hcju/isabelleps huggingface dataset:
SUP_eqI: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) x"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) y) \<Longrightarrow> less_eq x y" shows "Sup (f ` A) = x"
commute: fixes a :: "'h"   and b :: "'h" shows "a * b = b * a"
4: shows "sat_precond_as (fmrestrict_set vs s) (rem_condless_act (fmrestrict_set vs s) [] as')"
poly_lift_hom_extends_hom': fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and r :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "r \<in> carrier R" shows "poly_lift_hom R S \<phi> (monom P r 0) = monom (UP S) (\<phi> r) 0"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'g list" shows "enumerate n xs = zip [n..<n + length xs] xs"
sgn_scaleC: fixes r :: "complex"   and x :: "'b" shows "sgn (r *\<^sub>C x) = sgn r *\<^sub>C sgn x"
Ipol_closed: fixes R :: "('c, 'd) ring_scheme"   and l :: "'c list"   and p :: "pol" assumes "cring R"   and "cring.in_carrier R l" shows "cring.Ipol R l p \<in> carrier R"
h_def: shows "h \<equiv> fds (\<lambda>n. fds_nth (fds_ln l f) n * g n)"
gen_cong_def: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and j1 :: "'a"   and j2 :: "'a" assumes "cong rel eval retr" shows "cong.gen_cong rel eval R j1 j2 \<equiv> \<forall>R'. R \<le> R' \<and> cong.cong rel eval R' \<longrightarrow> R' j1 j2"
aforms_relp_internal: fixes R :: "('a list \<times> 'b) set" shows "aforms_relp R = aforms_rel O \<langle>R\<rangle>set_rel"
not_arr_null: shows "\<not> B.VV.arr B.VV.null"
rev_conv_rev_it: fixes xs :: "'a list" shows "rev xs = rev_it xs"
csib: shows "s\<inverse> O b \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
append_eq_appendI: fixes xs :: "'a list"   and xs1 :: "'a list"   and zs :: "'a list"   and ys :: "'a list"   and us :: "'a list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
ChamberComplex_\<Sigma>: shows "ChamberComplex \<Sigma>"
get_offending_flows_def: fixes M :: "'a SecurityInvariant_configured list"   and G :: "'a graph" shows "get_offending_flows M G = (\<Union>m\<in>set M. c_offending_flows m G)"
theother: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and n :: "nat" assumes "validk P c Q n" shows "\<Turnstile>\<^sub>3\<^sub>' {embP3 P \<and>* $ n} c { embP2 Q}"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst_unlabel: fixes S :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)) = unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t S) \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>"
sel_2: fixes x :: "quat"   and y :: "quat" shows "Im1 (x + y) = Im1 x + Im1 y"
opaque_equiv: fixes x :: "'a itrm"   and y :: "'a itrm" assumes "x \<simeq> y" shows "opaque x = opaque y"
det_nfa_accepts: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and w :: "bool list list" assumes "subset_DFS A n"   and "list_all (is_alph n) w" shows "dfa_accepts (det_nfa A) w = nfa_accepts A w"
simps_2: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "real_alg_1 (Irrational n rai) = rai"
assoc'_simps_4: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.dom (S.\<a>' f g h) = S.dom f \<star>\<^sub>S S.dom g \<star>\<^sub>S S.dom h"
sup_ge2: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
BOX_def: shows "BOX \<equiv> \<lambda>h x. cbox x (x + h *\<^sub>R One)"
filter_min_aux_minimal: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and y :: "'b" assumes "transp rel"   and "x \<in> set (General.filter_min_aux rel xs ys)"   and "y \<in> set (General.filter_min_aux rel xs ys)"   and "rel x y"   and "\<And>a b. \<lbrakk>a \<in> set xs \<union> set ys; b \<in> set ys; rel a b\<rbrakk> \<Longrightarrow> a = b" shows "x = y"
comp_min_basis_subset: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (comp_min_basis xs) \<subseteq> set xs"
sum_list_eq_0_iff: fixes ns :: "'a multiset list" shows "(monoid_add.sum_list (+) {#} ns = {#}) = (\<forall>n\<in>set ns. n = {#})"
f_obj: fixes CT :: "nat \<Rightarrow> classDef option" shows "fields(CT,Object) = []"
somegcd_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" shows "somegcd G a b = (SOME x. x \<in> carrier G \<and> x gcdof\<^bsub>G\<^esub> a b)"
nonzero_abs_inverse: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "a \<noteq> zero" shows "abs (inverse a) = inverse (abs a)"
map_simp_ide: fixes a :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "VV.ide a" shows "cmp\<^sub>D\<^sub>N a = \<Psi>\<^sub>o a"
ex_un_null: shows "\<exists>!n. \<forall>f. VxV.comp n f = n \<and> VxV.comp f n = n"
lang\<^sub>M\<^sub>2\<^sub>L_def: fixes n :: "nat"   and \<phi> :: "'a formula" shows "lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) \<phi> \<and> (w, I) \<Turnstile> \<phi>}"
connected_graph_axioms_def: fixes G :: "('a, 'b) graph" shows "connected_graph_axioms G \<equiv> \<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. nodes_connected G v v'"
B'B: fixes x :: "'a" shows "B' x \<subseteq> B x"
vwalk_arcs_Cons: fixes p :: "'a list"   and u :: "'a" assumes "p \<noteq> []" shows "vwalk_arcs (u # p) = (u, hd p) # vwalk_arcs p"
is_monic_setI: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>b. \<lbrakk>b \<in> B; b \<noteq> 0\<rbrakk> \<Longrightarrow> lc b = (1::'c)" shows "is_monic_set B"
singleton_add_means_equal: fixes A :: "'a"   and \<Gamma> :: "'a multiset"   and B :: "'a" assumes "\<LM> A  \<RM> = \<Gamma> \<oplus> B" shows "A = B"
simps_2: fixes v :: "char"   and va :: "char list" shows "lleq_charlist [] (v # va) = True"
r_e2stack_prim: shows "prim_recfn 1 r_e2stack"
quot_to_fract_def: shows "quot_to_fract = (\<lambda>(a, b). Fract a b)"
list_succ_in_conv: fixes xs :: "'b list"   and x :: "'b" shows "(list_succ xs x \<in> set xs) = (x \<in> set xs)"
bisim_inv_\<tau>s2_inv: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and s2' :: "'x2 \<times> 'm2" assumes "bisim_inv"   and "t \<turnstile> s1 \<approx> s2"   and "r2.silent_moves t s2 s2'" obtains s1' :: "'x1 \<times> 'm1" where "t \<turnstile> s1' \<approx> s2'"
leadsTo_UN: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'d set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto> B" shows "F \<in> \<Union> (A ` I) \<longmapsto> B"
keys_subset_indets: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys p" shows "keys t \<subseteq> indets p"
DataRefinement_def: fixes S1 :: "'e \<Rightarrow> 'f"   and R :: "'f \<Rightarrow> 'g"   and R' :: "'e \<Rightarrow> 'h"   and S2 :: "'h \<Rightarrow> 'g" shows "DataRefinement S1 R R' S2 = (R \<circ> S1 \<sqsubseteq> S2 \<circ> R')"
strategyproof: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent" assumes "is_pref_profile R"   and "total_preorder_on alts Ri'"   and "i \<in> agents" shows "\<not> scf R \<prec>[set_ext (R i)] scf (R(i := Ri'))"
ta_bisim_\<epsilon>_1: fixes b :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta' :: "('f, 'a, 'd, 'e, 'g, 'h) thread_action" shows "ta_bisim b \<lbrace>\<rbrace> ta' = (ta' = \<lbrace>\<rbrace>)"
rename_literal_def: fixes A :: "'at set"   and L :: "'at Literal" shows "rename_literal A L = (if atom L \<in> A then complement L else L)"
graph_homomorphism_empty: fixes G :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" shows "graph_homomorphism (LG {} {}) G f = (f = {} \<and> graph G)"
fun_dual1: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f = g \<circ> \<partial>) = (f \<circ> \<partial> = g)"
matchC_MC_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_MC theta d c = (\<forall>s t d' t'. s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> (\<exists>c' s'. (c, s) \<rightarrow>*c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta))"
iteration_finite_conjunctive_axioms: shows "iteration_finite_conjunctive (;) nil"
assms_2: shows "upper_asymptotic_density (B - A) = 0"
Int_subset_iff: fixes C :: "'d set"   and A :: "'d set"   and B :: "'d set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
unit_dvd_iff_3: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (times c b) = dvd.dvd times a c"
add_less_same_cancel1: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b + a \<subset># b) = (a \<subset># \<Empt>)"
assms: shows "\<forall>x y. f x \<bullet> y = x \<bullet> g y"
rel_FGl_neg_distr_cond_def: fixes Co1 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Co1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'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 Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra4 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra4' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and uu :: "('ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao \<times> 'ap \<times> 'aq \<times> 'ar \<times> 'as \<times> 'at \<times> 'au) itself" shows "rel_FGl_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' uu = (\<forall>L1 L1' L2 L2' L3 L3'. rel_FGl (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4') \<le> rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 OO rel_FGl L1' L2' L3' Co1' Co2' Co3' Co4' Contra1' Contra2' Contra3' Contra4')"
proots_within_card_zero_iff: fixes p :: "'a poly"   and A :: "'a set" assumes "p \<noteq> 0" shows "(card (proots_within p A) = 0) = (\<forall>x\<in>A. poly p x \<noteq> (0::'a))"
Int_absorb1: fixes B :: "'c set"   and A :: "'c set" assumes "B \<subseteq> A" shows "A \<inter> B = B"
assms_2: fixes n :: "nat" shows "z n \<in> A"
oconf_fupd: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and h :: "heap"   and v :: "val"   and fs :: "Objects.fields" assumes "P \<turnstile> C has F,NonStatic:T in D"   and "P,h \<turnstile> v :\<le> T"   and "P,h \<turnstile> (C, fs) \<surd>" shows "P,h \<turnstile> (C, fs((F, D) \<mapsto> v)) \<surd>"
cond_case_prod_eta: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'d \<times> 'e \<Rightarrow> 'f" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
leg_in_hom_2: shows "\<guillemotleft>S.leg1 : S.apex \<rightarrow> S.trg\<guillemotright>"
qPsubstAll_preserves_alphaAll2: fixes rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and rho'' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "rho' &= rho''"   and "qGoodEnv rho'"   and "qGoodEnv rho''" shows "(qGood X \<longrightarrow> X #[[rho']] #= X #[[rho'']]) \<and> (qGoodAbs A \<longrightarrow> A $[[rho']] $= A $[[rho'']])"
DynCom: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>c s \<down> Normal s" shows "\<Gamma>\<turnstile>DynCom c \<down> Normal s"
root_conjug: fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r \<cdot> u" shows "u\<inverse>\<^sup>>(r \<cdot> u) \<sim> r"
simps_6: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Neg x2 of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = f2 x2"
simps: fixes v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "left (T v l r) = l"
assms_1: shows "3 \<le> rk {A, A', C'}"
tauLaw2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> \<tau>.P \<approx> \<tau>.P"
discounted_integrable: fixes N :: "bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and r :: "real" assumes "integrable N (X n)"   and "- 1 < r"   and "space N = space M" shows "integrable N (discounted_value r X n)"
IH: fixes x2a :: "('fun, 'atom) Term.term" assumes "x2a \<in> set Y"   and "\<And>f T. Fun f T \<sqsubseteq> x2a \<Longrightarrow> 0 < arity f"   and "wf\<^sub>t\<^sub>r\<^sub>m x2a" shows "\<exists>t. \<Gamma> t = x2a \<and> wf\<^sub>t\<^sub>r\<^sub>m t"
neg_less_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "c < 0" shows "(a < b /\<^sub>R c) = (b < c *\<^sub>R a)"
simps: fixes x :: "nat"   and y :: "nat" shows "(LnChain x < LnChain y) = (x < y)"
weakCongSym: 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 P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P Q" shows "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' Q P"
neutr_eq_iff: fixes a :: "'d"   and b :: "'d" shows "(top = inf a b) = (a = top \<and> b = top)"
action_needed_vars_subset_sys_needed_vars_subset: fixes a :: "('a, 'b) fmap \<times> 'c"   and PROB :: "(('a, 'b) fmap \<times> 'c) set"   and s :: "('a, 'b) fmap" assumes "a \<in> PROB" shows "action_needed_vars a s \<subseteq> system_needed_vars PROB s"
ofilter_under_UNION: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wo_rel r"   and "ofilter r A" shows "A = \<Union> (under r ` A)"
progress_sat_cong: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<sigma>' :: "(char list \<times> 'a list) trace"   and i :: "nat"   and v :: "'a list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < pprogress \<phi> \<pi>" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat \<sigma>' v i \<phi>"
less_admS_def: fixes x :: "'a admS"   and y :: "'a admS" shows "x < y \<equiv> unlr x \<subset> unlr y"
dim_vec_first: fixes v :: "'a vec"   and n :: "nat" shows "dim_vec (vec_first v n) = n"
analz_Un_implConfidSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "G \<subseteq> payload"   and "Enc_keys_clean H" shows "analz ({implConfid A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
apply_swap_same: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<noteq> a"   and "c \<noteq> b" shows "\<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> c = c"
times_eq_iff: fixes A :: "'a set"   and B :: "'b set"   and C :: "'a set"   and D :: "'b set" shows "(A \<times> B = C \<times> D) = (A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
subresultant_prs_gcd_def: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" shows "subresultant_prs_gcd F n \<delta> f k \<beta> G1 G2 \<equiv> subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"
assms_1: shows "\<forall>i j. P i j \<longrightarrow> Q i j"
eSuc_minus_1: fixes n :: "eint" shows "n + 1 - 1 = n"
power2_le_iff_abs_le: fixes y :: "real"   and x :: "real" assumes "0 \<le> y" shows "(x\<^sup>2 \<le> y\<^sup>2) = (\<bar>x\<bar> \<le> y)"
w2: assumes "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k" shows "h - k \<in> orthogonal_complement M \<and> k \<in> M"
no_state_change_if_nonregular_event: fixes ev :: "('p, 's, 'm) event"   and c :: "('p, 's, 'm) configuration"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p" assumes "\<not> regular_event ev"   and "c \<turnstile> ev \<mapsto> c'" shows "states c p = states c' p"
neq_subtrees_empty: fixes t :: "'c tree" shows "subtrees t \<noteq> {}"
nth_drop_0: fixes ss :: "'a list" assumes "0 < length ss" shows "ss ! 0 # drop (Suc 0) ss = ss"
mult_matrix_zero_closed: fixes fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul :: "'c \<Rightarrow> 'd \<Rightarrow> 'b" assumes "fadd (0::'b) (0::'b) = (0::'b)"   and "zero_closed fmul" shows "zero_closed (mult_matrix fmul fadd)"
bw_diff_const_pow: fixes k :: "nat"   and c :: "'a" shows "(bw_diff ^^ k) (\<lambda>_. c) = (if k = 0 then \<lambda>_. c else (\<lambda>_. 0::'a))"
space_as_set_inject: fixes x :: "'b ccsubspace"   and y :: "'b ccsubspace" shows "(space_as_set x = space_as_set y) = (x = y)"
Inf_inners_Inf_inner: shows "(Inf_inners, Inf_inner) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> lv_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
assms_1: shows "graph_homomorphism (fst R) (LG E {0..<n}) f"
RedFAss: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and h' :: "'heap"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "heap_write h a (CField D F) v h'" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>F{D} := Val v,(h, l)\<rangle> -\<lbrace>WriteMem a (CField D F) v\<rbrace>\<rightarrow> \<langle>unit,(h', l)\<rangle>"
hyps: shows "\<T> k u \<subseteq> TokensAt k (\<J> k u)"
uints_bitlen_eq: fixes n :: "nat" shows "uints n = {i. 0 \<le> i \<and> bitlen i \<le> int n}"
star_1l: fixes x :: "'a" shows "x\<^sup>+ \<le> x\<^sup>\<star>"
soundness: fixes r :: "nat rexp"   and s :: "nat rexp" assumes "check_eqv r s" shows "lang r = lang s"
senior_states_psubset_card_mono: fixes A :: "'b set"   and q :: "'b"   and n :: "nat" assumes "A \<subset> senior_states q n" shows "card A < card (senior_states q n)"
q: shows "extreme_bound A (\<sqsubseteq>) {f^n \<bottom> |. n} q"
efficient_code_aux: fixes n :: "nat"   and ps :: "(nat \<times> nat) list" assumes "0 < n"   and "set ps = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"   and "distinct ps" shows "f n = (\<Prod>(p, d)\<leftarrow>ps. f_prime p ^ Suc d)"
symp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "symp r = (\<forall>x y. r x y \<longrightarrow> r y x)"
Process_eq_spec: fixes P :: "'b process"   and Q :: "'b process" shows "(P = Q) = (F P = F Q \<and> D P = D Q)"
median: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "length xs \<le> Suc (2 * d)" shows "is_median (med xs) xs"
fps_hypergeo_def: fixes as :: "'a list"   and bs :: "'a list"   and c :: "'a" shows "fps_hypergeo as bs c = Abs_fps (\<lambda>n. foldl (\<lambda>r a. r * pochhammer a n) (1::'a) as * c ^ n / (foldl (\<lambda>r b. r * pochhammer b n) (1::'a) bs * of_nat (fact n)))"
ord_Zp_one: shows "ord_Zp \<one> = 0"
is_unifier_subst_atm_eqI: fixes AA :: "'a set"   and \<sigma> :: "'s"   and A :: "'a"   and B :: "'a" assumes "finite AA"   and "is_unifier \<sigma> AA"   and "A \<in> AA"   and "B \<in> AA" shows "A \<cdot>a \<sigma> = B \<cdot>a \<sigma>"
assms: shows "continuous_on (cbox (a, c) (b, d)) f"
assms_2: fixes x :: "'b"   and y :: "'b" shows "bob x y \<longrightarrow> bob' x y"
seq_comp_def: fixes P :: "'c option process"   and Q :: "'c option process" shows "P ; Q \<equiv> Abs_process (seq_comp_failures P Q, {})"
tendsto_powr': fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "a \<noteq> 0 \<or> 0 < b \<and> (\<forall>\<^sub>F x in F. 0 \<le> f x)" shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "O[F](f) = O[F](g)"
nderivs_def: shows "nderivs \<equiv> foldl (\<lambda>r a. nderiv a r)"
nat_pow_comm: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> m \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> n"
not_le_imp_less: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
runit_char_3: fixes a :: "'d" assumes "D.ide a" shows "\<exists>!f. \<guillemotleft>f : a \<otimes>\<^sub>D \<I>\<^sub>D \<rightarrow>\<^sub>D a\<guillemotright> \<and> f \<otimes>\<^sub>D \<I>\<^sub>D = (a \<otimes>\<^sub>D \<iota>\<^sub>D) \<cdot>\<^sub>D \<a>\<^sub>D[a, \<I>\<^sub>D, \<I>\<^sub>D]"