hcju/isabelleps huggingface dataset:
contract_extensionality: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "K \<div> \<phi> = K \<div> \<psi>"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list"   and b :: "'c list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'b" and x2 :: "'b list" and y1 :: "'c" and y2 :: "'c list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
special_coset_eq_imp_eq_gensets: fixes T1 :: "'w set"   and T2 :: "'w set"   and w1 :: "'w"   and w2 :: "'w" assumes "T1 \<in> Pow S"   and "T2 \<in> Pow S"   and "w1 +o \<langle>T1\<rangle> = w2 +o \<langle>T2\<rangle>" shows "T1 = T2"
set_rw: shows "{\<chi>ia. Gram_Schmidt_upt_k A k $ ia $ i |i. i < mod_type_class.from_nat (Suc k)} = {\<chi>ia. Gram_Schmidt_upt_k A k $ ia $ i |i. mod_type_class.to_nat i \<le> k}"
stop_cxt: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "c = Stop" shows "\<Gamma> =:\<^sub>P \<Gamma>' \<and> \<S>' = \<S> \<and> P \<turnstile> P' \<and> (\<forall>mds. tyenv_wellformed mds \<Gamma> \<S> P \<longrightarrow> tyenv_wellformed mds \<Gamma>' \<S> P')"
simps_1: fixes k :: "nat"   and T :: "POPLmark.type" shows "[][k \<mapsto>\<^sub>\<tau> T]\<^sub>e = []"
restrict_apply: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'c" shows "restrict f A x = (if x \<in> A then f x else undefined)"
prod_fun_imageE: fixes c :: "'b \<times> 'c"   and f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and R :: "('d \<times> 'e) set" assumes "c \<in> map_prod f g ` R" obtains x :: "'d" and y :: "'e" where "c = (f x, g y)" and "(x, y) \<in> R"
up_set_idem: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder_with_dual dual less_eq less" shows "ord.upset_set less_eq \<circ> ord.upset_set less_eq = ord.upset_set less_eq"
restrict_relation_def: fixes A :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "restrict_relation A R = (\<lambda>x y. x \<in> A \<and> y \<in> A \<and> R x y)"
bre'_cond_def: fixes Qi :: "'d set" shows "bre'_cond Qi \<equiv> {(Q, W, rcm). W \<noteq> {} \<and> Qi \<inter> Q = {}}"
that: fixes et :: "(char list \<Rightarrow> val option) edge_kind" assumes "prog \<turnstile> (_ l _) -et\<rightarrow> (_ l' _)"   and "transfer et s = s'"   and "pred et s" shows "thesis"
add_mono_thms_linordered_field_4: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i \<le> j \<and> k < l" shows "i + k < j + l"
eng: fixes h1 :: "'e"   and G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and h2 :: "'e" assumes "h1 \<in> generate G H"   and "h2 \<in> generate G H" shows "h1 \<otimes>\<^bsub>G\<^esub> h2 \<in> generate G H"
gt_eq_assm: shows "dim_row (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T \<le> dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T"
simps_3: fixes A :: "dbfm"   and B :: "dbfm" shows "quot_dbfm (DBDisj A B) = Q_Disj (quot_dbfm A) (quot_dbfm B)"
set_update_by_pair_subset: fixes kv :: "'a \<times> 'c"   and xs :: "('a \<times> 'c) list" shows "set (update_by_pair kv xs) \<subseteq> insert kv (set xs)"
bintrunc_pos_eq: fixes x :: "int"   and n :: "nat" assumes "0 \<le> x" shows "(take_bit n x = x) = (x < 2 ^ n)"
fag_gen_carrier: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "inv_ipp z i f A"   and "commute_bpp f (aug_pm_set z i A)"   and "zeroA z i f A z" shows "carrier (fag_gen_by A f i z) = addition_set f (aug_pm_set z i A)"
split_sorted_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and xs :: "('b btree \<times> 'b) list"   and p :: "'b"   and ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list" assumes "split split"   and "split xs p = (ls, (sub, sep) # rs)"   and "sorted_less (separators xs)" shows "p \<le> sep"
simps: fixes k :: "name"   and s :: "name"   and y :: "tm" assumes "atom k \<sharp> (s, y)"   and "atom s \<sharp> y" shows "FormP y = SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) y))"
nth_rotate: fixes n :: "nat"   and xs :: "'a list"   and m :: "nat" assumes "n < |xs|" shows "rotate m xs ! n = xs ! ((m + n) mod |xs|)"
subst_fml: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "Fadmit \<sigma> \<phi>"   and "fsafe \<phi>"   and "ssafe \<sigma>" shows "(\<nu> \<in> fml_sem I (Fsubst \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (local.adjoint I \<sigma> \<nu>) \<phi>)"
type: shows "type_definition Rep Abs {0..<int CARD('a)}"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<rho>.map a = \<r>[a]"
invertible: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> G" shows "symmetric.sub.invertible u"
lappend_code_2: fixes xa :: "'b"   and x :: "'b llist"   and ys :: "'b llist" shows "lappend (LCons xa x) ys = LCons xa (lappend x ys)"
eq_Abs_ExpList: fixes z :: "exp list" obtains Us :: "freeExp list" where "z = Abs_ExpList Us"
diamond_confluent: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "diamond R" shows "confluent R"
F: fixes x :: "complex" assumes "x \<in> K" shows "open (Z x) \<and> x \<in> Z x \<and> (\<forall>h z'. h \<in> Y \<and> z' \<in> Z x \<longrightarrow> cmod (h z') \<le> F x)"
simps: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "(a \<in> SAT \<Gamma> f) = ((\<exists>x. a = x \<and> x \<in> A \<Gamma>) \<or> (\<exists>x. a = x \<and> weight \<Gamma> x \<le> d_IN f x))"
reflexive_complement_irreflexive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.reflexive x" shows "matrix_stone_relation_algebra.irreflexive (\<ominus> x)"
elims_2: fixes x :: "real_max"   and xa :: "real_max" assumes "x \<le> xa" obtains x :: "real" and y :: "real" where "xa__ = Rmax x" and "xa = Rmax y" and "x \<le> y"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "classDef" assumes "a1 \<turnstile> a2 OK" obtains CDef :: "classDef" and C :: "nat" and D :: "nat" and KDef :: "constructorDef" and M :: "methodDef list" and Dg :: "varDef list" and Cf :: "varDef list" and CT :: "nat \<Rightarrow> classDef option" where "a1 = CT" and "a2 = CDef" and "cName CDef = C" and "cSuper CDef = D" and "cConstructor CDef = KDef" and "cMethods CDef = M" and "kName KDef = C" and "kParams KDef = Dg @ Cf" and "kSuper KDef = varDefs_names Dg" and "kInits KDef = varDefs_names Cf" and "fields(CT,D) = Dg" and "CT \<turnstile>+ M OK IN C"
pochhammer_of_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "nat"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero (semiring_1.of_nat one plus zero x) n = semiring_1.of_nat one plus zero (pochhammer x n)"
hom_zero: fixes a :: "'c" shows "poly 0 a = (0::'c)"
coset_mult_inv2: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M #> x = M #> y"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M #> x \<otimes> inv y = M"
hyps: fixes y :: "nat" assumes "y < m" shows "(f ^^ (y mod n)) x = (f ^^ y) x"
Union_connected_component: fixes S :: "'b set" shows "\<Union> (connected_component_set S ` S) = S"
solve_ratfps: fixes f :: "complex ratfps"   and sol :: "complex poly \<times> (complex poly \<times> complex) list" assumes "solve_ratfps f = Some sol" shows "Abs_fps (interp_ratfps_solution sol) = fps_of_ratfps f"
emeasure_lborel_box_finite: fixes a :: "'c"   and b :: "'c" shows "emeasure lborel (box a b) < \<infinity>"
RQRM_2: shows "rqrm_prop \<delta> rqrm"
simps_169: fixes x9 :: "Term.var"   and x71 :: "binop"   and x72 :: "expr"   and x73 :: "expr" shows "Acc x9 \<noteq> BinOp x71 x72 x73"
PreSheafCat: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "Category (CatExp (Op C) SET)"
Cod_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Cod (Cod t) = Cod t"
pre_splitFace_oldF: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and nvs :: "nat list" assumes "pre_splitFace g ram1 ram2 oldF nvs" shows "oldF \<in> \<F> g"
vreal_uminus_uminus: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "x = -\<^sub>\<real> (-\<^sub>\<real> x)"
upper_3_tendsto: fixes x :: "real_alg_3" shows "upper_3 x \<longlonglongrightarrow> real_of_3 x"
Cons_eq_append_conv: fixes x :: "'f"   and xs :: "'f list"   and ys :: "'f list"   and zs :: "'f list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
assms_1: shows "LeftDerivationLadder \<alpha> D L \<gamma>"
LOr1: fixes b :: "coname"   and a :: "coname"   and M :: "trm"   and N1 :: "trm"   and N2 :: "trm"   and y :: "name"   and x1 :: "name"   and x2 :: "name" assumes "b \<sharp> ([a].M, N1, N2, a)"   and "y \<sharp> ([x1].N1, [x2].N2, M, x1, x2)"   and "x1 \<sharp> (M, N2)"   and "x2 \<sharp> (M, N1)"   and "a \<sharp> (N1, N2)"   and "x1 \<noteq> x2" shows "Cut <b>.OrR1 <a>.M b y.OrL x1.N1 x2.N2 y \<longrightarrow>\<^sub>l Cut <a>.M x1.N1"
simps_53: fixes x12 :: "qtname"   and x2 :: "expr" shows "Init x12 \<noteq> Expr x2"
d: shows "[d2 = q + q + e2 * b + q * b - s2 - (q - r) + (q - r)] (mod q)"
pfun_minus_plus_commute: fixes g :: "('a, 'b) pfun"   and h :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun" assumes "pdom g \<inter> pdom h = {}" shows "f - g + h = f + h - g"
add_divide_eq_if_simps_3: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and z :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus (uminus (divide a z)) b = (if z = zero then b else divide (plus (uminus a) (times b z)) z)"
simps_72: fixes x8 :: "'a OCL_Types.type"   and x6 :: "'a OCL_Types.type" shows "Sequence x8 \<noteq> OrderedSet x6"
lin_redI: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f \<in> F"   and "red_single p q f (0::'a)" shows "lin_red F p q"
id_subst: fixes v :: "('a, 'b) uexpr" shows "id \<dagger> v = v"
AF_subset_N: fixes k :: "nat"   and i :: "nat" shows "AF k i \<subseteq> N"
decseq_ge: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and n :: "nat" assumes "decseq X"   and "X \<longlonglongrightarrow> L" shows "L \<le> X n"
l_identity: fixes \<alpha> :: "'a"   and \<beta> :: "'a"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[[\<alpha> \<^bold>= \<beta> \<^bold>\<rightarrow> (\<phi> \<alpha> \<^bold>\<rightarrow> \<phi> \<beta>)]]"
is_disj_big_or_if: fixes fs :: "'b formula list" assumes "\<forall>f\<in>set fs. is_lit_plus f" shows "is_disj \<^bold>\<Or>fs"
simps_4: fixes uv :: "Unboxed.type"   and x :: "'d" shows "generalize_instr (IStoreUbx uv x) = IStore x"
zero_adds: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus zero a"
put_option_lgeom: fixes T :: "nat"   and K :: "real"   and w :: "bool stream" shows "put_option T K w = max (K - lgeom_proc u d init (stake T w)) 0"
hnorm_eq_zero: fixes x :: "'a star" shows "(hnorm x = 0) = (x = 0)"
production_subseteq_INTERP: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.production S N C \<subseteq> ground_resolution_with_selection.INTERP S N"
c2: shows "c2 \<le> (\<Sum>j = 0..<j - 1. 2 * n + 4 + 3 * Suc j)"
Rep_hierauto: fixes x :: "('b, 'c, 'a) hierauto" shows "Rep_hierauto x \<in> hierauto"
d_1: shows "poincare_distance u v = poincare_distance u' v'"
is_const_sum_mod_2: assumes "is_const" shows "(f 0 + f 1) mod 2 = 0"
degree_ge_iff: fixes k :: "nat"   and p :: "'a mpoly"   and x :: "nat" assumes "0 < k" shows "(k \<le> MPoly_Type.degree p x) = (\<exists>m\<in>monomials p. k \<le> lookup m x)"
hom_sum_list: fixes hom :: "'b \<Rightarrow> 'c"   and xs :: "'b list" assumes "monoid_add_hom hom" shows "hom (sum_list xs) = sum_list (map hom xs)"
rbt_greater_simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "ord.rbt_greater less k Empty = True"
Heine_Borel_imp_Bolzano_Weierstrass: fixes s :: "'c set"   and t :: "'c set" assumes "compact s"   and "infinite t"   and "t \<subseteq> s" shows "\<exists>x\<in>s. x islimpt t"
simps: fixes final :: "'a \<Rightarrow> bool"   and status :: "status"   and x :: "'a" shows "final_thread.init_fin_final final (status, x) = (status = Finished \<and> final x)"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}\<langle>fg\<rangle> p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
uncarr: shows "un \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
sp_append_one: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es :: "'c edge list"   and e :: "'c edge"   and v2 :: "'c" shows "Graph.subpath g v1 (es @ [e]) v2 = (Graph.subpath g v1 es (src e) \<and> e \<in> edges g \<and> tgt e = v2)"
cases: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" assumes "a1 cd-a2\<rightarrow>\<^sub>d* a3" obtains n :: "'node SDG_node" where "a1 = na__" and "a2 = []" and "a3 = na__" and "valid_SDG_node na__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a1 = n" and "a2 = ns @ [n'']" and "a3 = n'" and "n cd-ns\<rightarrow>\<^sub>d* n''" and "n'' \<longrightarrow>\<^bsub>cd\<^esub> n'"
get_min_rest_get_min_same_root: fixes ts :: "'b tree list"   and t' :: "'b tree"   and ts' :: "'b tree list" assumes "ts \<noteq> []"   and "get_min_rest ts = (t', ts')" shows "tree.root t' = get_min ts"
poly_eval_times: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a (p * q) = poly_eval a p * poly_eval a q"
hextI: fixes h :: "heap"   and h' :: "heap" assumes "\<forall>a C fs. h a = \<lfloor>(C, fs)\<rfloor> \<longrightarrow> (\<exists>fs'. h' a = \<lfloor>(C, fs')\<rfloor>)" shows "h \<unlhd> h'"
simps: fixes ts :: "'b tree list" shows "list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> \<langle>\<rangle>) ts in if us = [] then [] else map value us @ list_fast_rec (map left us @ map right us))"
ca_lrE: fixes d :: "ValD"   and P :: "db" assumes "d \<triangleleft> P" obtains "d = \<bottom>" and "closed P"   | "d = ValTT" and "closed P" and "P \<Down> DBtt"   | "d = ValFF" and "closed P" and "P \<Down> DBff"   | n :: "nat" where "d = ValN\<cdot>n" and "closed P" and "P \<Down> DBNum n"   | f :: "ValD \<rightarrow> ValD" and M :: "db" where "d = ValF\<cdot>f__" and "closed P" and "P \<Down> DBAbsN M__" and "\<And>x X. x \<triangleleft> X \<Longrightarrow> f__\<cdot>x \<triangleleft> M__<X/0>"   | f :: "ValD \<rightarrow> ValD" and M :: "db" where "d = ValF\<cdot>f" and "f\<cdot>\<bottom> = \<bottom>" and "closed P" and "P \<Down> DBAbsV M" and "\<And>x X V. \<lbrakk>x \<triangleleft> X; X \<Down> V\<rbrakk> \<Longrightarrow> f\<cdot>x \<triangleleft> M<V/0>"
gcs_def: fixes s :: "'a"   and t :: "'a" shows "gcs s t = s + t - lcs s t"
mset_subset_eq_add_right: fixes B :: "'a multiset"   and A :: "'a multiset" shows "B \<subseteq># A + B"
cmp_pi_i: shows "cmp (heap (parent l i)) (heap i)"
confluent_unique_normal_form: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "confluent R"   and "R\<^sup>*\<^sup>* a b"   and "R\<^sup>*\<^sup>* a c"   and "\<not> Domainp R b"   and "\<not> Domainp R c" shows "b = c"
tauLawPsiCong2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> \<tau>.P \<doteq> \<tau>.P"
cnvalid_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and n :: "nat"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
openin_open: fixes U :: "'b set"   and S :: "'b set" shows "openin (top_of_set U) S = (\<exists>T. open T \<and> S = U \<inter> T)"
finite_supp: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.fs permute" shows "finite (pt.supp permute x)"
Derives1_take: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "take i a = take i b"
term_pp_rel_refl: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a" assumes "reflp r" shows "term_pp_rel r a a"
con_comp_failures: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "failures (P \<parallel> Q <p, q>) = con_comp_failures P Q p q"
WHILET_refine': fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R :: "('d \<times> 'c) set"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE\<^sub>T b f x \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I'\<^esup> b' f' x')"
isDiagonal: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and a :: "'d"   and b :: "'c" assumes "isDiagonal A"   and "mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b" shows "A $h a $h b = (0::'b)"
minus_insert_fset: fixes x :: "'b"   and xs :: "'b fset"   and ys :: "'b fset" shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
lossless_init: fixes h :: "'a \<times> 'a"   and w :: "nat" shows "lossless_spmf (init h w)"
ns: shows "non_speculative P (w_values P (w_values P (\<lambda>_. {}) (map snd (E @ concat (map (\<lambda>(t, ta). map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))) (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas'))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))"
init_item_origin: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_origin (init_item r k) = k"
advantage1_def: fixes \<A> :: "('x \<times> 'hash, 'x, 'hash) gpv" shows "advantage1 \<A> = spmf (game1 \<A>) True"
less_diff_eq: 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 c :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (minus c b) = less (plus a b) c"
simps_2: fixes bbs :: "bool list"   and bs :: "int list" shows "Ifm bbs bs F = False"
do_con_check_build_conv: fixes env :: "'a sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "'b list"   and vs :: "v list" assumes "do_con_check (c env) cn (length es)" shows "build_conv (c env) cn vs \<noteq> None"
method_Nat_add: shows "method Nat add = Some \<lparr>par = Class Nat, res = Class Nat, lcl = [], bdy = If (LAcc This..pred) Res :== {Nat}LAcc This..pred..add({Nat}LAcc Par..suc(<>)) Else Res :== LAcc Par\<rparr>"
weakCongI: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> \<sigma>. wellFormedSubst \<sigma> \<Longrightarrow> \<Psi> \<rhd> P[<\<sigma>>] \<doteq> Q[<\<sigma>>]" shows "P \<doteq>\<^sub>c Q"
abs_eq: fixes x :: "nat" shows "Aeta_expand (to_Arity x) = eta_expand x"
complete_union: fixes CC :: "'b set set"   and CC' :: "'b set set"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "(CC \<union> CC')-complete A r = (CC-complete A r \<and> CC'-complete A r)"
cl10_d_var3: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d (x \<inter> NC) = 1\<^sub>\<sigma> \<inter> (x \<cdot> Multirelations.U)"
cases: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "CardinalityArityTransformation Aexp Aheap"   and "CardinalityArityTransformation.consistent Aexp a1 a2" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and a :: "Arity" and as :: "Arity list" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "a1 = (ae, a, as)" and "a2 = (\<Gamma>, e, S)" and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, e, S)" and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0"