hcju/isabelleps huggingface dataset:
rsp: shows "eq_onp (\<lambda>v. v \<noteq> vec_zero) \<infinity>\<^sub>v \<infinity>\<^sub>v"
zero_neq_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> numeral.numeral one plus n"
dependent_zero: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" assumes "module scale"   and "(0::'e) \<in> A" shows "module.dependent scale A"
Ord_mem_iff_lt: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" shows "(k \<in>\<^sub>\<circ> l) = (k \<subset>\<^sub>\<circ> l)"
composite_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'd" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
total_preorder_on_restrict: fixes A :: "'a set" shows "total_preorder_on (carrier \<inter> A) (restrict_relation A le)"
mult_1_left: fixes a :: "'a" shows "(1::'a) * a = a"
keys_sum_subset: fixes b :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and I :: "'e set" shows "keys (sum b I) \<subseteq> (\<Union>i\<in>I. keys (b i))"
class_wf: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and c :: "char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool" assumes "class P C = \<lfloor>c\<rfloor>"   and "wf_prog wf_md P" shows "wf_cdecl wf_md P (C, c)"
S_subst: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "fst (kbo s t)" shows "fst (kbo (subst_atm_abbrev s \<sigma>) (subst_atm_abbrev t \<sigma>))"
less_prod_def: fixes p1 :: "'a \<times> 'b"   and p2 :: "'a \<times> 'b" shows "(p1 < p2) = (p1 \<le> p2 \<and> \<not> p2 \<le> p1)"
add_mset_eq_single: fixes b :: "'b"   and M :: "'b multiset"   and a :: "'b" shows "(add_mset b M = \<LM> a  \<RM>) = (b = a \<and> M = \<Empt>)"
exhaust_sel: fixes rexp :: "'b rexp" obtains "rexp = Zero"   | "rexp = One"   | "rexp = Atom (un_Atom rexp)"   | "rexp = Plus (un_Plus1 rexp) (un_Plus2 rexp)"   | "rexp = Times (un_Times1 rexp) (un_Times2 rexp)"   | "rexp = Star (un_Star rexp)"
P2: shows "\<lfloor>\<lambda>w. \<exists>x. (\<^bold>\<box>Abstract x) w\<rfloor>"
Plus_mono: fixes r1 :: "'a language"   and s1 :: "'a language"   and r2 :: "'a language"   and s2 :: "'a language" assumes "r1 \<le> s1"   and "r2 \<le> s2" shows "Plus r1 r2 \<le> Plus s1 s2"
jpTraces_step_length_inv_aux: fixes t :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol"   and n :: "nat" assumes "t \<in> {t \<in> jpTraces jp. tLength t = Suc n}" shows "\<exists>t' s. t = t' \<leadsto> s \<and> t' \<in> jpTraces jp \<and> tLength t' = n \<and> (\<exists>eact\<in>set (envAction (tLast t')). \<exists>aact. (\<forall>a. aact a \<in> set (actJP jp t' a)) \<and> s = envTrans eact aact (tLast t'))"
fps_inverse_power: fixes a :: "'a fps"   and n :: "nat" shows "inverse (a ^ n) = inverse a ^ n"
mC_C_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_C aval tval cval indis theta c d s t P F \<equiv> PL_Indis.mC_C_part c d P F \<and> inj_on F P \<and> PL_Indis.mC_C_wt aval tval cval c d s t P F \<and> PL_Indis.mC_C_eff_cont aval tval cval indis theta c d s t P F"
NonInterferenceIntraGraph_axioms: shows "NonInterferenceIntraGraph sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val backward_slice (_Exit_) H L (_High_) (_Low_)"
return_option_def: fixes return :: "'d option \<Rightarrow> 'e"   and x :: "'d" shows "return_option return x = OptionT (return (Some x))"
add_0_right: fixes m :: "nat" shows "m + 0 = m"
to_Gromov_completion: shows "x = to_Gromov_completion a"
assms_2: fixes \<psi> :: "'a ltl" assumes "G \<psi> \<in> \<G>\<^sub>F\<^sub>G \<phi> w" shows "threshold \<psi> w (\<G>\<^sub>F\<^sub>G \<phi> w) = Some 0"
trans: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" assumes "prefix a b"   and "prefix b c" shows "prefix a c"
inject_1: fixes x11 :: "'c"   and x12 :: "'d"   and y11 :: "'c"   and y12 :: "'d" shows "(Add x11 x12 = Add y11 y12) = (x11 = y11 \<and> x12 = y12)"
rel_U_witness1: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "(('z, 'n, 'q, 't, 'w, 'aa) U \<times> (('z, 'n, 'q, 't, 'w, 'aa) U \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and L1 :: "'z \<Rightarrow> 'ab \<Rightarrow> bool"   and x :: "('z, 'n, 'q, 't, 'w, 'aa) U"   and y :: "('ab, 'p, 's, 'v, 'y, 'aa) U" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G"   and "rel_U L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_U (\<lambda>x (x', y). x' = x \<and> L1 x y) Co1 Co2 Contra1 Contra2 x (rel_U_witness L1 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y))"
sees_method_idemp: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "P \<turnstile> D sees M: Ts\<rightarrow>T = m in D"
one_F: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "\<^bold>1 \<^bold>* F g A = F g A"
"3_2": assumes "x \<noteq> y"   and "rel x y" shows "set (merge_wrt rel xs (y # ys)) = set xs \<union> set (y # ys)"
star_plus_without_loops: fixes x :: "'a" shows "x\<^sup>\<star> \<sqinter> - (1::'a) = x\<^sup>+ \<sqinter> - (1::'a)"
mult_right_cancel: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> E"   and "x \<noteq> (0::'a)" shows "(a \<cdot> x = b \<cdot> x) = (a = b)"
that: fixes c' :: "String.literal" assumes "c' \<in> s"   and "class_les cs c' c" shows "thesis"
smc_SemiCAT_obj_initialI: shows "obj_initial (smc_SemiCAT \<alpha>) smc_0"
prefixclosed_def: fixes tr :: "'a list set" shows "prefixclosed tr \<equiv> \<forall>l1\<in>tr. \<forall>l2. l2 \<preceq> l1 \<longrightarrow> l2 \<in> tr"
intro: fixes \<Gamma> :: "'b inference set" assumes "\<And>N. satisfiable N \<Longrightarrow> satisfiable (N \<union> concls_of (inference_system.inferences_from \<Gamma> N))" shows "sat_preserving_inference_system \<Gamma>"
expr_has_density_sound_aux: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and t :: "pdf_type" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : t"   and "density_context V V' \<Gamma> \<delta>"   and "free_vars e \<subseteq> V \<union> V'" shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t) f"
inv_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Cop.iso f"   and "Cop.in_hom f a b" shows "Cop.in_hom (Cop.inv f) b a"
top_greatest: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and top :: "'d"   and a :: "'d" assumes "class.order_top less_eq less top" shows "less_eq a top"
emeasure_lborel_cbox_eq: fixes l :: "'b"   and u :: "'b" shows "emeasure lborel (cbox l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
H: shows "D @ TVarB X Q # G \<turnstile> t : T"
pred_stream_sfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "alw (ev (holds P)) xs" shows "pred_stream P (sfilter P xs)"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (b = a\<midarrow>a)"
is_expansion_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "trimmed F"   and "is_expansion F basis" shows "is_expansion (inverse F) basis"
classes_above_subcls: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C \<preceq>\<^sup>* C'" shows "P' \<turnstile> C \<preceq>\<^sup>* C'"
T_intersec_meas_4: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat" assumes "A \<in> sets M"   and "B \<in> sets M" shows "(T ^^ n) -` A \<inter> B \<in> sets M"
hyps: assumes "Suc 0 \<le> n" shows "stirling (Suc n) 2 = sum ((div) (fact n)) {1..n}"
\<pi>_lower_bound: fixes x :: "real" assumes "2 \<le> x" shows "1 / 6 * (x / ln x) < \<pi> x"
fmapWT_def: shows "fmapWT = (\<Lambda> f m. bindWT\<cdot>m\<cdot>(\<Lambda> x. unitWT\<cdot>(f\<cdot>x)))"
finsum_cong: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier R) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum R f A = finsum R g B"
power2_eq_square_octo: fixes x :: "octo" shows "x\<^sup>2 = x * x"
l1_implements_a0i: shows "implements med01ia a0i l1"
prj_chine_assoc_3: shows "\<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>0 \<cdot> \<mu>\<nu>\<pi>.chine_assoc = \<mu>\<nu>\<pi>.Prj\<^sub>0"
induct: fixes x :: "'b list"   and C :: "'b list set"   and b :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> no_head C b"   and "P \<epsilon>"   and "\<And>u. \<lbrakk>u \<in> C; u \<noteq> b\<rbrakk> \<Longrightarrow> P u"   and "\<And>u v. \<lbrakk>u \<noteq> \<epsilon>; u \<in> no_head C b; P u; v \<in> \<langle>C\<rangle>\<rbrakk> \<Longrightarrow> P (u \<cdot> v)" shows "P x"
tc: fixes x :: "'a list set" assumes "p_prod x x \<subseteq> x" shows "p_prod (p_star x) x = x"
compose_shift: fixes j :: "nat"   and k :: "nat"   and c :: "nat"   and \<tau> :: "ty" shows "shift (j + k) c \<tau> = shift j c (shift k c \<tau>)"
1: shows "\<guillemotleft>\<r>[a] : a \<otimes> \<I> \<rightarrow> a\<guillemotright> \<and> \<r>[a] \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
simps_1: fixes x21 :: "'b"   and x22 :: "'b list"   and y21 :: "'b"   and y22 :: "'b list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
succeed_membership: fixes x :: "nat" shows "token_succeeds x = (\<exists>i. x \<in> succeed i)"
simps_22: shows "cospan_shape.comp cospan_shape.BT cospan_shape.Null = cospan_shape.Null"
dist_eq_dist_code_lt: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int"   and p\<^sub>2 :: "int \<times> int"   and p\<^sub>3 :: "int \<times> int" shows "(dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>2 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>3 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) = (dist_code p\<^sub>0 p\<^sub>1 < dist_code p\<^sub>2 p\<^sub>3)"
ex_dense_closure_interE: fixes S :: "'a set" assumes "S \<subseteq> carrier" obtains D :: "'a set" and C :: "'a set" where "D \<subseteq> carrier" and "C \<subseteq> carrier" and "D dense" and "C closed" and "S = D \<inter> C"
elims_2: fixes x :: "atom" assumes "is_Less x" obtains i :: "nat" and j :: "nat" where "x = Less i j"
matchC_M_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchC_M tval aval indis theta c d \<equiv> \<forall>s t c' s'. (s, t) \<in> indis \<and> PL.transC tval aval (c, s) (c', s') \<longrightarrow> (\<exists>d' t'. PL.MtransC tval aval (d, t) (d', t') \<and> (s', t') \<in> indis \<and> (c', d') \<in> theta) \<or> (\<exists>t'. PL.MtransT tval aval (d, t) t' \<and> (s', t') \<in> indis \<and> PL_Indis.discr tval aval indis c')"
set_Macaulay_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (Macaulay_list ps) = set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) - {0}"
cases: fixes x :: "('c, 'd) Node list" obtains a :: "('c, 'd) Node" where "x = [a__]"   | a :: "('c, 'd) Node" and b :: "('c, 'd) Node" where "x = [aa__, b__]"   | a :: "('c, 'd) Node" and b :: "('c, 'd) Node" and c :: "('c, 'd) Node" where "x = [ab__, ba__, c__]"   | a :: "('c, 'd) Node" and b :: "('c, 'd) Node" and c :: "('c, 'd) Node" and d :: "('c, 'd) Node" where "x = [a, b, c, d]"   | "x = []"   | v :: "('c, 'd) Node" and vb :: "('c, 'd) Node" and va :: "('c, 'd) Node" and vc :: "('c, 'd) Node" and vd :: "('c, 'd) Node" and vf :: "('c, 'd) Node list" where "x = v # vb # va # vc # vd # vf"
irreducible_reflect_poly: fixes f :: "'a poly" assumes "coeff f 0 \<noteq> (0::'a)" shows "irreducible (reflect_poly f) = irreducible f"
Span_mem_imp_non_trivial_combine: fixes K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "a \<in> Span K Us" obtains k :: "'a" and Ks :: "'a list" where "k \<in> K - {\<zero>}" and "set Ks \<subseteq> K" and "length Ks = length Us" and "combine (k # Ks) (a # Us) = \<zero>"
modProcArg_def: fixes x :: "procArg \<times> integer" shows "modProcArg x = (case x of (ProcArg ty name, val) \<Rightarrow> if varType_inv ty then let init = checkVarValue ty val in (name, Var ty init) else abortv STR ''Invalid proc arg (varType_inv failed)'' name (\<lambda>_. (name, Var VTChan 0)))"
P: shows "finer P1 P \<and> finer P2 P \<and> finite P \<and> {} \<notin> P \<and> part {..<brn c} P"
simps_27: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "'b"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "'c"   and x26 :: "'d" shows "(case Send x21 x22 x23 x24 x25 x26 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25 x26"
measurable_T1: fixes s :: "'a cfg"   and M :: "'b measure" shows "MC.T s \<rightarrow>\<^sub>M M = stream_space (count_space UNIV) \<rightarrow>\<^sub>M M"
im_sheaf_is_presheaf: shows "presheaf_of_rings S is_open imS.im_sheaf imS.im_sheaf_morphisms b imS.add_im_sheaf imS.mult_im_sheaf imS.zero_im_sheaf imS.one_im_sheaf"
annotate_f: fixes I :: "nat \<Rightarrow> bool" shows "f = f_inv I"
simps_2: fixes x :: "nat" shows "Pi (Suc x) = Np (Pi x)"
remove_weak_ops_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (remove_weak_ops \<phi>) \<subseteq> atoms_ltln \<phi>"
rel_inject: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'f"   and y :: "'g" shows "rel_optionT' R1 R2 (OptionT x) (OptionT y) = R2 x y"
semIntAll_prOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prOp (semInt SEM) (semIntAbs SEM) SEM"
final_read_Lm: fixes a :: "'b"   and r :: "(bool \<times> 'b) rexp" shows "final (read a r) = ([a] \<in> Lm r)"
add_diff_cancel_left: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and c :: "'r"   and a :: "'r"   and b :: "'r" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
simps_7: fixes \<tau> :: "ty" shows "erase (AuthT \<tau>) = erase \<tau>"
vbrelation_vinsertI: fixes a :: "V"   and b :: "V" shows "vbrelation (vinsert \<langle>a, b\<rangle> xs)"
code: fixes L :: "'b list set" shows "to_language L = Lang ([] \<in> L) (\<lambda>a. to_language {w. a # w \<in> L})"
Sigma: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" assumes "comm_monoid_set f z"   and "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "comm_monoid_set.F f z (\<lambda>x. comm_monoid_set.F f z (g x) (B x)) A = comm_monoid_set.F f z (\<lambda>(x, y). g x y) (Sigma A B)"
jkbpCn_length: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> Clock.jkbpCn n" shows "tLength t = n"
cls_catt: fixes c :: "CTypeId"   and f :: "AttId" assumes "CClassT c \<le> dtype f" shows "cls (catt c f) = c"
strict_trans: fixes b :: "'f"   and a :: "'f"   and c :: "'f" assumes "b < a"   and "c < b" shows "c < a"
map_iterator_key_filter_def: fixes P :: "'e \<Rightarrow> bool" shows "map_iterator_key_filter P \<equiv> set_iterator_filter (P \<circ> fst)"
mult_eq_first_row: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and m :: "nat"   and V :: "'b mat" assumes "A \<in> carrier_mat 1 n"   and "B \<in> carrier_mat m n"   and "m \<noteq> 0"   and "Matrix.row A 0 = Matrix.row B 0" shows "Matrix.row (A * V) 0 = Matrix.row (B * V) 0"
pp_fusion_leftneutral: fixes x :: "'a ppath" shows "pp_fusion (Node (pp_first x)) x = x"
strict_monoI_Suc: fixes f :: "nat \<Rightarrow> 'c" assumes "\<And>n. f n < f (Suc n)" shows "strict_mono f"
reindex_cong: fixes l :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and h :: "'c \<Rightarrow> 'e" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
Next'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration" shows "Next' c0 c1 = (NextPerformOp c0 c1 \<or> NextSendUpd c0 c1 \<or> NextRecvUpd c0 c1 \<or> NextPropagate c0 c1 \<or> NextRecvCap c0 c1 \<or> c1 = c0)"
payload_Hash: fixes X :: "msg" assumes "Hash X \<in> payload" obtains "X \<in> payload"
collapse: fixes lit :: "lit" shows "Lit (pos lit) (var lit) = lit"
eq: fixes x :: "'s"   and y :: "'s" assumes "x \<in> C"   and "(x, y) \<in> acc" shows "y \<in> C"
P1_real_ddh0_\<sigma>_false: fixes \<sigma> :: "bool"   and n :: "nat"   and M :: "'grp \<times> 'grp"   and z :: "'e"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)"   and D :: "'d \<times> 'grp \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "P1_real_model n M \<sigma> z \<A> \<bind> D = local.ddh.DDH0 n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D)"
P3_2: shows "s1 \<subseteq>\<^sub>f s2"
vcons_vsubset: fixes xs :: "V"   and x :: "V" shows "xs \<subseteq>\<^sub>\<circ> xs #\<^sub>\<circ> x"
cod_simps_2: shows "fg.cod.leg0 = fg.leg0"
alloc_def: fixes xs :: "'a list"   and h :: "heap" shows "Array.alloc xs h = (let l = lim h; r = Array l in Let (Array.set r xs (h\<lparr>lim := l + 1\<rparr>)) (Pair r))"
t_def: shows "t \<equiv> \<lambda>y. THE x. y = absc x \<and> x \<in> set_pmf (K_cfg cfg)"
limXn: shows "\<forall>i<m. \<forall>j<m. (\<lambda>n. X n $$ (i, j)) \<longlonglongrightarrow> A $$ (i, j)"
domain_eq: shows "Domainp (pcr_fpxs (=)) = is_fpxs"
Simulation_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Simulation A B sim \<equiv> \<forall>a b a'. A a b \<longrightarrow> sim a a' \<longrightarrow> (\<exists>b'. B a' b' \<and> sim b b')"
second_summand_disjointness: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "\<not> O z (x \<oplus> y)" shows "\<not> O z y"
insertion_aux_def: shows "insertion_aux \<equiv> map_fun id (map_fun poly_mapping.lookup id) insertion_fun"
symKeys_neq_imp_neq: fixes K :: "key"   and K' :: "key" assumes "(K \<in> symKeys) \<noteq> (K' \<in> symKeys)" shows "K \<noteq> K'"
prappend_signed_list_assoc_conv_snoc2Cons: fixes xs :: "('a \<times> bool) list"   and y :: "'a \<times> bool"   and ys :: "('a \<times> bool) list" assumes "proper_signed_list (xs @ [y])"   and "proper_signed_list (y # ys)" shows "prappend_signed_list (xs @ [y]) ys = prappend_signed_list xs (y # ys)"
eval_ring_hom: fixes K :: "'a list set"   and a :: "'a list" assumes "subring K (K [X])"   and "a \<in> carrier (K [X])" shows "ring_hom_ring (K [X]\<^bsub>K [X]\<^esub>) (K [X]) (\<lambda>p. UP.eval p a)"
maximalD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and c :: "real" assumes "maximal t"   and "0 \<le> c" shows "t (\<lambda>_. c) = (\<lambda>_. c)"
inputLengthProc: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "inputLength (M\<lparr>\<lambda>*xvec N\<rparr>.P) = length xvec"
t2_range: shows "interpret_floatarith r xs \<in>\<^sub>i range_tm e t2"
avl_delete_2: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "height t \<in> {height (delete x t), height (delete x t) + 1}"
inf_sup_absorb: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"