hcju/isabelleps huggingface dataset:
separated_tmps_nth_update: fixes i :: "nat"   and t :: "nat"   and s :: "stmt"   and sb :: "(stmt \<times> nat) memref list"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "valid_sops_stmt t s"   and "\<forall>(s', t')\<in>prog_configs sb. valid_sops_stmt t' s'"   and "\<forall>i\<in>load_tmps is. i < t"   and "\<forall>i\<in>read_tmps sb. i < t"   and "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"   and "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"   and "dom \<theta> \<union> load_tmps is = {i. i < t}" shows "separated_tmps (ts[i := ((s, t), is, \<theta>, sb, \<D>, \<O>)])"
stut_action: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" assumes "NSTUTINV P" shows "STUTINV \<box>[P]_v"
m_anti_mono: fixes x :: "'av"   and y :: "'av" assumes "x \<le> y" shows "m y \<le> m x"
castmsg_def: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d node_action" shows "castmsg P a \<equiv> case a of x:*cast(m) \<Rightarrow> P m | _ \<Rightarrow> True"
sin_gt_zero: fixes x :: "real" assumes "0 < x"   and "x < pi" shows "0 < sin x"
insertion_single: fixes f :: "nat \<Rightarrow> 'a"   and v :: "nat"   and n :: "nat"   and a :: "'a" shows "insertion f (MPoly_Type.monom (Poly_Mapping.single v n) a) = a * f v ^ n"
simps_3: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Integer) = (P,E,h \<turnstile> e : Integer)"
Us_4: shows "R.Span K Us = a_kernel (R\<lparr>carrier := E\<rparr>) S h"
stim_idem: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set"   and x :: "'c" assumes "class.stimuli leq add seq_comp neutral deactivation basic" shows "add x x = x"
ref_two_step: fixes A :: "'c nres"   and R :: "('c \<times> 'd) set"   and B :: "'d nres"   and C :: "'d nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
le_succ_iff: fixes i :: "V"   and j :: "V" assumes "Ord i"   and "Ord j" shows "(ZFC_in_HOL.succ i \<le> ZFC_in_HOL.succ j) = (i \<le> j)"
composite_simps_2: shows "src ((f \<star> src f) \<star> src f) = src (src f)"
inst_cswitch_independent_of_state: fixes s :: "rstate_t"   and t :: "rstate_t"   and n :: "nat" assumes "rcurrent s = rcurrent t" shows "rcurrent (rcswitch n s) = rcurrent (rcswitch n t)"
assm: shows "\<forall>x\<in>X. 0 \<le> u x"
l_span_sub_submodule: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and N :: "'e set"   and H :: "'e set" assumes "R module M"   and "ideal R A"   and "submodule R M N"   and "H \<subseteq> N" shows "linear_span R M A H \<subseteq> N"
elts_sup_iff: fixes x :: "V"   and y :: "V" shows "elts (x \<union>\<^sub>\<circ> y) = elts x \<union> elts y"
Heap_execute: fixes f :: "'a Heap" shows "Heap (execute f) = f"
d1: shows "distinct (map (unity_root k) (map int [0..<k]))"
openin_interior_of: fixes X :: "'d topology"   and S :: "'d set" shows "openin X (X interior_of S)"
T_App: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and T\<^sub>1\<^sub>2 :: "POPLmarkRecord.type"   and t\<^sub>2 :: "trm" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<^sub>1 \<rightarrow> T\<^sub>1\<^sub>2"   and "\<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<bullet> t\<^sub>2 : T\<^sub>1\<^sub>2"
simps_3: fixes m :: "bool"   and x :: "'b" shows "strip2 (Atom2 m x) = Atom (m, x)"
gbinomial_int_negated_upper: fixes a :: "int"   and k :: "nat" shows "a gchoose k = (- 1) ^ k * (int k - a - 1 gchoose k)"
eq_class_of_def: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d" shows "class_of\<^bsub>S\<^esub> x = {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}"
set_less_aux_finiteD: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubset>' B" shows "finite A \<and> B \<in> infinite_complement_partition"
path_simps_2: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and u' :: "'b"   and l :: "'b list"   and w :: "'b" shows "path E u (u' # l) w = (u' = u \<and> (\<exists>v. (u, v) \<in> E \<and> path E v l w))"
path_det: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b"   and n'' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "CFG.path sourcenode targetnode valid_edge n as n''" shows "n' = n''"
c: shows "punit.lt (rep_list f2) adds punit.lt (rep_list q)"
lemma_2_5_11: fixes b :: "'b"   and a :: "'b" shows "b * a \<le> a \<sqinter> b"
eval_binop_arg2_indep: fixes binop :: "binop"   and v1 :: "val"   and x :: "val"   and y :: "val" assumes "\<not> need_second_arg binop v1" shows "eval_binop binop v1 x = eval_binop binop v1 y"
subset_eq: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
rel_refl_strong: fixes x :: "'a fset"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> fset x \<Longrightarrow> Ra z z" shows "rel_fset Ra x x"
finite_filter: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite S" shows "finite (Set.filter P S)"
encL: fixes S :: "'procS" shows "indRelEQp (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)"
image_is_empty: fixes f :: "'l \<Rightarrow> 'k"   and A :: "'l set" shows "(f ` A = {}) = (A = {})"
baIANow: fixes \<phi> :: "'cmp \<Rightarrow> bool"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'cmp" assumes "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "eval c t t' n [\<phi>]\<^sub>b"
strictify_conv_if: fixes f :: "'c \<rightarrow> 'b"   and x :: "'c" shows "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
assms_2: shows "a $ 0 * x = (1::'a)"
v_TS_defs_1: fixes Quorum :: "process set set" assumes "quorum_process Quorum" shows "quorum_process.v_TS Quorum = \<lparr>init = v_init, trans = quorum_process.v_trans Quorum\<rparr>"
list_all_rev: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_all P (rev xs) = list_all P xs"
in_nodes_lt_n: fixes x :: "nat" assumes "x \<in> set (g_V G)" shows "x < n"
sq_norm_vec_le_linf_norm: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "\<parallel>v\<parallel>\<^sup>2 \<le> of_nat n * \<parallel>v\<parallel>\<^sub>\<infinity>\<^sup>2"
orthogonal_clauses_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner a x"   and "real_inner.orthogonal inner a y" shows "real_inner.orthogonal inner a (plus x y)"
det_rows_mul: fixes a :: "nat \<Rightarrow> 'b vec"   and n :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "a \<in> {0..<n} \<rightarrow> carrier_vec n" shows "det (mat\<^sub>r n n (\<lambda>i. c i \<cdot>\<^sub>v a i)) = prod c {0..<n} * det (mat\<^sub>r n n a)"
smcf_comp_assoc: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and \<HH> :: "V"   and \<BB> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<HH> : \<CC> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<DD>"   and "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M (\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)"
triangle_diff_paths: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<not> (\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q)"
verit_negate_coefficient_2: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "- b < - a"
simple_path_joinE: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "simple_path (g1 +++ g2)"   and "pathfinish g1 = pathstart g2" obtains "arc g1" and "arc g2" and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
mod_tree_lemma2: fixes x :: "'a tree"   and y :: "'a tree" shows "(x + y) mod y = x mod y"
subst_closed_id: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c"   and env :: "(name, 'c) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "pre_term.closed_except frees t {||}" shows "subst t env = t"
minit0_And_Not: fixes \<psi> :: "'a MFOTL.formula"   and \<phi> :: "'a MFOTL.formula"   and n :: "nat" assumes "safe_formula \<psi> \<and> fv \<psi> \<subseteq> fv \<phi>" shows "minit0 n (MFOTL.And_Not \<phi> \<psi>) = MAnd (minit0 n \<phi>) False (minit0 n \<psi>) ([], [])"
lesspoll_not_refl: fixes i :: "'a set" shows "\<not> i \<prec> i"
prj1_simps_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1" shows "partial_magma.arr C (prj1 f g) = (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g)"
dw_le: fixes y :: "'b" assumes "y \<in> frontier (f ` S)" shows "norm (f a - w) \<le> norm (f a - y)"
simps: fixes a1 :: "nat \<Rightarrow> Syntax.type"   and a2 :: "nat \<Rightarrow> Syntax.type"   and a3 :: "trm"   and a4 :: "Syntax.type" shows "(a1 , a2 \<turnstile>\<^sub>T a3 : a4) = ((\<exists>\<Gamma> x T \<Delta>. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = `x \<and> a4 = T \<and> \<Gamma> x = T) \<or> (\<exists>\<Gamma> \<Delta> t T1 T2 s. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = t \<degree> s \<and> a4 = T2 \<and> \<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1 \<rightarrow> T2 \<and> \<Gamma> , \<Delta> \<turnstile>\<^sub>T s : T1) \<or> (\<exists>\<Gamma> T1 \<Delta> t T2. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = \<lambda>T1:t \<and> a4 = T1 \<rightarrow> T2 \<and> \<Gamma>\<langle>0:T1\<rangle> , \<Delta> \<turnstile>\<^sub>T t : T2) \<or> (\<exists>\<Gamma> \<Delta> T c. a1 = \<Gamma> \<and> a2 = \<Delta> \<and> a3 = \<mu>T:c \<and> a4 = T \<and> \<Gamma> , \<Delta>\<langle>0:T\<rangle> \<turnstile>\<^sub>C c))"
memb_equiv_1: fixes f :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "arr f"   and "t \<in> f"   and "u \<in> f" shows "Par t u"
lossless_cond_spmf_fst: fixes p :: "('i \<times> 'h) spmf"   and x :: "'i" shows "lossless_spmf (cond_spmf_fst p x) = (x \<in> fst ` set_spmf p)"
isGuard_not_satL_intT: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and l :: "('a, 'b) lit"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "Signature.wtL wtFsym wtPsym arOf resOf parOf l"   and "\<not> Struct.satL (ModelIkPolMcalc2C.I_intF arOf intT intF) (ModelIkPolMcalc2C.I_intP parOf pol intT intP) \<xi> l"   and "ProblemIkPol.isGuard pol x l"   and "Struct.wtE (ModelIkPolMcalc2C.I_intT infTp intT) \<xi>" shows "intT (Signature.tpOfV x) (\<xi> x)"
simps_3: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
elims_3: fixes x :: "('b \<times> color) tree" assumes "\<not> invh x" obtains l :: "('b \<times> color) tree" and xa :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (xa, c), r\<rangle>" and "\<not> (bheight l = bheight r \<and> invh l \<and> invh r)"
delete_simps_1: fixes k :: "'d" shows "delete k [] = []"
STCal_step_2: fixes T :: "'procT"   and P' :: "('procS, 'procT) Proc" shows "TargetTerm T \<longmapsto>STCal Source Target P' = (\<exists>T'. T' \<in>T P' \<and> T \<longmapsto>Target T')"
oprod_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "r *o r' = {((x1, y1), x2, y2). (y1, y2) \<in> r' - Id \<and> x1 \<in> Field r \<and> x2 \<in> Field r \<or> (y1, y2) \<in> Restr Id (Field r') \<and> (x1, x2) \<in> r}"
fps_lr_inverse_fps_X_plus1_2: shows "fps_right_inverse (1 + fps_X) (1::'b) = Abs_fps ((^) (- (1::'b)))"
O_rtrancl_O_O: fixes R :: "('a \<times> 'b) set"   and S :: "('b \<times> 'a) set" shows "R O (S O R)\<^sup>* O S = (R O S)\<^sup>+"
run_put_writer: fixes put :: "'g \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'g"   and m :: "('e, 'f, 'd) writerT" shows "run_writer (put_writer put s m) = put s (run_writer m)"
hyps: fixes ms :: "mregex" assumes "ok m mr"   and "ms \<in> LPD mr" shows "ok m ms"
BinOpSetGroup_normal_quotient: fixes H :: "'g set" assumes "Group H \<and> order.greater_eq G H"   and "normal H" shows "BinOpSetGroup (G // LCoset_rel H) (+) H"
seqSubstBang: fixes P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "!P[<\<sigma>>] = !(P[<\<sigma>>])"
proj_proj_set_subset_4: fixes n :: "'e"   and m :: "'e"   and A :: "('e strand_label \<times> 'd) list" shows "set (proj_unl n (proj m A)) \<subseteq> set (proj_unl m A)"
process_charn: fixes P :: "'b process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<notin> F P \<or> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<notin> F P \<or> \<not> X \<subseteq> Y \<or> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<notin> D P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> D P) \<and> (\<forall>s X. s \<notin> D P \<or> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<notin> D P \<or> s \<in> D P)"
is_run_def: fixes G :: "('a, 'b) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" shows "graph_defs.is_run G r \<equiv> r 0 \<in> g_V0 G \<and> ipath (g_E G) r"
to_fun_diff: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "UP_cring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "a \<in> carrier R" shows "UP_cring.to_fun R (p \<ominus>\<^bsub>UP R\<^esub> q) a = UP_cring.to_fun R p a \<ominus>\<^bsub>R\<^esub> UP_cring.to_fun R q a"
is_rbt_rbt_sorted: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "ord.is_rbt less t" shows "ord.rbt_sorted less t"
enumAppend_def: fixes nmax :: "nat"   and iss :: "nat list list" shows "enumAppend nmax iss \<equiv> concat (map (\<lambda>is. map (\<lambda>n. is @ [n]) [last is..<Suc nmax]) iss)"
subseteq_zmset_def: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" shows "(A \<subseteq>#\<^sub>z B) = (\<forall>a. zcount A a \<le> zcount B a)"
int_pow_pow: fixes x :: "'a"   and m :: "int"   and n :: "int" assumes "x \<in> carrier R" shows "[m] \<cdot> ([n] \<cdot> x) = [(n * m)] \<cdot> x"
hurwitz_formula_bound2: fixes s :: "complex" obtains C :: "real" where "0 \<le> C" and "\<And>r z. \<lbrakk>0 < r; r < pi; z \<in> sphere 0 r\<rbrakk> \<Longrightarrow> cmod (f s z) \<le> C * r powr (- Re s - 1)"
partition_on_le_set_elements: fixes A :: "'a set"   and P :: "'a set set" assumes "finite A"   and "partition_on A P" shows "card P \<le> card A"
finfun_single_neq_bot: fixes x :: "'a" shows "finfun_single x \<noteq> {}\<^sub>f"
intros_9: fixes \<sigma>' :: "nat \<Rightarrow> 'c"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and i' :: "nat"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), connect(i, i'), \<sigma>', NodeS i s (R \<union> {i'})) \<in> onode_sos S"
lepoll_UNIV_imp_small: fixes X :: "'a set" assumes "X \<lesssim> UNIV" shows "small X"
vproduct_vdoubletonE: fixes f :: "V"   and a :: "V"   and b :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {a, b}) A" obtains "vsv f" and "f\<lparr>a\<rparr> \<in>\<^sub>\<circ> A a" and "f\<lparr>b\<rparr> \<in>\<^sub>\<circ> A b" and "\<D>\<^sub>\<circ> f = set {a, b}" and "f = set {\<langle>a, f\<lparr>a\<rparr>\<rangle>, \<langle>b, f\<lparr>b\<rparr>\<rangle>}"
comm_semiring_1_cancel_ow_axioms_def: fixes U :: "'c set"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_semiring_1_cancel_ow_axioms U minus times \<equiv> \<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times a (minus b c) = minus (times a b) (times a c)"
convex_lincomb_def: fixes c :: "'a vec \<Rightarrow> 'a"   and Vs :: "'a vec set"   and b :: "'a vec" shows "n.convex_lincomb c Vs b = (n.nonneg_lincomb c Vs b \<and> sum c Vs = (1::'a))"
rep_eq: fixes x :: "uint64" shows "nat_of_uint64 x = unat (Rep_uint64 x)"
If_bury: fixes b :: "bexp"   and bc1 :: "com"   and bc2 :: "com"   and c :: "com"   and X :: "char list set" shows "(IF b THEN bc1 ELSE bc2 = bury c X) = (\<exists>c1 c2. c = IF b THEN c1 ELSE c2 \<and> bc1 = bury c1 X \<and> bc2 = bury c2 X)"
intro: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" assumes "Pivot eq_idx_for_lvar pivot"   and "Update update" shows "PivotUpdate eq_idx_for_lvar pivot update"
unique_convergence: fixes x :: "'a"   and y :: "'a"   and F :: "'a set set" assumes "x \<in> carrier"   and "y \<in> carrier"   and "F \<in> Filters"   and "F \<longlongrightarrow> x"   and "F \<longlongrightarrow> y" shows "x = y"
length_concat_sequences: fixes xs :: "'a list" shows "length (concat (sequences key xs)) = length xs"
up_filter: fixes x :: "'a" shows "ord_class.filter (\<up>x)"
cnf2bhv_ge_llength: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n" shows "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 \<le> \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n"
clop_dual: shows "clop = coclop \<circ> \<partial>\<^sub>F"
polyfun_submatrix: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a mat"   and m :: "nat"   and n :: "nat"   and N :: "nat set"   and i :: "nat"   and I :: "nat set"   and j :: "nat"   and J :: "nat set" assumes "\<And>x. A x \<in> carrier_mat m n"   and "\<And>x i j. \<lbrakk>i < m; j < n\<rbrakk> \<Longrightarrow> polyfun N (\<lambda>x. A x $$ (i, j))"   and "i < card {i. i < m \<and> i \<in> I}"   and "j < card {j. j < n \<and> j \<in> J}"   and "infinite I"   and "infinite J" shows "polyfun N (\<lambda>x. submatrix (A x) I J $$ (i, j))"
map_comp: fixes g1 :: "'c \<Rightarrow> 'a"   and g2 :: "'d \<Rightarrow> 'b"   and f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and v :: "('e, 'f) tllist" shows "tmap g1 g2 (tmap f1 f2 v) = tmap (g1 \<circ> f1) (g2 \<circ> f2) v"
rbt_def: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "rbt t = (invc t \<and> invh t \<and> invpst t \<and> color t = Black)"
getEnv_idEnv: fixes xs :: "'b"   and x :: "'c" shows "idEnv xs x = None"
id1: shows "set (take i (map of_int_hom.vec_hom fs)) = (\<lambda>x. of_int_hom.vec_hom (fs ! x)) ` {0..<i}"
lookup_def: fixes F :: "'a Abs_State.st"   and x :: "char list" shows "lookup F x = (if x \<in> set (Abs_State.dom F) then fun F x else \<top>)"
lookupEvalHeap: fixes v :: "'d"   and h :: "('d \<times> 'e) list"   and f :: "'e \<Rightarrow> 'f" assumes "v \<in> domA h" shows "evalHeap h f v = f (the (map_of h v))"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b \<Rightarrow> 'a" assumes "\<And>x. Ra x x" shows "rel_fun (=) Ra x x"
card_eq_Suc_0_ex1: fixes A :: "'b set" shows "(card A = Suc 0) = (\<exists>!x. x \<in> A)"
third_mertens_const_def: shows "third_mertens_const = exp (- (\<Sum>p. if prime p then - ln (1 - 1 / real p) - 1 / real p else 0) - meissel_mertens)"
homotopy_eqv_cohomotopic_triviality_null: fixes S :: "'a set"   and T :: "'b set"   and U :: "'c set" assumes "S homotopy_eqv T" shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S U f (\<lambda>x. c))) = (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. c)))"
that: fixes f' :: "'a \<Rightarrow> 'a" assumes "bij_betw f' C C"   and "\<And>x. x \<in> A \<Longrightarrow> f' x = f x"   and "\<And>x. x \<in> C - A \<Longrightarrow> f' x \<in> C - B" shows "thesis"
disjnt_empty2: fixes A :: "'b set" shows "disjnt A {}"
mem_single_l_spanTr: fixes A :: "'b set"   and h :: "'a"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "h \<in> carrier M" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> {h} \<and> l_comb R M n s f \<in> linear_span R M A {h} \<longrightarrow> (\<exists>a\<in>A. l_comb R M n s f = a \<cdot>\<^sub>s h)"
eq_fin_end: fixes u\<^sub>1 :: "'item list"   and u\<^sub>2 :: "'item list"   and v\<^sub>1 :: "'item list"   and v\<^sub>2 :: "'item list" assumes "u\<^sub>1 =\<^sub>F u\<^sub>2"   and "u\<^sub>1 @ v\<^sub>1 =\<^sub>F u\<^sub>2 @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
successor_set_finite: fixes n :: "'b" assumes "valid_node n" shows "finite {n'. \<exists>a'. valid_edge a' \<and> sourcenode a' = n \<and> targetnode a' = n'}"
box_Int_box: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "box a b \<inter> box c d = box (\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i)"