hcju/isabelleps huggingface dataset:
asBinp_pickBinp: fixes binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" assumes "goodBinp binp" shows "asBinp (pickBinp binp) = binp"
subst_list2_eq_map_substs: fixes l :: "exp list" shows "\<forall>\<sigma>. subst_list2 \<sigma> l = map (substs \<sigma>) l"
exec_call_0': fixes \<sigma> :: "'a control_state_scheme"   and M :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and M' :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> call_0\<^sub>C M ;- M') = (\<sigma> \<Turnstile> M')"
ext: fixes g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.comp g f \<noteq> Cop_S.null" shows "Cop_S.seq g f"
simps_12: fixes p :: "nat"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p floatarith.Pi vs = Some (ivl_err (real_interval (pi_float_interval p)))"
set_sel_2: fixes a :: "('f, 'g, 'h) Resumption.resumption"   and xa :: "('f, 'g, 'h) Resumption.resumption"   and xb :: "'f" assumes "\<not> Resumption.resumption.is_Done a"   and "xa \<in> range (Resumption.resumption.resume a)"   and "xb \<in> results xa" shows "xb \<in> results a"
dim_def: fixes V :: "'b set" shows "local.dim V = (if \<exists>b. local.independent b \<and> local.span b = local.span V then card (SOME b. local.independent b \<and> local.span b = local.span V) else 0)"
simps_2: fixes \<sigma> :: "'tp" shows "GE_resOf (Wit \<sigma>) = \<sigma>"
order_gt_0: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_comm_group G" shows "0 < order G"
find: fixes s :: "'b" assumes "list.has_next g s"   and "P (fst (list.next g s))" shows "Q s"
simps_2: fixes x :: "'b"   and q :: "'b queue" shows "do_act (Enq x) q = enq x q"
cycle_permutes: fixes cs :: "'a list" shows "cycle_of_list cs permutes set cs"
y_def: shows "y \<equiv> (1 - e) *\<^sub>R x + e *\<^sub>R z"
runit_hcomp_4: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<a>\<^sub>D[f, g, src\<^sub>D g] \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = f \<star>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[g]"
Some: fixes t :: "('d, 'c) bintree"   and xs :: "('d, 'c) bintree option list" shows "pqueue (Some t # xs) = bt_augment t (pqueue xs)"
hom_unique_on_span: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and gens :: "'e set"   and h :: "'e \<Rightarrow> 'g"   and h' :: "'e \<Rightarrow> 'g" assumes "Group.group G"   and "Group.group H"   and "gens \<subseteq> carrier G"   and "h \<in> hom G H"   and "h' \<in> hom G H"   and "\<forall>g\<in>gens. h g = h' g" shows "\<forall>x\<in>\<langle>gens\<rangle>\<^bsub>G\<^esub>. h x = h' x"
real_vector_space_on_def: fixes S :: "'c set" shows "real_vector_space_on S \<equiv> subspace S"
set_list_of_A: fixes t :: "'b tree" shows "set (list_of_A t) = set_tree t"
induced_automorphism_sym: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X g f = OpposedThinChamberComplexFoldings.induced_automorphism X f g"
left_lesseq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a r\<rightarrow> b = (1::'a))"
step_invariant_weaken_with_invariantE: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and R :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "A \<TTurnstile>\<^sub>A (I \<rightarrow>) Q"   and "\<And>s a s'. \<lbrakk>P s; P s'; Q (s, a, s'); I a\<rbrakk> \<Longrightarrow> R (s, a, s')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) R"
has_integral_mult_left: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and S :: "'b set"   and c :: "'c" assumes "(f has_integral y) S" shows "((\<lambda>x. f x * c) has_integral y * c) S"
div_mult_mult1: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
Wax_def: shows "Wax \<equiv> {wax \<sigma> |\<sigma>. \<not> unprot \<sigma> \<and> (\<not> isRes \<sigma> \<or> protCl \<sigma>)}"
path_weight_path: fixes s :: "'lbl"   and loc1 :: "'vtx"   and loc2 :: "'vtx" assumes "s \<in>\<^sub>A path_weight loc1 loc2" obtains xs :: "('vtx \<times> 'lbl \<times> 'vtx) list" where "path loc1 loc2 xs \<and> (\<forall>ys. path loc1 loc2 ys \<longrightarrow> \<not> sum_weights (map (\<lambda>(s, l, t). l) ys) < sum_weights (map (\<lambda>(s, l, t). l) xs))" and "distinct xs" and "sum_weights (map (\<lambda>(s, l, t). l) xs) = s"
arg_min_if_finite_2: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "\<not> (\<exists>x\<in>S. f x < f (arg_min_on f S))"
simps_3: fixes I :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a" shows "sub\<^sub>1 ({I} WHILE b DO c {P}) = c"
Dup_branch_nominals: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs" shows "branch_nominals (omit_branch xs branch) = branch_nominals branch"
eventually_eq_max: fixes f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "\<forall>\<^sub>F x in at_top. f x = g x" shows "\<forall>\<^sub>F x in at_top. max (f x) (g x) = f x"
hom_relboundary_def: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "hom_relboundary p X S T = hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id \<circ> hom_boundary p X S"
InSync_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and a :: "'addr" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (insync\<^bsub>V\<^esub> (a) e, xs) (insync\<^bsub>V\<^esub> (a) e', xs')"
igVarIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igVarIPresIGWls MOD \<equiv> \<forall>xs x. igWls MOD (asSort xs) (igVar MOD xs x)"
exhaust: fixes y :: "('f, 'g, 'h) env" obtains x1 :: "'f" where "y = Val x1"   | x21 :: "'g" and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option" where "y = Env x21 x22"
thunks_domA: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "thunks \<Gamma> \<subseteq> domA \<Gamma>"
RelS_Eg1Eg2_secure_refinement_simple_\<R>: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "secure_refinement_simple (A.\<R> \<Gamma> \<S> P) RelS_Eg1Eg2 abs_steps"
distinct_23: fixes x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> Await x71 x72"
ccpo_rel_prodI: shows "class.ccpo (prod_lub luba lubb) (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
denyAllDom: fixes x :: "'b" shows "x \<in> dom D\<^sub>U"
fresh_fmempty: fixes a :: "atom" shows "a \<sharp> {$$}"
autoref_list_eq_aux: fixes R :: "('c \<times> 'd) set" shows "(local.list_eq, local.list_eq) \<in> (R \<rightarrow> R \<rightarrow> bool_rel) \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> bool_rel"
monad_altc_envT': assumes "monad_altc Monad_Overloading.return Monad_Overloading.bind altc" shows "monad_altc Monad_Overloading.return Monad_Overloading.bind altc"
2191: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> (- y \<sqinter> x) = - y \<sqinter> x"
suff_ptop_nhd_base: fixes u :: "'a llist set"   and t :: "'a llist" assumes "u \<in> nhds t" obtains r :: "'a llist" where "r \<in> pfinpref A t" and "suff A r \<subseteq> u"
rep_eq: fixes x :: "'b regular"   and xa :: "'b regular" shows "Rep_regular (x * xa) = Rep_regular x * Rep_regular xa"
M_def: shows "M = hausdorff_distance A B + hausdorff_distance B C"
msg_inject_7: fixes X :: "ltkey"   and X' :: "ltkey" shows "(LtK X = LtK X') = (X = X')"
size_mset_SucE: fixes A :: "'c multiset"   and n :: "nat" assumes "size A = Suc n" obtains a :: "'c" and B :: "'c multiset" where "A = {#a#} + B" and "size B = n"
min_sls_def: shows "min_sls \<equiv> Abs_semilattice_set min"
F2: shows "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)"
nonzero_norm_inverse: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "norm (inverse a) = inverse (norm a)"
intro: fixes n :: "nat"   and fs :: "'a vec list" assumes "\<not> module.lin_dep class_ring (module_vec TYPE('a) n) (set fs)"   and "distinct fs" shows "gram_schmidt_fs_lin_indpt_axioms n fs"
axioms_4: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S' :: "'d set"   and is_open' :: "'d set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map S is_open S' is_open' f" shows "continuous_map_axioms S is_open is_open' f"
multpw_mult_commute: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns" shows "mult s O multpw ns \<subseteq> multpw ns O mult s"
add_le_less_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" assumes "a \<subseteq># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
pmc_completion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A" shows "Tarskian_logic.meet_contraction K (Tarskian_logic.completion Cn selection) \<phi> = Tarskian_logic.meet_contraction K selection \<phi>"
name_set_fresh_fresh: fixes pi :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and x :: "'d" assumes "set pi \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "pi \<bullet> x = x"
execute_parallel_operator_positive_effect_if: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator"   and v :: "'b" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (add_effects_of op)" shows "execute_parallel_operator s ops v = Some True"
iso_hcomp: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr" assumes "S.iso \<mu>"   and "S.iso \<nu>"   and "S.src \<nu> = S.trg \<mu>" shows "S.iso (\<nu> \<star>\<^sub>S \<mu>)"
equality: fixes r :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) NatTrans_scheme"   and r' :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) NatTrans_scheme" assumes "NTDom r = NTDom r'"   and "NTCod r = NTCod r'"   and "NatTransMap r = NatTransMap r'"   and "NatTrans.more r = NatTrans.more r'" shows "r = r'"
case_prod_app: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "'b \<times> 'c"   and y :: "'d" shows "(case x of (x, xa) \<Rightarrow> f x xa) y = (case x of (l, r) \<Rightarrow> f l r y)"
fun_sum_commute: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "f (0::'e) = (0::'d)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
case_1: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
bs_def: shows "bs = insert (1::'a) {b ^ Suc x * of_nat (Suc x) |x. x \<le> x0}"
sr_imp_eigen_vector_main: fixes u :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "u \<in> t.X"   and "t.r u = t.sr" shows "t.sr *s u = Finite_Cartesian_Product.transpose A *v u"
isometry_quasi_isometry_on: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "isometry_on X f" shows "1 0 -quasi_isometry_on X f"
ord_elems_inf_carrier: fixes a :: "'a" assumes "a \<in> carrier G"   and "ord a \<noteq> 0" shows "{a [^] x |x. x \<in> UNIV} = {a [^] x |x. x \<in> {0..ord a - 1}}"
ErrorT_runErrorT: fixes m :: "'e\<cdot>('f, 'g) errorT" shows "ErrorT\<cdot>(runErrorT\<cdot>m) = m"
list_all2_induct: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list"   and R :: "'g list \<Rightarrow> 'h list \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "R [] []"   and "\<And>x xs y ys. \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)" shows "R xs ys"
Always_LeadsToI: fixes F :: "'c program"   and INV :: "'c set"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> Always INV"   and "F \<in> INV \<inter> A \<longmapsto>w A'" shows "F \<in> A \<longmapsto>w A'"
r: shows "r (Suc k) (1::'a) = (1::'a)"
div_exp_mod_exp_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) = divide (modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (n + m))) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
unsat: shows "\<G>_Fset (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)) \<TTurnstile>e {{#}}"
globEA: fixes n :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and \<gamma> :: "(nat \<Rightarrow> 'ND) \<Rightarrow> nat \<Rightarrow> bool"   and n' :: "nat" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<box>\<^sub>b\<gamma>)"   and "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "eval c t t' n' \<gamma>"
singleton_Un_iff: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" shows "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "size1 \<langle>l, x, r\<rangle> = size1 l + size1 r"
bind_gpv_sel': fixes r :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "the_gpv (bind_gpv r f) = bind_spmf (the_gpv r) (\<lambda>generat. if is_Pure generat then the_gpv (f (generat.result generat)) else return_spmf (IO (generat.output generat) (\<lambda>input. bind_gpv (continuation generat input) f)))"
hyps: shows "N = concls_of (ord_FO_resolution.inferences_between Q C)"
wf_prog_wwf_prog: fixes P :: "(char list list \<times> char list exp) prog" assumes "wf_J_prog P" shows "wwf_J_prog P"
Ioc_subset_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<or> less_eq c a \<and> less_eq b d)"
Bernstein_coeffs_01_smult: fixes P :: "real poly"   and p :: "nat"   and a :: "real"   and i :: "nat" assumes "degree P \<le> p" shows "nth_default 0 (Bernstein_coeffs_01 p (smult a P)) i = a * nth_default 0 (Bernstein_coeffs_01 p P) i"
poly_map_poly: fixes p :: "rat poly"   and x :: "rat" shows "rpoly p (of_rat x) = of_rat (poly p x)"
ex_eqa: fixes a1 :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and a2 :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "\<exists>A B C. A B C Ang a1 \<and> A B C Ang a2" shows "a1 = a2"
snoc: shows "nths (nths xs A) (- ind_in_set A ` B) = nths xs (A - B)"
d_charpoly: shows "n + 1 = degree (charpoly A)"
cl_ident_iff: fixes L :: "'b set set"   and r :: "'b set" assumes "lattice L" shows "(cl L r = r) = (r \<in> L)"
integral_zero: fixes M :: "'d measure" shows "LINT x|M. (0::'c) = (0::'c)"
fmap_ext: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
assms: shows "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
not_model_mono_imp_addedge_mono: fixes G :: "'v graph"   and a1 :: "'v"   and a2 :: "'v"   and X :: "('v \<times> 'v) set"   and Y :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph G"   and "(a1, a2) \<in> edges G"   and "X \<subseteq> Y"   and "\<not> sinvar (add_edge a1 a2 (delete_edges G Y)) nP" shows "\<not> sinvar (add_edge a1 a2 (delete_edges G X)) nP"
that: fixes x :: "'b"   and heap' :: "heap" assumes "execute b heap = Some (x, heap')"   and "P heap'"   and "Q heap'" shows "thesis"
collapse_1: fixes generat :: "('g, 'h, 'i) generat" assumes "is_Pure generat" shows "Pure (generat.result generat) = generat"
hinsert_nonempty: fixes a :: "hf"   and A :: "hf" shows "A \<triangleleft> a \<noteq> 0"
basic_trans_rules_31: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
functor_implies_equals_dom: assumes "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) \<tau>" shows "F = \<tau>"
spmf_of_set_singleton: fixes x :: "'b" shows "spmf_of_set {x} = return_spmf x"
countable_dense_setE: obtains D :: "'c set" where "countable D" and "\<And>X. \<lbrakk>open X; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
abundant_number_def: fixes n :: "nat" shows "abundant_number n \<equiv> n < aliquot_sum n"
conjugation_is_antihomography: shows "is_antihomography conjugate"
lm002: fixes R :: "('a \<times> 'b) set" assumes "runiq R" shows "graph (Domain R) (toFunction R) \<subseteq> R"
isomorphic_sym: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W" shows "\<exists>T. FGModuleIso G smult' W smult T V"
assms_4: fixes k :: "nat" assumes "k \<in> set js" shows "k \<le> j"
halfspace_rel_internal: fixes R :: "('c \<times> 'd) set" shows "halfspaces_rel R = \<langle>\<langle>R\<rangle>sctn_rel\<rangle>list_set_rel O br below_halfspaces top"
hyps: fixes t :: "('a, 'b) tm" shows "position_of (emb_step_at (p @ [d1] @ q) d2 t) (p @ [d1]) = position_of t (p @ [d1])"
fempty_not_finsert: fixes a :: "'a"   and A :: "'a fset" shows "{||} \<noteq> finsert a A"
subset_of_opens: fixes U :: "'a set set" assumes "U \<in> top.neighborhoods \<pp>" shows "is_zariski_open U"
intros: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set"   and c :: "'d"   and s :: "('c \<times> 'd) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
ideD_1: fixes a :: "'a arrow_of_spans_data" assumes "V.ide a" shows "V.arr a"
ip_cidr_intersect: fixes b2 :: "'a word"   and m2 :: "nat"   and b1 :: "'a word"   and m1 :: "nat" assumes "\<not> ipset_from_cidr b2 m2 \<subseteq> ipset_from_cidr b1 m1"   and "\<not> ipset_from_cidr b1 m1 \<subseteq> ipset_from_cidr b2 m2" shows "ipset_from_cidr b1 m1 \<inter> ipset_from_cidr b2 m2 = {}"
initI: fixes e :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "0 < e_length e"   and "\<And>x. x < e_length e \<Longrightarrow> f x \<down>= e_nth e x" shows "f \<triangleright> (e_length e - 1) = e"
mgu_subst_domain_range_vars_disjoint: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<sigma>" shows "subst_domain \<sigma> \<inter> range_vars \<sigma> = {}"
wp_choice: fixes x :: "'b"   and y :: "'b" shows "wp (x \<sqinter> y) = wp x \<sqinter> wp y"
subst_pp_single: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and x :: "'d"   and e :: "nat" shows "subst_pp f (monomial e x) = f x ^ e"
reachable_target_elim: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" assumes "q \<in> greachable A w V p" obtains r :: "(nat \<times> 'a) list" where "gpath A w V r p" and "q = gtarget r p"
smult_minus_right: fixes a :: "'b"   and p :: "'b poly" shows "Polynomial.smult a (- p) = - Polynomial.smult a p"
distinct_7: fixes x1 :: "'c" shows "Suspend x1 \<noteq> Notified"
need_second_arg_CondOr: fixes b :: "bool" shows "need_second_arg CondOr (Bool b) = (\<not> b)"
hyps: shows "fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n"
nat_of_natural_cases: fixes y :: "nat" assumes "y \<in> UNIV" obtains x :: "natural" where "y = nat_of_natural x"
SPN_def: fixes \<mu> :: "'a" shows "SPN \<mu> \<equiv> if arr \<mu> then \<lparr>Chn = \<lbrakk>\<lbrakk>spn \<mu>\<rbrakk>\<rbrakk>, Dom = \<lparr>Leg0 = \<lbrakk>\<lbrakk>tab\<^sub>0 (local.dom \<mu>)\<rbrakk>\<rbrakk>, Leg1 = \<lbrakk>\<lbrakk>tab\<^sub>1 (local.dom \<mu>)\<rbrakk>\<rbrakk>\<rparr>, Cod = \<lparr>Leg0 = \<lbrakk>\<lbrakk>tab\<^sub>0 (cod \<mu>)\<rbrakk>\<rbrakk>, Leg1 = \<lbrakk>\<lbrakk>tab\<^sub>1 (cod \<mu>)\<rbrakk>\<rbrakk>\<rparr>\<rparr> else Span.null"
linear_inner2: fixes n :: "'b"   and i :: "'b" shows "linear (\<lambda>x. inner2 x n i)"
A: shows "sp_impl_subj_subj s1 (Step.partition (case dir of SEND \<Rightarrow> tid | RECV \<Rightarrow> partner)) (Step.partition (case dir of SEND \<Rightarrow> partner | RECV \<Rightarrow> tid)) = sp_impl_subj_subj s2 (Step.partition (case dir of SEND \<Rightarrow> tid | RECV \<Rightarrow> partner)) (Step.partition (case dir of SEND \<Rightarrow> partner | RECV \<Rightarrow> tid))"
inv_D_def: shows "inv_D = diagvector (\<lambda>j. cis (- arg (y $h j)))"
lemma_3_5_iv_1: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "d1 b c * d1 a b * d1 b c \<le> d1 a c"
feasible_def: fixes c :: "('c, 'd) conf"   and ls :: "('c, 'd) label list" shows "feasible c ls \<equiv> \<exists>c'. se_star c ls c' \<and> Conf.sat c'"
setUp_def: fixes ast :: "AST.module list" shows "setUp ast = (let (decls, procs, uu_) = preprocess ast; assertVar = Var (VTBounded 0 1) 0; pre_procs = map (\<lambda>(x, y). toProcess x y) (List.enumerate 1 procs); procs = IArray ((0, Index 0, [], []) # map (\<lambda>(uu_, uu_, uu_, uu_, p). p) pre_procs); labels = IArray (lm.empty () # map (\<lambda>(uu_, uu_, uu_, l, uu_). l) pre_procs); states = IArray (IArray [(0, [])] # map (\<lambda>(s, uu_). s) pre_procs); names = IArray (STR ''invalid'' # map (\<lambda>(uu_, uu_, n, uu_). n) pre_procs); proc_data = lm.to_map (map (\<lambda>(uu_, uu_, n, uu_, idx, uu_). (n, idx)) pre_procs); prog = \<lparr>processes = procs, labels = labels, states = states, proc_names = names, proc_data = proc_data\<rparr>; g = \<lparr>gState.vars = lm.sng STR ''__assert__'' assertVar, channels = [InvChannel], timeout = False, procs = []\<rparr>; g' = foldl (\<lambda>g d. fst (mkVarChannel d (apfst \<circ> gState.vars_update) g emptyProc)) g decls in Let (foldl (\<lambda>g (uu_, a, name, uu_). foldl (\<lambda>g name. fst (runProc name [] prog g emptyProc)) g (replicate a name)) g' pre_procs) (Pair prog))"
scalar_vec_one: fixes v :: "'a vec" shows "(1::'a) \<cdot>\<^sub>v v = v"
h: shows "h \<in> M \<rightarrow>\<^sub>M M2"
assms_1: shows "Some \<delta> = mgu (Fun f T) t"
interval_axioms: fixes x0 :: "'a" shows "interval (existence_ivl0 x0)"
rat_to_normalized_int_poly_1: fixes p :: "rat poly"   and d :: "rat"   and q :: "int poly" assumes "rat_to_normalized_int_poly p = (d, q)" shows "p = smult d (of_int_poly q)"