hcju/isabelleps huggingface dataset:
dense: fixes l :: "'a"   and y :: "'a" assumes "\<forall>y\<in>{l<..<x}. y \<notin> L"   and "l < x"   and "l < y"   and "y \<le> x" shows "P y"
face_cycle_succ_no_arc: fixes a :: "'b" assumes "a \<notin> arcs G" shows "face_cycle_succ a = a"
eq_equals: fixes \<phi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" shows "eq \<phi> \<psi> \<pi>l = (\<phi> \<pi>l = \<psi> \<pi>l)"
negate_f: assumes "has_disc_negated disc False m" shows "\<forall>pts. \<not> has_disc_negated disc False (negate_f C pts)"
\<delta>_no_C\<V>i: shows "\<delta> \<upharpoonleft> C\<^bsub>\<V>i\<^esub> = []"
card_of_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "|f ` A| \<le>o |A|"
le_numeral_extra_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
ass_function_euclidean_def: fixes p :: "'b" shows "ass_function_euclidean p = normalize p"
rel_inject_2: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x21 :: "'g"   and x22 :: "'g list"   and y21 :: "'h"   and y22 :: "'h list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
lower_rank_of_def: fixes carrier' :: "'a set" shows "\<E>.rank_of carrier' \<equiv> Min {card B |B. \<E>.basis_in carrier' B}"
nv0: shows "nv \<noteq> 0\<^sub>v n"
trisuma_perm_312: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C TriSumA D E F" shows "C A B TriSumA D E F"
totalValuationForClauseDefinesAllItsLiterals: fixes valuation :: "Literal list"   and clause :: "Literal list"   and literal :: "Literal" assumes "CNF.total valuation (vars clause)"   and "literal el clause" shows "literalTrue literal valuation \<or> literalFalse literal valuation"
prems_7: shows "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> (\<nexists>u u' m. t ! k = Send cid p q u u' m)"
sub_incidence_system_axioms: shows "sub_incidence_system \<U> \<A> \<V> \<B>"
im_f: shows "f ` carrier (G\<lparr>carrier := H\<rparr>) = carrier (G\<lparr>carrier := N <#> H\<rparr> Mod N)"
restrict_PiE: fixes f :: "'h \<Rightarrow> 'i"   and I :: "'h set"   and S :: "'h \<Rightarrow> 'i set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
bounded: fixes x :: "'a" assumes "x \<in> S" shows "norm (f x) \<le> B * norm x"
one_ne_\<omega>\<^sub>z: shows "1 \<noteq> \<omega>\<^sub>z"
assms_3: fixes x :: "'b" assumes "x \<in> A" shows "f x \<noteq> {}"
mbisim_flip: fixes bisim :: "'a \<Rightarrow> 'd \<times> 'e \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and bisim_wait :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set" shows "FWbisimulation_base.mbisim (\<lambda>t. flip (bisim t)) (flip bisim_wait) s2 s1 = FWbisimulation_base.mbisim bisim bisim_wait s1 s2"
infm: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and N :: "'b" assumes "infinitely_many2 P N" shows "\<forall>i\<ge>N. \<exists>j>i. P i j"
fold1: fixes s1 :: "('e, 'd, 'f) F1 \<Rightarrow> 'd"   and s2 :: "('e, 'd, 'f) F2 \<Rightarrow> 'f"   and x :: "('e, 'e IF1, 'e IF2) F1" shows "fold1 s1 s2 (ctor1 x) = s1 (F1map id (fold1 s1 s2) (fold2 s1 s2) x)"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e stream" shows "smap g (smap f v) = smap (g \<circ> f) v"
res: shows "(if degree p = 0 then factorize_rat_poly_main (c * coeff p 0) irr ps else case rational_proper_factor p of None \<Rightarrow> factorize_rat_poly_main c (p # irr) ps | Some q \<Rightarrow> factorize_rat_poly_main c irr (q # p div q # ps)) = (d, qs)"
\<F>: fixes S :: "'a set" assumes "S \<in> \<F>" shows "\<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"
gderiv: shows "(g has_field_derivative gd) (at z within S)"
code: fixes z :: "quat" shows "Quaternions.cnj z = Quat (quat.Re z) (- Im1 z) (- Im2 z) (- Im3 z)"
strongest_intuitionistic: fixes P :: "'a \<Rightarrow> bool" shows "\<nexists>Q. (\<forall>h. Q h \<longrightarrow> (P \<and>* (\<lambda>s. True)) h) \<and> intuitionistic Q \<and> Q \<noteq> (P \<and>* (\<lambda>s. True)) \<and> (\<forall>h. P h \<longrightarrow> Q h)"
cut_less_not_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>< t \<noteq> {}) = (\<exists>x\<in>I. x < t)"
elims_2: fixes x :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list"   and xa :: "'a \<Rightarrow> 'c"   and xb :: "'b" assumes "postList x xa xb" obtains l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = []" and "xa = l__" and "xb = s__"   | xc :: "'a" and e :: "'b \<Rightarrow> 'c" and v :: "'d" and xs :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list" and l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = (xc, e, v) # xs" and "xa = l" and "xb = s" and "l xc = e s \<and> postList xs l s"
simps_3: fixes int :: "int" shows "\<infinity> \<noteq> eint int"
token_succeeds_def: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and F :: "'c set"   and x :: "nat" shows "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x = (\<exists>n. semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<in> F)"
sets_completionI_sets: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "A \<in> sets (completion M)"
induct: fixes x1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and x2 :: "nat"   and x3 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and P :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> nat \<Rightarrow> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool" assumes "x1 \<leadsto>\<^bsup>x2\<^esup> x3"   and "\<And>x y. x = y \<Longrightarrow> P x 0 y"   and "\<And>x y n z. \<lbrakk>x \<leadsto> y; y \<leadsto>\<^bsup>n\<^esup> z; P y n z\<rbrakk> \<Longrightarrow> P x (Suc n) z" shows "P x1 x2 x3"
valuate_diff: fixes p :: "linear_poly"   and v1 :: "nat \<Rightarrow> 'a"   and v2 :: "nat \<Rightarrow> 'a" shows "p \<lbrace> v1 \<rbrace> - p \<lbrace> v2 \<rbrace> = p \<lbrace> \<lambda>x. v1 x - v2 x \<rbrace>"
has_field_derivative_at_within: fixes f :: "'b \<Rightarrow> 'b"   and f' :: "'b"   and x :: "'b"   and s :: "'b set" assumes "(f has_field_derivative f') (at x)" shows "(f has_field_derivative f') (at x within s)"
set_intros_1: fixes x12 :: "'a"   and x11 :: "nat" shows "x12 \<in> set_tree (Leaf x11 x12)"
Max_in: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "finite A"   and "A \<noteq> {}" shows "ordered_powerprod_lin.Max A \<in> A"
nres_monad_laws_1: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c nres" shows "RETURN x \<bind> f = f x"
objI_trg: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V a"   and "trg a = a" shows "horizontal_homs.obj V src a"
DERIV_minus: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c"   and s :: "'c set" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. - f x) has_field_derivative - D) (at x within s)"
cases: fixes x :: "'b DBMEntry" obtains c :: "'b" where "x = Le c__"   | c :: "'b" where "x = Lt c"   | "x = \<infinity>"
pimage: shows "\<forall>p\<in>path_image \<gamma>. Im z \<le> Im p"
weak_le_iff_meet: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq> y) = (x \<squnion> y .= y)"
has_derivative_at_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. (f y - f x - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> (0::'b)) (at x within s))"
listDiffMap: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b list"   and b :: "'b list" assumes "\<forall>x y. x \<noteq> y \<longrightarrow> f x \<noteq> f y" shows "map f (list_diff a b) = list_diff (map f a) (map f b)"
simps_3: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "removeLeaf (T v E (T va vb vc)) = (fst (removeLeaf (T va vb vc)), T v E (snd (removeLeaf (T va vb vc))))"
R2_S2_True: fixes \<sigma> :: "bool"   and b0 :: "bool"   and b1 :: "bool"   and D2 :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf" assumes "(if \<sigma> then b0 else b1) = True"   and "\<forall>a. lossless_spmf (D2 a)" shows "\<bar>spmf (R2 (b0, b1) \<sigma> \<bind> D2) True - spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). S2 \<sigma> out2 \<bind> D2)) True\<bar> = \<bar>2 * (spmf (local.etp.HCP_game \<A> \<sigma> (if \<sigma> then b1 else b0) D2) True - 1 / 2)\<bar>"
trans_letter_less: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (letter_less r)"
assms: shows "x \<in> results'_gpv (catch_gpv gpv)"
path_valid_edges: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'" shows "\<forall>a\<in>set as. valid_edge a"
classes_changed_class: fixes cn :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "cn \<notin> classes_changed P P'" shows "class P cn = class P' cn"
fconverse_fconst_on: fixes A :: "V"   and c :: "V" shows "(fconst_on A c)\<inverse>\<^sub>\<bullet> = set {c} \<times>\<^sub>\<bullet> A"
minus_dvd_iff: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times (uminus x) y = dvd.dvd times x y"
inject: fixes x1 :: "real"   and y1 :: "real" shows "(ereal x1 = ereal y1) = (x1 = y1)"
simps_4: fixes x2 :: "'c"   and x1 :: "'b" shows "Inr x2 \<noteq> Inl x1"
G_af\<^sub>G_simp: fixes \<phi> :: "'a ltl"   and w :: "'a set list" shows "\<^bold>G (af\<^sub>G \<phi> w) = \<^bold>G \<phi>"
cfs_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "(p \<oplus>\<^bsub>P\<^esub> q) n = p n \<oplus> q n"
simps_1: fixes uu :: "'b set" shows "difference_list_opt uu [] = []"
isomorphic_def: fixes a :: "'d"   and a' :: "'d" shows "D.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f)"
map_comp_empty_2: fixes m :: "'c \<Rightarrow> 'e option" shows "Map.empty \<circ>\<^sub>m m = Map.empty"
card_of_least: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "well_order_on A r" shows "|A| \<le>o r"
prime_elem_powerD: fixes p :: "'a"   and n :: "nat" assumes "prime_elem (p ^ n)" shows "prime_elem p \<and> n = 1"
closed_except_simps_3: fixes cs :: "(Term_Class.term \<times> sterm) list"   and S :: "name fset" shows "closed_except (Sabs cs) S = list_all (\<lambda>(pat, t). closed_except t (S |\<union>| frees pat)) cs"
simps_2: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr list"   and m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" shows "normalize_rules f (Rule m a # rs) = map (\<lambda>m. Rule m a) (f m) @ normalize_rules f rs"
"2_4": shows "(poincare_map p2 (flow0 x0 h), D2 o\<^sub>L (Dflow x0 h o\<^sub>L d0)) \<in> P2"
triangle_not_betw_bac: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<not> [[b a c]]"
substc_crename_comm': fixes c :: "coname"   and a :: "coname"   and b :: "coname"   and P :: "trm"   and M :: "trm"   and x :: "name" assumes "c \<noteq> a"   and "c \<noteq> b"   and "a \<sharp> P" shows "M{c:=x.P}[a\<turnstile>c>b] = M[a\<turnstile>c>b]{c:=x.P}"
eq: shows "sum \<chi> {1..x} = (\<Sum>d | d dvd n. moebius_mu d * \<Phi> d * sum \<Phi> {1..x div d})"
prems: shows "\<I>' \<turnstile>g gpv \<surd>"
next_empty: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "emptyseq s" shows "next n s = s"
tuple_pr: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "(\<r>[a] \<cdot> (a \<otimes> \<t>[b]) \<otimes> \<l>[b] \<cdot> (\<t>[a] \<otimes> b)) \<cdot> \<d>[a \<otimes> b] = a \<otimes> b"
simps_2: fixes s :: "char" shows "iface_name_is_wildcard [s] = (s = CHR ''+'')"
refl_ge_eq: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. R x x" shows "(=) \<le> R"
sym_iff_eq_refl: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "antisymmetric A less_eq"   and "x \<in> A"   and "y \<in> A" shows "sympartp less_eq x y = (x = y \<and> less_eq y y)"
simps_1: fixes worlds :: "'b set"   and relations :: "'c \<Rightarrow> ('b \<times> 'b) set"   and valuation :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "worlds \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = worlds"
tminus_add_mono: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "a + b \<ominus> c + d \<le> (a \<ominus> c) + (b \<ominus> d)"
keys_subset_singleton_imp_monomial: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c" assumes "keys p \<subseteq> {t}" shows "monomial (lookup p t) t = p"
refl_symcl_of_refl_rel: fixes A :: "'a set"   and Rel :: "('a \<times> 'a) set" assumes "refl_on A Rel" shows "refl_on A (symcl Rel)"
infl_mono: shows "\<forall>a\<in>A. a \<sqsubseteq> f a \<or> (\<forall>b\<in>A. b \<sqsubseteq> a \<longrightarrow> f b \<sqsubseteq> f a)"
nandci_def: fixes e1 :: "nat"   and e2 :: "nat"   and s :: "bddi" shows "nandci e1 e2 s \<equiv> andci e1 e2 s \<bind> (\<lambda>(r, s). notci r s)"
le_iff_add: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
in_abs_rangeI: fixes s :: "(nat \<times> nat) list"   and v :: "nat"   and a :: "nat" assumes "wf_partial_state s"   and "(v, a) \<in> set s" shows "a \<in> set (the (abs_range_map v))"
toy_ex_label: fixes p :: "(state, msg, pseqp, pseqp label) seqp" shows "\<exists>l. l \<in> labels \<Gamma>\<^sub>T\<^sub>O\<^sub>Y p"
span_singleton: fixes x :: "'a" shows "cspan {x} = range (\<lambda>k. k *\<^sub>C x)"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x .=\<^bsub>L\<^esub> y"   and "y .=\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x .=\<^bsub>L\<^esub> z"
sum_var_list_monom_mult_list: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c \<times> nat) list"   and n :: "('c \<times> nat) list"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.sum_var_list (linorder.monom_mult_list less m n) x = linorder.sum_var_list m x + linorder.sum_var_list n x"
option_while'_inj: fixes s :: "'d option"   and s' :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option"   and s'' :: "'d option" assumes "(s, s') \<in> option_while' C B"   and "(s, s'') \<in> option_while' C B" shows "s' = s''"
zero: fixes K :: "'a set"   and x :: "'a" shows "\<zero> \<in> simple_extension K x"
in_listsD: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<in> lists A" shows "\<forall>x\<in>list.set xs. x \<in> A"
dual_neg: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "inf (times x top) (times (dual x) bot) = bot"
simps_4: fixes \<alpha>1 :: "'g"   and t1 :: "('e, 'f, 'g) Tree"   and \<alpha>2 :: "'g"   and t2 :: "('e, 'f, 'g) Tree" shows "(tAct \<alpha>1 t1 =\<^sub>\<alpha> tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
simps_3: fixes f :: "Core.term"   and t :: "Core.term"   and lev :: "nat"   and u :: "Core.term" shows "subst_bv1 (f $ t) lev u = subst_bv1 f lev u $ subst_bv1 t lev u"
wf_mbufn'_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and n :: "nat"   and R :: "event_data list set"   and r :: "Formula.formula Regex.regex"   and buf :: "event_data option list set list list" shows "wf_mbufn' \<sigma> P V j n R r buf = (case to_mregex r of (mr, \<phi>s) \<Rightarrow> wf_mbufn (progress_regex \<sigma> P r j) (map (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) \<phi>s) (map (\<lambda>\<phi> i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s) buf)"
residue_pole_order: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "is_pole f z" shows "residue f z = (deriv ^^ (nat (- zorder f z) - 1)) (zor_poly f z) z / fact (nat (- zorder f z) - 1)"
simps_4: fixes p :: "('a, 'b, 'c) formula" shows "SIGP (? p) = SIGF p"
eq: fixes k :: "nat" shows "A k = (if k = 0 then 1 else (- 1) ^ (k + 1) * stieltjes_gamma (k - 1) / fact (k - 1))"
inj_implies_step_forwards_backwards: fixes x :: "'a" assumes "is_inj x" shows "x\<^sup>\<star> ; (x\<^sup>+ \<cdot> 1') ; 1 \<le> x\<^sup>T ; 1"
brec_iqm_correct: fixes \<delta> :: "('a, 'b) ta_rule dlist" assumes "ls.invar \<delta>" shows "hm.\<alpha> (brec_iqm \<delta>) \<in> brw_iq (ls.\<alpha> \<delta>)"
writeSucceedCorrect: fixes cfile1 :: "CFile"   and cfile2 :: "CFile"   and index :: "nat"   and "value" :: "byte" assumes "nextFreeBlockInvariant cfile1"   and "unallocatedBlocksInvariant cfile1"   and "lastBlockInvariant cfile1"   and "Some cfile2 = cfWrite cfile1 index value" shows "Some (abstFn cfile2) = afWrite (abstFn cfile1) index value"
flists_nonempty: fixes A :: "'b fset" shows "\<exists>xs. xs \<in> flists A"
filter_distinct_at: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and us :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> P v) xs = u # us" shows "filter P bs = us \<and> filter P as = []"
induct: 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 x1 :: "char list"   and x2 :: "char list list"   and Pa :: "char list \<Rightarrow> char list list \<Rightarrow> bool" assumes "Subobjs P x1 x2"   and "\<And>C Cs. Subobjs\<^sub>R P C Cs \<Longrightarrow> Pa C Cs"   and "\<And>C C' D Cs. \<lbrakk>P \<turnstile> C \<preceq>\<^sup>* C'; P \<turnstile> C' \<prec>\<^sub>S D; Subobjs\<^sub>R P D Cs\<rbrakk> \<Longrightarrow> Pa C Cs" shows "Pa x1 x2"
simps_2: fixes x21 :: "'b"   and x22 :: "real"   and x23 :: "'b kdt"   and x24 :: "'b kdt"   and y21 :: "'b"   and y22 :: "real"   and y23 :: "'b kdt"   and y24 :: "'b kdt" shows "(Node x21 x22 x23 x24 = Node y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
power_not_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero"   and "a \<noteq> zero" shows "power.power one times a n \<noteq> zero"
c_sum_is_pr: shows "c_sum \<in> PrimRec1"
wf_trancl_conv: fixes r :: "('a \<times> 'a) set" shows "wf (r\<^sup>+) = wf r"
rank_ins2: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.rank_invar bq" shows "BinomialHeapStruc.rank t \<le> BinomialHeapStruc.rank (hd (BinomialHeapStruc.ins t bq)) \<or> BinomialHeapStruc.rank (hd (BinomialHeapStruc.ins t bq)) = BinomialHeapStruc.rank (hd bq) \<and> bq \<noteq> []"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "C.arr f"
canE_unitor_2: fixes f :: "'a E.term" assumes "E.Ide f" shows "B.can f (E.Trg f \<^bold>\<star> f) = B.lunit (EVAL f)"
minus_sat_def: fixes x :: "'b sat"   and y :: "'b sat" shows "x - y = Abs_sat' (nat_of x - nat_of y)"
member_concat: fixes LL :: "'a list list"   and x :: "'a" shows "List.member (concat LL) x = (\<exists>L. List.member LL L \<and> List.member L x)"
map_add_dom_app_simps_3: fixes m :: "'d"   and l2 :: "'d \<Rightarrow> 'e option"   and l1 :: "'d \<Rightarrow> 'e option" assumes "m \<notin> dom l2" shows "(l2 \<Oplus> l1) m = l1 m"
match_3: fixes g :: "'a"   and f :: "'a"   and h :: "'a" assumes "seq g f"   and "seq h g" shows "seq (h \<cdot> g) f"
has_derivative_eq_rhs: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') F"   and "f' = g'" shows "(f has_derivative g') F"
plossless_iff_colossless_pfinite: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv \<I> gpv = (colossless_gpv \<I> gpv \<and> pfinite_gpv \<I> gpv)"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_quasi_ordered_set A less_eq" shows "connex A less_eq"
crev: fixes t :: "real"   and x :: "real"   and y :: "real" shows "t \<in> point_ivl (1 / 8) \<longrightarrow> (x, y) \<in> point_ivl (2, 0) \<longrightarrow> t \<in> c.rev.existence_ivl0 (x, y) \<and> c.rev.flow0 (x, y) t \<in> {(515 / 10\<^sup>2, - (651 / 10 ^ 3))..(518 / 10\<^sup>2, - (647 / 10 ^ 3))}"
unify_list_wf_trm: fixes E :: "(('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) list"   and B :: "('b \<times> ('fun, 'b) Term.term) list"   and U :: "('b \<times> ('fun, 'b) Term.term) list" assumes "unify E B = Some U"   and "\<forall>(s, t)\<in>set E. wf\<^sub>t\<^sub>r\<^sub>m s \<and> wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<forall>(v, y)\<in>set B. wf\<^sub>t\<^sub>r\<^sub>m y" shows "\<forall>(v, y)\<in>set U. wf\<^sub>t\<^sub>r\<^sub>m y"
comp_runit_runit'_1: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.runit a \<cdot> \<F>C.runit' a = a"
ord_distinct: shows "\<forall>a b c. ord a b c \<longrightarrow> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
T_slow_select_def: fixes k :: "nat"   and xs :: "'a list" shows "T_slow_select k xs = T_isort xs + T_nth (isort xs) k + 1"
os_out_os: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and C :: "'p"   and D :: "'p"   and C' :: "'p" assumes "Col A B P"   and "A B OS C D"   and "P Out C C'" shows "A B OS C' D"
irbd_set_var: fixes \<phi> :: "'c set \<Rightarrow> 'd set" shows "bd\<^sup>-\<^sub>\<R> \<phi> = {(x, y). s2p (\<phi> (\<eta> x)) y}"