hcju/isabelleps huggingface dataset:
cyclic_group_axioms_def: fixes G :: "('c, 'd) cyclic_group_scheme" shows "cyclic_group_axioms G \<equiv> \<^bold>g\<^bsub>G\<^esub> \<in> carrier G \<and> carrier G \<subseteq> range (([^]\<^bsub>G\<^esub>) \<^bold>g\<^bsub>G\<^esub>)"
index_of_id_def: shows "index_of_id = pair_by_index index_of_c_fst index_of_c_snd"
gcd_mult_right_right_cancel: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "gcd a (times b c) = gcd a b"
ia_less_length': shows "ia < length [0..<IArray.length (vec_to_iarray A)]"
measurable_on_scaleR_const: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and c :: "real" assumes "f measurable_on S" shows "(\<lambda>x. c *\<^sub>R f x) measurable_on S"
TUA_sub: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<theta> :: "('a, 'c) trm"   and B :: "('c + 'c) set"   and A :: "('c + 'c) set" assumes "TUadmit \<sigma> \<theta> B"   and "A \<subseteq> B" shows "TUadmit \<sigma> \<theta> A"
proots_line_card_def: fixes p :: "complex poly"   and st :: "complex"   and tt :: "complex" shows "proots_line_card p st tt = card (proots_within p (open_segment st tt))"
ring_iso_memI: fixes R :: "('e, 'f) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and S :: "('g, 'h) ring_scheme" assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"   and "bij_betw h (carrier R) (carrier S)" shows "h \<in> ring_iso R S"
xivp_2: shows "\<nu> (X t\<^sub>0) < \<mu> (X t\<^sub>0)"
comp_converter_id_right: fixes conv :: "('m, 'n, 'o, 'p) converter" shows "conv \<odot> 1\<^sub>C = conv"
ring_simprules_24: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
oreturn_NF_wp: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d" shows "ovalidNF (P x) (oreturn x) P"
Re_Ln_pos_le: fixes z :: "complex" assumes "z \<noteq> 0" shows "(\<bar>Im (Ln z)\<bar> \<le> pi / 2) = (0 \<le> Re z)"
nres_order_simps_10: fixes X :: "'b nres set" assumes "FAIL \<in> X" shows "Sup X = FAIL"
5: shows "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
1: shows "\<guillemotleft>\<a>[f, g, h] : (f \<star> g) \<star> h \<Rightarrow> f \<star> g \<star> h\<guillemotright>"
assms_3: shows "\<bar>A - A'\<bar> \<le> \<delta>_A"
monofun_0: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and h :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "h \<in> H" shows "monofun R M N f H h \<zero> = \<zero>\<^bsub>N\<^esub>"
cond_2: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "movement.s a\<^sub>o v\<^sub>o s\<^sub>o (movement.t_stop a\<^sub>o v\<^sub>o) \<le> movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e)" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0..}"
fTo_conditional_Un_Chain_mem1: fixes C :: "'a set"   and Ca :: "'a set set" assumes "C \<in> carrier fTo D"   and "Chain (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca"   and "Ca \<noteq> {}" shows "\<Union> Ca \<in> upper_bounds (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S}) Ca"
frgv_impl_rel_ext_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "\<langle>Rm, Rv\<rangle>frgv_impl_rel_ext \<equiv> \<langle>Rm, \<langle>Rv\<rangle>list_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
has_bochner_integral_imp_has_integral: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and I :: "'b" assumes "has_bochner_integral lebesgue (\<lambda>x. indicat_real S x *\<^sub>R f x) I" shows "(f has_integral I) S"
single_non_qAFreshAll_ordLess_var: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> |{x. \<not> qAFresh xs x X}| <o |UNIV|) \<and> (FixVars.qGoodAbs A \<longrightarrow> |{x. \<not> qAFreshAbs xs x A}| <o |UNIV|)"
soldAllocationVariantEquivalence: fixes N :: "integer set"   and \<Omega> :: "'c set" shows "soldAllocations'' N \<Omega> = soldAllocations''' N \<Omega>"
ln_one: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and ln :: "'c \<Rightarrow> 'c" assumes "class.ln dist uniformity open minus norm plus zero uminus scaleR sgn times one ln" shows "ln one = zero"
emeasure_pair_measure: fixes X :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M M2)" shows "emeasure (N \<Otimes>\<^sub>M M2) X = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M2 \<partial>N"
results_DONE: fixes x :: "'f" shows "results (DONE x) = {x}"
step_Normal_elim_cases_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], css, Stuck) \<rightarrow> u" obtains nrms :: "('e, 'd, 'f) com list" and abrs :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" where "css = (nrms, abrs) # cssa" and "u = (nrms, cssa, Stuck)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool" assumes "\<And>h. type_wf h \<Longrightarrow> DocumentClass.type_wf h" shows "l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf"
n_mod_cas: shows "gr.INTERP M N \<TTurnstile>m mset CAs"
rec_le_def: shows "rec_le = Cn (Suc (Suc 0)) rec_disj [rec_less, rec_eq]"
lm042: fixes x :: "'c"   and X :: "'d set" shows "{(x, X)} - {(x, {})} \<in> partitionValuedUniverse"
Output_def: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "x3{x2}.x1 \<equiv> Abs_pi (Output_Rep x3 x2 (Rep_pi x1))"
E_state_fix: fixes \<T> :: "'a stateful_policy" assumes "wf_stateful_policy \<T>" shows "flows_state \<T> \<subseteq> flows_fix \<T>"
unit_dvd_iff_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "(a * b dvd c) = (b dvd c)"
ipresIGOp_def: fixes h :: "'o \<Rightarrow> 'p"   and hA :: "'q \<Rightarrow> 'r"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 's, 'o, 'q, 't) model_scheme"   and MOD' :: "('index, 'bindex, 'u, 'v, 'opSym, 'w, 'p, 'r, 'x) model_scheme" shows "ipresIGOp h hA MOD MOD' \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> h (igOp MOD delta inp binp) = igOp MOD' delta (lift h inp) (lift hA binp)"
has_parametrized_subprob_density_def: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" shows "has_parametrized_subprob_density M N R f \<equiv> (\<forall>x\<in>space M. has_subprob_density (N x) R (f x)) \<and> (\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)"
bounded_C1: shows "\<exists>K. \<forall>x. norm (C1 x) \<le> norm x * K"
CDPlayer_CTRL_Action1_def: shows "CDPlayer_CTRL_Action1 = ({}, CDPlayer_CTRL_Update1)"
le_trans_refl: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "le_trans x x"
mapping_eqI: fixes m :: "('b, 'c) mapping"   and m' :: "('b, 'c) mapping" assumes "\<And>x. Mapping.lookup m x = Mapping.lookup m' x" shows "m = m'"
ask_ask: fixes f :: "'r \<Rightarrow> 'r \<Rightarrow> 'm" shows "ask (\<lambda>r. ask (f r)) = ask (\<lambda>r. f r r)"
comp_arr_dom: fixes f :: "(bool, unit) arr"   and a :: "(bool, unit) arr" assumes "arr f"   and "local.dom f = a" shows "COMP f a = f"
foldl_conv_foldr: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'g"   and a :: "'g"   and xs :: "'h list" shows "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
Spec: fixes s :: "'d"   and t :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> (Spec r, Normal s) \<rightarrow> (Skip, Normal t)"
lookup_raw_map2_val_raw: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and g :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'g) list \<times> 'o"   and h :: "('a \<times> 'f) list \<times> 'o \<Rightarrow> ('a \<times> 'g) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g"   and k0 :: "'a" assumes "oalist_inv xs"   and "oalist_inv ys"   and "map2_val_compat' g"   and "map2_val_compat' h"   and "\<And>zs. oalist_inv zs \<Longrightarrow> g zs = map_val_raw (\<lambda>k v. f k v (0::'f)) zs"   and "\<And>zs. oalist_inv zs \<Longrightarrow> h zs = map_val_raw (\<lambda>k. f k (0::'e)) zs"   and "\<And>k. f k (0::'e) (0::'f) = (0::'g)" shows "lookup_raw (map2_val_raw f g h xs ys) k0 = f k0 (lookup_raw xs k0) (lookup_raw ys k0)"
paths_single: fixes x :: "'a" shows "paths (single x) = {[], [x]}"
bounded_iff: fixes S :: "'a set" shows "bounded S = (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
atS_mono: fixes p :: "'g"   and ls :: "'h set"   and s :: "('i, 'h, 'g, 'j, 'k) system_state"   and ls' :: "'h set" assumes "atS p ls s"   and "ls \<subseteq> ls'" shows "atS p ls' s"
cut_less_inext_nth_card_eq1: fixes n :: "nat"   and I :: "nat set" assumes "n < card I \<or> infinite I" shows "card (I \<down>< (I \<rightarrow> n)) = n"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a1 :: "'b SDG_node"   and a2 :: "'b SDG_node list"   and a3 :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None None False n') \<or> (\<exists>n ns n'' V n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' (Some V) None False n' \<and> n'' \<noteq> n') \<or> (\<exists>n ns n'' p n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None (Some (p, True)) True n'))"
polynomial_sum_monoms_1: fixes p :: "'a mpoly" shows "p = (\<Sum>x\<in>keys (mapping_of p). MPoly_Type.monom x (MPoly_Type.coeff p x))"
simps_4: fixes v :: "nat"   and x :: "'b"   and f' :: "'b list"   and r' :: "'b list" shows "exec (App (Suc v) (x # f') r') = App (Suc v - 1) f' (x # r')"
simps_2: shows "rexp_ewp 1\<^sub>r = True"
semiring_norm_84: fixes n :: "num" shows "(num.One = num.Bit1 n) = False"
simps_2: fixes "is" :: "'b set"   and ia :: "'b interrupt_action"   and ias :: "'b interrupt_action list" shows "redT_updIs is (ia # ias) = redT_updIs (redT_updI is ia) ias"
zero_prod_def: shows "0 = (0::'d, 0::'e)"
I: shows "(l\<^sub>1, l\<^sub>2) \<in> imp_graph cnf"
indep_sets_mono_index: fixes J :: "'c set"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'a set set" assumes "J \<subseteq> I"   and "indep_sets F I" shows "indep_sets F J"
simps_15: fixes e :: "('f, 'g, 'h) exp"   and M :: "String.literal"   and es :: "('f, 'g, 'h) exp list"   and A :: "'f set option" shows "\<D> (e\<bullet>M(es)) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
ex_un_null: shows "\<exists>!n. \<forall>f. COMP n f = n \<and> COMP f n = n"
irreducible\<^sub>d_m_def: fixes f :: "int poly" shows "pl.irreducible\<^sub>d_m f \<equiv> 0 < pl.degree_m f \<and> (\<forall>g h. pl.degree_m g < pl.degree_m f \<longrightarrow> pl.degree_m h < pl.degree_m f \<longrightarrow> pl.Mp f \<noteq> pl.Mp (g * h))"
Fubini_integrable: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"   and "integrable M1 (\<lambda>x. LINT y|M2. norm (f (x, y)))"   and "AE x in M1. integrable M2 (\<lambda>y. f (x, y))" shows "integrable (M1 \<Otimes>\<^sub>M M2) f"
to_nat_mono: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "mod_type.to_nat Rep a < mod_type.to_nat Rep b"
dom_restrict_def: fixes S :: "'c set"   and p :: "'c \<Rightarrow> 'd decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
tagged_fun_relD_none: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
cbox_Pair_iff: fixes x :: "'b"   and y :: "'c"   and a :: "'b"   and c :: "'c"   and b :: "'b"   and d :: "'c" shows "((x, y) \<in> cbox (a, c) (b, d)) = (x \<in> cbox a b \<and> y \<in> cbox c d)"
elims_3: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<not> binrelchain x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" and y :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # y # xs" and "\<not> (P xb y \<and> binrelchain P (y # xs))"
if_all_smaller_are_false_then_cl_not_valid: fixes t :: "'a trm"   and \<sigma> :: "('a \<times> 'a trm) list"   and C :: "'a literal set"   and S :: "'a eclause set"   and L1 :: "'a literal"   and s :: "'a trm" assumes "smaller_lits_are_false (t \<lhd> \<sigma>) (subst_cl C \<sigma>) S"   and "fo_interpretation (same_values (\<lambda>t. trm_rep t S))"   and "orient_lit_inst L1 t s pos \<sigma>"   and "maximal_literal (subst_lit L1 \<sigma>) (subst_cl C \<sigma>)"   and "ground_clause (subst_cl C \<sigma>)"   and "subst_lit L1 \<sigma> \<in> subst_cl C \<sigma>"   and "\<not> equivalent_eq_exists t s C (same_values (\<lambda>t. trm_rep t S)) \<sigma> L1"   and "trm_rep (t \<lhd> \<sigma>) S = trm_rep (s \<lhd> \<sigma>) S" shows "\<not> validate_ground_clause (same_values (\<lambda>t. trm_rep t S)) (subst_cl (C - {L1}) \<sigma>)"
drinks2_vend_sufficient: fixes r :: "nat \<Rightarrow>f value option"   and x1 :: "int" assumes "r $ 2 = Some (Num x1)"   and "\<not> x1 < 100" shows "possible_steps drinks2 2 r STR ''vend'' [] = {|(3, vend)|}"
that_2: fixes c :: "('a, 'b) chart" assumes "c \<in> charts" shows "k-smooth_compat c2 c"
feasible_fixes_top: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "feasible t" shows "t (\<lambda>s. 1) \<tturnstile> \<lambda>s. 1"
Dynkin_system_trivial: fixes A :: "'c set" shows "Dynkin_system A (Pow A)"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" shows "slow.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> slow.check_eqvRE enum_class.enum n (Plus (rexp_of'' n (norm \<phi>)) One) (Plus (rexp_of'' n (norm \<psi>)) One))"
wlp_close: fixes S :: "com"   and P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "is_quantum_predicate (wlp S P)"
xpoint: shows "x \<in> A\<^sup>\<spadesuit>"
fps_left_inverse: fixes x :: "'a"   and f :: "'a fps" assumes "x * fps_nth f 0 = (1::'a)" shows "fps_left_inverse f x * f = 1"
subfrmlsn_finite: fixes \<phi> :: "'a ltln" shows "finite (subfrmlsn \<phi>)"
apply_cltn2_imp_mult: fixes p :: "proj2"   and C :: "cltn2"   and q :: "proj2" assumes "apply_cltn2 p C = q" shows "\<exists>k. k \<noteq> 0 \<and> proj2_rep p v* cltn2_rep C = k *\<^sub>R proj2_rep q"
vbrelation_axioms: shows "vbrelation (\<NN>\<lparr>NTMap\<rparr>)"
subst_def: fixes xs :: "'varSort"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and x :: "'var" shows "subst xs X x \<equiv> psubst (idEnv [x \<leftarrow> X]_xs)"
subset_antisym: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
x0_hb_bound0': fixes x :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> x"   and "b \<in> set bs" shows "hb / ln x powr (1 + e) < b"
add_less_same_cancel1: fixes b :: "'a"   and a :: "'a" shows "(b + a < b) = (a < (0::'a))"
2: shows "y \<le> gfp (\<lambda>y. snd (F (gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y)))))), y)))"
dbm_not_lt_eq: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "\<not> a \<prec> b"   and "\<not> b \<prec> a" shows "a = b"
face_cycle_set_def: fixes a :: "'b" shows "G.face_cycle_set a = orbit G.face_cycle_succ a"
push_down_oldest_token_configuration: fixes q :: "'b"   and n :: "nat"   and x :: "nat" assumes "oldest_token q n = Some x" shows "x \<in> configuration q n"
bdd_above_image_mono: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "mono f"   and "bdd_above A" shows "bdd_above (f ` A)"
Cod_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Cod (cod f) = Cod f"
poly_minus: fixes p :: "'a Polynomial.poly"   and x :: "'a" shows "poly (- p) x = - poly p x"
equalityE: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
is_acc_class_is_class: fixes G :: "prog"   and P :: "pname"   and C :: "qtname" assumes "is_acc_class G P C" shows "is_class G C"
free_tv_app_subst_scheme_list: fixes S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list" shows "free_tv ($ S A) \<subseteq> cod S \<union> free_tv A"
hom_gauss_jordan: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and nc2 :: "nat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc2" shows "gauss_jordan (mat\<^sub>h A) (mat\<^sub>h B) = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan A B)"
zero_le_divide_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
s: shows "s = 1 - \<i> * complex_of_real a"
to_fun_closed: fixes R :: "('a, 'b) ring_scheme"   and f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "x \<in> carrier R" shows "UP_cring.to_fun R f x \<in> carrier R"
term_ok_eq_ax_5: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "is_std_sig (sig \<Theta>)" shows "term_ok \<Theta> eq_elim_ax"
tendsto_Pair: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'f"   and b :: "'f" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
xs: fixes xs :: "'a list" shows "(xs, xs) \<in> \<langle>Id\<rangle>list_rel"
outstanding_refs_takeWhile_opposite: fixes P :: "'b memref \<Rightarrow> bool"   and xs :: "'b memref list" shows "outstanding_refs P (takeWhile (Not \<circ> P) xs) = {}"
leastPD2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "leastP P x" shows "x <=* Collect P"
that: fixes xs :: "'a list" assumes "set xs \<subseteq> A"   and "distinct xs"   and "length xs = n" shows "thesis"
rel_spmf_return_spmf2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and x :: "'d" shows "rel_spmf R p (return_spmf x) = (lossless_spmf p \<and> (\<forall>a\<in>set_spmf p. R a x))"
locally_faithful_def: shows "F.locally_faithful \<equiv> \<forall>f g \<mu> \<mu>'. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> \<guillemotleft>\<mu>' : f \<Rightarrow>\<^sub>C g\<guillemotright> \<and> F \<mu> = F \<mu>' \<longrightarrow> \<mu> = \<mu>'"
Comm1: fixes P :: "pi"   and b :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>lb in P''\<rightarrow>a<x> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P' \<parallel> Q'"
ord_isom_segment_segment: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "Worder E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "ord_isom (Iod D (segment D a)) (Iod E (segment E (f a))) (restrict f (carrier (Iod D (segment D a))))"
chamber_system_image: shows "f \<turnstile> folding_g.\<C> = folding_g.\<C> \<inter> (f \<turnstile> X)"