hcju/isabelleps huggingface dataset:
homeomorphism_cong: fixes X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and X' :: "'a set"   and Y' :: "'b set"   and f' :: "'a \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'a" assumes "homeomorphism X Y f g"   and "X' = X"   and "Y' = Y"   and "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"   and "\<And>y. y \<in> Y \<Longrightarrow> g' y = g y" shows "homeomorphism X' Y' f' g'"
arr_LHS: shows "Span.seq (SPN \<a>[f, g, h]) (\<Phi>.map (f \<star> g, h) \<bullet> (\<Phi>.map (f, g) \<circ> SPN h))"
okmaxsync_def: fixes i :: "nat"   and x :: "real" shows "okmaxsync i x = (\<forall>p q. correct p (max (te p i) (te q i)) \<and> correct q (max (te p i) (te q i)) \<longrightarrow> \<bar>IC p i (max (te p i) (te q i)) - IC q i (max (te p i) (te q i))\<bar> \<le> x)"
strong_inducts: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and lam :: "lam" assumes "\<And>name z. P z (Var name)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (App lam1 lam2)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z (Lam [name].lam)" shows "P z lam"
domA_def: fixes h :: "('c \<times> 'd) list" shows "domA h = fst ` set h"
Collect_case_prod: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
spr_simAbs_def: shows "spr_simAbs \<equiv> \<lambda>(cec, aec). rel_ext (\<lambda>uu_. \<exists>s s'. uu_ = \<lparr>sprFst = s, sprLst = s', sprCRel = toSet cec\<rparr> \<and> (s, s') \<in> toSet aec)"
alphaAbs_all_distinct_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_distinct_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
mult_closed: fixes G :: "('c, 'd) Group_scheme"   and a :: "'c"   and b :: "'c" assumes "Group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<cdot>\<^bsub>G\<^esub> b \<in> carrier G"
sgn_eq: fixes z :: "complex" shows "sgn z = z / cor (cmod z)"
proots_count_0_imp_empty: fixes p :: "'a poly"   and s :: "'a set" assumes "proots_count p s = 0"   and "p \<noteq> 0" shows "proots_within p s = {}"
language_ranking: fixes A :: "('d, 'c) nba"   and w :: "'d stream" assumes "finite (nodes A)"   and "w \<notin> language A" shows "ranking A w (rank A w)"
const_sum_list: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and a :: "nat" assumes "\<forall>x\<in>set xs. f x = a" shows "sum_list (map f xs) = a * length xs"
lin_def: fixes x :: "'c" shows "lin x = {a \<cdot> x |a. True}"
empty_subst_def: shows "empty_subst = (SOME s. (\<forall>f. subst s f = f) \<and> subst_lconsts s = {})"
normalize_mult: fixes a :: "'b poly"   and b :: "'b poly" shows "smult (inverse (lead_coeff (a * b))) (a * b) = smult (inverse (lead_coeff a)) a * smult (inverse (lead_coeff b)) b"
eq_fin_ind': fixes a :: "'item"   and u :: "'item list"   and u\<^sub>1 :: "'item list"   and u\<^sub>2 :: "'item list" assumes "[a] @ u =\<^sub>F u\<^sub>1 @ [a] @ u\<^sub>2"   and "a \<notin> set u\<^sub>1" shows "Ind {a} (set u\<^sub>1)"
connected_space_connected_component_set: fixes X :: "'b topology" shows "connected_space X = (\<forall>x\<in>topspace X. connected_component_of_set X x = topspace X)"
card_le_dim_spanning: fixes B :: "'a poly set"   and V :: "'a poly set" assumes "B \<subseteq> V"   and "V \<subseteq> poly_vs.span B"   and "finite B"   and "card B \<le> poly_vs.dim V" shows "poly_vs.independent B"
prefix_conc_snd: fixes u :: "'b list"   and n :: "nat"   and v :: "nat \<Rightarrow> 'b" assumes "length u \<le> n" shows "prefix n (u \<frown> v) = u @ prefix (n - length u) v"
upto_enum_step_red: fixes sz :: "nat"   and us :: "nat" assumes "sz < LENGTH('b)"   and "us \<le> sz" shows "[0 , 2 ^ us .e. 2 ^ sz - 1] = map (\<lambda>x. word_of_nat x * 2 ^ us) [0..<2 ^ (sz - us)]"
disc_iff: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'c llist"   and a :: "'b" shows "\<not> lnull (lscan f w a)"
maintain_invar2_ctd_2: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and d :: "nat" assumes "Prim2.prim_invar2_ctd w g r Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" shows "Prim2.T_measure2 g r (Prim2.Q' w g r Q \<pi> u) (Prim2.\<pi>' w g r Q \<pi> u) < Prim2.T_measure2 g r Q \<pi>"
unify_list_distinct: fixes E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and B :: "('a \<times> ('b, 'a) Term.term) list"   and U :: "('a \<times> ('b, 'a) Term.term) list" assumes "unify E B = Some U"   and "distinct (map fst B)"   and "(\<Union>x\<in>set E. fv (fst x) \<union> fv (snd x)) \<inter> set (map fst B) = {}" shows "distinct (map fst U)"
span_mul: fixes x :: "'a"   and S :: "'a set"   and c :: "real" assumes "x \<in> span S" shows "c *\<^sub>R x \<in> span S"
infinite_subset: fixes A :: "'d set"   and B :: "'d set" assumes "infinite A"   and "A \<subseteq> B" shows "infinite B"
openI: fixes S :: "'a set" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S" shows "open S"
simps_3: fixes la :: "'a list"   and acc2 :: "'a list list" shows "merge_list (la # acc2) [] = merge_list [] (la # acc2)"
div_subset_cbox: fixes K :: "'a set" assumes "K \<in> \<D>" shows "K \<subseteq> cbox a b"
ring_irreducibleE_5: fixes r :: "'a"   and a :: "'a"   and b :: "'a" assumes "r \<in> carrier R"   and "ring_irreducible r"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "r = a \<otimes> b" shows "a \<in> Units R \<or> b \<in> Units R"
game2a_bad_bound: shows "spmf game2_a_bad True \<le> real q1 / 2 ^ len"
lookup2: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and b :: "coname"   and a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list" assumes "\<not> x \<sharp> idn \<Gamma> b" shows "lookup x a (idn \<Gamma> b) \<theta>_c = lookupb x a \<theta>_c b (Ax x b)"
protFw_ntsem: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "protFw \<sigma>" shows "ntsem \<sigma> a = a"
ladder_n_last_is_length: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list" assumes "is_ladder D L" shows "ladder_n L (length L - 1) = length D"
map_fst: fixes x :: "'b"   and xs :: "'a list" shows "map fst (map (\<lambda>pc. (pc, x)) xs) = xs"
pickEff_def: fixes r :: "'rule"   and s :: "'state" shows "pickEff r s \<equiv> if enabled r s then SOME sl. effG eff' r s sl else the None"
mgu_gives_wellformed_MGU: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<theta>" shows "wf\<^sub>M\<^sub>G\<^sub>U \<theta> s t"
share_all_until_volatile_write_update_sb: fixes sb' :: "'b memref list"   and sb :: "'b memref list"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<S> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e" assumes "\<And>S. share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "share_all_until_volatile_write ts \<S> = share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')]) \<S>"
a: shows "Sup_lattice.Infs Inf (\<lambda>x y. y \<le> x) = Supi"
augment_flow_presv_con: fixes u :: "nat" assumes "u \<in> V - {s, t}" shows "sum (augment f') (outgoing u) = sum (augment f') (incoming u)"
carrier_append_cols: fixes A :: "'b mat"   and nr :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat" assumes "A \<in> carrier_mat nr nc1"   and "B \<in> carrier_mat nr nc2" shows "A @\<^sub>c B \<in> carrier_mat nr (nc1 + nc2)"
Cod_cod: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Cod (cod f) = Cod f"
padic_add_comm0: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "prime p" shows "padic_add p x y = padic_add p y x"
S_trans: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" assumes "S s t"   and "S t u" shows "S s u"
Compl1_set_less_aux_conv_Compl_set_less_aux: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_aux less_eq (- set xs) (set ys) = proper_intrvl.Compl_set_less_aux less proper_interval None xs ys"
Max_ge: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "x \<in> A" shows "x \<le>s signed.Max A"
neg_one_even_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "even n" shows "power.power one times (uminus one) n = one"
oprod_Linear_order: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Linear_order r"   and "Linear_order r'" shows "Linear_order (r *o r')"
scast_ucast_id: fixes x :: "'b word" shows "SCAST('b signed \<rightarrow> 'b) (UCAST('b \<rightarrow> 'b signed) x) = x"
assms_1: shows "has_density M (stock_measure (PRODUCT INTEG INTEG)) \<delta>"
assms_3: shows "p = monom P \<one> (Suc n) \<ominus>\<^bsub>P\<^esub> monom P (b [^] Suc n) 0"
equal_foldr: assumes "(case (r, m) of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b)) = foldr (div_rat_poly_step (of_int_poly q)) (map rat_of_int (coeffs p)) (case (0, 0) of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b))" shows "Some (r, m) = foldr (div_int_poly_step q) (coeffs p) (Some (0, 0))"
negate_inject: fixes x :: "lit"   and y :: "lit" shows "(negate x = negate y) = (x = y)"
DBFair: shows "\<turnstile> \<box>[DBNext]_(inp, mid, out, q1, q2) \<and> WF(DBPass)_(inp, mid, out, q1, q2) \<and> WF(DBDeq)_(inp, mid, out, q1, q2) \<longrightarrow> WF(Deq inp qc out)_(inp, qc, out)"
simps_22: fixes vc :: "'b"   and v :: "char list"   and va :: "aexp"   and vb :: "'b" shows "(SKIP {vc} \<sqsubseteq> v ::= va {vb}) = False"
imp_igSubstInpIPresIGWlsInp: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSubstIPresIGWls MOD" shows "igSubstInpIPresIGWlsInp MOD"
simps_1: fixes uu :: "'a \<Rightarrow> bool" shows "find uu [] = None"
Delta_On_CTRL: shows "Delta On_CTRL = On_CTRL_Delta"
bind_map_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf"   and g :: "'d \<Rightarrow> 'c spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
vec_index_vCons_Suc: fixes a :: "'a"   and v :: "'a vec"   and n :: "nat" shows "vCons a v $ Suc n = v $ n"
tauSeqSubst: fixes \<sigma> :: "(name list \<times> 'a list) list"   and P :: "('a, 'b, 'c) psi" assumes "wellFormedSubst \<sigma>" shows "\<tau>.P[<\<sigma>>] = \<tau>.P[<\<sigma>>]"
weak_formula_eqvt: fixes x :: "('idx, 'pred, 'act) formula"   and p :: "perm" assumes "weak_formula x" shows "weak_formula (p \<bullet> x)"
sat_def: fixes \<AA> :: "interp"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" shows "WS1S.sat \<AA> \<phi> \<equiv> WS1S.satisfies_gen (\<lambda>k P n. restrict k P \<and> True) \<AA> \<phi> \<and> (\<forall>k\<in>UNIV. \<forall>x\<in>FV \<phi> k. restrict k (x\<^bsup>\<AA>\<^esup>k))"
bal_nil: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "(c \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>[]\<^esub> c') = (c = c')"
rep_eq: fixes x :: "('e, 'd) fmap"   and xa :: "('c, 'e) fmap" shows "fmlookup (x \<circ>\<^sub>f xa) = fmlookup x \<circ>\<^sub>m fmlookup xa"
LeadsETo_UN: fixes I :: "'d set"   and F :: "'e program"   and A :: "'d \<Rightarrow> 'e set"   and CC :: "'e set set"   and B :: "'e set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i LeadsTo[CC] B" shows "F \<in> \<Union> (A ` I) LeadsTo[CC] B"
p1_def: shows "p1 = quadratic_part_1 var a b d (Less p)"
multr_strict: fixes x :: "'a" shows "\<bottom> \<cdot> x = \<bottom>"
tautology: fixes p :: "('b, 'a) fm"   and E :: "nat \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c list \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c list \<Rightarrow> bool" assumes "tautology p" shows "\<lbrakk>E, F, G\<rbrakk> p"
l_get_dom_component_attach_shadow_root\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_attach_shadow_root\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM_DOM_Components.get_dom_component Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root get_disconnected_nodes Shadow_DOM_DOM_Components.is_strongly_dom_component_safe Shadow_DOM_DOM_Components.is_weakly_dom_component_safe get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs"
assms_2: shows "pseudominimal_in (P .\<^bold>\<le> y) w"
insort_is_Cons: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c" assumes "class.linorder less_eq less"   and "\<forall>x\<in>set xs. less_eq (f a) (f x)" shows "linorder.insort_key less_eq f a xs = a # xs"
step: fixes a :: "'b"   and b :: "'b" assumes "(a, b) \<in> r" shows "(f a, f b) \<in> r'\<^sup>+"
mt_sym: fixes a :: "'b \<Rightarrow> 'c option"   and b :: "'b \<Rightarrow> 'd option" assumes "dom a \<inter> dom b = {}" shows "dom b \<inter> dom a = {}"
normalize_power_normalize: fixes x :: "'a"   and n :: "nat" shows "normalize (normalize x ^ n) = normalize (x ^ n)"
zip_rev: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys" shows "zip (rev xs) (rev ys) = rev (zip xs ys)"
unfold: shows "unwrapS = (\<Lambda> f. f\<cdot>0 && unwrapS\<cdot>(f oo (\<Lambda> x. 1 + x)))"
exhaust_sel: fixes prod :: "'g \<times> 'h" obtains "prod = (fst prod, snd prod)"
lossless_S2_12: fixes c :: "bool"   and out2 :: "bool" shows "lossless_spmf (S2_OT12 c out2)"
gamma_Step_subcomm: fixes f1 :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and f2 :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> 'av) option \<Rightarrow> (char list \<Rightarrow> 'av) option"   and g1 :: "bexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and g2 :: "bexp \<Rightarrow> (char list \<Rightarrow> 'av) option \<Rightarrow> (char list \<Rightarrow> 'av) option"   and S :: "(char list \<Rightarrow> 'av) option"   and C :: "(char list \<Rightarrow> 'av) option acom" assumes "\<And>x e S. f1 x e (\<gamma>\<^sub>o S) \<subseteq> \<gamma>\<^sub>o (f2 x e S)"   and "\<And>b S. g1 b (\<gamma>\<^sub>o S) \<subseteq> \<gamma>\<^sub>o (g2 b S)" shows "Step f1 g1 (\<gamma>\<^sub>o S) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c (Step f2 g2 S C)"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "ide u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> r \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> g' \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<theta>) \<cdot> \<a>[r, f', w] \<cdot> (\<rho>' \<star> w)) \<cdot> \<nu> = \<omega>"
not_add_less_right: fixes y :: "V"   and x :: "V" assumes "Ord y" shows "\<not> x + y \<subset>\<^sub>\<circ> x"
transfer_operations1: fixes x :: "'c"   and y :: "'c"   and xi :: "real list"   and yi :: "real list" assumes "SIDE_PRECOND (x \<le> y)"   and "(xi, x) \<in> lv_rel"   and "(yi, y) \<in> lv_rel" shows "(appr_of_ivl ops xi yi, op_atLeastAtMost_appr $ x $ y) \<in> appr_rel"
assms_5: shows "(d, t) \<rightarrow>*t t'"
coset_def: fixes xs :: "'g list" shows "List.coset xs = - set xs"
sw_def: fixes ys :: "'b"   and y1 :: "'a"   and y2 :: "'a"   and xs :: "'b"   and x :: "'a" shows "x @xs[y1 \<and> y2]_ys \<equiv> if ys = xs then if x = y1 then y2 else if x = y2 then y1 else x else x"
simps_3: fixes ts' :: "t list"   and ts :: "ct list" shows "produce (Type ts') (TopType ts) = TopType ts"
refine: fixes G :: "('a, 'b) b_graph_rec_scheme" shows "blue_dfs G \<equiv> NDFS_SI_Statistics.start_nres \<bind> (\<lambda>_. blue_dfs_fe G \<bind> (\<lambda>(uu_, uu_, cyc). NDFS_SI_Statistics.stop_nres \<bind> (\<lambda>_. RETURN (extract_res cyc))))"
de_Morgan_conj: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "compl (conj x y) = disj (compl x) (compl y)"
rep_eq: fixes x :: "'a vector"   and xa :: "'a vector" shows "Rep_vector (x \<sqinter> xa) = Rep_vector x \<sqinter> Rep_vector xa"
imageI: fixes x :: "'b::type"   and A :: "'b::type set"   and f :: "'b::type \<Rightarrow> 'c::type" assumes "(x::'b::type) \<in> (A::'b::type set)" shows "(f::'b::type \<Rightarrow> 'c::type) (x::'b::type) \<in> f ` (A::'b::type set)"
assms_2: shows "norm2 \<equiv> vec_of_basis_enum a \<bullet>c vec_of_basis_enum a"
natural_extension_proj_def: shows "natural_extension_proj = (\<lambda>x. x 0)"
sum_norm_le: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (comm_monoid_add.sum plus zero f S) \<le> sum g S"
funpow_simps_right_1: fixes f :: "'b \<Rightarrow> 'b" shows "f^0 = id"
swap_simps: fixes a :: "'b"   and b :: "'a" shows "swap (a, b) = (b, a)"
real_greaterThanLessThan_inter: fixes N :: "ereal"   and M :: "ereal" shows "real_of_ereal ` {N<..<M} = real_of_ereal ` {- \<infinity><..<M} \<inter> real_of_ereal ` {N<..<\<infinity>}"
signs_smash_def: fixes s1 :: "'b list list"   and s2 :: "'b list list" shows "signs_smash s1 s2 = concat (map (\<lambda>l1. map ((@) l1) s2) s1)"
norm_mult_vec: fixes x :: "'b" shows "norm (x \<bullet> x) = norm x * norm x"
measure_zero_union: fixes s :: "'a set"   and p :: "'a spmf"   and t :: "'a set" assumes "s \<in> sets (measure_spmf p)"   and "t \<in> sets (measure_spmf p)"   and "Sigma_Algebra.measure (measure_spmf p) t = 0" shows "Sigma_Algebra.measure (measure_spmf p) (s \<union> t) = Sigma_Algebra.measure (measure_spmf p) s"
sup_set_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B = Collect ((\<lambda>x. x \<in> A) \<squnion> (\<lambda>x. x \<in> B))"
ground_resolution_with_selection_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset" shows "ground_resolution_with_selection S \<equiv> selection S"
member_pred_def: fixes P :: "'a Predicate.pred"   and x :: "'a" shows "local.member_pred P x = (\<exists>y. eq x y \<and> pred.eval P y)"
peak_greek_less: fixes as :: "(accent \<times> 'a) list"   and r :: "('a \<times> 'a) set"   and a :: "'a"   and b' :: "(accent \<times> 'a) list"   and b :: "'a"   and cs :: "(accent \<times> 'a) list"   and a' :: "(accent \<times> 'a) list"   and bs :: "(accent \<times> 'a) list" assumes "snd ` set as \<subseteq> under r a"   and "b' \<in> {[(Grave, b)], []}"   and "snd ` set cs \<subseteq> under r a \<union> under r b"   and "a' \<in> {[(Acute, a)], []}"   and "snd ` set bs \<subseteq> under r b" shows "(as @ b' @ cs @ a' @ bs, [(Acute, a), (Grave, b)]) \<in> greek_less r"
inverse_mod: fixes x :: "int"   and m :: "int" assumes "comm_monoid_mult_class.coprime x m"   and "1 < m" shows "inverse_mod x m * x mod m = 1"
in_indets_subst_ppE: fixes x :: "'b"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat" assumes "x \<in> indets (subst_pp f t)" obtains y :: "'d" where "y \<in> keys t" and "x \<in> indets (f y)"
assms_4: fixes dir :: "('i, 'a) Direction"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and l\<^sub>i :: "'a" assumes "dir = (if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative)"   and "\<not> \<U> s"   and "min_lvar_not_in_bounds s = Some x\<^sub>i"   and "\<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i)"   and "min_rvar_incdec dir s x\<^sub>i = Inr x\<^sub>j"   and "l\<^sub>i = the (LB dir s x\<^sub>i)"   and "check' dir x\<^sub>i s = pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s" shows "P (check (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s))"
not_equals_and_not_equals_not_in: fixes y :: "'a"   and x :: "'a"   and x' :: "'a"   and phi :: "bool" shows "(y \<noteq> x \<and> y \<noteq> x' \<and> phi) = (y \<notin> {x, x'} \<and> phi)"
dagger_unfoldl_eq: fixes x :: "'a" shows "1' + x\<^sup>+ = x\<^sup>\<star>"
assms_3: shows "\<forall>\<^sub>F x in net. f x $ i \<le> b"
Carmichael_mult_coprime: fixes m :: "nat"   and n :: "nat" assumes "coprime m n" shows "Carmichael (m * n) = lcm (Carmichael m) (Carmichael n)"
list_orderI: fixes ops :: "('c \<times> 'c option) list"   and xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and y :: "'c"   and zs :: "'c list" assumes "interp_ins ops = xs @ [x] @ ys @ [y] @ zs" shows "Insert_Spec.list_order ops x y"
range_enumerate: fixes S :: "nat set" assumes "infinite S" shows "range (enum S) = S"
unlabel_Cons_1: fixes l :: "'e"   and a :: "'d"   and A :: "('e \<times> 'd) list" shows "unlabel ((l, a) # A) = a # unlabel A"
sorted_wrt_mono_rel: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
analz_initState: fixes C :: "agent" shows "analz (initState C) = initState C"
HSem_fresh_cong: fixes S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value"   and \<rho>' :: "Vars.var \<Rightarrow> Value" assumes "\<rho> f|` (S \<union> fv \<Gamma> - domA \<Gamma>) = \<rho>' f|` (S \<union> fv \<Gamma> - domA \<Gamma>)" shows "(\<lbrace>\<Gamma>\<rbrace>\<rho>) f|` S = (\<lbrace>\<Gamma>\<rbrace>\<rho>') f|` S"