hcju/isabelleps huggingface dataset:
prj_simps_2: shows "C.seq \<mu>\<nu>.prj\<^sub>0 \<p>\<^sub>1[\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>.leg1]"
ipcidr_conjunct_correct: fixes b1 :: "'a word"   and m1 :: "nat"   and b2 :: "'a word"   and m2 :: "nat" shows "(case ipcidr_conjunct (b1, m1) (b2, m2) of None \<Rightarrow> {} | Some (bx, mx) \<Rightarrow> ipset_from_cidr bx mx) = ipset_from_cidr b1 m1 \<inter> ipset_from_cidr b2 m2"
changes_le_smods_ext_def: fixes b :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_le_smods_ext b p q = (let ps = smods_ext p q in changes_poly_neg_inf ps - changes_poly_at ps b)"
a: shows "a \<notin> \<real>\<^sub>\<le>\<^sub>0"
abs_eq: fixes x :: "int" shows "Code_Numeral.dup (integer_of_int x) = integer_of_int (x + x)"
is_ubI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and u :: "'a" assumes "class.po below"   and "\<And>x. x \<in> S \<Longrightarrow> below x u" shows "po.is_ub below S u"
And_int_preservation: fixes n :: "nat"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "dbm_int M1 n"   and "dbm_int M2 n" shows "dbm_int (And M1 M2) n"
eq_Eq_conv: fixes x :: "'a"   and y :: "'a" shows "(comparator_of x y = Eq) = (x = y)"
eNorm_triangular_ineq: fixes N :: "'a quasinorm"   and x :: "'a"   and y :: "'a" shows "eNorm N (x + y) \<le> ennreal (defect N) * eNorm N x + ennreal (defect N) * eNorm N y"
fold_set_fold: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and y :: "'h"   and xs :: "'g list" assumes "comp_fun_idem f" shows "Finite_Set.fold f y (set xs) = fold f xs y"
last_ConsR: fixes xs :: "'a list"   and x :: "'a" assumes "xs \<noteq> []" shows "last (Cons x xs) = last xs"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "linear f"   and "bounded_linear_axioms f" shows "bounded_linear f"
steps_empty_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and cs :: "(char list \<Rightarrow> int) set acom"   and cs' :: "(char list \<Rightarrow> int) set acom"   and n :: "nat" assumes "step S cs = cs"   and "cs' \<le> cs" shows "(step {} ^^ n) cs' \<le> cs"
atLeastAtMost_shift_bounds: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..n + k} = sum (g \<circ> (+) k) {m..n}"
union_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
I_def: shows "I \<equiv> $sem = #1 \<and> $pc1 = #a \<and> $pc2 = #a \<or> $sem = #0 \<and> ($pc1 = #a \<and> $pc2 \<in> {#b, #g} \<or> $pc2 = #a \<and> $pc1 \<in> {#b, #g})"
gFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshCls MOD \<equiv> gFreshGVar MOD \<and> gFreshGAbs1 MOD \<and> gFreshGAbs2 MOD \<and> gFreshGOp MOD"
Apps_lam_induct: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and t :: "lam" assumes "\<And>n ts z. (\<And>z. \<forall>t\<in>set ts. P z t) \<Longrightarrow> P z (Var n \<degree>\<degree> ts)"   and "\<And>x u ts z. \<lbrakk>x \<sharp> z; \<And>z. P z u; \<And>z. \<forall>t\<in>set ts. P z t\<rbrakk> \<Longrightarrow> P z ((Lam [x].u) \<degree>\<degree> ts)" shows "P z t"
mx: shows "max \<bar>c\<bar> (maxcoeff t) = 0"
Nats_less_whn: fixes n :: "nat star" assumes "n \<in> \<nat>" shows "n < whn"
group_part: fixes p :: "'a"   and xs :: "'a list" shows "group_part p xs = (rev (filter (\<lambda>x. f x < f p) xs), fold merge (filter (\<lambda>x. f x = f p) xs) p, rev (filter (\<lambda>x. f p < f x) xs))"
npClose: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<in> carrier R"
comp_assoc: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'g"   and h :: "'d \<Rightarrow> 'f" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
extract_IPSets_length: fixes rs :: "'b simple_rule list" shows "length (extract_IPSets rs) \<le> 2 * length rs"
simps: fixes al :: "abc_inst list"   and bl :: "abc_inst list" shows "al [+] bl = (let al_len = length al in al @ abc_shift bl al_len)"
weight_w0: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "w0 \<le> weight_fun.weight w w0 scf t"
INFM_conjI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x"   and "\<forall>\<^sub>\<infinity>x. Q x" shows "\<exists>\<^sub>\<infinity>x. P x \<and> Q x"
SystemKCut_def: shows "SystemKCut = (([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]](TT && Geq (f0 fid1) (Const 0)) \<rightarrow> [[DiffAssign vid1 (f0 fid1)]]Geq (Differential (Var vid1)) (Differential (Const 0))) \<rightarrow> ([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]]TT && Geq (f0 fid1) (Const 0)) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]][[DiffAssign vid1 (f0 fid1)]]Geq (Differential (Var vid1)) (Differential (Const 0)))"
intro: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symmetric A equiv"   and "partial_equivalence_axioms A equiv" shows "partial_equivalence A equiv"
backflows_def: fixes E :: "('b \<times> 'b) set" shows "backflows E \<equiv> {(r, s). (s, r) \<in> E}"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (C.tensor (cod \<iota>))"
distinct_zipunzip: fixes xs :: "'a list"   and ys :: "'b list" assumes "distinct xs" shows "distinct (map fst (zip xs ys))"
cases: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and a1 :: "(nat \<times> 'b) list"   and a2 :: "nat \<times> 'b" assumes "gpath A w V a1 a2" obtains p :: "nat \<times> 'b" where "a1 = []" and "a2 = p__"   | a :: "nat \<times> 'b" and p :: "nat \<times> 'b" and r :: "(nat \<times> 'b) list" where "a1 = a # r" and "a2 = p" and "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" and "gpath A w V r a"
frac_le: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "(0::'a) \<le> x"   and "x \<le> y"   and "(0::'a) < w"   and "w \<le> z" shows "x / z \<le> y / w"
desargues_config_2D_non_collinear_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, C, C'} = 3"
distinct_277: fixes x13 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Roll x13 \<noteq> Unroll x14"
process_rule_2_traces: fixes xs :: "'c list"   and xs' :: "'c list"   and P :: "'c process" assumes "xs @ xs' \<in> traces P" shows "xs \<in> traces P"
extreme_bound_bound: fixes A :: "'b set"   and X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "extreme_bound A less_eq X y"   and "x \<in> X" shows "less_eq x y"
shiftl_x_0: fixes x :: "'a word" shows "x << 0 = x"
enum_Suc: fixes i :: "nat" assumes "i < n" shows "enum (Suc i) = (enum i)(upd i := Suc (enum i (upd i)))"
strictly_maximal_literal_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and L :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.strictly_maximal_literal trm_ord C L \<sigma> = (\<forall>x\<in>cl_ecl C - {L}. (subst_lit x \<sigma>, subst_lit L \<sigma>) \<in> basic_superposition.lit_ord trm_ord)"
rep_eq: fixes x :: "'d"   and xa :: "'d cset" shows "cin x xa = (x \<in> rcset xa)"
mult_zero_l: fixes a :: "nat \<Rightarrow> int" shows "a \<otimes> \<zero> = \<zero>"
2: shows "f ` set (x # xs) \<inter> f ` set ys = {}"
sum_nonneg: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> (0::'a) \<le> f x" shows "(0::'a) \<le> sum_0 f A"
finite_combination_2: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" assumes "finite (LID\<^sub>S \<sigma>)"   and "finite (LID\<^sub>S \<tau>)" shows "finite (LID\<^sub>S (\<sigma>;;\<tau>))"
ordinal_oLog_0: fixes b :: "ordinal" shows "oLog b 0 = 0"
distinct_3: fixes x1 :: "nat"   and x3 :: "nat" shows "Regions.intv.Const x1 \<noteq> Regions.intv.Greater x3"
inconsistent_imp_rank_less: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "inconsistent A b" shows "rank A < (if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b)) + 1 else 0)"
lm085: fixes X :: "'d set" shows "X = (\<lambda>x. x \<in> X) -` {True}"
matrix_mul_rid: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "A ** mat (1::'a) = A"
Im_sgn: fixes z :: "complex" shows "Im (sgn z) = Im z / cmod z"
Upper_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "Upper L A \<subseteq> carrier L"
zero_spaceN_iff: fixes x :: "'c"   and N :: "'c quasinorm" shows "(x \<in> zero_space\<^sub>N N) = (eNorm N x = 0)"
lift_propagate_inv_propagate_all: fixes P :: "('loc, 't) Propagate.configuration \<Rightarrow> bool"   and c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration" assumes "\<And>c0 c1 loc t. \<lbrakk>P c0; next_propagate' c0 c1 loc t\<rbrakk> \<Longrightarrow> P c1"   and "P c0"   and "propagate_all c0 = Some c1" shows "P c1"
cover_sum: fixes C :: "nat set"   and V :: "'a set"   and c :: "'a \<Rightarrow> real" assumes "sc C V"   and "\<forall>i. 0 \<le> c i" shows "sum c V \<le> (\<Sum>i\<in>C. sum c (S i))"
M: fixes i :: "'d" assumes "i \<in> I" shows "sup_continuous (M i)"
assms: shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A"
MOST_nat: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>n. P n) = (\<exists>m. \<forall>n>m. P n)"
compEs2_append: fixes es :: "'a expr1 list"   and es' :: "'a expr1 list" shows "compEs2 (es @ es') = compEs2 es @ compEs2 es'"
fps_power_zeroth: fixes a :: "'a fps"   and n :: "nat" shows "a ^ n $ 0 = (a $ 0) ^ n"
fusion_character: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "\<forall>y. O y (\<sigma>v. F v) = (\<exists>x. F x \<and> O y x)"
weak_reflection_of_barbs_in_barbed_encoding: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and a :: "'barbs" assumes "rel_weakly_reflects_barbs Rel (STCalWB SWB TWB)"   and "(P, Q) \<in> Rel"   and "(\<exists>S. S \<in>S Q \<and> S\<Down><SWB>a) \<or> (\<exists>T. T \<in>T Q \<and> T\<Down><TWB>a)" shows "(\<exists>S. S \<in>S P \<and> S\<Down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<Down><TWB>a)"
histories_distinct: fixes history :: "nat \<Rightarrow> 'c list"   and i :: "nat" assumes "node_histories history" shows "distinct (history i)"
exists_trace_for_any_i: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "distributed_system channel"   and "\<exists>c'. distributed_system.trace channel trans send recv c t c'" shows "distributed_system.trace channel trans send recv c (take i t) (distributed_system.s channel trans send recv c t i)"
ec: shows "clock_simAbs ec = Clock.sim_equiv_class a t"
ground_img_if_ground_subst: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<And>v t. s v = t \<Longrightarrow> fv t = {}" shows "range_vars s = {}"
UNpart1_cases: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart1 cl dl P" obtains n :: "nat" and I :: "nat set" where "n \<in> theNFTBoth cl dl" and "I \<in> P n" and "II = local.shift cl n I"
that: fixes op :: "('a \<times> 'b) strips_operator" assumes "op \<in> set \<pi>"   and "ops = [\<phi>\<^sub>O\<inverse> \<Pi> op]" shows "thesis"
OO_eq: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "R OO (=) = R"
eqpoll_sym: fixes A :: "'a set"   and B :: "'b set" assumes "A \<approx> B" shows "B \<approx> A"
maxchain_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and C :: "'c set" shows "pred_on.maxchain A P C = (pred_on.chain A P C \<and> (\<nexists>S. pred_on.chain A P S \<and> C \<subset> S))"
ld: fixes i :: "'n"   and j :: "'n" assumes "i < j" shows "A i j = (0::'a)"
hoare_comp_wp: fixes p :: "'b Assertion"   and x :: "'b"   and y :: "'b"   and q :: "'b Assertion" shows "hoare p (x * y) q = hoare p x (wp y q)"
o_def: fixes f :: "'t \<Rightarrow> 's"   and g :: "'r \<Rightarrow> 't" shows "f \<circ> g = (\<lambda>x. f (g x))"
assms_1: shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>a'"
ord_le_eq_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
domintros_1: fixes c :: "nat" shows "stepC_dom (c, C [])"
bst_diff: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.bst (diff t1 t2)"
src_inv: fixes \<mu> :: "'c" assumes "C.iso \<mu>" shows "src\<^sub>C (C.inv \<mu>) = src\<^sub>C \<mu>"
gauss_jordan_check_invertable: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and b :: "'c" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "(A \<in> Units (ring_mat TYPE('b) n b)) = (fst (gauss_jordan A B) = 1\<^sub>m n)"
hypreal_sqrt_gt_zero_pow2: fixes x :: "real star" assumes "0 < x" shows "((*f* sqrt) x)\<^sup>2 = x"
Cdom2: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators"   and a :: "((int \<times> int) set set, int) Combinators" assumes "x \<in> dom (C b)" shows "C (a \<oplus> b) x = C b x"
sorted_wrt_mergesort_by_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (mergesort_by_rel R xs)"
trivial_def: fixes x :: "'b set" shows "trivial x = (x \<subseteq> {the_elem x})"
tabulation_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b" shows "tabulation V H \<a> \<i> src trg r \<rho> f g \<equiv> tabulation_data V H \<a> \<i> src trg r \<rho> f g \<and> tabulation_axioms V H \<a> r \<rho> f g"
simps_1: fixes x :: "gauss_int" shows "ReZ (- x) = - ReZ x"
list_appendI: fixes a :: "'a list"   and n :: "nat"   and A :: "'a set"   and b :: "'a list"   and m :: "nat" assumes "a \<in> list n A"   and "b \<in> list m A" shows "a @ b \<in> list (n + m) A"
vector_matrix_mult_def: fixes v :: "('d, 'f) vec"   and m :: "(('d, 'e) vec, 'f) vec" shows "v v* m \<equiv> \<chi>j. \<Sum>i\<in>UNIV. m $ i $ j * v $ i"
vcomp_vconst_on_vid_on: fixes A :: "V"   and c :: "V" shows "vconst_on A c \<circ>\<^sub>\<circ> vid_on A = vconst_on A c"
cases: fixes x :: "'a \<times> 'b" obtains n :: "'a" and m :: "'b" where "x = (n, m)"
assms_1: shows "A \<Turnstile> (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) global P"
div_mult_mod_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times (divide a b) b) (modulo a b) = a"
simps_1: fixes A :: "complex Matrix.mat" shows "A \<otimes>\<^bsup>Suc 0\<^esup> = A"
Strong_Soundness_aux: fixes c :: "'b ann_com"   and s :: "'b"   and co :: "'b ann_com option"   and t :: "'b"   and q :: "'b set" assumes "(Some c, s) -*\<rightarrow> (co, t)"   and "s \<in> pre c"   and "\<turnstile> c  q" shows "if co = None then t \<in> q else t \<in> pre (the co) \<and> \<turnstile> the co  q"
test_comp_closed: fixes p :: "'a" assumes "test p" shows "test (!p)"
last_appendR: fixes ys :: "'e list"   and xs :: "'e list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
vcons_vdoubleton: fixes a :: "V"   and b :: "V" shows "[a, b]\<^sub>\<circ> = set {\<langle>0\<^sub>\<nat>, a\<rangle>, \<langle>1\<^sub>\<nat>, b\<rangle>}"
enabled'_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "enabled' problem name s \<equiv> case lookup_operator' problem name of None \<Rightarrow> False | Some \<pi> \<Rightarrow> enabled_opr' \<pi> s"
lih'_is_run_sim: fixes r :: "nat \<Rightarrow> (nat, local_state, global_state) Gen_Scheduler.global_config" assumes "lih'.sa.is_run r" shows "li'.s1.sa.is_run ((Some \<circ>\<circ> cprog.gc_\<alpha>) prog \<circ> r)"
sum_list_ge_mono: fixes as :: "'a list"   and bs :: "'a list" assumes "length as = length bs"   and "\<And>i. i < length bs \<Longrightarrow> bs ! i \<le> as ! i" shows "sum_list bs \<le> sum_list as"
terminates_on_append_raw_Inr: fixes h :: "'e \<Rightarrow> ('f, 'e) step"   and sh :: "'e"   and g :: "'g \<Rightarrow> ('f, 'g) step"   and sh_start :: "'e" assumes "terminates h" shows "Inr sh \<in> terminates_on (append_raw g h sh_start)"
vec_of_poly_as: fixes p :: "'b poly" shows "vec_of_poly_n p (Suc (degree p)) = vec_of_poly p"
widen_Class: fixes P :: "'a prog"   and T :: "ty"   and C :: "char list" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C))"
synth_Int1: fixes x :: "msg"   and A :: "msg set"   and B :: "msg set" assumes "x \<in> synth (A \<inter> B)" shows "x \<in> synth A"
lookup_pair_update_by_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k1 :: "'a"   and v :: "'b"   and k2 :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.lookup_pair comp (comparator.update_by_pair comp (k1, v) xs) k2 = (if k1 = k2 then v else comparator.lookup_pair comp xs k2)"
list_mod_cases: fixes xs :: "nat list"   and n :: "nat" assumes "length xs = n"   and "0 < n" shows "(\<exists>x\<in>set xs. x mod n = 0) \<or> (\<exists>i<length xs. \<exists>j<length xs. i \<noteq> j \<and> xs ! i mod n = xs ! j mod n)"
lem_6_1_3: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and k :: "nat"   and ps :: "((('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) \<times> 'c set) list"   and i :: "nat"   and h :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and U :: "'c set" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "pm_powerprod.standard_decomp k ps"   and "pm_powerprod.exact_decomp X 0 ps"   and "Suc 0 \<le> i"   and "i \<le> card X"   and "(h, U) \<in> set (pm_powerprod.pos_decomp ps)"   and "card U = i" shows "pm_powerprod.\<b> ps (Suc i) \<le> poly_deg h"
finite_set_of_finite_funs: fixes A :: "'c set"   and B :: "'d set"   and d :: "'d" assumes "finite A"   and "finite B" shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}"
abs_power_pos: fixes a :: "'a"   and n :: "nat" shows "((0::'a) < \<bar>a\<bar> ^ n) = (a \<noteq> (0::'a) \<or> n = 0)"
int_g: fixes f :: "'s stream \<Rightarrow> ennreal"   and cfg :: "'s cfg" assumes "f \<in> borel_measurable St" shows "\<integral>\<^sup>+ \<omega>. g (state cfg) (f \<omega>) \<partial>T cfg = g (state cfg) (integral\<^sup>N (T cfg) f)"
nsets_empty_iff: fixes r :: "nat" shows "[{}]\<^bsup>r\<^esup> = (if r = 0 then {{}} else {})"
WT_sub_gpvsI: fixes gpv :: "('f, 'd, 'e) gpv"   and \<Gamma> :: "('d, 'e) \<I>" assumes "\<And>out c. IO out c \<in> set_spmf (the_gpv gpv) \<Longrightarrow> out \<in> outs_\<I> \<Gamma>"   and "\<And>gpv'. gpv' \<in> sub_gpvs \<Gamma> gpv \<Longrightarrow> \<Gamma> \<turnstile>g gpv' \<surd>" shows "\<Gamma> \<turnstile>g gpv \<surd>"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b list" shows "list.set (map f v) = f ` list.set v"
elims: fixes x :: "'c list"   and xa :: "'d option \<Rightarrow> 'd"   and xb :: "('c, 'd) trie"   and y :: "('c, 'd) trie" assumes "update_with_trie x xa xb = y" obtains f :: "'d option \<Rightarrow> 'd" and v :: "'d option" and ps :: "('c \<times> ('c, 'd) trie) list" where "x = []" and "xa = f__" and "xb = Trie v__ psa__" and "y = Trie (Some (f__ v__)) psa__"   | k :: "'c" and ks :: "'c list" and f :: "'d option \<Rightarrow> 'd" and v :: "'d option" and ps :: "('c \<times> ('c, 'd) trie) list" where "x = k # ks" and "xa = f" and "xb = Trie v ps" and "y = Trie v (AList.update_with_aux empty_trie k (update_with_trie ks f) ps)"
float_defs_8: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "(a \<le> b) = fle a b"
included_project: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set" shows "included V (projectQuery V Q)"
per_rootI': fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<^sup>@ k"   and "r \<noteq> \<epsilon>" shows "u \<le>p r\<^sup>\<omega>"
length_rev: fixes xs :: "'f list" shows "length (rev xs) = length xs"
left_refines_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "(a \<Sqinter> b) \<sqsubseteq> a"
unity_root_sum_def: fixes k :: "nat"   and n :: "int" shows "unity_root_sum k n = (\<Sum>m<k. unity_root k (n * int m))"
msum_pdevs_raw_def: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'd"   and y :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "msum_pdevs_raw n x y i = (if i < n then x i else y (i - n))"
in_set_inner_verts_appendI_l: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list" assumes "u \<in> set (inner_verts p)" shows "u \<in> set (inner_verts (p @ q))"
coclop_dual: shows "coclop = clop \<circ> \<partial>\<^sub>F"
proj_nAct: fixes n :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat n < llength t" shows "\<pi>\<^bsub>c\<^esub>ltake (enat n) t = ltake \<langle>c #\<^bsub>enat n\<^esub>t\<rangle> (\<pi>\<^bsub>c\<^esub>t)"