondRed: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and e1 :: "('b, 'c, 'addr) exp"   and e2 :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (if (Val v) e1 else e2)"
lm076: shows "inj_on ((\<lambda>pseudo. {(x, pseudo ,,, x) |x. x \<in> Domain pseudo}) \<circ> pseudoAllocation) (runiqs \<inter> Pow (UNIV \<times> (UNIV - {{}})))"
ChiPlus_ChiStar: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(S, T) \<in> ChiPlus A" shows "(S, T) \<in> ChiStar A"
Gromov_product_nonneg: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "0 \<le> Gromov_product_at e x y"
left_add_mult_distrib_mset: fixes i :: "nat"   and u :: "'b multiset"   and j :: "nat"   and k :: "'b multiset" shows "repeat_mset i u + (repeat_mset j u + k) = repeat_mset (i + j) u + k"
integrableD_2: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity>"
no_await_trans: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "no_await c"   and "\<langle>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "no_await c'"
eventually_removed_from_N: fixes D :: "'a literal multiset"   and i :: "nat" assumes "D \<in> N_of_state (lnth Sts i)"   and "fair_state_seq Sts"   and "enat i < llength Sts" shows "\<exists>l. D \<in> N_of_state (lnth Sts l) \<and> D \<notin> N_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
qtable_filter: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and R :: "'a option list \<Rightarrow> bool"   and Q' :: "'a option list \<Rightarrow> bool" assumes "qtable n A P Q X"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> (Q x \<and> R x) = Q' x" shows "qtable n A P Q' (Set.filter R X)"
nat_mono_def: fixes x :: "nat" shows "nat_mono x \<equiv> x \<noteq> 0"
residual_obtain_orig_block: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>R" obtains bl2 :: "'a set" where "b = bl2 - bl" and "bl2 \<in># remove1_mset bl \<B>"
trapG1_subset: shows "trapG1 \<subseteq> cball 0 20"
assms: shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
approximation_preproc_push_neg_2: fixes a :: "real"   and b :: "real" shows "(\<not> a \<le> b) = (b < a)"
exp_ge_zero: fixes x :: "real" shows "0 \<le> exp x"
Model_axioms_def: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and Ax :: "('a, 'b) Axioms" shows "Model_axioms I Ax \<equiv> (\<forall>\<phi>. \<phi> \<in> aAxioms Ax \<longrightarrow> L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool True) \<and> aS\<^bsub>Ax\<^esub> = iS\<^bsub>I\<^esub>"
upshift_def: fixes m :: "nat" shows "upshift m \<equiv> m * 2"
assms: shows "(u, v) \<in> Partitions x \<and> u \<in> A \<and> v \<in> B"
stable_Int: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> stable A"   and "F \<in> stable A'" shows "F \<in> stable (A \<inter> A')"
LSC_cnf: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n"   and "is_nnf_mset \<Gamma>" shows "image_mset cnf_form_of \<Gamma> \<Rightarrow>\<^sub>n"
zero_not_primepow: shows "\<not> primepow (0::'a)"
bot_less: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order_bot bot less_eq less" shows "(a \<noteq> bot) = less bot a"
moebius_rotation_inv: fixes \<phi> :: "real" shows "moebius_inv (moebius_rotation \<phi>) = moebius_rotation (- \<phi>)"
norm_zero: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm zero = 0"
rvI: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set"   and V :: "'var" assumes "n -as\<rightarrow>* n'"   and "n' \<in> backward_slice S"   and "V \<in> Use n'"   and "\<forall>nx\<in>set (sourcenodes as). V \<notin> Def nx" shows "V \<in> rv S n"
path_Entry_target_1: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>* (_Entry_)" shows "n = (_Entry_)"
angles_12: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "angle b1 c1 a1 = angle b2 c2 a2"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> enat) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>2 {x1} x2 {x3}"   and "\<And>P. P (\<lambda>s. eSuc (P s)) SKIP P"   and "\<And>P a x. P (\<lambda>s. eSuc (P (s[a/x]))) (x ::= a) P"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>2 {\<lambda>s. P s + \<up> (bval b s)} c\<^sub>1 {Q}; P (\<lambda>s. P s + \<up> (bval b s)) c\<^sub>1 Q; \<turnstile>\<^sub>2 {\<lambda>s. P s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}; P (\<lambda>s. P s + \<up> (\<not> bval b s)) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P (\<lambda>s. eSuc (P s)) (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>2 {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>2 {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>I b c. \<lbrakk>\<turnstile>\<^sub>2 {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}; P (\<lambda>s. I s + \<up> (bval b s)) c (\<lambda>t. I t + 1)\<rbrakk> \<Longrightarrow> P (\<lambda>s. I s + 1) (WHILE b DO c) (\<lambda>s. I s + \<up> (\<not> bval b s))"   and "\<And>P c Q P' Q'. \<lbrakk>\<turnstile>\<^sub>2 {P} c {Q}; P P c Q; \<And>s. P s \<le> P' s; \<And>s. Q' s \<le> Q s\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
pred_succ: fixes x :: "hf" shows "pred (succ x) = x"
expr: fixes accC :: "qtname"   and L :: "lenv"   and e :: "expr"   and eT :: "ty"   and s0 :: "st"   and E :: "assigned"   and v :: "val"   and s1 :: "state" assumes "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-eT"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd (Norm s0))) \<guillemotright>In1l e\<guillemotright> E"   and "P L accC (Norm s0) (In1l e) (In1 v) s1" shows "P L accC (Norm s0) (In1r (Expr e)) \<diamondsuit> s1"
Lcm_fin_eq_Lcm: fixes A :: "'a set" assumes "finite A" shows "Lcm\<^sub>f\<^sub>i\<^sub>n A = Lcm A"
redundant_code_def: fixes g :: "'g" shows "redundant_code g = (trivial_phis g \<noteq> {})"
supteq_supt_conv: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "s \<unrhd> t = (s \<rhd> t \<or> s = t)"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "S.arr f"   and "S.dom f = a" shows "S.comp f a = f"
eval_fps_divide': fixes f :: "complex fps"   and g :: "complex fps"   and r :: "ereal"   and z :: "complex" assumes "0 < fps_conv_radius f"   and "0 < fps_conv_radius g"   and "0 < r"   and "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"   and "ereal (cmod z) < Min {r, fps_conv_radius f, fps_conv_radius g}" shows "eval_fps (f / g) z = eval_fps f z / eval_fps g z"
fps_inverse_zero_conv_fps_const: shows "inverse 0 = fps_const (inverse (0::'a))"
supset_ld_is_ld: fixes A :: "'c set"   and B :: "'c set" assumes "lin_dep A"   and "A \<subseteq> B" shows "lin_dep B"
g_1: fixes v :: "'c" assumes "v \<in> C" shows "g v \<in> B"
right: fixes c21 :: "'a list" assumes "c1' = c1 @ qs # c21"   and "c2 = c21 @ qs' # c2'" shows "thesis"
Singleton_def: fixes x :: "ZF" shows "Singleton x \<equiv> Upair x x"
accepting_pair\<^sub>G\<^sub>R_simp: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and F :: "('b \<times> 'c \<times> 'b) set"   and \<I> :: "('b \<times> 'c \<times> 'b) set set"   and w :: "nat \<Rightarrow> 'c" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, \<I>) w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> F = {} \<and> (\<forall>I\<in>\<I>. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {})"
cc_def: shows "cc = 1 / (complex_of_real (2 * pi) * \<i>)"
hyps_2: shows "(\<Gamma> @ \<Delta>)\<turnstile>T1<:S1"
add_diff_cancel_left': fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
fin: shows "(\<Sum>i. f i) \<noteq> \<infinity>"
A_splay_ub3: fixes t :: "'b tree"   and x :: "'b" assumes "bst t" shows "A_splay x t \<le> 3 * \<phi> t + 1"
assms_11: shows "Ball (set (tl ms\<^sub>1)) return_node"
vec_nth_inject: fixes x :: "('c, 'b) Finite_Cartesian_Product.vec"   and y :: "('c, 'b) Finite_Cartesian_Product.vec" shows "(($h) x = ($h) y) = (x = y)"
pelims: fixes idx0 :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and idx1 :: "'b bound \<Rightarrow> 'b \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat"   and x :: "('a \<times> 'b) bound"   and xa :: "'a \<times> 'b"   and y :: "nat" assumes "prod_index_def.idx idx0 idx1 size1 x xa = y"   and "Wellfounded.accp prod_index_def.idx_rel (x, xa)" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" and a :: "'a" and b :: "'b" where "x = Bound (l0, r0) (l1, r1)" and "xa = (a, b)" and "y = idx0 (Bound l0 l1) a * size1 (Bound r0 r1) + idx1 (Bound r0 r1) b" and "Wellfounded.accp prod_index_def.idx_rel (Bound (l0, r0) (l1, r1), a, b)"
cubic_root_radical_sqrt_lemma: fixes a :: "real"   and b :: "real"   and c :: "real"   and e :: "expr" assumes "a \<in> \<rat>"   and "b \<in> \<rat>"   and "c \<in> \<rat>"   and "radicals e \<noteq> {}"   and "\<lbrace>e\<rbrace> ^ 3 + a * \<lbrace>e\<rbrace>\<^sup>2 + b * \<lbrace>e\<rbrace> + c = 0" shows "\<exists>e1. radicals e1 \<subset> radicals e \<and> \<lbrace>e1\<rbrace> ^ 3 + a * \<lbrace>e1\<rbrace>\<^sup>2 + b * \<lbrace>e1\<rbrace> + c = 0"
pred_alt_def_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vclosed (Vrecabs css name \<Gamma>) = (closed_venv \<Gamma> \<and> pred_fmap (list_all (\<lambda>(pat, t). closed_except t (fmdom \<Gamma> |\<union>| frees pat))) css \<and> name |\<in>| fmdom css \<and> True)"
P'''Chain: shows "P''' \<Longrightarrow>\<^sub>\<tau> P'"
sub_right_inj_on: fixes k :: "nat" shows "inj_on ((-) k) {..k}"
select_convs_3: fixes g_V :: "'c set"   and g_E :: "('c \<times> 'c) set"   and g_V0 :: "'c set"   and more :: "'d" shows "g_V0 \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V0"
nat_pow_0: fixes x :: "nat \<Rightarrow> 'a" shows "[0] \<cdot>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
insert_def: fixes x :: "'a"   and t :: "('a \<times> 'b) tree" shows "local.insert x t = (let (l, uu_, r) = split t x in join l x r)"
Max_wrt_among_preorder: fixes A :: "'alt set" shows "Max_wrt_among (R i) A = {x \<in> alts \<inter> A. \<forall>y\<in>alts \<inter> A. R i x y \<longrightarrow> R i y x}"
x_1: shows "(0::'a) < \<bar>x\<bar>"
homotopy_equivalent_space_def: fixes X :: "'a topology"   and Y :: "'b topology" shows "X homotopy_equivalent_space Y \<equiv> \<exists>f g. continuous_map X Y f \<and> continuous_map Y X g \<and> homotopic_with (\<lambda>x. True) X X (g \<circ> f) id \<and> homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id"
INT_simps_1: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'b set" shows "(\<Inter>x\<in>C. A x \<inter> B) = (if C = {} then UNIV else \<Inter> (A ` C) \<inter> B)"
assms_2: assumes "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = \<langle>a, b\<rangle>" shows "P"
comm_monoid_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "Group.comm_monoid_axioms G \<equiv> \<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
\<alpha>sc_def: shows "\<alpha>sc \<equiv> map_prod qw.\<alpha> mr.\<alpha>"
simps_1: fixes A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and s :: "('state \<times> 'b) pmf" shows "T_on_rand' A s [] = 0"
samet_imp_iso_seteq: fixes gsopt :: "(nat, nat list list list) trie option"   and arch :: "nat list list list" assumes "\<And>gs g. \<lbrakk>gsopt = Some gs; g \<in> set_tries gs\<rbrakk> \<Longrightarrow> pre_iso_test g"   and "\<And>g. g \<in> set arch \<Longrightarrow> pre_iso_test g"   and "\<And>gs. gsopt = Some gs \<Longrightarrow> invar_trie gs"   and "samet gsopt arch" shows "\<exists>gs. gsopt = Some gs \<and> set_tries gs =\<^sub>\<simeq> set arch"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "subseq a b"   and "subseq b c" shows "subseq a c"
hyps: shows "isOK (existsM f xs) = (\<exists>x\<in>set xs. isOK (f x))"
S2_def: fixes H :: "'a \<times> 'a"   and c :: "nat" shows "S2 H c = (let (h, h') = H in sample_uniform (order \<G>) \<bind> (\<lambda>z. let a = \<^bold>g [^] z \<otimes> inv (h [^] c); a' = g' [^] z \<otimes> inv (h' [^] c) in return_spmf ((a, a'), z)))"
cod_comp: fixes g :: "'a1"   and f :: "'a1" assumes "C1.seq g f" shows "C1.cod (g \<cdot>\<^sub>1 f) = C1.cod g"
pair_sym_digraph_axioms: shows "pair_sym_digraph (subdivide G e w)"
dw_closed_def: fixes A :: "'a set" shows "dw_closed A = (\<forall>a\<in>A. \<forall>b. a \<rightarrow> b \<longrightarrow> b \<in> A)"
amult_pos_mono_r: fixes w :: "int"   and x :: "ant"   and y :: "ant" assumes "0 < w" shows "(x * ant w \<le> y * ant w) = (x \<le> y)"
verit_sko_ex_indirect: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "x = (SOME x. P x)" shows "(\<exists>x. P x) = P x"
S\<^sub>r_def: shows "S\<^sub>r = S - (S\<^sub>e \<union> S2)"
abs_mult: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'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 a :: "'a"   and b :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (times a b) = times (abs a) (abs b)"
assms: shows "proj2_no_3_Col (insert a3 (range (($) a)))"
assms_1: shows "\<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
cases: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" assumes "(a1, a2) \<in> g3mod2" obtains A :: "(C, BD) form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Aa__  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> \<diamond> Aa__  \<RM>)"   | A :: "(C, BD) form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)"
ide_MkIde: fixes A :: "bool" assumes "A \<in> UNIV" shows "ide (MkIde A)"
hyper_comm: fixes A :: "'a set option"   and B :: "'a set option"   and C :: "'a set option" shows "A \<squnion> B = B \<squnion> A \<and> A \<squnion> B \<squnion> C = B \<squnion> A \<squnion> C"
generalized_sfw_join_set: fixes a :: "'b simple_match"   and b1 :: "'c"   and b2 :: "'d"   and f1 :: "('b simple_match \<times> 'c) list"   and f2 :: "('b simple_match \<times> 'd) list" shows "((a, b1, b2) \<in> set (generalized_fw_join f1 f2)) = (\<exists>a1 a2. (a1, b1) \<in> set f1 \<and> (a2, b2) \<in> set f2 \<and> simple_match_and a1 a2 = Some a)"
szip_smap_snd: fixes xs :: "'d stream"   and ys :: "'c stream" shows "smap snd (xs ||| ys) = ys"
T: shows "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
stake_szip: fixes n :: "nat"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "stake n (szip s1 s2) = zip (stake n s1) (stake n s2)"
simps_2: fixes st :: "'St"   and u :: "'U"   and c :: "'C"   and ucl :: "('U \<times> 'C) list" shows "doo st ((u, c) # ucl) = doo (do st u c) ucl"
low_digits_def: fixes ds :: "nat list"   and i :: "nat" shows "low_digits ds i = (i < prod_list ds \<and> (\<forall>i0\<in>set (digit_encode ds i). i0 < m))"
simps_9: fixes var :: "nat"   and uu :: "atom fm" shows "grab_atoms var (Neg uu) = None"
wf_syscls_is_class_sub_Throwable: fixes P :: "'b prog"   and C :: "String.literal" assumes "wf_syscls P"   and "P \<turnstile> C \<preceq>\<^sup>* Throwable" shows "is_class P C"
n_kat_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<cdot> \<tt>\<tt> z = (0::'a)) = \<lparr>!x\<rparr>y\<lparr>y \<cdot> !z\<rparr>"
gfp_mono: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "\<And>Z. f Z \<le> g Z" shows "gfp f \<le> gfp g"
state_update_vs_allInstances_generic_including: fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and oid :: "nat"   and Object :: "'d"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a"   and "\<And>x. \<sigma>' oid = \<lfloor>x\<rfloor> \<Longrightarrow> x = Object"   and "Type Object \<noteq> None" shows "OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = \<lambda>_. OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>', assocs = A\<rparr>)->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>\<lfloor>\<lceil>Type Object\<rceil>\<rfloor>\<rfloor>) (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>)"
mlt_def: shows "mlt = (\<lambda>q. ord_term_lin.Max (lt ` {q g \<odot> g |g. g \<in> G \<and> q g \<odot> g \<noteq> 0}))"
swapAbs_psubstAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "A $[rho] $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[rho &[z1 \<and> z2]_zs]"
complete_indiff_not: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "complete A r"   and "hasw [x, y] A"   and "\<not> x \<^bsub>r\<^esub>\<approx> y" shows "x \<^bsub>r\<^esub>\<prec> y \<or> y \<^bsub>r\<^esub>\<prec> x"