hcju/isabelleps huggingface dataset:
weak_complete_lattice_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_complete_lattice_axioms L \<equiv> (\<forall>A\<subseteq>carrier L. \<exists>s. is_lub L s A) \<and> (\<forall>A\<subseteq>carrier L. \<exists>i. is_glb L i A)"
smallc_def: shows "smallc = contour_integral (part_circlepath 0 r pi 0) (f s) - cnj (contour_integral (part_circlepath 0 r pi 0) (f (cnj s)))"
intro: fixes X :: "'d set"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'f set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'f"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'f"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'g \<Rightarrow> 'g"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'g"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'g \<Rightarrow> 'f" assumes "\<And>x V. \<lbrakk>x \<in> X; is_open\<^sub>Y V; f x \<in> V\<rbrakk> \<Longrightarrow> ind_mor_btw_stalks.is_local X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f x V \<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub>" shows "morphism_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
fsubset_refl: fixes A :: "'a fset" shows "A |\<subseteq>| A"
assms_7: shows "sum f1 A1 = sum f2 A2"
canI_associator_hcomp_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "\<a>[f, g, h \<star> k] = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>) ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>)"
bar: fixes A :: "'a multiset set"   and B :: "'a multiset \<Rightarrow> 'a multiset set" assumes "A \<subseteq> ((mult (r\<^sup>+))\<^sup>*)\<inverse> `` {X}"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> ((mult (r\<^sup>+))\<^sup>*)\<inverse> `` {X}" shows "mult r \<inter> Sigma A B \<subseteq> mult (Restr (r\<^sup>+) (((r\<^sup>+)\<^sup>*)\<inverse> `` set_mset X))"
EmptyMap_RootEx: fixes SA :: "('d, 'e, 'f) seqauto" shows "RootEx {SA} (EmptyMap (States SA))"
omax_ordLess: fixes R :: "('c \<times> 'c) set set"   and p :: "('d \<times> 'd) set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. r <o p" shows "omax R <o p"
create_opt_sol_in_LP_3: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "y \<bullet> b \<le> x \<bullet> c"
Reals_minus_iff: fixes a :: "'b" shows "(- a \<in> \<real>) = (a \<in> \<real>)"
correctCompositionDiffLevelsS1: shows "correctCompositionDiffLevels sS1"
Q_transitive: fixes q :: "L"   and G :: "Var \<Rightarrow> L" shows "transitive (Q q G)"
Arden_def: fixes X :: "'b list set"   and rhs :: "'b trm set" shows "Arden X rhs \<equiv> Append_rexp_rhs (rhs - {Trn X r |r. Trn X r \<in> rhs}) (Star (\<Uplus>{r. Trn X r \<in> rhs}))"
deriv: fixes w :: "real" assumes "w \<in> {0..1}" shows "(g has_real_derivative g' w) (at w within {0..1})"
exhaust: fixes y :: "'a + 'b" obtains x1 :: "'a" where "y = Inl x1"   | x2 :: "'b" where "y = Inr x2"
tfinite_2: fixes t' :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" assumes "t' |\<in>| cont t"   and "local.wf t'" shows "\<exists>it. local_iwf it (fst (tree.root t'))"
map_is_simplicial_morph: fixes f :: "'a \<Rightarrow> 'c" shows "SimplicialComplex (f \<turnstile> X)"
ivl_disj_un_one_2: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{..<l} \<union> {l..<u} = {..<u}"
weight_stable_lt: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('f, 'd) Term.term" assumes "weight s < weight t"   and "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)" shows "weight (s \<cdot> \<sigma>) < weight (t \<cdot> \<sigma>)"
sFFalse: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and n :: "nat" assumes "\<not> pbval b ps"   and "vars b \<subseteq> dom ps" shows "wp\<^sub>3 (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
derivs_snoc: fixes xs :: "'b list"   and x :: "'b"   and r :: "'b rexp" shows "derivs (xs @ [x]) r = deriv x (derivs xs r)"
FunctorCompPreserved: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f \<in> mor\<^bsub>CatCod F\<^esub>"
filterlim_mono_eventually: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter"   and G :: "'b filter"   and F' :: "'c filter"   and G' :: "'b filter"   and f' :: "'b \<Rightarrow> 'c" assumes "filterlim f F G"   and "F \<le> F'"   and "G' \<le> G"   and "\<forall>\<^sub>F x in G'. f x = f' x" shows "filterlim f' F' G'"
regular_graph_def: fixes G :: "('a, 'b) pre_digraph"   and \<r> :: "int" shows "regular_graph G \<r> \<equiv> graph G \<and> regular_digraph G \<r>"
Hii_Kii: shows "\<forall>i. H $ i $ i = K $ i $ i"
unl_pref_imp: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and w :: "'a" assumes "|~ F \<longrightarrow> G"   and "F w" shows "G w"
simps_2: fixes x2 :: "'b" shows "None \<noteq> Some x2"
simps_31: shows "VAR sS6 = {stA2, stA4}"
prv_eqv_prv: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv \<phi>"   and "prv (eqv \<phi> \<chi>)" shows "prv \<chi>"
gx3: fixes x :: "nat" assumes "gx3 \<le> x" shows "g x \<le> gc1 * g' (real x)"
exhaust: fixes y :: "I" obtains "y = init"   | "y = loop"   | "y = final"
simps: fixes key :: "'c \<Rightarrow> 'd"   and u :: "nat"   and ns :: "nat list"   and xs :: "'c list" shows "sort_inv key (u, ns, xs) = (\<forall>i<length ns. \<forall>j<offs ns 0 ! i. \<forall>k\<in>{offs ns 0 ! i..<length xs}. key (xs ! j) \<le> key (xs ! k))"
eq: fixes x :: "nat" assumes "0 < x" shows "(\<Sum>n = 1..2 * x. (- 1) ^ (n + 1) * ln (real n) ^ k / real n) = ln 2 ^ (k + 1) / real (k + 1) - (\<Sum>t<k. real (k choose t) * ln 2 ^ (k - t) * stieltjes_gamma t) + h k (2 * x) - h' x"
CX5h_singular: fixes h :: "H2"   and X :: "X5 set" shows "inj_on X5d (CX5h h X)"
supp: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "supp T \<subseteq> M"
HT_partial_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT_partial \<pi> P c Q \<equiv> \<forall>s\<^sub>0. P s\<^sub>0 \<longrightarrow> wlp \<pi> c (Q s\<^sub>0) s\<^sub>0"
handshake_sweep_mark_endD: fixes s :: "(('f, 'g, 'h) response, char list, 'e process_name, 'e process_name \<times> ('f, 'e, 'g, 'h) request_op, ('f, 'e, 'g, 'h) Model.local_state) system_state"   and m :: "'e" assumes "atS gc no_grey_refs_locs s"   and "handshake_invL s"   and "handshake_phase_inv s\<down>" shows "ghost_hs_phase (s\<down> (mutator m)) = hp_IdleMarkSweep \<and> All (ghost_hs_in_sync (s\<down> sys))"
suminf_cmult_indicator: fixes A :: "nat \<Rightarrow> 'a set"   and x :: "'a"   and i :: "nat"   and f :: "nat \<Rightarrow> ennreal" assumes "disjoint_family A"   and "x \<in> A i" shows "(\<Sum>n. f n * indicator (A n) x) = f i"
bra_def: fixes v :: "complex Matrix.mat" shows "\<langle>v| \<equiv> Matrix.mat 1 (dim_row v) (\<lambda>(i, j). cnj (v $$ (j, i)))"
finite_measure_def: fixes M :: "'b measure" shows "finite_measure M \<equiv> sigma_finite_measure M \<and> finite_measure_axioms M"
COMPARE_def: fixes x :: "real"   and y :: "real" shows "COMPARE x y = (if x < y then LT else if y < x then GT else EQ)"
lex_pp_def: shows "lex_pp \<equiv> map_fun pp.mapping_of (map_fun pp.mapping_of id) lex_pm"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "\<forall>y. (P[x::=y], Q[x::=y]) \<in> Rel"   and "eqvt Rel" shows "a<x>.P \<leadsto>\<^sup>^<Rel> a<x>.Q"
opt: fixes F :: "atom fm"   and xs :: "real list" shows "eval (opt F) xs = eval F xs"
unit_l_cancel: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Dmonoid prod one"   and "Dmonoid.unit prod one x" shows "(prod x y = prod x z) = (y = z)"
null_space_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "null_space A = {x. A *v x = 0}"
ord_dvd_group_order: fixes a :: "nat" assumes "a \<in> carrier (units_of R)" shows "units.ord a dvd order (units_of R)"
assms_1: shows "0 \<le> c \<and> c \<noteq> \<infinity>"
digraphD_2: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vfsequence \<CC>"
hext_typeof_mono: fixes h :: "'heap"   and h' :: "'heap"   and v :: "'addr val"   and T :: "ty" assumes "h \<unlhd> h'"   and "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h'\<^esub> v = \<lfloor>T\<rfloor>"
simps_4: fixes f1 :: "'e"   and \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd, 'e) Tree"   and f2 :: "'e"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd, 'e) Tree" shows "(tAct f1 \<alpha>1 t1 =\<^sub>\<alpha> tAct f2 \<alpha>2 t2) = (f1 = f2 \<and> (\<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)))"
pair_map_pmf1: fixes f :: "'e \<Rightarrow> 'c"   and A :: "'e pmf"   and B :: "'d pmf" shows "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
Signature_axioms: shows "Signature TYPE('tp) GE_wtFsym GE_wtPsym"
simIFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "x \<sharp> \<Psi>"   and "x \<sharp> P"   and "x \<sharp> Q"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* subject \<alpha>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* C; x \<sharp> \<alpha>; x \<sharp> Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
order_iff_strict: fixes b :: "('c, 'd) Term.term"   and a :: "('c, 'd) Term.term" shows "(b \<sqsubseteq> a) = (b \<sqsubset> a \<or> a = b)"
chamber_is_singleton: fixes x :: "'w set set" assumes "chamber x" shows "\<exists>w\<in>W. dual_order.poset_simplex_map \<P> {w} = x"
inductive_invariantI: fixes P :: "('r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option) \<Rightarrow> bool" assumes "\<And>s. \<exists>e r. s = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<and> program_expr e \<Longrightarrow> P s"   and "\<And>s s'. \<lbrakk>s \<leadsto> s'; P s\<rbrakk> \<Longrightarrow> P s'" shows "inductive_invariant P"
span_diff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x - y \<in> module.span scale S"
strassen_even_def: fixes A :: "'b mat"   and B :: "'c mat" shows "strassen_even A B \<equiv> even (dim_row A) \<and> even (dim_col A) \<and> even (dim_col B)"
fset_cases: fixes y :: "'a set" assumes "y \<in> {A. finite A}" obtains x :: "'a fset" where "y = fset x"
sep_disj_commuteI': fixes x :: "'b"   and y :: "'b" assumes "x ## y" shows "y ## x"
rel_S_eq: shows "rel_S (=) (=) (=) (=) (=) (=) = (=)"
less_asym: fixes a :: "'a"   and b :: "'a"   and C :: "bool" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec> b"   and "b \<prec> a" shows "C"
connectedI: assumes "verts G \<noteq> {}"   and "\<And>u v. \<lbrakk>u \<in> verts G; v \<in> verts G\<rbrakk> \<Longrightarrow> u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v" shows "Digraph_Component.connected G"
CallParamsThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and es :: "nat exp list"   and es' :: "nat exp list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and ex :: "nat exp"   and es\<^sub>2 :: "nat exp list"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   and "es' = map Val vs @ throw ex # es\<^sub>2" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
smult_one: fixes c :: "'a" shows "smult c 1 = [:c:]"
mult_sheaf_spec_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "mult_sheaf_spec U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.mult_rel (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> (s \<pp>) (s' \<pp>)"
isomorphic_designs_def: fixes \<D> :: "'c set \<times> 'c set multiset"   and \<D>' :: "'d set \<times> 'd set multiset" shows "(\<D> \<cong>\<^sub>D \<D>') = (\<exists>\<pi>. design_isomorphism (fst \<D>) (snd \<D>) (fst \<D>') (snd \<D>') \<pi>)"
ring_1_ow: shows "class.ring_1 = ring_1_ow UNIV"
abs_diff_triangle_ineq2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<bar>a - b\<bar> \<le> \<bar>a - c\<bar> + \<bar>c - b\<bar>"
IH_2: fixes ys :: "('a \<times> 'b) list"   and xs :: "'a list" assumes "ys = map (\<lambda>x. (x, p x)) xs"   and "inj_on p (set xs)"   and "filter ((<) z) zs = sort_key p xs" shows "map_tree fst (treap_of (set ys)) = bst_of_list (filter ((<) z) zs)"
tabulate2_nth: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i < nr"   and "j < nc" shows "tabulate2 nr nc f !! i !! j = f i j"
group_l_invI: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
pivot'_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and s :: "('b, 'a) state" shows "pivot' x\<^sub>i x\<^sub>j s \<equiv> \<T>_update (pivot_tableau' x\<^sub>i x\<^sub>j (\<T> s)) s"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "0 ^ n = 0"
M: fixes n :: "nat" assumes "M \<le> n" shows "real (card (B \<inter> {..<n})) < e / (4 * (C + 1)) * real n"
retr_eval: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "rel_fun (rel R) R eval eval"   and "rel (inf R (retr R)) x y" shows "retr R (eval x) (eval y)"
m_inv_consistent: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x = inv\<^bsub>G\<^esub> x"
elims_3: fixes x :: "('c, 'd) BinomialTree list" assumes "\<not> rank_invar x" obtains t :: "('c, 'd) BinomialTree" and t' :: "('c, 'd) BinomialTree" and bq :: "('c, 'd) BinomialTree list" where "x = t # t' # bq" and "\<not> (rank t < rank t' \<and> rank_invar (t' # bq))"
operative_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b set \<Rightarrow> 'a" shows "operative f z g \<equiv> comm_monoid_set f z \<and> operative_axioms f z g"
continuous_minus: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. - f x)"
nonneg_weights: fixes v :: "'V"   and w :: "'W"   and v' :: "'V" assumes "(v, w, v') \<in> hlgv.E" shows "(0::'W) \<le> w"
that: fixes k :: "nat" assumes "i \<le> k"   and "k < j"   and "l ! k \<le> x"   and "x < l ! (k + 1)" shows "thesis"
DISJ_EFF_DISJ_PROJ_EFF: fixes a :: "('d, 'f) fmap \<times> ('d, 'e) fmap"   and s :: "'d set"   and vs :: "'d set" assumes "fmdom' (snd a) \<inter> s = {}" shows "fmdom' (snd (action_proj a vs)) \<inter> s = {}"
correctCompositionKS_exprChannel_s_P: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and secret :: "Secrets"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "sKS secret \<notin> LocalSecrets PQ"   and "ch \<in> ins P"   and "exprChannel ch (sE secret)"   and "sKS secret \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "ch \<in> ins PQ \<and> exprChannel ch (sE secret)"
is_subprob_density_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "is_subprob_density M f \<equiv> f \<in> borel_measurable M \<and> space M \<noteq> {} \<and> (\<forall>x\<in>space M. 0 \<le> f x) \<and> integral\<^sup>N M f \<le> 1"
word_cat_bl: fixes a :: "'b word"   and b :: "'c word" shows "word_cat a b = of_bl (to_bl a @ to_bl b)"
add_mset_not_empty: fixes a :: "'b"   and A :: "'b multiset" shows "add_mset a A \<noteq> \<Empt>"
cones_map_comp: fixes f :: "'c"   and g :: "'c" assumes "C.seq f g" shows "D.cones_map (f \<cdot> g) = restrict (D.cones_map g \<circ> D.cones_map f) (D.cones (C.cod f))"
"1_5": shows "(poincare_map p1 x0, D1 o\<^sub>L d0) \<in> P1"
cInf_eq: fixes X :: "'b set"   and a :: "'b" assumes "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" shows "Inf X = a"
double_add_le_zero_iff_single_add_le_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (plus a a) zero = less_eq a zero"
nprv_exiI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "nprv F (subst \<phi> t x)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "nprv F (exi x \<phi>)"
proj_tuple_in_join_def: fixes pos :: "bool"   and bs :: "bool list"   and as :: "'a option list"   and t :: "'a option list set" shows "proj_tuple_in_join pos bs as t = join_cond pos t (proj_tuple bs as)"
elims_3: fixes x :: "'b tree"   and xa :: "'b list" assumes "\<not> braun_list x xa" obtains xs :: "'b list" where "x = \<langle>\<rangle>" and "xa = xs__" and "xs__ \<noteq> []"   | l :: "'b tree" and xb :: "'b" and r :: "'b tree" and xs :: "'b list" where "x = \<langle>l, xb, r\<rangle>" and "xa = xs" and "\<not> (xs \<noteq> [] \<and> xb = hd xs \<and> braun_list l (take_nths 1 1 xs) \<and> braun_list r (take_nths 2 1 xs))"
is_extensional: fixes f :: "'a" assumes "\<not> Right_a.arr f" shows "H\<^sub>R a f = Right_a.null"
constr_rep_1: fixes x3 :: "'d"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) psi" shows "Rep_residual (RIn x3 x2 x1) = RIn_Rep x3 x2 x1"
Pair_Collect: fixes x :: "'b"   and y :: "'c"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "((x, y) \<in> {(x, y). P x y}) = P x y"
less_b_least2_impl_zero: fixes z :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "z < b_least2 f x y" shows "f x z = 0"
subset_eq: fixes A :: "'n set"   and B :: "'n set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
A: shows "c\<^sub>0 = c\<^sub>0' \<and> c\<^sub>1 = c\<^sub>1'"
tmod3_3: shows "r = Rerr (Rabort Rtype_error)"
dim_zero_eq': fixes A :: "('a, 'b) vec set" assumes "A = {} \<or> A = {0}" shows "vec.dim A = 0"
assms_6: shows "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt c"
Sbis_trans: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and e :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>s d"   and "d \<approx>s e" shows "c \<approx>s e"
empty_P0: shows "P_of_wstate (lhd Sts) = {}"
defs_uses_disjoint: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> uses g n = {}"
SD_efficient_lottery_exists: fixes p :: "'b pmf" assumes "p \<in> lotteries_on alts" obtains q :: "'b pmf" where "q \<in> lotteries_on alts" and "Pareto (SD \<circ> R) p q" and "SD_efficient R q"
map_fst_zip_take: fixes xs :: "'c list"   and ys :: "'d list" shows "map fst (xs || ys) = take (min (length xs) (length ys)) xs"
k_empt_def: shows "k_empt \<equiv> \<lparr>aActs = MapOps.empty aOps, aTrans = MapOps.empty tOps\<rparr>"
Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf_def: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f" shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf \<equiv> Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf \<and> Minimal_Truth_Soundness_HBL1iff_Compl_Pf_Compl_NegPf_axioms var fmla Var FvarsT Fvars subst imp fls num enc bprv isTrue Pf"
PB: shows "P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n"
refl_onI: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
trees_sub_lfp: fixes S :: "'d set" shows "trees S \<subseteq> lfp (\<lambda>T. T \<union> {\<langle>\<rangle>} \<union> (\<Union>l\<in>T. \<Union>v\<in>S. \<Union>r\<in>T. {\<langle>l, v, r\<rangle>}))"