hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and a2 :: "('a, 'b) Language"   and a3 :: "('c, 'd) IType" assumes "L\<lbrakk>a2\<rbrakk>\<^bsub>a1\<^esub> \<rightarrow> a3" obtains I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme" and A :: "'a" where "a1 = I__" and "a2 = \<turnstile> A__ Type" and "a3 = IObj Ty\<lbrakk>A__\<rbrakk>\<^bsub>I__\<^esub>" and "Sig iS\<^bsub>I__\<^esub> \<triangleright> \<turnstile> A__ Type"   | I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme" and A :: "'a" and c :: "'c" where "a1 = Ia__" and "a2 = Vx : Aa__ \<turnstile> Vx : Aa__" and "a3 = IMor (id\<^bsub>iC\<^bsub>Ia__\<^esub>\<^esub> c)" and "L\<lbrakk>\<turnstile> Aa__ Type\<rbrakk>\<^bsub>Ia__\<^esub> \<rightarrow> IObj c"   | I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme" and A :: "'a" and e :: "('a, 'b) Expression" and B :: "'a" and f :: "'b" and C :: "'a" and g :: "'d" where "a1 = Ib__" and "a2 = Vx : Ab__ \<turnstile> f E@ e : C" and "a3 = IMor (g ;;\<^bsub>iC\<^bsub>Ib__\<^esub>\<^esub> Fn\<lbrakk>f\<rbrakk>\<^bsub>Ib__\<^esub>)" and "Sig iS\<^bsub>Ib__\<^esub> \<triangleright> Vx : Ab__ \<turnstile> e : B__" and "f \<in> Sig iS\<^bsub>Ib__\<^esub> : B__ \<rightarrow> C" and "L\<lbrakk>Vx : Ab__ \<turnstile> e : B__\<rbrakk>\<^bsub>Ib__\<^esub> \<rightarrow> IMor g"   | I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme" and A :: "'a" and e1 :: "('a, 'b) Expression" and B :: "'a" and g1 :: "'d" and e2 :: "('a, 'b) Expression" and g2 :: "'d" where "a1 = I" and "a2 = Vx : A \<turnstile> e1\<equiv>e2 : B" and "a3 = IBool (g1 = g2)" and "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g1" and "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g2"
det_scalar: fixes wa :: "'a vec" assumes "wa \<in> carrier_vec n" shows "det (A wa) = wa \<bullet> nv"
X: fixes x :: "'a set" assumes "x \<in> X" shows "x \<in> sets (sigma UNIV (F ` A))"
bdd_aboveI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and M :: "'d" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x M" shows "preorder.bdd_above less_eq A"
simps_1: fixes h :: "'b hp option" shows "merge h None = h"
Suc_Suc_div: fixes s :: "nat"   and n :: "nat" assumes "0 < s"   and "s mod 2 = Suc 0"   and "Suc (Suc s) \<le> 2 * n" shows "Suc (Suc (s div 2)) \<le> n"
separate_closed_compact: fixes s :: "'b set"   and t :: "'b set" assumes "closed s"   and "compact t"   and "s \<inter> t = {}" shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
neg_closed: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "- x \<in> V"
iso_class_def: fixes G :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "group.iso_class G = Abs_group_iso (Abs_group (monoid.truncate G))"
currentD_weight_OUT: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f" shows "d_OUT f x \<le> weight \<Gamma> x"
SSN_intro: fixes k :: "stack" assumes "\<And>k'.  k \<mapsto> k'  \<Longrightarrow> SSN k'" shows "SSN k"
Diff_idemp: fixes A :: "'e set"   and B :: "'e set" shows "A - B - B = A - B"
skip_doesnt_access: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and x :: "'Var" assumes "lc \<in> loc_reach \<langle>Skip \<otimes> annos, mds, mem\<rangle>"   and "lc = \<langle>c', mds', mem'\<rangle>" shows "doesnt_read c' x \<and> doesnt_modify c' x"
judge_weak_fresh_env_fresh_term: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty"   and a :: "Syntax.var" assumes "\<Gamma> \<turnstile>\<^sub>W e : \<tau>"   and "atom a \<sharp> \<Gamma>" shows "atom a \<sharp> e"
sep_disj_addD2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x z"
hom_induced_compose_empty: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and Z :: "'c topology"   and g :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "continuous_map X Y f"   and "continuous_map Y Z g" shows "hom_induced p X {} Z {} (g \<circ> f) = hom_induced p Y {} Z {} g \<circ> hom_induced p X {} Y {} f"
msg_fresh_def: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" shows "msg_fresh \<sigma> m \<equiv> case m of Rreq hopsc x xa xb xc oipc osnc sipc xd \<Rightarrow> 1 \<le> osnc \<and> (sipc \<noteq> oipc \<longrightarrow> oipc \<in> kD (rt (\<sigma> sipc)) \<and> osnc \<le> nsqn (rt (\<sigma> sipc)) oipc \<and> (nsqn (rt (\<sigma> sipc)) oipc = osnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) oipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) oipc) = Aodv_Basic.inv)) | Rrep hopsc dipc dsnc x sipc \<Rightarrow> 1 \<le> dsnc \<and> (sipc \<noteq> dipc \<longrightarrow> dipc \<in> kD (rt (\<sigma> sipc)) \<and> dsnc \<le> nsqn (rt (\<sigma> sipc)) dipc \<and> (nsqn (rt (\<sigma> sipc)) dipc = dsnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) dipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) dipc) = Aodv_Basic.inv)) | Rerr destsc sipc \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> kD (rt (\<sigma> sipc)) \<and> the (destsc ripc) - 1 \<le> nsqn (rt (\<sigma> sipc)) ripc | _ \<Rightarrow> True"
mult_le_mono: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i \<le> j"   and "k \<le> l" shows "i * k \<le> j * l"
partition_implies_permutes: fixes A :: "'c set"   and P :: "'c set set"   and P' :: "'c set set" assumes "finite A"   and "partition_on A P"   and "partition_on A P'"   and "image_mset card (mset_set P') = image_mset card (mset_set P)" obtains p :: "'c \<Rightarrow> 'c" where "p permutes A" and "P' = (`) p ` P"
winding_number_zero: assumes "simply_connected S" shows "connected S \<and> (\<forall>\<gamma> z. path \<gamma> \<and> path_image \<gamma> \<subseteq> S \<and> pathfinish \<gamma> = pathstart \<gamma> \<and> z \<notin> S \<longrightarrow> winding_number \<gamma> z = 0)"
finite_card_of_iff_card3: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(|A| <o |B|) = (card A < card B)"
fun_ext: fixes f :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V"   and g :: "V" assumes "f \<in> elts (VPi A B)"   and "g \<in> elts (VPi A B)"   and "\<And>x. x \<in> elts A \<Longrightarrow> app f x = app g x" shows "f = g"
assms_1: shows "d2 = ((e2 + q - x) mod q * b + (q - r)) mod q"
iso_iff_section_and_epi: fixes f :: "'a \<times> 'a" shows "VV.iso f = (VV.section f \<and> VV.epi f)"
abs_eq: fixes xa :: "\<kappa>"   and x :: "\<Pi>\<^sub>1" shows "\<lbrace>xa,x\<rbrace> = make\<o> (\<lambda>s w. proper xa \<and> (case rep xa of \<omega>\<nu> \<omega> \<Rightarrow> False | \<alpha>\<nu> \<alpha> \<Rightarrow> x \<in> \<alpha>))"
permute_set_eq: fixes p :: "perm"   and X :: "'c set" shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
sigma_algebra_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "sigma_algebra \<Omega> M \<equiv> algebra \<Omega> M \<and> sigma_algebra_axioms M"
WT_red_external_list_conv: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and U :: "ty"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "wf_prog wf_md P"   and "P,h \<turnstile> a\<bullet>M(vs) : U"   and "P,h \<turnstile> t \<surd>t" shows "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle> = ((ta, va, h') \<in> red_external_aggr P t a M vs h)"
pir1J1: shows "r1 \<inter> path_image J1 = {}"
size_roots_le_degree: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)" shows "size (roots p) \<le> degree p"
Fvars_all: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "Fvars (all x \<phi>) = Fvars \<phi> - {x}"
verts: shows "g_V G = [0..<length (g_V G)]"
ltl_step_some: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and s' :: "nat"   and t :: "transition"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option" assumes "possible_steps e s r l i = {|(s', t)|}"   and "evaluate_outputs t i r = p"   and "evaluate_updates t i r = r'" shows "ltl_step e (Some s) r (l, i) = (Some s', p, r')"
is_final_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" shows "relation.is_final r a \<equiv> \<nexists>b. r a b"
Induction_axP_graph_aux_def: shows "Induction_axP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>Induction_axP_aux ax p v w x x0 xw xevw allw allvw. x1 = p \<and> x2 = SyntaxN.Ex v (SyntaxN.Ex w (SyntaxN.Ex x (SyntaxN.Ex x0 (SyntaxN.Ex xw (SyntaxN.Ex xevw (SyntaxN.Ex allw (SyntaxN.Ex allvw (SyntaxN.Ex ax (Var v NEQ Var w AND VarNonOccFormP (Var w) (Var x) AND SubstFormP (Var v) Zero (Var x) (Var x0) AND SubstFormP (Var v) (Var w) (Var x) (Var xw) AND SubstFormP (Var v) (Q_Eats (Var v) (Var w)) (Var x) (Var xevw) AND AbstFormP (Var w) Zero (Q_Imp (Var x) (Q_Imp (Var xw) (Var xevw))) (Var allw) AND AbstFormP (Var v) Zero (Q_All (Var allw)) (Var allvw) AND AbstFormP (Var v) Zero (Var x) (Var ax) AND p EQ Q_Imp (Var x0) (Q_Imp (Q_All (Var allvw)) (Q_All (Var ax)))))))))))) \<and> atom ax \<sharp> (p, v, w, x, x0, xw, xevw, allw, allvw) \<and> atom allvw \<sharp> (p, v, w, x, x0, xw, xevw, allw) \<and> atom allw \<sharp> (p, v, w, x, x0, xw, xevw) \<and> atom xevw \<sharp> (p, v, w, x, x0, xw) \<and> atom xw \<sharp> (p, v, w, x, x0) \<and> atom x0 \<sharp> (p, v, w, x) \<and> atom x \<sharp> (p, v, w) \<and> atom w \<sharp> (p, v) \<and> atom v \<sharp> p)"
contract_conj_inclusion_variant: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "K \<div> (\<phi> .\<and>. \<psi>) \<subseteq> K \<div> \<phi> \<or> K \<div> (\<phi> .\<and>. \<psi>) \<subseteq> K \<div> \<psi>"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and list :: "'c list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
Liminf_llist_LCons: fixes X :: "'b set"   and Xs :: "'b set llist" shows "Liminf_llist (LCons X Xs) = (if lnull Xs then X else Liminf_llist Xs)"
Gibbard_Satterthwaite: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt" assumes "GS f" shows "\<exists>i. dict f i"
prems_1: shows "pairwise coprime (set (y # ys))"
mult_scalar_distrib_left: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and r :: "'b \<Rightarrow>\<^sub>0 'd"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_scalar pair_of_term term_of_pair r (p + q) = term_powerprod.mult_scalar pair_of_term term_of_pair r p + term_powerprod.mult_scalar pair_of_term term_of_pair r q"
lub_is_join: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "{x, y} <<| z" shows "x \<squnion> y = z"
eq_implies: fixes x :: "'c"   and y :: "'c" assumes "x = y" shows "y \<le> x"
flip_commute: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
nz_le_conv_less: fixes k :: "nat"   and m :: "nat" assumes "0 < k"   and "k \<le> m" shows "k - Suc 0 < m"
per_union_subs: fixes x :: "'b"   and S :: "'b set"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "x \<in> S"   and "y \<in> S"   and "R \<subseteq> S \<times> S" shows "per_union R x y \<subseteq> S \<times> S"
lim: fixes f :: "nat \<Rightarrow> ordinal" assumes "OrdinalInduct.strict_mono f" shows "F (oLimit f) = oLimit (\<lambda>n. F (f n))"
length_le_list_ex: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "\<exists>ys. length (xs @ ys) = n"
assms_7: shows "domain.min_gallery (C # Cs @ [D])"
UN_extend_simps_7: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "A - \<Inter> (B ` C) = (\<Union>x\<in>C. A - B x)"
div_combine: 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 p :: "'a"   and r :: "nat"   and n :: "'a"   and a :: "nat"   and k :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times (power.power one times p (Suc r)) n"   and "dvd.dvd times (power.power one times p (a + r)) (times n k)" shows "dvd.dvd times (power.power one times p a) k"
dord_fun_plus_monotone: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "ord s t \<Longrightarrow> ord (s + u) (t + u)"   and "finite (supp_fun s)"   and "finite (supp_fun t)"   and "finite (supp_fun u)"   and "dord_fun ord s t" shows "dord_fun ord (s + u) (t + u)"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>B c\<guillemotright>" shows "B.seq g f"
foldg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ThinChamberComplexFolding X g"
natUnion_superset: fixes g :: "nat \<Rightarrow> 'a set"   and i :: "nat" shows "g i \<subseteq> natUnion g"
vinsert_vinsert: fixes a :: "V"   and b :: "V"   and C :: "V" shows "vinsert a (vinsert b C) = set {a, b} \<union>\<^sub>\<circ> C"
euclidean_size_gcd_less1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a b" shows "euclidean_size (gcd a b) < euclidean_size a"
ns: shows "non_speculative P (\<lambda>_. {}) (ltake (enat i) (lmap snd E))"
tail_is_vert: fixes G :: "nat graph_rec"   and u :: "nat"   and v :: "nat" assumes "cfg_doms G"   and "(u, v) \<in> g_E G" shows "u \<in> set (g_V G)"
p_max_def: shows "other.p_max = other.p other.t_stop"
join_equality: fixes x :: "'a"   and y :: "'a"   and s :: "'a" assumes "is_sup x y s" shows "x \<squnion> y = s"
sinh_real_nonneg_iff: fixes x :: "real" shows "(0 \<le> sinh x) = (0 \<le> x)"
poincare_bendixson: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "(0::'a) \<notin> f ` \<omega>_limit_set x" obtains y :: "'a" where "periodic_orbit y" and "range (flow0 y) = \<omega>_limit_set x"
tendsto_mult: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'f"   and F :: "'e filter"   and g :: "'e \<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"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (- x) = (\<lambda>xa. - (x *\<^sub>V xa))"
less_fun_trichotomy: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {k. f k \<noteq> g k}" shows "less_fun f g \<or> f = g \<or> less_fun g f"
1: fixes name :: "String.literal"   and ars :: "(String.literal \<times> String.literal set list) list"   and c1 :: "String.literal"   and ss1 :: "String.literal set list"   and c2 :: "String.literal"   and ss2 :: "String.literal set list" assumes "(name, ars) \<in> set arss"   and "(c1, ss1) \<in> set ars"   and "(c2, ss2) \<in> set ars"   and "class_leq (set cs) c1 c2" shows "list_all2 (sort_leq (set cs)) (the (lookup (\<lambda>x. x = c1) ars)) (the (lookup (\<lambda>x. x = c2) ars))"
IH: assumes "const_list ves" shows "\<exists>vs. ves = $$* vs"
size_fset_of_list: fixes l :: "'b list" shows "size (fset_of_list l) = length (remdups l)"
order_trans_rules_23: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
interval_integral_sum: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal"   and f :: "real \<Rightarrow> 'b" assumes "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f" shows "interval_lebesgue_integral lborel a b f + interval_lebesgue_integral lborel b c f = interval_lebesgue_integral lborel a c f"
denseE: fixes T :: "'b set set"   and A :: "'b set" assumes "A dense\<^bsub>T\<^esub>" obtains "carrier\<^bsub>T\<^esub> \<subseteq> closure\<^bsub>T\<^esub> A"
pre: shows "pre_subdivFace' g f u v n (ov # ovs)"
chaum_ped_\<Sigma>_def: fixes \<G> :: "'a cyclic_group" shows "chaum_ped_\<Sigma> \<G> \<equiv> chaum_ped_\<Sigma>_base \<G> \<and> Cyclic_Group.cyclic_group \<G>"
tiling_UnI: fixes t :: "'a set"   and A :: "'a set set"   and u :: "'a set" assumes "t \<in> tiling A"   and "u \<in> tiling A"   and "t \<inter> u = {}" shows "t \<union> u \<in> tiling A"
FWbisimulation_base_aux_flip: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics" assumes "FWbisimulation_base_aux final1 r1 final2 r2" shows "FWbisimulation_base_aux final2 r2 final1 r1"
poly_exp_constant_bound: fixes c :: "'a"   and deg :: "nat" shows "\<exists>p. \<forall>x. c * b ^ x * of_nat x ^ deg \<le> p"
presSubst_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSubst h (fromMOD MOD)" shows "presSubst (fromIMor h) MOD"
cInf_greatest: fixes X :: "'a multiset set"   and z :: "'a multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "z \<subseteq># \<Sqinter> X"
char_matrix_def: fixes A :: "'b mat"   and e :: "'b" shows "char_matrix A e = A + - e \<cdot>\<^sub>m 1\<^sub>m (dim_row A)"
step_1_def: fixes A :: "'b mat" shows "step_1 A = step_1_main (dim_row A) 0 0 A"
star_of_simps_14: fixes y :: "'a" shows "(0 < star_of y) = ((0::'a) < y)"
assms_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R x y"   and "R y z" shows "R x z"
BS4: fixes g :: "'a"   and w :: "'a" assumes "is_left_adjoint g"   and "ide w"   and "is_left_adjoint (g \<star> w)" shows "is_left_adjoint w"
parallel_HSem_ind: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and \<rho>\<^sub>1 :: "'b \<Rightarrow> 'c"   and \<Gamma>\<^sub>1 :: "('b \<times> 'a) list"   and \<rho>\<^sub>2 :: "'b \<Rightarrow> 'c"   and \<Gamma>\<^sub>2 :: "('b \<times> 'a) list" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>1 \<^bold>\<rbrakk>\<^bsub>y\<^esub>) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>2 \<^bold>\<rbrakk>\<^bsub>z\<^esub>)" shows "P (\<lbrace> \<Gamma>\<^sub>1 \<rbrace>\<rho>\<^sub>1) (\<lbrace> \<Gamma>\<^sub>2 \<rbrace>\<rho>\<^sub>2)"
segment_altdef: fixes b :: "'c"   and f :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "b \<in> orbit f a" shows "segment f a b = (\<lambda>n. (f ^^ n) a) ` {1..<funpow_dist1 f a b}"
connect_def: fixes d :: "('i, 'j) distinguisher"   and res :: "('i, 'j) resource" shows "connect d res = run_gpv run_resource d res"
comp_runit_runit'_2: fixes a :: "'a" assumes "ide a" shows "C\<^sub>1.runit' a \<cdot> \<r>\<^sub>1[a] = a \<otimes>\<^sub>1 \<I>\<^sub>1"
Ionescu_Tulcea_axioms: shows "Ionescu_Tulcea (K' x) (\<lambda>_. M)"
orthogonal_col_of_unitary: fixes U :: "complex mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "unitary U"   and "dim_col U = n"   and "i < n"   and "j < n"   and "i \<noteq> j" shows "\<langle>col U i|col U j\<rangle> = 0"
UP_pre_univ_prop_fact: shows "UP_pre_univ_prop R R (\<lambda>x. x)"
AGM_Contraction_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set" shows "AGM_Contraction Cn contraction \<equiv> Tarskian_logic Cn \<and> AGM_Contraction_axioms Cn contraction"
assms: shows "\<langle>r, A\<rangle> \<in>\<^sub>\<circ> D"
dgrad_p_set_closed_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m" shows "p + q \<in> dgrad_p_set d m"
prj_simps_4: shows "C.arr \<mu>\<nu>.prj\<^sub>0"
safe_def: fixes c :: "('loc, 't) configuration" shows "safe c \<equiv> \<forall>loc1 loc2 t s. 0 < zcount (c_pts c loc1) t \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<longrightarrow> (\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c loc2))"
simps_20: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x41 :: "'b"   and x42 :: "'c" shows "(case In x41 x42 of Fo x \<Rightarrow> f1 x | Z x \<Rightarrow> f2 x | Less x xa \<Rightarrow> f3 x xa | In x xa \<Rightarrow> f4 x xa) = f4 x41 x42"
subst_ground_ident: fixes t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv t = {}" shows "t \<cdot> s = t"
mset_le_add_mset_decr_left2: fixes c :: "'c"   and a :: "'c multiset"   and b :: "'c multiset" assumes "add_mset c a \<subseteq># b" shows "{#c#} \<subseteq># b"
cofinite_extremally_disconnected_aux: fixes X :: "nat cofinite set" shows "closure (interior (closure X)) \<subseteq> interior (closure X)"
restrict_count_space: fixes B :: "'c set"   and A :: "'c set" shows "restrict_space (count_space B) A = count_space (A \<inter> B)"
valid_abs: fixes M :: "msg" assumes "M \<in> valid" shows "\<exists>M'. M' \<in> local.abs {M}"
test_propI: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "test x"   and "P x" shows "P (t x)"
alw_ev_cong: fixes P :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and S :: "'a stream \<Rightarrow> bool"   and R :: "'a stream \<Rightarrow> bool" assumes "alw P xs"   and "\<And>x. P x \<Longrightarrow> S x = R x" shows "alw (ev S) xs = alw (ev R) xs"