hcju/isabelleps huggingface dataset:
inducts: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'c"   and P :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "(r OO s) x1 x2"   and "\<And>a b c. \<lbrakk>r a b; s b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
codD: fixes v :: "nat"   and S :: "nat \<Rightarrow> typ" assumes "v \<in> cod S" shows "v \<in> free_tv S"
exP: shows "\<exists>x\<in>lset ys. P x"
simps_1: fixes b :: "bool" shows "val_type (BoolVal b) = BOOL"
H_cond: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "H (t p \<cdot> t r) x q"   and "H (t p \<cdot> !r) y q" shows "H p (if r then x else y fi) q"
gbinomial_int_mult_fact: fixes k :: "nat"   and a :: "int" shows "fact k * (a gchoose k) = (\<Prod>i = 0..<k. a - int i)"
euclidean_eqI: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and x :: "'e"   and y :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
chain_index: fixes f :: "nat \<Rightarrow> 'b set"   and F :: "'b set" assumes "is_chain f"   and "finite F"   and "F \<subseteq> \<Union> (range f)" shows "\<exists>n. F \<subseteq> f n"
rel_FGl_pos_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co3 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co3' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co4 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co4' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra3 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra4 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra4' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and tytok_F :: "(('ah, 'ai, 'p, 'aj, 'y, 'ak, 'al) G \<times> ('am, 'an, 'q, 'ao, 'z, 'ap, 'al) G \<times> ('aq, 'ar, 'r, 'as, 'aa, 'at, 'al) G \<times> 'ah \<times> 'am \<times> 'aq \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and Co2 :: "'aj \<Rightarrow> 'ao \<Rightarrow> bool"   and Co2' :: "'ao \<Rightarrow> 'as \<Rightarrow> bool"   and Contra2 :: "'ak \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra2' :: "'ap \<Rightarrow> 'at \<Rightarrow> bool"   and tytok_G :: "('ah \<times> 'am \<times> 'aq \<times> 'ai \<times> 'an \<times> 'ar \<times> 'al) itself"   and tytok_FGl :: "('ah \<times> 'am \<times> 'aq \<times> 'ai \<times> 'an \<times> 'ar \<times> 'au \<times> 'av \<times> 'aw \<times> 'al \<times> 'ax) itself" assumes "rel_F_pos_distr_cond Co1 Co1' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra3 Contra3' Contra4 Contra4' tytok_F"   and "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_FGl_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' tytok_FGl"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
inputRenameSubject: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and M :: "'d"   and N :: "'d"   and P' :: "('d, 'e, 'f) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'e"   and K :: "'d" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (M\<lparr>N\<rparr> \<prec> P')"   and "assertionAux.extractFrame SCompose' SBottom' P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "SImp' (SCompose' \<Psi> \<Psi>\<^sub>P) (SChanEq' M K)"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* K" shows "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (K\<lparr>N\<rparr> \<prec> P')"
hyps_2: shows "(t, u) \<in> timpl_closure'_step TI"
h1: shows "\<forall>csa. (csa \<in> consistent_sign_vectors (fst (factorize_polys qs)) UNIV \<and> \<not> has_no_zeros csa) = (csa \<in> set (find_sgas_aux (cast_rat_list (fst (factorize_polys qs)))))"
Gamma_times_zeta_has_integral: fixes z :: "complex" assumes "1 < Re z" shows "((\<lambda>x. complex_of_real x powr (z - 1) / (complex_of_real (exp x) - 1)) has_integral Gamma z * zeta z) {0<..}"
simps_6: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'a atom formula"   and va :: "'a atom formula" shows "wf_fmla_atom ty_ent (v \<^bold>\<or> va) = False"
DirProds_one_cong_sym: fixes f :: "'e \<Rightarrow> ('c, 'd) monoid_scheme"   and G :: "'e" shows "f G \<cong> DirProds f {G}"
that: fixes r :: "rat" assumes "q = of_rat r" shows "thesis"
i: shows "has_bochner_integral (lebesgue_on {a..c}) f (integral\<^sup>L (lebesgue_on {a..c}) f)"
findMin_xlate_aux: fixes q :: "('c, 'd) BsSkewBinomialTree list" assumes "q \<noteq> []" shows "(findMin q, eprio (findMin q)) = SkewBinomialHeapStruc.findMin (map bsmapt q)"
set_pders_nzero: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "q \<in> set (pders p)" shows "q \<noteq> 0"
abs_le_zero_iff: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) zero = (a = zero)"
redT_thread_not_disappear: fixes P :: "'addr J_prog"   and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr expr \<times> (String.literal \<Rightarrow> 'addr val option)) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) J_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr expr \<times> (String.literal \<Rightarrow> 'addr val option)) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t' :: "'thread_id" assumes "P \<turnstile> s -t\<triangleright>ta\<rightarrow> s'"   and "thr s' t' = None" shows "thr s t' = None"
cinner_canonical_basis_eq: fixes F :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and G :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" assumes "\<And>u v. \<lbrakk>u \<in> set canonical_basis; v \<in> set canonical_basis\<rbrakk> \<Longrightarrow> v \<bullet>\<^sub>C (F *\<^sub>V u) = v \<bullet>\<^sub>C (G *\<^sub>V u)" shows "F = G"
erg: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "INV l s" shows "\<exists>t p. (WHILE b DO c, s) \<Rightarrow> p \<Down> t \<and> p \<le> (k + 1) * e s \<and> INV l t \<and> \<not> bval b t"
assms_1: shows "constructive_security_obsf' real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 (\<lambda>\<eta>. absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>)))))"
assms: fixes x :: "'a"   and u :: "nat \<Rightarrow> 'a" assumes "x \<in> S"   and "\<forall>n. u n \<in> S" shows "u \<longlonglongrightarrow> x = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f x"
dg_prod_is_arrD_6: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "\<D>\<^sub>\<circ> b = I"
collapse: fixes ast_effect :: "'a ast_effect" shows "Effect (adds ast_effect) (dels ast_effect) = ast_effect"
lt_dbm_le: fixes a :: "'d"   and b :: "'d" assumes "Lt a \<preceq> Le b" shows "a \<le> b"
D'': shows "[D'']\<^bsub>v,n\<^esub> = {u. u \<turnstile> g}"
del_edge_undirected_degree_minus: fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "finite (edges g)"   and "(v, e, v') \<in> edges g"   and "(v', e, v) \<in> edges g" shows "degree v (del_unEdge v e v' g) = degree v g - 1"
F_comp_func: fixes A :: "'o"   and B :: "'o"   and C :: "'o"   and g :: "'a"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "C \<in> Ob"   and "g \<in> Hom A B"   and "f \<in> Hom B C" shows "set_func (F \<^bsub>\<a>\<^esub> (f \<bullet> g)) = compose (F \<^bsub>\<o>\<^esub> A) (set_func (F \<^bsub>\<a>\<^esub> f)) (set_func (F \<^bsub>\<a>\<^esub> g))"
set_homE_4: fixes f :: "'b set_arrow"   and U :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "f \<in> Hom\<^bsub>set_cat U\<^esub> A B" shows "set_func f \<in> A \<rightarrow> B"
sets_eq_imp_space_eq: fixes M :: "'d measure"   and M' :: "'d measure" assumes "sets M = sets M'" shows "space M = space M'"
hyps_4: shows "p\<^sub>4 = ell_field_class.add a p\<^sub>1 p\<^sub>2"
closed_sum_is_sup: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "closed_csubspace C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "A +\<^sub>M B \<subseteq> C"
g'_in_hhom: shows "\<guillemotleft>g' : trg f \<rightarrow> src f\<guillemotright>"
map_pmf_def: fixes f :: "'i \<Rightarrow> 'h"   and M :: "'i pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
exhaust: fixes y :: "('a, 'b) blindable\<^sub>m" obtains x1 :: "'a" where "y = Unblinded x1"   | x2 :: "'b blindable\<^sub>h" where "y = Blinded x2"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p. (\<And>x. P p) \<Longrightarrow> P (E p)"   and "\<And>p. (\<And>x. P (NOT p)) \<Longrightarrow> P (A p)"   and "\<And>p. (\<And>x. P p) \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
that: fixes y :: "'a" assumes "y \<in> set (local.filter_min_aux xs ys)"   and "rel y x" shows "thesis"
true_cls_def: fixes I :: "'a set"   and C :: "'a literal multiset" shows "(I \<Turnstile> C) = (\<exists>L\<in>#C. I \<Turnstile>l L)"
abelian_group_mat: fixes nr :: "nat"   and nc :: "nat" shows "abelian_group (module_mat TYPE('b) nr nc)"
real_cond_exp_cong: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "AEeq M f g"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AEeq M (real_cond_exp M F f) (real_cond_exp M F g)"
inducts: fixes g :: "'g"   and x :: "'node list"   and P :: "'node list \<Rightarrow> bool" assumes "local.path g x"   and "\<And>n. \<lbrakk>n \<in> set (\<alpha>n g); invar g\<rbrakk> \<Longrightarrow> P [n]"   and "\<And>ns n'. \<lbrakk>local.path g ns; P ns; n' \<in> set (predecessors g (hd ns))\<rbrakk> \<Longrightarrow> P (n' # ns)" shows "P x"
abs_leI: fixes abs :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq (uminus a) b" shows "less_eq (abs a) b"
simps_3: fixes x2 :: "'b" shows "\<lfloor>x2\<rfloor> \<noteq> None"
sup_option_def: fixes x :: "'b option"   and y :: "'b option" shows "x \<squnion> y = (case x of None \<Rightarrow> y | Some x' \<Rightarrow> case y of None \<Rightarrow> x | Some y \<Rightarrow> Some (x' \<squnion> y))"
antisymmetric_axioms: shows "antisymmetric A (\<lambda>x y. y \<sqsubseteq> x)"
RETURN_ref_RETURND: fixes c :: "'b"   and R :: "('b \<times> 'c) set"   and a :: "'c" assumes "RETURN c \<le> \<Down> R (RETURN a)" shows "(c, a) \<in> R"
assms: shows "(\<lambda>x. if x \<in> S then f x else (0::'b)) \<in> borel_measurable lebesgue"
inj_Poly_list_of_vec: fixes n :: "nat" shows "inj_on (Poly \<circ> list_of_vec) (carrier_vec n)"
subsetCardinality: fixes Z :: "'c set"   and X :: "'c set"   and Y :: "'c set" assumes "Z \<subseteq> X \<union> Y"   and "finite Z" shows "sum (X <| Y) X - sum (X <| Y) Z = card X - card (Z \<inter> X)"
map: fixes f :: "'e \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'e tree list" shows "map_tree f (NODE x1 x2) = NODE (f x1) (map (map_tree f) x2)"
pareto_losersE: fixes x :: "'alt pmf" assumes "x \<in> pareto_losers (SD \<circ> R)" obtains y :: "'alt pmf" where "y \<in> lotteries" and "x \<prec>[Pareto (SD \<circ> R)] y"
Tgs_authenticates_A: fixes authK :: "nat"   and A :: "agent"   and T2 :: "nat"   and evs :: "event list"   and Ta :: "nat" assumes "Crypt authK \<lbrace>Agent A, Number T2\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>B. Says A Tgs \<lbrace>Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"
m_s2: fixes X :: "char list set"   and S1 :: "'a st"   and S2 :: "'a st" assumes "finite X"   and "fun S1 = fun S2 on - X"   and "S1 < S2" shows "m\<^sub>s S2 X < m\<^sub>s S1 X"
image_vimage_subset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" shows "f ` f -` A \<subseteq> A"
in_face_cycle_setD: fixes y :: "'b"   and x :: "'b" assumes "y \<in> face_cycle_set x"   and "x \<in> arcs G" shows "y \<in> arcs G"
regularCard_stable: fixes r :: "('b \<times> 'b) set" assumes "Card_order r"   and "infinite (Field r)"   and "regularCard r" shows "stable r"
add_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b"   and b' :: "'b" shows "cblinfun_apply a (b + b') = cblinfun_apply a b + cblinfun_apply a b'"
pasting_lemma_closed: fixes I :: "'b set"   and X :: "'c topology"   and T :: "'b \<Rightarrow> 'c set"   and Y :: "'d topology"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x" shows "continuous_map X Y g"
graph_union_def: fixes G\<^sub>1 :: "('b, 'c) labeled_graph"   and G\<^sub>2 :: "('b, 'c) labeled_graph" shows "graph_union G\<^sub>1 G\<^sub>2 = LG (edges G\<^sub>1 \<union> edges G\<^sub>2) (vertices G\<^sub>1 \<union> vertices G\<^sub>2)"
uncountable_infinite: fixes A :: "'c set" assumes "uncountable A" shows "infinite A"
VC_disjoint_def: fixes v :: "'a V_rec" shows "VC_disjoint v \<equiv> V\<^bsub>v\<^esub> \<inter> C\<^bsub>v\<^esub> = {}"
not_ordLess_ordIso: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r <o r'" shows "(r, r') \<notin> ordIso"
iso_is_arr: fixes f :: "'a2" assumes "C2.iso f" shows "C2.arr f"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "agents \<noteq> {}"   and "alts \<noteq> {}"   and "\<And>i. i \<in> agents \<Longrightarrow> finite_total_preorder_on alts (R i)"   and "\<And>i x y. i \<notin> agents \<Longrightarrow> \<not> R i x y" shows "pref_profile_wf agents alts R"
diff: shows "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0::'a} \<and> y \<in> T'}"
arg_mult_2kpi: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 * z2 \<noteq> 0" shows "\<exists>x. arg (z1 * z2) = arg z1 + arg z2 + 2 * real_of_int x * pi"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "ide (UP a)"
subgroup_inverse_iff: fixes x :: "'a" assumes "IA.additive.invertible x"   and "x \<in> A" shows "(IA.additive.inverse x \<in> I) = (x \<in> I)"
Succ_def: fixes Kl :: "'e list set"   and kl :: "'e list" shows "BNF_Greatest_Fixpoint.Succ Kl kl = {k. kl @ [k] \<in> Kl}"
hom_one: shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
BT: shows "subset_dfs A (nfa_startnode A) = (bt, ls)"
dia_diff_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> y \<le>  |x\<rangle> (d y \<cdot> ad z) +  |x\<rangle> z"
SK_failS_failK: shows "SK\<cdot>failS = failK"
Enc_Pair: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Enc X Y \<noteq> \<langle>X', Y'\<rangle>"
action_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<box>[P]_v \<equiv> %s. \<forall>i. P (s |\<^sub>s i) \<or> (Unchanged v) (s |\<^sub>s i)"
bind_advantage_def: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "okamoto_\<Sigma>_commit.abstract_com.bind_advantage \<eta> \<A> \<equiv> spmf (okamoto_\<Sigma>_commit.abstract_com.bind_game \<eta> \<A>) True"
res: shows "(if i < m \<and> True then basis_reduction_main (basis_reduction_step upw i fs) else fs) = fs'"
one_chain_line_integral_eq_line_integral_on_sudivision_2: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_chain one_chain1 subdiv"   and "boundary_chain one_chain1"   and "boundary_chain subdiv"   and "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"   and "finite one_chain1"   and "finite basis" shows "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
abel_semigroup_axioms: shows "abel_semigroup (lift_option (+))"
add_mono_neutl: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "Le (0::'c) \<preceq> b" shows "a \<preceq> dbm_add b a"
dom_fst_netgmap: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and n :: "'f net_state" shows "dom (fst (netgmap sr n)) = net_ips n"
FrQ: shows "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"
netmap_none_not_in_net_ips: fixes ns :: "'c net_state"   and i :: "nat" assumes "netmap ns i = None" shows "i \<notin> net_ips ns"
simps_19: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a"   and x4 :: "ty" shows "(case ArrayT x4 of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f4 x4"
msg4_Says_imp_Notes: fixes i :: "nat"   and M :: "agent"   and NM2 :: "nat"   and merSK :: "nat"   and merEK :: "nat"   and evs :: "event list" assumes "Says (CA i) M \<lbrace>sign (priSK (CA i)) \<lbrace>Agent M, Nonce NM2, Agent (CA i)\<rbrace>, cert M merSK onlySig (priSK (CA i)), cert M merEK onlyEnc (priSK (CA i)), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace> \<in> set evs"   and "evs \<in> set_mr" shows "Notes (CA i) (Key merSK) \<in> set evs \<and> Notes (CA i) (Key merEK) \<in> set evs"
F: fixes i :: "'i"   and j :: "'j" shows "F i j \<in> sets borel"
elims_3: fixes x :: "real mpoly" assumes "\<not> check_nonzero_const x" obtains p :: "real mpoly" where "x = p" and "\<not> (case get_if_const p of None \<Rightarrow> False | Some x \<Rightarrow> x \<noteq> 0)"
less: shows "\<forall>a\<in>set as. is_Less a"
sym_lr_mono: fixes F :: "'a dual \<times> 'a \<Rightarrow> 'a" assumes "DomSol F" shows "mono (DomSol.sym_lr F)"
distinct_1: fixes x1 :: "'a"   and x21 :: "char list"   and x22 :: "aexp"   and x23 :: "'a" shows "SKIP {x1} \<noteq> x21 ::= x22 {x23}"
rep_eq: fixes x :: "uint" shows "Rep_uint (- x) = - Rep_uint x"
inter_guards_Basic: fixes f :: "'d \<Rightarrow> 'd"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Basic f \<inter>\<^sub>g c2) = Some c) = (c2 = Basic f \<and> c = Basic f)"
WWHBL22: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}"   and "Fvars \<psi> = {}"   and "prv (imp \<phi> (imp \<chi> \<psi>))" shows "prv (imp (PP \<langle>\<phi>\<rangle>) (imp (PP \<langle>\<chi>\<rangle>) (PP \<langle>\<psi>\<rangle>)))"
lcm_left_idem: fixes a :: "'a"   and b :: "'a" shows "lcm a (lcm a b) = lcm a b"
finprod_mono_neutral_cong: fixes B :: "'d set"   and A :: "'d set"   and h :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "finite B"   and "finite A"   and "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>"   and "\<And>i. i \<in> A - B \<Longrightarrow> g i = \<one>"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> g x = h x"   and "g \<in> A \<rightarrow> carrier G"   and "h \<in> B \<rightarrow> carrier G" shows "finprod G g A = finprod G h B"
degree_min_aform: fixes x :: "(real \<times> real pdevs) \<times> real"   and d :: "nat"   and y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "degree_aform_err x \<le> d"   and "degree_aform_err y \<le> d" shows "degree_aform_err (min_aform_err p x y) \<le> d"
pow_generator_mod: fixes G :: "('d, 'e) cyclic_group_scheme"   and k :: "nat" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> (k mod order G) = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> k"
gp: fixes a :: "'b fps" assumes "a $ 0 = (0::'b)" shows "Abs_fps (\<lambda>n. 1::'b) oo a = 1 / (1 - a)"
inj_on_contraD: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
Interp_def: fixes C :: "'a literal multiset" shows "Interp C = interp C \<union> production C"
partial_heap_added_first_el: fixes l :: "int"   and m :: "int"   and r :: "int"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a" assumes "l \<le> m"   and "m \<le> r"   and "is_partial_heap cmp heap l (m + 1) r" shows "is_partial_heap_except_down cmp heap l m r m"