hcju/isabelleps huggingface dataset:
a1: shows "get_delayed_pool s = [] \<and> get_trap_set s = {} \<and> snd (execute_instruction () s) = False \<and> s' = snd (fst (execute_instruction () s)) \<and> get_S (cpu_reg_val PSR s) = 0"
op_out12: shows "open (outside (path_image c1 \<union> path_image c2))"
RedT_lift_preserveD: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ttas :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and P :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> bool" assumes "s -\<triangleright>ttas\<rightarrow>* s'"   and "P s"   and "\<And>s t tas s'. \<lbrakk>s -t\<triangleright>tas\<rightarrow> s'; P s\<rbrakk> \<Longrightarrow> P s'" shows "P s'"
hyps_1: shows "ell_field_class.add a p p = Point x\<^sub>1 y\<^sub>1"
pathin_canon_iff: fixes T :: "'a set"   and g :: "real \<Rightarrow> 'a" shows "pathin (top_of_set T) g = (path g \<and> g ` {0..1} \<subseteq> T)"
Bignorabimus_finite: fixes \<alpha> :: "game" shows "finite (Bignorabimus \<alpha>)"
p'm_def: shows "p'm = root (traverse_tree path' (stern_brocot_iterate_aux I))"
inject: fixes nat :: "nat"   and nat' :: "nat" shows "(Suc (nat::nat) = Suc (nat'::nat)) = (nat = nat')"
intros_1: fixes ord :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "ord_option ord None x"
empty: fixes g :: "'b \<Rightarrow> 'a" shows "prod g {} = (1::'a)"
freshAbs_simp: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "freshAbs ys y (Abs xs x X) = (ys = xs \<and> y = x \<or> fresh ys y X)"
PolysD_2: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "p \<in> P[X]" shows "indets p \<subseteq> X"
f: shows "f \<in> Group.iso (relative_homology_group p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0}) (relative_homology_group p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0})"
invmfun_mHom: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "bijec\<^bsub>M,N\<^esub> f"   and "f \<in> mHom R M N" shows "invmfun R M N f \<in> mHom R N M"
sign_simps_5: fixes ah :: "'a"   and bh :: "'a" shows "ah * bh = bh * ah"
ordIso_imp_ordLeq: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "r =o r'" shows "r \<le>o r'"
fmlookup_default_fmmap: fixes d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and M :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup_default d (fmmap f M) x = (if x \<in> fmdom' M then f (fmlookup_default d M x) else d)"
If: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and e1 :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> bval b s} c\<^sub>1 { e1 \<Down> Q}"   and "\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> \<not> bval b s} c\<^sub>2 { e1 \<Down> Q}" shows "\<turnstile>\<^sub>1 {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 { \<lambda>s. e1 s + Suc 0 \<Down> Q}"
action': shows "a' \<in> actions (justifying_exec (J n))"
lprefix_lhdD: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys"   and "\<not> lnull xs" shows "lhd xs = lhd ys"
homology_additivity_explicit: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set"   and p :: "int" assumes "openin X S"   and "openin X T"   and "disjnt S T"   and "S \<union> T = topspace X" shows "(\<lambda>(a, b). hom_induced p (subtopology X S) {} X {} id a \<otimes>\<^bsub>homology_group p X\<^esub> hom_induced p (subtopology X T) {} X {} id b) \<in> Group.iso (homology_group p (subtopology X S) \<times>\<times> homology_group p (subtopology X T)) (homology_group p X)"
zero_def: fixes r :: "('d \<times> 'd) set" assumes "wo_rel r" shows "zero r = minim r (Field r)"
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "C.hseq (fst f) (snd f) = C.VV.arr f"
assms: shows "q $ 0 \<noteq> (0::'a)"
generator_pow_order: fixes G :: "('c, 'd) cyclic_group_scheme" assumes "cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> order G = \<one>\<^bsub>G\<^esub>"
fds_abs_converges_subseries: fixes f :: "'b fds"   and s :: "'b"   and P :: "nat \<Rightarrow> bool" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_subseries P f) s"
sub_addI2: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> \<lambda>s. t P s + t Q s \<tturnstile> t (\<lambda>s. P s + Q s)" shows "sub_add t"
multiset_inter_count: fixes A :: "'k multiset"   and B :: "'k multiset"   and x :: "'k" shows "count (A \<inter># B) x = min (count A x) (count B x)"
Suc: shows "(\<exists>\<^sub>\<infinity>i. P (i - n)) = (\<exists>\<^sub>\<infinity>i. P i)"
Red_dom_lcl: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"
INF_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
meas: fixes i :: "'a"   and j :: "'a" assumes "i \<in> A"   and "j \<in> A" shows "{h \<in> A \<rightarrow>\<^sub>E UNIV. h i = h j} \<in> sets (Pi\<^sub>M A (\<lambda>_. lborel))"
simps_2: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and uw :: "(variable \<times> typ) list"   and ux :: "variable set"   and Hs :: "Core.term list"   and n :: "nat" shows "replay' thy uw ux Hs (PBound n) = partial_nth Hs n"
toSet_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b odlist" shows "toSet (ODList.filter P xs) = rel_ext (\<lambda>x. x \<in> toSet xs \<and> P x)"
comp_retract_closed: fixes G :: "('d, 'e, unit, unit) galcon"   and F :: "('f, 'd, unit, unit) galcon" assumes "retract G"   and "retract F"   and "\<Y>\<^bsub>F\<^esub> = \<X>\<^bsub>G\<^esub>" shows "retract (G \<circ>\<^sub>g F)"
filterlim_filtermap: fixes f :: "'b \<Rightarrow> 'c"   and F1 :: "'c filter"   and g :: "'d \<Rightarrow> 'b"   and F2 :: "'d filter" shows "filterlim f F1 (filtermap g F2) = (LIM x F2. f (g x) :> F1)"
leg_simps_3: shows "C.arr \<epsilon>.cod.leg1"
hcomp_in_hhomE: fixes \<nu> :: "'b"   and \<mu> :: "'b"   and a :: "'b"   and c :: "'b" assumes "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>" obtains "D.arr \<mu>" and "D.arr \<nu>" and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" and "src\<^sub>D \<mu> = a" and "trg\<^sub>D \<nu> = c"
SequentialRacing_def: fixes U :: "'b pupdate set" shows "SequentialRacing U = {u. FoldSet UpdateOverride DefaultUpdate U u}"
remove_vertices_v0_v1_Digraph: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and v :: "'c" assumes "v0_v1_Digraph G v0 v1"   and "v \<noteq> v0"   and "v \<noteq> v1" shows "v0_v1_Digraph (Digraph.remove_vertex G v) v0 v1"
fps_zerol: fixes f :: "('c, 'd) fps" shows "f + 0 = f"
x_fresh_qrho: shows "qFreshEnv xs x (pickE rho)"
strict_not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "x \<prec>[Pareto R'] y" shows "x \<in> alts'"
construct_outside: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B :: "'e set"   and v :: "'e"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "\<not> module.dependent s1 B"   and "v \<in> module.span s1 (vector_space.extend_basis s1 B - B)" shows "vector_space_pair.construct s1 s2 B f v = (0::'f)"
ordertype_infinite_ge_\<omega>: fixes A :: "V set" assumes "infinite A"   and "small A" shows "\<omega> \<le> tp A"
h2: shows "t2 \<noteq> Tip \<and> sortedTree h t2 \<longrightarrow> (\<forall>l\<in>setOf (wrm h t2). h l < h (rm h t2))"
subdg_is_arr_vsubset: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b"
denom_gt_1: fixes x :: "real"   and p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int"   and n :: "nat" assumes "liouville x p q" shows "1 < q n"
base: fixes F :: "'d frame" shows "\<lparr>\<nu>*[]\<rparr>F = F"
Horn_renamable_formula_def: fixes S :: "'at Literal set set" shows "Horn_renamable_formula S = (\<exists>I. all_fulfill Horn (rename_formula I S))"
rel_envT_def: fixes R :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and M :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "local.rel_envT R M = BNF_Def.vimage2p run_env run_env (R ===> M)"
cUNION_cempty: fixes f :: "'d \<Rightarrow> 'c cset" shows "cUnion (cimage f cempty) = cempty"
chine_assoc_in_hom: shows "\<guillemotleft>\<mu>_\<nu>_\<pi>\<rho>.chine_assoc : HH\<mu>\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C H\<mu>H\<nu>H\<pi>\<rho>.chine\<guillemotright>"
x0_le_x1: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p" shows "x\<^sub>0 \<le> x\<^sub>1"
Let_fresh: fixes a :: "atom"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "a \<sharp> Terms.Let \<Gamma> e = (a \<sharp> e \<and> a \<sharp> \<Gamma> \<or> a \<in> atom ` domA \<Gamma>)"
up_add_closed: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "ring R"   and "p \<in> up R"   and "q \<in> up R" shows "(\<lambda>i. p i \<oplus>\<^bsub>R\<^esub> q i) \<in> up R"
pre_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d" shows "pre f a = {x. f x = a}"
invertible_mat: shows "invertible_mat (M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])])"
dom_hcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "local.dom (\<nu> \<star> \<mu>) = local.dom \<nu> \<star> local.dom \<mu>"
inverse_arrows_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "inverse_arrows (assoc a b c) (assoc' a b c)"
exhaust: fixes y :: "'a convnet" obtains x1 :: "nat" where "y = Input x1"   | x21 :: "'a" and x22 :: "'a convnet" where "y = Conv x21 x22"   | x31 :: "'a convnet" and x32 :: "'a convnet" where "y = Pool x31 x32"
geometric_observation_2: shows "(dist X M)\<^sup>2 = (dist C X)\<^sup>2 - (dist M C)\<^sup>2"
mod_sum_eq: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and A :: "'b set" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (comm_monoid_add.sum plus zero (\<lambda>i. modulo (f i) a) A) a = modulo (comm_monoid_add.sum plus zero f A) a"
finite_lattice_complete_Inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "\<forall>x\<in>A. z \<le> x" shows "z \<le> Inf A"
subset_refl: fixes A :: "'a set" shows "order.greater_eq A A"
OrL: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset"   and G :: "'a formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta>"
rel_fundefs_compile_env: fixes e :: "'fenv_std" shows "rel_fundefs (Fstd_get e) (Finca_get (compile_env e))"
identity\<^sub>\<nu>_def: shows "(\<^bold>=) \<equiv> \<lambda>x y. x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P \<^bold>\<or> \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>)"
cons: fixes a :: "'action"   and p :: "'state"   and r :: "'action list" assumes "en a p"   and "path r (ex a p)" shows "path (a # r) p"
unitary_def: fixes P :: "'b \<Rightarrow> real" shows "unitary P = (sound P \<and> bounded_by 1 P)"
span2: shows "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
fun_sum_list_commute: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "f (0::'c) = (0::'b)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
cmp_components_are_iso: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi> (f, g))"
dist: shows "distinct as = (\<forall>a. (\<Sum>x\<leftarrow>as. if x = a then 1 else 0) \<le> Suc 0)"
Loss_L2: fixes t :: "nat"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and lose :: "nat \<Rightarrow> bool list"   and a :: "nat \<Rightarrow> aType list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list" assumes "\<forall>t2<t. i2 t2 = []"   and "Loss lose a i2 y i"   and "ts lose" shows "\<forall>t2<t. i t2 = []"
gen_sa_impl_rel_eext_def: fixes Rm :: "('j \<times> 'm) set"   and Rl :: "('i \<times> ('k \<Rightarrow> 'l)) set" shows "\<langle>Rm, Rl\<rangle>gen_sa_impl_rel_eext = {(\<lparr>sai_L = Li, \<dots> = mi\<rparr>, \<lparr>sa_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
l_prod_elim: fixes w :: "'a list"   and X :: "'a list set"   and Y :: "'a list set" shows "(w \<in> X \<cdot> Y) = (\<exists>u v. w = u @ v \<and> u \<in> X \<and> v \<in> Y)"
list_all2_Cons: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list"   and y :: "'h"   and ys :: "'h list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
exact_decompD_1: fixes m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "exact_decomp m ps"   and "(h, U) \<in> set ps" shows "h \<in> P[X]"
that: fixes x :: "'a" assumes "s x \<noteq> t x"   and "\<And>y. s y \<noteq> t y \<Longrightarrow> x \<le> y" shows "thesis"
induct: fixes P2 :: "'a hterm list \<Rightarrow> bool"   and P1 :: "'a hterm \<Rightarrow> bool"   and hterm_list :: "'a hterm list" assumes "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (HApp x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 hterm_list"
generalized_sfw_def: fixes l :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme" shows "generalized_sfw l p \<equiv> find (\<lambda>(m, a). simple_matches m p) l"
residue_prime_mult_group_has_gen: fixes p :: "nat" assumes "prime p" shows "\<exists>a\<in>{1..p - 1}. {1..p - 1} = {a ^ i mod p |i. i \<in> UNIV}"
wf_exists_minimal: fixes r :: "('b \<times> 'b) set"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "wf r"   and "Q x" shows "\<exists>x. Q x \<and> (\<forall>y. (f y, f x) \<in> r \<longrightarrow> \<not> Q y)"
mkIde_as_mkArr: fixes A :: "'t set" assumes "A \<subseteq> S'.Univ"   and "|A| <o \<AA>" shows "S'.mkArr A A (\<lambda>x. x) = S'.mkIde A"
thetaSeqW_Wretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqW tval aval indis \<subseteq> PL_Indis.Wretr tval aval indis (PL_Indis.thetaSeqW tval aval indis \<union> PL_Indis.Wbis tval aval indis)"
a_i_pos: fixes m :: "int" assumes "0 < m" shows "\<infinity> * ant m = \<infinity>"
star_of_less: fixes x :: "'a"   and y :: "'a" shows "(star_of x < star_of y) = (x < y)"
ereal_times_divide_eq: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" shows "a * (b / c) = a * b / c"
verit_sko_ex_indirect: fixes x :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "x = (SOME x. P x)" shows "(\<exists>x. P x) = P x"
rtranclp_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
filter_min_relE: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "transp rel"   and "reflp rel"   and "x \<in> set xs" obtains y :: "'b" where "y \<in> set (General.filter_min rel xs)" and "rel y x"
enn2ereal_inject: fixes x :: "ennreal"   and y :: "ennreal" shows "(enn2ereal x = enn2ereal y) = (x = y)"
disj_sets_disj_Edges2: fixes ys :: "nat list"   and xs :: "nat list" assumes "set ys \<inter> set xs = {}" shows "Edges xs \<inter> Edges ys = {}"
interaction_bound_Fail: fixes "consider" :: "'a \<Rightarrow> bool" shows "interaction_bound consider Fail = 0"
bit_and_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (and a b) n = (bit a n \<and> bit b n)"
agree_supset: fixes B :: "('a + 'a) set"   and A :: "('a + 'a) set"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<nu>' :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "B \<subseteq> A"   and "Vagree \<nu> \<nu>' A" shows "Vagree \<nu> \<nu>' B"
collapse_2: fixes unboxed :: "('a, 'b, 'c) unboxed" assumes "is_OpUbx1 unboxed" shows "OpUbx1 (un_OpUbx1 unboxed) = unboxed"
tendsto_0_le: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'd"   and K :: "real" assumes "(f \<longlongrightarrow> (0::'c)) F"   and "\<forall>\<^sub>F x in F. norm (g x) \<le> norm (f x) * K" shows "(g \<longlongrightarrow> (0::'d)) F"
lb_set_def: fixes X :: "'b set" shows "lb_set X = {y. \<forall>x\<in>X. y \<le> x}"
bs_sub: shows "fst ` set bs \<subseteq> dgrad_p_set d m"
flow_eq_rev_2: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "flow0 x0 (t - t0) = ll_on_open.flow (range (preflect t0)) (\<lambda>t. - f) X t0 x0 (preflect t0 t)"
defined_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<delta> X \<equiv> \<lambda>\<tau>. if X \<tau> = \<bottom> \<tau> \<or> X \<tau> = null \<tau> then false \<tau> else true \<tau>"
rel_uprod_simps: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'d"   and d :: "'d" shows "rel_uprod R (Upair a b) (Upair c d) = (R a c \<and> R b d \<or> R a d \<and> R b c)"
PR2: fixes p :: "'a pfm"   and i :: "'a" assumes "\<turnstile>\<^sub>! p" shows "\<turnstile>\<^sub>! K\<^sub>! i p"
ContractsWithUnilateralSubstitutesAndIRCAndLAD_axioms: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"
TESL_interp_left_idem: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
less_induct: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
21: shows "padic_mult p f g \<noteq> padic_zero p"
Ln_inverse: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Ln (inverse z) = - Ln z"
Eq\<^sub>2_prop: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2"   and \<phi> :: "\<Pi>\<^sub>2 \<Rightarrow> \<o>" assumes "[F \<^bold>=\<^sub>2 G in v]" shows "[\<phi> F in v] = [\<phi> G in v]"
lsided_eq_var1: fixes x :: "'a" shows "(x \<le> \<top> \<rightarrow> x) = (x = \<top> \<rightarrow> x)"
Rep_slist_inverse: fixes x :: "('d, 'e) slist" shows "Abs_slist (Rep_slist x) = x"
ldistinct_conv_lnth: fixes xs :: "'b llist" shows "ldistinct xs = (\<forall>i j. enat i < llength xs \<longrightarrow> enat j < llength xs \<longrightarrow> i \<noteq> j \<longrightarrow> xs $ i \<noteq> xs $ j)"
disc_8: shows "\<not> is_SAT_PUSH' NONSAT_PUSH'"
pmf_None_eq_weight_spmf: fixes p :: "'b spmf" shows "pmf p None = 1 - weight_spmf p"
tree_rank_height: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.tree_invar (BinomialHeapStruc.Node e a r ts)" shows "BinomialHeapStruc.height_tree (BinomialHeapStruc.Node e a r ts) = r"
cases: fixes x :: "int poly \<times> rat \<times> rat" obtains p :: "int poly" and uu_ :: "rat" and uv_ :: "rat" where "x = (p, uu_, uv_)"
replace_col_def: fixes A :: "'a mat"   and b :: "'a vec"   and k :: "nat" shows "replace_col A b k = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if j = k then b $ i else A $$ (i, j))"
cases: fixes ps :: "'d \<times> 'b"   and a :: "'d" assumes "scopep ps a" obtains v :: "'d" where "a = v" and "v |\<in>| vertices" and "\<And>pth v'. \<lbrakk>path v v' pth; terminal_vertex v'\<rbrakk> \<Longrightarrow> ps \<in> snd ` set pth"