hcju/isabelleps huggingface dataset:
sum_square: fixes a :: "'c \<Rightarrow> 'b"   and I :: "'c set" shows "(sum a I)\<^sup>2 = (\<Sum>i\<in>I. \<Sum>j\<in>I. a i * a j)"
inf_compl_bot_right: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (y \<sqinter> - x) = bot"
is_lifting_def: fixes f :: "'c \<Rightarrow> 'd"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list" shows "is_lifting f basis basis' = (\<forall>C. eval (f C) = eval C \<and> (is_expansion C basis \<longrightarrow> is_expansion (f C) basis') \<and> trimmed (f C) = trimmed C \<and> fst (dominant_term (f C)) = fst (dominant_term C))"
simps_3: fixes A :: "('b, 'c) ioa"   and s :: "'b" shows "is_exec_frag_of A (s, []) = True"
B_def: shows "B = (\<lambda>x. x * 2) ` A"
second_vertices_first_edge: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "xs \<in> Q"   and "H.first_edge_of_walk xs = (v, w)" shows "w \<in> Q.second_vertices"
quorum_non_empty: fixes Q :: "process set" assumes "Q \<in> Quorum" shows "Q \<noteq> {}"
2: shows "Fvars \<chi> = Fvars \<phi> - {x} \<union> (if x \<in> Fvars \<phi> then FvarsT t else {})"
assms: shows "\<forall>i. (\<lambda>n. f n i) sums (\<Sum>n. f n i)"
maxR_absorb1: fixes y :: "'a"   and x :: "'a" assumes "(y, x) \<in> r" shows "local.MaxR.maxR x y = x"
runit_naturality: fixes f :: "'a" assumes "arr f" shows "\<r>[cod f] \<cdot> (f \<otimes> \<I>) = f \<cdot> \<r>[local.dom f]"
typing_elims_3: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type" assumes "e \<turnstile> Abs t : T" obtains T :: "LambdaType.type" and U :: "LambdaType.type" where "Ta__ = T \<Rightarrow> U" and "e\<langle>0:T\<rangle> \<turnstile> t : U"
assms_3: shows "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
tuple_pr: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and h :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "\<guillemotleft>h : local.dom h \<rightarrow> hfsetcat.prod a b\<guillemotright>" shows "hfsetcat.tuple (local.comp (hfsetcat.pr1 a b) h) (local.comp (hfsetcat.pr0 a b) h) = h"
case_eq_if: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
that_2: shows "(X, l') \<in> set_pmf \<mu>"
hmmM: shows "M = HMSet (replicate_mset n 0)"
prems_1: shows "\<forall>x. 0 \<le> f x"
add_closed_converse_left: fixes g :: "'g"   and x :: "'g" assumes "g + x \<in> G"   and "x \<in> G" shows "g \<in> G"
luckiestFind_eval': fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (luckiestFind var L F) (xs @ \<Gamma>))"
has_codomain_iff_arr: fixes f :: "'a" shows "(codomains f \<noteq> {}) = arr f"
minus_one_power_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus one) n = (if even n then one else uminus one)"
suminf_enn2ereal: fixes f :: "nat \<Rightarrow> ennreal" shows "(\<Sum>i. enn2ereal (f i)) = enn2ereal (suminf f)"
poly_add_comm: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" shows "poly_add p q =p poly_add q p"
step1_mat_size: shows "reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)) \<in> carrier_mat (length signs) (length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))"
simps: fixes "oracle" :: "'b \<Rightarrow> 'grp \<Rightarrow> (bool list \<times> 'b) spmf"   and \<sigma> :: "'b"   and \<A> :: "('grp \<Rightarrow> ((bool list \<times> bool list) \<times> 'c, 'grp, bool list) gpv) \<times> ('grp \<times> bool list \<Rightarrow> 'c \<Rightarrow> (bool, 'grp, bool list) gpv)" shows "advantage (oracle, \<sigma>) \<A> = \<bar>spmf (run_gpv oracle (game \<A>) \<sigma>) True - 1 / 2\<bar>"
subst_cls_def: fixes AA :: "('c, 'd) Term.term literal multiset"   and \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "subst_cls AA \<sigma> = {#subst_lit A \<sigma>. A \<in># AA#}"
star_def_var1: fixes x :: "'a" shows "quantale_class.star x = Sup {y. \<exists>i. y = x ^ i}"
LIMSEQ_NSLIMSEQ_iff: fixes f :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "f \<longlonglongrightarrow> L = f \<longlonglongrightarrow>\<^sub>N\<^sub>S L"
mset_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "mset (map f xs) = image_mset f (mset xs)"
problem_plan_bound_works': fixes PROB :: "(('d, bool) fmap \<times> ('d, bool) fmap) set"   and s :: "('d, bool) fmap"   and as :: "(('d, bool) fmap \<times> ('d, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> problem_plan_bound PROB \<and> sat_precond_as s as'"
assms_1: shows "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>"
prems_2: fixes i' :: "nat" assumes "i' < length (x # xs)"   and "\<not> tr_lookup T (i' + j) i" shows "(x # xs) ! i' = None"
assms: shows "nat \<lceil>log (real_of_rat (4 * \<alpha> / (4 + \<alpha>))) (real N)\<rceil> \<le> Lg"
ANR_imp_locally_connected: fixes S :: "'b set" assumes "ANR S" shows "locally connected S"
list_all2_cons_def: fixes sg :: "'s1"   and sh :: "'s2" shows "local.list_all2_cons sg sh = list_all2 P (unstream g sg) (unstream h sh)"
box_equals: fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "a = b"   and "a = c"   and "b = d" shows "c = d"
associativity: fixes M1 :: "int poly fract list list"   and M2 :: "int poly fract list list"   and M3 :: "int poly fract list list" assumes "mat (rat_poly.row_length M1) (length M1) M1 \<and> mat (rat_poly.row_length M2) (length M2) M2 \<and> mat (rat_poly.row_length M3) (length M3) M3" shows "M1 \<otimes> (M2 \<otimes> M3) = M1 \<otimes> M2 \<otimes> M3"
is_singletonE: fixes A :: "'g set" assumes "is_singleton A" obtains x :: "'g" where "A = {x}"
image_insert: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and B :: "'e set" shows "f ` insert a B = insert (f a) (f ` B)"
matrix_inv_idempotent: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "invertible A" shows "A\<^sup>-\<^sup>1\<^sup>-\<^sup>1 = A"
f_inv_into_f: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
is_processT6_S2: fixes X :: "'b event set"   and P :: "'b process" assumes "tick \<notin> X"   and "[tick] \<in> T P" shows "([], X) \<in> F P"
SATFlagAfterSolveLoop: fixes state :: "State"   and Vbl :: "nat set"   and F0' :: "Literal list list"   and F0 :: "Literal list list" assumes "solve_loop_dom state Vbl"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantUniqQ (getQ state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "getSATFlag state = UNDEF \<longrightarrow> InvariantEquivalentZL (getF state) (getM state) F0'"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"   and "getSATFlag state = FALSE \<longrightarrow> \<not> satisfiable F0'"   and "getSATFlag state = TRUE \<longrightarrow> satisfiable F0'"   and "finite Vbl"   and "vars F0' \<subseteq> vars F0"   and "vars F0 \<subseteq> Vbl"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl" shows "let state' = solve_loop state Vbl in getSATFlag state' = FALSE \<and> \<not> satisfiable F0' \<or> getSATFlag state' = TRUE \<and> satisfiable F0'"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : s\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<pi>.chine\<guillemotright>"
viewnres_cases: fixes r :: "(('c, 'd) Node \<times> ('c, 'd) FingerTreeStruc) option" obtains "r = None"   | a :: "('c, 'd) Node" and t :: "('c, 'd) FingerTreeStruc" where "r = Some (a, t)"
Pi'_mono: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and C :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi' A B \<subseteq> Pi' A C"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "D.J.ide (J g f)" obtains "D.J.seq g f" and "D.J.seq f g" and "J g f = D.J.dom f" and "J g f = D.J.cod g"
RetrT_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "RetrT theta \<equiv> {(c, d). matchC_TMC theta c d \<and> matchT_TMT c d}"
ring_iso_set_trans: fixes f :: "'g \<Rightarrow> 'h"   and R :: "('g, 'i) ring_scheme"   and S :: "('h, 'j) ring_scheme"   and g :: "'h \<Rightarrow> 'k"   and Q :: "('k, 'l) ring_scheme" assumes "f \<in> ring_iso R S"   and "g \<in> ring_iso S Q" shows "g \<circ> f \<in> ring_iso R Q"
nb\<^sub>9_2: shows "\<forall>(v, a)\<in>\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a}. map_of (effect_to_assignments op') (v, a) = Some False"
invariant: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
components_are_iso: fixes a :: "'c" assumes "Cop.ide a" shows "S.iso (\<Phi>'\<Psi>.map a)"
distinct_card: fixes xs :: "'a list" assumes "cycle xs" shows "card (set xs) = length xs"
lock_ok_las_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and las :: "'a lock_actions" shows "lock_ok_las ls t las \<equiv> \<forall>l. lock_actions_ok (ls $ l) t (las $ l)"
seq_extract_Cons_atLeastLessThan: fixes m :: "nat"   and n :: "nat"   and x :: "'b"   and xs :: "'b list" assumes "m < n" shows "{m..<n} \<upharpoonleft>\<^sub>l (x # xs) = (if m = 0 then x # {0..<n - 1} \<upharpoonleft>\<^sub>l xs else {m - 1..<n - 1} \<upharpoonleft>\<^sub>l xs)"
shiftr_shiftl1: fixes c :: "nat"   and b :: "nat"   and a :: "'b word" assumes "c \<le> b" shows "a >> b << c = (a >> b - c) AND NOT (mask c)"
arr_cod_iff_arr: fixes f :: "'b" shows "B.arr (B.cod f) = B.arr f"
simps: fixes acc :: "'a list"   and xs :: "'a list" shows "permutations_of_set_aux_list acc xs = (if xs = [] then [acc] else List.bind xs (\<lambda>x. permutations_of_set_aux_list (x # acc) (remove1 x xs)))"
sccs_verts_conv_scc_of: shows "sccs_verts = scc_of ` verts G"
r2f_Inf_pres: shows "Inf_pres \<F>"
space_select_sets: fixes M :: "'c measure measure"   and N :: "'c measure" assumes "sets M = sets (subprob_algebra N)" shows "space (select_sets M) = space N"
x: shows "2 * 2 ^ n < 2 * 2 ^ m"
mult_2_right: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "class.semiring_numeral one times plus" shows "times z (numeral.numeral one plus (num.Bit0 num.One)) = plus z z"
subdiv1_contr_axioms: shows "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"
simps_2: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprLst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprLst"
zero_in: shows "zero \<in> topspace (Euclidean_space n)"
assms: shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
h_def: shows "h \<equiv> \<lambda>x s t. complex_of_real x * \<i> * cis t * exp (complex_of_real a * (complex_of_real x * cis t) - (Ln (complex_of_real x) + \<i> * complex_of_real t) * s) / (1 - exp (complex_of_real x * cis t))"
exhaust: fixes y :: "thread" obtains "y = INIT"   | x2 :: "nat" where "y = WAIT x2"   | x3 :: "nat" where "y = HOLD x3"   | x4 :: "nat" where "y = REL x4"
True: fixes F :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
resultant_as_nonzero_poly_weak: fixes p :: "'b poly"   and q :: "'b poly" assumes "0 < degree p"   and "0 < degree q"   and "resultant p q \<noteq> (0::'b)" shows "\<exists>p' q'. degree p' < degree q \<and> degree q' < degree p \<and> [:resultant p q:] = p' * p + q' * q \<and> p' \<noteq> 0 \<and> q' \<noteq> 0"
BuildSeq3_def: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf"   and y' :: "hf"   and y'' :: "hf" shows "BuildSeq3 B C s k y y' y'' \<equiv> BuildSeq (\<lambda>p. \<exists>x x' x''. p = \<langle>x, x', x''\<rangle> \<and> B x x' x'') (\<lambda>p q r. \<exists>x x' x'' y y' y'' z z' z''. p = \<langle>x, x', x''\<rangle> \<and> q = \<langle>y, y', y''\<rangle> \<and> r = \<langle>z, z', z''\<rangle> \<and> C x x' x'' y y' y'' z z' z'') s k \<langle>y, y', y''\<rangle>"
pmf_geometric: fixes p :: "real"   and n :: "nat" assumes "0 < p"   and "p \<le> 1" shows "pmf (geometric_pmf p) n = (1 - p) ^ n * p"
card_of_nlists_infinite: fixes A :: "'a set"   and n :: "nat" assumes "infinite A" shows "|nlists A n| \<le>o |A|"
nodes_of_uses'_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and phiVals :: "'val set"   and nodes_of_uses :: "('val, 'node set) mapping" shows "uninst_code.nodes_of_uses' g next next' phiVals nodes_of_uses = (let users = case Mapping.lookup nodes_of_uses (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> id x in if next' \<in> phiVals then Mapping.map_default next' {} (\<lambda>ns. ns \<union> users) (Mapping.delete (snd next) nodes_of_uses) else Mapping.delete (snd next) nodes_of_uses)"
tr_par_iff_tr'_par: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j"   and "\<forall>(l, t, s)\<in>set D. (fv t \<union> fv s) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}" shows "(\<exists>B\<in>set (tr\<^sub>p\<^sub>c A D). \<lbrakk>M; unlabel B\<rbrakk>\<^sub>d \<I>) = (\<exists>C\<in>set (tr'\<^sub>p\<^sub>c A D). \<lbrakk>M; unlabel C\<rbrakk>\<^sub>d \<I>)"
rel_FGcontra_neg_distr_cond_def: fixes Co1 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Co3 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Co3' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Co4 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Co4' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Co5 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Co5' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra1 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra2 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and Contra2' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra3 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and Contra3' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra4 :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra4' :: "'ar \<Rightarrow> 'as \<Rightarrow> bool"   and Contra5 :: "'at \<Rightarrow> 'au \<Rightarrow> bool"   and Contra5' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and uu :: "('aw \<times> 'ax \<times> 'ay \<times> 'az \<times> 'ba) itself" shows "rel_FGcontra_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' Contra5 Contra5' uu = (\<forall>L1 L1'. rel_FGcontra (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Co5 OO Co5') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4') (Contra5 OO Contra5') \<le> rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 OO rel_FGcontra L1' Co1' Co2' Co3' Co4' Co5' Contra1' Contra2' Contra3' Contra4' Contra5')"
intro: fixes \<AA>\<rr>\<ii>\<tt>\<yy> :: "'b \<Rightarrow> nat"   and \<OO>\<pp> :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a" assumes "\<And>xs op. length xs = \<AA>\<rr>\<ii>\<tt>\<yy> op \<Longrightarrow> \<exists>y. \<OO>\<pp> op xs = y" shows "nary_operations \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>"
shr_start_state: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'b \<Rightarrow> 'addr val list \<Rightarrow> 'a"   and P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list" shows "shr (h.start_state f P C M vs) = h.start_heap"
mono_on_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "mono_on f A"   and "B \<subseteq> A" shows "mono_on f B"
sum_list_log: fixes b :: "real"   and xs :: "nat list" assumes "0 < b"   and "b \<noteq> 1"   and "\<forall>x\<in>set xs. b \<le> real x" shows "sum_list (map (log b) (map real xs)) = log b (real (prod_list xs))"
diff_gt_0_iff_gt: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (minus a b) = less b a"
simps_1: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uu :: "'d"   and uv :: "'d"   and F :: "'c" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.Map (concrete_category.MkArr uu uv F) = F"
poly_mult: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and p :: "('d monom \<times> 'c) list"   and q :: "('d monom \<times> 'c) list" shows "eval_poly \<alpha> (poly_mult p q) = eval_poly \<alpha> p * eval_poly \<alpha> q"
collapse_2: fixes val :: "val" assumes "is_IntVal val" shows "IntVal (extract_int val) = val"
assms_2: shows "\<exists>z. \<forall>x\<in>S. x \<le> z"
simps_1: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "hosts \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = hosts"
ex_set_conv: fixes xs :: "'f list" shows "(\<exists>x. x \<in> set xs) = (xs \<noteq> [])"
filtermap_nhds_open_map: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "isCont f a"   and "\<And>S. open S \<Longrightarrow> open (f ` S)" shows "filtermap f (nhds a) = nhds (f a)"
awalk_verts_reachable_to: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "awalk u p v"   and "w \<in> set (awalk_verts u p)" shows "w \<rightarrow>\<^sup>* v"
mk_finite_char_def: fixes C :: "'b set set" shows "mk_finite_char C \<equiv> {S. \<forall>S'. finite S' \<longrightarrow> S' \<subseteq> S \<longrightarrow> S' \<in> C}"
ccard_diff_lgth: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "card {i. n < i \<and> i \<le> n' \<and> hmining i} \<le> hbc n' - hbc n"
m_expr_2: shows "m2 = m2' * a mod n \<and> m2' \<in> kernel G H f"
OFMS_from_rel: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and q0 :: "'a" assumes "OFSM_rel rel q0" shows "OFSM (from_rel rel q0)"
zero_sheaf_spec_def: fixes U :: "('a \<times> 'a) set set set" shows "local.local.zero_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.zero_rel (local.carrier_quotient_ring\<setminus>\<pp>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel local.one_rel"
mat_row_code: fixes k :: "'e"   and i :: "'d" shows "($) (mat_row k i) = (\<lambda>j. if i = j then k else (0::'e))"
Pasch_deg: fixes x :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "\<not> distinct [x, y, z, u, v] \<or> poincare_collinear {x, y, z}"   and "poincare_between x u z"   and "poincare_between y v z" shows "\<exists>a. a \<in> unit_disc \<and> poincare_between u a y \<and> poincare_between x a v"
i: fixes n :: "nat" assumes "n \<le> N" shows "min (dist (x (from_nat n)) (y (from_nat n))) 1 \<le> M"
is_TNil_tSup: fixes A :: "('d, 'b) tllist set" shows "is_TNil (local.tSup A) = (\<forall>x\<in>A. is_TNil x)"
lprefix_lfiniteD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "lfinite ys" shows "lfinite xs"
Delay_L2: fixes t :: "nat"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and d :: "nat"   and x :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list" assumes "\<forall>t1<t. i1 t1 = []"   and "Delay y i1 d x i2" shows "\<forall>t2<t + d. i2 t2 = []"
rep_eq: fixes x :: "'c cset" shows "Sequence.seq.Rep_seq (cset2seq x) = (\<lambda>i. if i \<in> cset_count x ` rcset x then inv_into (rcset x) (cset_count x) i else SOME xa. xa \<in>\<^sub>c x)"
directed_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.directed less_eq X = (\<forall>Y. finite Y \<and> Y \<subseteq> X \<longrightarrow> (\<exists>x\<in>X. \<forall>y\<in>Y. less_eq y x))"
B: shows "is_InSet u \<and> is_Var (the_elem_term u) \<and> is_Fun_Set (the_set_term u) \<and> the_Set (the_Fun (the_set_term u)) = s \<and> the_Var (the_elem_term u) = x \<and> the_check u = a"
assms_1: shows "\<forall>N. sum f {..<N} \<le> x"
tauChainBangI: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<parallel> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "guarded P" obtains Q :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> !P \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q" and "\<Psi> \<rhd> Q \<sim> P' \<parallel> !P"
clop_ext: fixes x :: "'b" shows "x \<le> cl_op x"
acute_chara: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and C :: "'p" assumes "Bet A B A'"   and "B \<noteq> A'" shows "(Acute A B C) = (A B C LtA A' B C)"
assms_4: shows "x \<notin> Fvars \<phi>"
CK_nf: fixes f :: "'a set \<Rightarrow> 'a set" shows "CK f = CK_nf f"
subset_refl: fixes A :: "'b set" shows "A \<subseteq> A"
neq_ring: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "max_ideal R I addition multiplication zero unit" shows "I \<noteq> R"
equalsffemptyI: fixes A :: "'b fset" assumes "\<And>y. y |\<in>| A \<Longrightarrow> False" shows "A = {||}"