hcju/isabelleps huggingface dataset:
freshOutputAction_3: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
select_memberI: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "x \<in> S"   and "f x = Some y" shows "y \<in> select f S"
not_lta_and_conga: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<not> (Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F \<and> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F)"
Fo_def: fixes y :: "'d" shows "Fo y = (SOME x. \<exists>u. initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y x u)"
strict_boundedE: fixes b :: "'a multiset"   and c :: "'a multiset"   and a :: "'a multiset" assumes "b \<union># c \<subset># a" obtains "b \<subset># a" and "c \<subset># a"
NewFail: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and 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 C :: "char list"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = None" shows "P,E \<turnstile> \<langle>new C,(h, l)\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory, [OutOfMemory]),(h, l)\<rangle>"
norm_matrix_le_mult_op_norm: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "\<parallel>A *v x\<parallel> \<le> (\<parallel>A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>x\<parallel>)"
red_self: fixes p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "p \<noteq> 0" shows "punit.red {p} p 0"
Sup_atLeastAtMost: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq x y" shows "Sup (ord.atLeastAtMost less_eq x y) = y"
region_cover_V: fixes v :: "'a \<Rightarrow> real" assumes "v \<in> V" shows "\<exists>R. R \<in> \<R> \<and> v \<in> R"
has_integral_iff_nn_integral_lebesgue: fixes f :: "'c \<Rightarrow> real"   and r :: "real" assumes "\<And>x. 0 \<le> f x" shows "(f has_integral r) UNIV = (f \<in> borel_measurable lebesgue \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>lebesgue = ennreal r \<and> 0 \<le> r)"
replay_def: fixes prog :: "program"   and g\<^sub>1 :: "gState"   and g\<^sub>2 :: "gState" shows "replay prog g\<^sub>1 g\<^sub>2 = (let g\<^sub>1 = to\<^sub>I g\<^sub>1; check = \<lambda>g. from\<^sub>I g = g\<^sub>2 in REC\<^sub>T (\<lambda>D g. executable (states prog) g \<bind> (\<lambda>E. if E = [] then if check g then RETURN [] else if \<not> timeout g then D (g\<lparr>timeout := True\<rparr>) else abort STR ''Stuttering should not occur on replay'' (\<lambda>_. RETURN []) else let g = reset\<^sub>I g in nfoldli E (\<lambda>E. E = []) (\<lambda>(e, p) _. applyEdge prog e p g \<bind> (\<lambda>g'. if handshake g' \<noteq> 0 \<or> isAtomic e then D g' \<bind> (\<lambda>E\<^sub>R. if E\<^sub>R = [] then if check g' then RETURN [(e, p)] else RETURN [] else RETURN ((e, p) # E\<^sub>R)) else if check g' then RETURN [(e, p)] else RETURN [])) [])) g\<^sub>1)"
assms_2: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<X> Q"
IH: fixes As :: "'a set set" assumes "disjoint As" shows "disjoint (partitioning ss As)"
card_UN_disjoint: fixes I :: "'g set"   and A :: "'g \<Rightarrow> 'h set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "card (\<Union> (A ` I)) = (\<Sum>i\<in>I. card (A i))"
indV: shows "indV \<equiv> BaseRep.indspace G FH V"
gauss_cnj_eq_0_iff: fixes z :: "gauss_int" shows "(gauss_cnj z = 0) = (z = 0)"
invariant_def: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" shows "(A \<TTurnstile> (I \<rightarrow>) P) = (\<forall>s\<in>reachable A I. P s)"
4: shows "s1 \<sqinter> s2 \<preceq> s1"
alphaAbs_qAbs_iff: fixes A :: "('f, 'g, 'h, 'i, 'j) qAbs"   and xs' :: "'h"   and x' :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "(A $= qAbs xs' x' X') = (\<exists>x y X. A = qAbs xs' x X \<and> y \<notin> {x, x'} \<and> qAFresh xs' y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> x]]_xs' #= X' #[[y \<and> x']]_xs')"
nonempty_imp_ex_extremal: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'c set" assumes "well_founded A less"   and "Q \<subseteq> A"   and "Q \<noteq> {}" shows "\<exists>z\<in>Q. \<forall>y\<in>Q. \<not> less y z"
induced_subgraph_altdef: fixes H :: "('a, 'b) pre_digraph" shows "induced_subgraph H G = (subgraph H G \<and> (\<forall>H'. subgraph H' G \<longrightarrow> verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H))"
GF_advice_subst: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X]\<^sub>\<nu> = subst \<phi> (\<lambda>\<psi>. Some (\<psi>[X]\<^sub>\<nu>))"
subst_typ'_rename_tvar_bind_fv2: fixes y :: "variable"   and t :: "Core.term"   and b :: "variable"   and S :: "String.literal set"   and \<tau> :: "typ"   and a :: "variable"   and i :: "nat"   and x :: "variable" assumes "y \<notin> fst ` fv t"   and "(b, S) \<notin> tvs t"   and "(b, S) \<notin> tvsT \<tau>" shows "bind_fv2 (y, subst_typ [((a, S), Tv b S)] \<tau>) i (subst_typ' [((a, S), Tv b S)] (subst_term [((x, \<tau>), Fv y \<tau>)] t)) = subst_typ' [((a, S), Tv b S)] (bind_fv2 (x, \<tau>) i t)"
list_exhaust2: fixes y :: "'a list"   and ya :: "'b list" obtains "y = []" and "ya = []"   | x21 :: "'b" and x22 :: "'b list" where "y = []" and "ya = x21__ # x22__"   | x21 :: "'a" and x22 :: "'a list" where "y = x21a__ # x22a__" and "ya = []"   | x21 :: "'a" and x22 :: "'a list" and x21a :: "'b" and x22a :: "'b list" where "y = x21 # x22" and "ya = x21a # x22a"
vimage_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "(-`) dual \<circ> (-`) dual = id"
1: shows "[\<^bold>\<box>((\<^bold>\<exists>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr>)) in v]"
continuous_map_product_then_coordinatewise_2: fixes T1 :: "'c topology"   and T :: "'d \<Rightarrow> 'e topology"   and I :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and i :: "'d"   and x :: "'c" assumes "continuous_map T1 (product_topology T I) f"   and "i \<notin> I"   and "x \<in> topspace T1" shows "f x i = undefined"
pow_chain: fixes x :: "'a"   and n :: "nat" shows "p_power (1\<^sub>\<sigma> + x) n \<le> p_power (1\<^sub>\<sigma> + x) (n + 1)"
naturality: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "\<phi> (\<phi>.A.cod f) \<cdot>\<^sub>B F f = G f \<cdot>\<^sub>B \<phi> (\<phi>.A.dom f)"
assms: shows "\<Gamma> \<turnstile> e, eP, eV : \<tau>"
mat_extension_mult: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)" shows "mat_extension dims vars (m1 * m2) = mat_extension dims vars m1 * mat_extension dims vars m2"
envInit_def: shows "envInit \<equiv> concat (map (\<lambda>psf. map (\<lambda>esf. \<lparr>es = esf, ps = psf, pubActs = (default, \<lambda>_. default)\<rparr>) (fst envInitBits)) (listToFuns (snd envInitBits) agents))"
image_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<P> f A = {y. \<exists>x\<in>A. y = f x}"
NSLIM_LIM: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "f \<midarrow>a\<rightarrow> L"
of_real_0: shows "of_real 0 = (0::'b)"
weakening_4: fixes lholed :: "Lholed" shows "\<not> Lfilled 0 lholed [$Return] (cs @ ($* es))"
fst_apfst: fixes f :: "'d \<Rightarrow> 'a"   and x :: "'d \<times> 'c" shows "fst (apfst f x) = f (fst x)"
inject: fixes x :: "'a \<Rightarrow> bool"   and ya :: "'a \<Rightarrow> bool" shows "(pred.Pred x = pred.Pred ya) = (x = ya)"
commute_multiples_identity: fixes A :: "'b mat"   and n :: "nat"   and k :: "'b" assumes "A \<in> carrier_mat n n" shows "A * (k \<cdot>\<^sub>m 1\<^sub>m n) = k \<cdot>\<^sub>m 1\<^sub>m n * A"
case_of_C_below: fixes x :: "'a"   and r :: "C" shows "(case r of C\<cdot>y \<Rightarrow> x) \<sqsubseteq> x"
lex_pm_refl: fixes s :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm s s"
vimage2p_rel_fun: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (BNF_Def.vimage2p f g R) R f g"
simps_2: fixes xs :: "['b]"   and y :: "'b"   and ys :: "['b]" shows "xs \\ (y : ys) = delete\<cdot>y\<cdot>xs \\ ys"
brouwer_compactness_lemma: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact s"   and "continuous_on s f"   and "\<not> (\<exists>x\<in>s. f x = (0::'b))" obtains d :: "real" where "0 < d" and "\<forall>x\<in>s. d \<le> norm (f x)"
Units_inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<otimes> y = \<one>"   and "x \<in> Units G"   and "y \<in> Units G" shows "y \<otimes> x = \<one>"
le_supI1: fixes x :: "'a \<times> 'b"   and a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" assumes "less_eq_prod x a" shows "less_eq_prod x (sup_prod a b)"
connected_contains_Ioo: fixes A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "connected A"   and "a \<in> A"   and "b \<in> A" shows "{a<..<b} \<subseteq> A"
Cs0'_Cs: shows "Cs0' \<cdot>\<cdot>cl \<eta>s0' = Cs"
sepDA: fixes p :: "('b, 'c) Combinators list" assumes "DenyAll \<notin> set p" shows "DenyAll \<notin> set (separate p)"
span_minimal: fixes S :: "'d set"   and T :: "'d set" assumes "S \<subseteq> T"   and "csubspace T" shows "cspan S \<subseteq> T"
A_n_def: fixes n :: "nat" shows "A_n n = from_nat_into (A \<Gamma>) ` {..n}"
finite_measure_mono: fixes A :: "bool stream set"   and B :: "bool stream set" assumes "A \<subseteq> B"   and "B \<in> events" shows "prob A \<le> prob B"
eint_ord_plus_one: fixes n :: "int"   and x :: "eint"   and y :: "eint" assumes "eint n \<le> x"   and "x < y" shows "eint (n + 1) \<le> y"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys x = {k. lookup x k \<noteq> (0::'d)}"
inverse_gf_U'_tendsto: fixes y :: "'s" assumes "recurrent y" shows "((\<lambda>x. - 1 / - gf_U' y y x) \<longlongrightarrow> enn2real (1 / U' y y)) (at_left 1)"
le: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "cbox a b \<subseteq> cbox c d" shows "norm (integral (cbox a b) f - integral (cbox c d) f) \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
vec_of_poly_rev_shifted_dim: fixes p :: "'b poly"   and n :: "nat"   and j :: "nat" shows "dim_vec (vec_of_poly_rev_shifted p n j) = n"
publicKey_notin_image_privateKey: fixes A :: "agent"   and AS :: "agent set" shows "pubK A \<notin> priK ` AS"
lim_def: shows "local.lim = extend_measure (space (Pi\<^sub>M I M)) generator (\<lambda>x. x) \<mu>G"
eq_dfa_accepts: fixes n :: "nat"   and bss :: "bool list list"   and ks :: "int list"   and l :: "int" assumes "list_all (is_alph n) bss" shows "dfa_accepts (eq_dfa n ks l) bss = (eval_dioph ks (nats_of_boolss n bss) = l)"
desargues_config_3D_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" shows "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma> \<equiv> rk {A, B, C} = 3 \<and> rk {A', B', C'} = 3 \<and> rk {A, A', P} = 2 \<and> rk {B, B', P} = 2 \<and> rk {C, C', P} = 2 \<and> 4 \<le> rk {A, B, C, A', B', C'} \<and> rk {B, C, \<alpha>} = 2 \<and> rk {B', C', \<alpha>} = 2 \<and> rk {A, C, \<beta>} = 2 \<and> rk {A', C', \<beta>} = 2 \<and> rk {A, B, \<gamma>} = 2 \<and> rk {A', B', \<gamma>} = 2"
simps_5: fixes S :: "'c ltl set" shows "eval\<^sub>G S false = false"
opex_def: shows "opex = open_exclude_set (f x) (range f)"
winding_number_join_pos_combined: fixes \<gamma>1 :: "real \<Rightarrow> complex"   and z :: "complex"   and \<gamma>2 :: "real \<Rightarrow> complex" assumes "valid_path \<gamma>1"   and "z \<notin> path_image \<gamma>1"   and "0 < Re (winding_number \<gamma>1 z)"   and "valid_path \<gamma>2"   and "z \<notin> path_image \<gamma>2"   and "0 < Re (winding_number \<gamma>2 z)"   and "pathfinish \<gamma>1 = pathstart \<gamma>2" shows "valid_path (\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image (\<gamma>1 +++ \<gamma>2) \<and> 0 < Re (winding_number (\<gamma>1 +++ \<gamma>2) z)"
Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>2 {\<lambda>s. eSuc (P (s[a/x]))} x ::= a {P}"
ff: fixes a :: "'a" assumes "a \<in> A" shows "f (f a) = a"
memb_imp_not_empty: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "S \<noteq> {}"
cod_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "cod f = (if arr f then f else null)"
max_0_1_6: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq (numeral.numeral one plus x) one = numeral.numeral one plus x"
Cod_Dom: shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr> = (op_dg \<AA> \<times>\<^sub>D\<^sub>G op_dg \<BB>)\<lparr>Dom\<rparr>"
stop_no_eval: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "(((Stop, mds), mem), (c', mds'), mem') \<notin> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
src_vcomp: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.seq \<nu> \<mu>" shows "src\<^sub>C (\<nu> \<cdot>\<^sub>C \<mu>) = src\<^sub>C \<nu>"
one_minus_fps_X_const_neg_power: fixes c :: "'a"   and n :: "nat" shows "inverse ((1 - fps_const c * fps_X) ^ n) = fps_binomial (- of_nat n) oo - fps_const c * fps_X"
app_eq: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "app xs ys zs = (zs = xs @ ys)"
labelAtOut_def: fixes v :: "'form \<times> nat list"   and p :: "('form, 'var) out_port" shows "labelAtOut v p = subst (inst v) (freshen (vidx v) (labelsOut (nodeOf v) p))"
dvd_div_iff_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "c \<noteq> zero"   and "dvd.dvd times c b" shows "dvd.dvd times a (divide b c) = dvd.dvd times (times a c) b"
not_is_unit_0: shows "\<not> is_unit (0::'a)"
odd_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) one"
subst_Var_simp2: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good Y" shows "Var xs x #[Y / x]_xs = Y"
normalize_mult_normalize_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize (times (normalize a) b) = normalize (times a b)"
iSince_singleton: fixes t :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t0. t0 \<S> t1 {t}. Q t1) = Q t"
Agreement: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and i :: "('a, 'c, 'b) configuration"   and c :: "('a, 'c, 'b) configuration" assumes "flpPseudoConsensus trans sends start" shows "flpSystem.agreementInit trans sends start i c"
fwi_neg_cycle_neg_cycle: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and xs :: "nat list" assumes "len (fwi m n k n n) i i xs < (0::'a)"   and "i \<le> n"   and "k \<le> n"   and "set xs \<subseteq> {0..n}" shows "\<exists>i\<le>n. \<exists>ys. set ys \<subseteq> set xs \<union> {k} \<and> len m i i ys < (0::'a)"
power_overflow: fixes n :: "nat" assumes "LENGTH('a) \<le> n" shows "2 ^ n = 0"
step: shows "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>w (m', sb', \<O>', \<R>', \<S>')"
push_bit_add: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "push_bit n (a + b) = push_bit n a + push_bit n b"
bot_not_empty: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "consequence_relation Bot entails" shows "Bot \<noteq> {}"
simps: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "qe_eq_one var L F = (case find_eq var L of ([], L') \<Rightarrow> list_conj (map fm.Atom L @ F) | (p # A, L') \<Rightarrow> Or (And (Neg (split_p var p)) (elimVar var L F (Eq p))) (And (split_p var p) (list_conj (map fm.Atom (map Eq A @ L') @ F))))"
ge_const_equiv: shows "(\<sim>\<^sub>C) \<le> (\<sim>)"
assms_4: shows "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)"
tl_top_zero: fixes x :: "real" assumes "x \<in> {0..1}" shows "local.tl 1 x = x"
image_mset_is_empty_iff: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e multiset" shows "(image_mset f M = {#}) = (M = {#})"
graph_homomorphism_Id: fixes a :: "('d, 'e) labeled_graph" shows "graph_homomorphism (restrict a) (restrict a) (Id_on (vertices a))"
arc_linepath: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "arc (linepath a b)"
prj_chine_composite_3: shows "cods.Prj\<^sub>0 \<cdot> \<mu>\<nu>_\<pi>.chine = \<pi>.chine \<cdot> doms.Prj\<^sub>0"
order_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "suffix x y"   and "suffix y z" shows "suffix x z"
subst_subst_compose: fixes t :: "('d, 'f) Term.term"   and \<sigma> :: "'f \<Rightarrow> ('d, 'g) Term.term"   and \<tau> :: "'g \<Rightarrow> ('d, 'e) Term.term" shows "t \<cdot> \<sigma> \<circ>\<^sub>s \<tau> = t \<cdot> \<sigma> \<cdot> \<tau>"
le_boolI': fixes P :: "bool"   and Q :: "bool" assumes "P \<longrightarrow> Q" shows "P \<le> Q"
pnormalize_eq: fixes p :: "'a list" assumes "last p \<noteq> (0::'a)" shows "pnormalize p = p"
d'_4: fixes K :: "'a set" assumes "K \<in> d" shows "\<exists>a b. K = cbox a b"
M_cong: shows "((\<lambda>x. return_val (op_sem oper x)) \<in> stock_measure t \<rightarrow>\<^sub>M subprob_algebra (stock_measure t')) = ((\<lambda>x. return (stock_measure t') (op_sem oper x)) \<in> stock_measure t \<rightarrow>\<^sub>M subprob_algebra (stock_measure t'))"
dom_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.dom f = (if VV.arr f then VxV.dom f else VxV.null)"
fls_lr_inverse_zero_1: fixes x :: "'a" shows "fls_left_inverse 0 x = fls_const x"
le_Suc': fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less_eq (plus a one) b"   and "mod_type.to_nat Rep a + 1 < CARD('a)" shows "less a b"
maps_category_axioms: shows "maps_category (\<cdot>) (\<star>) \<a> \<i> src trg"
i_hate_words_helper: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j - k" shows "i \<le> j"
semantics_stateful_intro_start: fixes \<sigma>\<^sub>0 :: "'a"   and \<sigma>' :: "'a"   and ps_processed :: "('b \<times> final_decision) list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma>\<^sub>\<sigma> :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and state_upate :: "'a \<Rightarrow> final_decision \<Rightarrow> 'b \<Rightarrow> 'a"   and built_in_chain :: "char list"   and default_policy :: "action"   and ps :: "'b list" assumes "\<sigma>\<^sub>0 = \<sigma>'"   and "ps_processed = []" shows "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) ps ps_processed \<sigma>'"
vpairs_iff_elts: fixes a :: "V"   and b :: "V"   and r :: "V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vpairs r) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
mult_sign_intros_8: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "b < (0::'a)" shows "(0::'a) < a * b"
set_rw: shows "{..<n} = insert a ({..<n} - {a})"
length_sums: fixes xs :: "'a list" shows "length (sums xs) = Suc (length xs)"
cauchy_def: fixes X :: "nat \<Rightarrow> rat" shows "Real.cauchy X = (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
Hash_imp_parts: fixes evs :: "event list"   and X :: "msg"   and N :: "nat" assumes "evs \<in> set_cr"   and "Hash \<lbrace>X, Nonce N\<rbrace> \<in> parts (knows Spy evs)" shows "Nonce N \<in> parts (knows Spy evs)"
extend_basis_superset: fixes B :: "'b set" assumes "local.independent B" shows "B \<subseteq> local.extend_basis B"
elims: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) rbt"   and y :: "('a, 'b) rbt" assumes "ins x xa xb = y" obtains xc :: "'a" and v :: "'b" where "x = xc__" and "xa = v__" and "xb = Leaf" and "y = Node Leaf R xc__ v__ Leaf"   | xc :: "'a" and v :: "'b" and l :: "('a, 'b) rbt" and c :: "color" and ya :: "'a" and w :: "'b" and r :: "('a, 'b) rbt" where "x = xc" and "xa = v" and "xb = Node l c ya w r" and "y = (if c = B then if xc = ya then Node l B xc v r else if xc < ya then balance (ins xc v l) ya w r else balance l ya w (ins xc v r) else if xc = ya then Node l R xc v r else if xc < ya then Node (ins xc v l) R ya w r else Node l R ya w (ins xc v r))"
sorted_wrt_tl: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "xs \<noteq> []"   and "sorted_wrt P xs" shows "sorted_wrt P (tl xs)"
unfold: shows "sdrop = (\<Lambda> i xs. If le\<cdot>i\<cdot>0 then xs else (case xs of [::] \<Rightarrow> [::] | y :# ys \<Rightarrow> sdrop\<cdot>(i - 1)\<cdot>ys))"