hcju/isabelleps huggingface dataset:
fixp_def: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a" assumes "class.ccpo Sup less_eq less" shows "ccpo.fixp Sup less_eq f = Sup (ccpo.iterates Sup less_eq f)"
nth_append_length: fixes xs :: "'d list"   and x :: "'d"   and ys :: "'d list" shows "(xs @ x # ys) ! length xs = x"
boundedE: fixes a :: "'g multiset"   and b :: "'g multiset"   and c :: "'g multiset" assumes "a \<subseteq># b \<inter># c" obtains "a \<subseteq># b" and "a \<subseteq># c"
prj_chine_assoc_2: shows "f_src_src.Prj\<^sub>1\<^sub>0 \<cdot> f_src_src.chine_assoc = f_src_src.Prj\<^sub>0\<^sub>1"
hseqI': fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>" shows "B.hseq \<nu> \<mu>"
cong: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_add.Sum_any plus zero"
reachableGD: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow>\<^sup>*\<^bsub>G\<^esub> y" shows "x \<rightarrow>\<^sup>*\<^bsub>H\<^esub> y"
ideD_1: fixes a :: "'s" assumes "S.ide a" shows "S.arr a"
zip_append1: fixes xs :: "'k list"   and ys :: "'k list"   and zs :: "'l list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
linear: fixes x :: "'b word"   and y :: "'b word" shows "x \<le>s y \<or> y \<le>s x"
codomains_null: shows "Right_a.codomains Right_a.null = {}"
subst_swp: fixes y :: "'a"   and A :: "'a trm"   and z :: "'a"   and M :: "'a trm" assumes "y \<notin> fvs A" shows "[y \<leftrightarrow> z] \<cdot> A[y ::= M] = A[z ::= M]"
wf_ruleset_stripfst: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and r :: "'b rule"   and rs :: "'b rule list" assumes "wf_ruleset \<gamma> p (r # rs)" shows "wf_ruleset \<gamma> p rs"
vimage2p_def: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
INFM_conjI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<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"
l9: shows "[prod Prod local.S2 = - 1] (mod int p)"
comp_arr_ide: fixes a :: "'d"   and f :: "'d" assumes "D.ide a"   and "D.seq f a" shows "f \<cdot>\<^sub>D a = f"
member_def: fixes xs :: "'c list"   and x :: "'c" shows "List.member xs x = (x \<in> set xs)"
col_mk_poly_sub: fixes l :: "nat"   and j' :: "nat"   and A :: "'b poly mat"   and j :: "nat" assumes "l \<noteq> j'"   and "j' < dim_col A" shows "col (mk_poly_sub A l j) j' = col A j'"
scaleR: fixes r :: "real"   and b :: "'n" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
vp_ideal: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "ideal (Vr K v) (vp K v)"
assms: shows "closed_segment p0 p1 \<inter> {p. fst p = x} = {}"
gen_remove_invar: fixes As :: "'s \<times> natural \<times> natural"   and bound :: "natural"   and n :: "nat" assumes "invar (fst As)" shows "invar (fst (local.gen_remove bound n As))"
Seq_MtransT_MtransC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>*t s'" shows "(c1 ;; c2, s) \<rightarrow>*c (c2, s')"
is_module: fixes W :: "'a vec set" assumes "subspace class_ring W V" shows "LinearCombinations.submodule class_ring W V"
distinct_2: fixes x2 :: "ltkey"   and x1 :: "fresh_t" shows "ltK x2 \<noteq> sesK x1"
simps: fixes cenum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "all_n_lists cenum_all P n = (if n = 0 then P [] else cenum_all (\<lambda>x. all_n_lists cenum_all (\<lambda>xs. P (x # xs)) (n - 1)))"
assms_2: shows "f \<in> A \<rightarrow> H"
trg_vcomp: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "seq \<nu> \<mu>" shows "trg (\<nu> \<bullet> \<mu>) = trg \<nu> \<bullet> trg \<mu>"
dense_ge: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and z :: "'k"   and y :: "'k" assumes "class.dense_linorder less_eq less"   and "\<And>x. less z x \<Longrightarrow> less_eq y x" shows "less_eq y z"
wa_intro: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and addi :: "('d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)) set" assumes "while_algo (bre_algo Qi \<delta>)"   and "wa_invar (bre'_algo Qi \<delta>) = addi \<inter> {s. br'_\<alpha> s \<in> wa_invar (bre_algo Qi \<delta>)}"   and "\<And>s s'. \<lbrakk>s \<in> addi; s \<in> wa_cond (bre'_algo Qi \<delta>); br'_\<alpha> s \<in> wa_invar (bre_algo Qi \<delta>); (s, s') \<in> wa_step (bre'_algo Qi \<delta>)\<rbrakk> \<Longrightarrow> s' \<in> addi"   and "wa_initial (bre'_algo Qi \<delta>) \<subseteq> addi" shows "while_algo (bre'_algo Qi \<delta>)"
hyps: shows "start_of ly e \<le> start_of ly (e + d)"
discrete_category_def: fixes Obj :: "'b set"   and Null :: "'b" shows "discrete_category Obj Null \<equiv> Null \<notin> Obj"
expr_inj_term_def: fixes e :: "expr" shows "\<langle>e\<rangle> = In1l e"
subtract_mask_1: fixes p :: "'b word"   and n :: "nat" shows "p - (p AND mask n) = p AND NOT (mask n)"
the_signed_int_def: shows "the_signed_int \<equiv> map_fun Word.rep id (signed_take_bit (LENGTH('k) - Suc 0))"
nvalidI: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'g, 'f) com option"   and c :: "('e, 'g, 'f) com"   and n :: "nat"   and P :: "'e set"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
bigEx: fixes s :: "real" assumes "s \<in> {0..t}" shows "\<exists>sola. sol 0 = sola 0 \<and> (\<exists>ta. (fst (mk_v I (OVar vid1) (sol 0, b) (sol s)), snd (mk_v I (OVar vid1) (sol 0, b) (sol s))) = mk_v I (OVar vid1) (sola 0, b) (sola ta) \<and> 0 \<le> ta \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..ta} {x. Predicates I vid1 (\<chi>i. dterm_sem I (local.empty i) (mk_v I (OVar vid1) (sol 0, b) x))})"
subst_pp_by_monomials: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and s :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>0 nat"   and c :: "'b \<Rightarrow> 'd" assumes "\<And>y. y \<in> keys t \<Longrightarrow> f y = monomial (c y) (s y)" shows "subst_pp f t = monomial (\<Prod>y\<in>keys t. c y ^ lookup t y) (\<Sum>y\<in>keys t. lookup t y \<cdot> s y)"
YFtor'Obj: fixes X :: "'d"   and C :: "('d, 'e, 'f) LSCategory_scheme" assumes "X \<in> obj\<^bsub>CatDom (YFtor' C)\<^esub>"   and "LSCategory C" shows "YFtor' C @@ X = Hom\<^bsub>C\<^esub>[\<midarrow>,X]"
correctCompositionDiffLevelsS11: shows "correctCompositionDiffLevels sS11"
A: shows "(\<lambda>_. 1) \<in> o[F](f)"
trading_strategy_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "disc_filtr_prob_space M F" shows "disc_filtr_prob_space.trading_strategy F p = (portfolio p \<and> (\<forall>asset\<in>support_set p. borel_predict_stoch_proc F (p asset)))"
ball_UNIV: fixes P :: "'d \<Rightarrow> bool" shows "Ball UNIV P = All P"
converse_relcompp: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'a \<Rightarrow> bool" shows "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
set_cases_2: fixes e :: "'e"   and a :: "('f, 'e) step" assumes "e \<in> set2_step a" obtains "a = Skip e"   | z1 :: "'f" where "a = Yield z1 e"
vector_component_4: fixes c :: "'b"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(c *s y) $h i = c * y $h i"
pred_rel_simps_1: fixes s :: "('a, 'f) simple_state_scheme"   and s' :: "('a, 'f) simple_state_scheme"   and u :: "'a" assumes "(s, s') \<in> rel" shows "is_discovered_impl u s = (is_discovered_impl u s' \<or> u \<in> R)"
Domain_Rel_cases: fixes x0 :: "'g"   and x1 :: "'a" assumes "(x0, x1) \<in> Domain Rel_OR" shows "(\<exists>w0. (x0, w0) \<in> Rel0 \<and> x1 \<in> valid_pub1) \<or> (\<exists>w1. (x1, w1) \<in> Rel1 \<and> x0 \<in> valid_pub0)"
intros: fixes i :: "nat"   and j :: "nat"   and xs :: "'b set list"   and x :: "'b"   and y :: "'b" assumes "i \<le> j"   and "i < length xs"   and "j < length xs"   and "x \<in> xs ! i"   and "y \<in> xs ! j" shows "of_weak_ranking xs y x"
dropWhile_rsorted_eq_filter: fixes xs :: "(nat \<times> 'd) list"   and m :: "nat" assumes "rsorted (map fst xs)" shows "dropWhile (\<lambda>(i, x). m \<le> i) xs = filter (\<lambda>(i, x). i < m) xs"
bounded_clinear_eq_on: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b"   and G :: "'a set"   and t :: "'a" assumes "bounded_clinear A"   and "bounded_clinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
hd_no_head: fixes \<C> :: "'b list set"   and u :: "'b list"   and b :: "'b list" assumes "code \<C>"   and "u \<in> \<langle>\<C>\<rangle>"   and "hd (Dec \<C> u) \<noteq> b" shows "u \<in> no_head \<C> b"
properfactor_cong_l: fixes x' :: "'a"   and x :: "'a"   and y :: "'a" assumes "x' \<sim> x"   and "properfactor G x y"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "y \<in> carrier G" shows "properfactor G x' y"
closed_segment_in_Reals: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> closed_segment a b"   and "a \<in> \<real>"   and "b \<in> \<real>" shows "x \<in> \<real>"
map\<^sub>1_simps_1: fixes f :: "'a" assumes "C.ide f" shows "D.seq \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<r>\<^sub>D[F f]"
from_nat_to_nat: fixes x :: "'a" shows "from_nat (to_nat x) = x"
open_fun_contains_ball_aux: fixes U :: "('a \<Rightarrow> 'b) set"   and x :: "'a \<Rightarrow> 'b" assumes "open U"   and "x \<in> U" shows "\<exists>e>0. {y. dist x y < e} \<subseteq> U"
prems_5: shows "Im (trace (\<rho>\<^sub>1 + \<rho>\<^sub>2)) = 0"
lseg_conc: fixes l1 :: "'a list"   and p :: "'a node ref option"   and q :: "'a node ref option"   and l2 :: "'a list"   and r :: "'a node ref option" shows "lseg l1 p q * lseg l2 q r \<Longrightarrow>\<^sub>A lseg (l1 @ l2) p r"
atMost_atLeast0: fixes n :: "nat" shows "{..n} = {0..n}"
build_rel_def: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and I :: "'d \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
strong_induct: fixes P :: "'e \<Rightarrow> 'd frame \<Rightarrow> bool"   and z :: "'e"   and frame :: "'d frame" assumes "\<And>assertion z. P z (\<langle>\<epsilon>, assertion\<rangle>)"   and "\<And>name frame z. \<lbrakk>name \<sharp> z; \<And>z. P z frame\<rbrakk> \<Longrightarrow> P z (\<lparr>\<nu>name\<rparr>frame)" shows "P z frame"
measurable_ident: fixes M :: "'c measure" shows "id \<in> M \<rightarrow>\<^sub>M M"
side: shows "r'' \<notin> RID\<^sub>G s\<^sub>1"
mono_onD: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and r :: "'c"   and s :: "'c" assumes "mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r \<le> s" shows "f r \<le> f s"
set3_F_bound: fixes x :: "('v, 'w, 'n, 'o, 'p, 'q, 'r, 's, 't, 'u) F" shows "|set3_F x| \<le>o bd_F"
mult2_s_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_s ns s \<equiv> multpw ns O mult s"
valid_to_valid_strip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"   and "F' \<subseteq> - F" shows "strip F' \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
p_ge_2: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "2 < p"
emeasure_eq_1_AE: fixes S :: "'b set"   and M :: "'b pmf" assumes "S \<in> measure_pmf.events M"   and "AE x in measure_pmf M. x \<in> S" shows "emeasure (measure_pmf M) S = 1"
sink_def: fixes q :: "'b" shows "sink q \<equiv> q\<^sub>0 \<noteq> q \<and> (\<forall>\<nu>\<in>\<Sigma>. \<delta> q \<nu> = q)"
elims: fixes x :: "nat"   and xa :: "'a"   and y :: "real" assumes "v x xa = y" obtains n :: "nat" and xb :: "'a" where "x = n" and "xa = xb" and "y = (if n = 0 then max (u 0 xb) 0 else if n = 1 then u 1 xb else min (u n xb) (MIN k\<in>{0<..<n}. v k xb + v (n - k) ((T ^^ k) xb)))"
inject_1: fixes x1 :: "('c, 'd) insert_message"   and y1 :: "('c, 'd) insert_message" shows "(Insert x1 = Insert y1) = (x1 = y1)"
Cons: fixes fsm :: "'b fsm"   and st :: "hf"   and x :: "'b"   and st'' :: "hf"   and xs :: "'b list"   and st' :: "hf" assumes "next fsm st x st''"   and "reaches fsm st'' xs st'"   and "st \<^bold>\<in> states fsm" shows "reaches fsm st (x # xs) st'"
concl_in: shows "concl_of \<iota> \<in> N"
simps_201: fixes x101 :: "Term.var"   and x102 :: "expr"   and x9 :: "Term.var" shows "x101:=x102 \<noteq> Acc x9"
discr_Par: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "discr c1"   and "discr c2" shows "discr (c1 | c2)"
rep_number_non_zero_system_point: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and x :: "'b" assumes "incidence_system point_set block_collection"   and "0 < block_collection rep x" shows "x \<in> point_set"
inv1: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([y, x], rev qs @ h)"
IH: fixes x :: "'b" assumes "vs2.independent (f ` b)"   and "inj_on f b"   and "x \<in> vs1.span b"   and "f x = (0::'c)" shows "x = (0::'b)"
conv_res: fixes x :: "'a"   and y :: "'a" shows "(x \<rhd> (1::'a)) \<cdot> - (x \<cdot> y) \<le> - y"
restrict_locale: fixes P :: "'a set" assumes "subgroup P G"   and "p ^ a dvd card P" shows "snd_sylow (G\<lparr>carrier := P\<rparr>) p a (card P div p ^ a)"
nprv_addImpLemmaE2: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "prv (imp \<phi>1 \<phi>2)"   and "nprv F \<phi>1"   and "\<psi> = \<phi>2 \<or> \<psi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
record_call_simp: fixes x :: "'a"   and f :: "'a \<Rightarrow> one\<^sub>\<bottom>"   and x' :: "'a" shows "(record_call x\<cdot>f) x' = (if x = x' then two_pred\<cdot>(f x') else f x')"
lookup_tabulate: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "distinct xs" shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
pts_symbolic_c_preserves_wf_is: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list"   and V :: "'var set" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "wf\<^sub>e\<^sub>s\<^sub>t V \<A>" shows "wf\<^sub>e\<^sub>s\<^sub>t V \<A>'"
to_mregex_to_mregex_exec_Nil: fixes r :: "Formula.formula Regex.regex" shows "to_mregex r = to_mregex_exec r []"
diff_fun_cong: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'f"   and g :: "'c \<Rightarrow> 'f" assumes "diff_fun k charts f"   and "\<And>x. x \<in> manifold.carrier charts \<Longrightarrow> f x = g x" shows "diff_fun k charts g"
matrix_to_lpolies_lambda_valuate_scalarP: fixes i :: "nat"   and A :: "rat mat"   and x :: "rat vec" assumes "i < dim_row A"   and "dim_col A = dim_vec x" shows "matrix_to_lpolies A ! i \<lbrace> \<lambda>i. if i < dim_vec x then x $ i else 0 \<rbrace> = row A i \<bullet> x"
not_ephemeralD_pos_period: fixes C :: "'s set" assumes "not_ephemeral C" shows "0 < period C"
edge_preserving_def: fixes h :: "('c \<times> 'd) set"   and e1 :: "('e \<times> 'c \<times> 'c) set"   and e2 :: "('e \<times> 'd \<times> 'd) set" shows "edge_preserving h e1 e2 \<equiv> \<forall>(k, v1, v2)\<in>e1. \<forall>v1' v2'. (v1, v1') \<in> h \<and> (v2, v2') \<in> h \<longrightarrow> (k, v1', v2') \<in> e2"
X_poly_plus_def: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e" shows "X_poly_plus R a = X_poly R \<oplus>\<^bsub>UP R\<^esub> to_polynomial R a"
r: shows "\<forall>p. \<not> cri.GlobNonposUpto (exchange_config c) p x"
bisim_inv1: 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"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and t :: "'b"   and s1 :: "'a \<times> 'c"   and s2 :: "'g \<times> 'h"   and ta1 :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s1' :: "'a \<times> 'c" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "bisim t s1 s2"   and "r1 t s1 ta1 s1'" obtains s2' :: "'g \<times> 'h" where "bisim t s1' s2'"
prog_configs_in1: fixes p\<^sub>1 :: "'a"   and p\<^sub>2 :: "'a"   and "is" :: "instr list"   and xs :: "'a memref list" assumes "Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is \<in> set xs" shows "p\<^sub>1 \<in> prog_configs xs"
assms: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and v :: "nat" assumes "lookup p m \<noteq> (0::'a)"   and "lookup m v \<noteq> 0" shows "f v = g v"
has_bochner_integral_completion: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and I :: "'b" assumes "has_bochner_integral M f I" shows "has_bochner_integral (completion M) f I"
1: shows "l \<circ> (h \<circ> g) \<le>\<le> g \<circ> h \<circ> l"
lset_ldropWhile_subset: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lset (ldropWhile P xs) \<subseteq> lset xs"
arc_expanded: fixes x :: "'a" shows "arc x = (coreflexive (x * top * x\<^sup>T) \<and> coreflexive (x\<^sup>T * top * x) \<and> times_top_class.total (top * x))"
rsum_diff_bound: fixes p :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "p tagged_division_of cbox a b"   and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" shows "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)) \<le> e * content (cbox a b)"
fbox_def: fixes R :: "('e \<times> 'f) set"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" shows "|R] f x = \<Sqinter>{f y |y. (x, y) \<in> R}"
in_hom_char: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "B.VV.in_hom f a b = (B.VV.arr a \<and> B.VV.arr b \<and> B.VV.arr f \<and> B.VxV.in_hom f a b)"
scale_left_commute: fixes a :: "'d"   and b :: "'d"   and x :: "'c \<Rightarrow>\<^sub>0 'd" shows "a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
intros: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<leadsto>\<^sup>^<weakBisimulation> Q"   and "Q \<approx> P" shows "P \<approx> Q"
assms: shows "vs \<in> LS\<^sub>i\<^sub>n M q V"
d\<^sub>\<kappa>_inject: fixes o\<^sub>1 :: "\<nu>"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> y" shows "x = y"
hom_imp_img_monoid: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom (add_monoid R) H" shows "Group.monoid (H\<lparr>carrier := h ` carrier R, one := h \<zero>\<rparr>)"
neg_equal_iff_equal: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
sort_key_simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f [] = []"
indicator_eq_0_iff: fixes A :: "'d set"   and x :: "'d" shows "(indicator A x = (0::'c)) = (x \<notin> A)"
mem_val_alt_10_mem_equal_3: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 3) s1 = mem_val_alt 10 ((pa AND 68719476732) + 3) s2"
is_SNF_Smith_mxn_ge_2: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n"   and "2 < m"   and "2 < n" shows "is_SNF A (Smith_mxn A)"
pmf_integral_def: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> real" shows "pmf_integral p f = measure_pmf.expectation p f"
mon_s_single: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and n :: "'g" shows "mon_s fg [n] = mon_n fg n"
Q4: shows "A B Perp P' P \<or> P' = P"
HPT_Source_Conf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "Source TS \<subseteq> Conf ST"
tendsto_at_topI_sequentially: fixes f :: "real \<Rightarrow> 'b"   and y :: "'b" assumes "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_top"
iso_is_arr: fixes f :: "'a" assumes "Left_a.iso f" shows "Left_a.arr f"
subst_sst_cons: fixes a :: "('d, 'e) stateful_strand_step"   and A :: "('d, 'e) stateful_strand_step list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) # (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
X1_def: shows "X1 = {x \<in> \<X>. \<exists>i. \<forall>j\<ge>i. unbounded x ([xs !! j]\<^sub>\<R>)}"
mapping_eqI: fixes m :: "('b, 'c) mapping"   and m' :: "('b, 'c) mapping" assumes "\<And>x. lookup m x = lookup m' x" shows "m = m'"
sopen_eq_Fvar': fixes n :: "nat"   and s :: "char list"   and p :: "char list"   and t :: "sterm"   and x :: "char list" assumes "{n \<rightarrow> [Fvar s,Fvar p]} t = Fvar x"   and "x \<noteq> s"   and "x \<noteq> p" shows "t = Fvar x"
differentiable_on_inverse: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f differentiable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> (0::'d)" shows "(\<lambda>x. inverse (f x)) differentiable_on S"
intro: fixes G :: "('e, 'f) monoid_scheme" assumes "finite_group G"   and "comm_group G" shows "finite_comm_group G"
snapshot_event: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a" assumes "trace init t final"   and "\<not> ps (S t i) p \<noteq> None"   and "ps (S t (i + 1)) p \<noteq> None" shows "isSnapshot (t ! i) \<or> isRecvMarker (t ! i)"
add_rel_carrier: fixes X :: "('a set \<times> 'b) set"   and Y :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim"   and "Y \<in> carrier_direct_lim" shows "add_rel X Y \<in> carrier_direct_lim"
spn_simps_3: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src (spn \<mu>) = src (tab\<^sub>0 (local.dom \<mu>))"
Lyndon_mono_sorted: fixes ws :: "'a list list" assumes "Lyndon_mono ws" shows "linorder.sorted (\<le>lex) (rev ws)"
rewrite_iter_slow_def: fixes \<phi> :: "'b ltln" shows "rewrite_iter_slow \<phi> \<equiv> iterate (rewrite_syn_imp \<circ> rewrite_modal \<circ> rewrite_X) \<phi> (size \<phi>)"
irrefl_diff_Id: fixes r :: "('a \<times> 'a) set" shows "irrefl (r - Id)"
sums_iff: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b" shows "f sums x = (summable f \<and> suminf f = x)"
jointfun_hom0: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and A :: "'c set"   and g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and B :: "'c set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {k. k \<le> m} \<rightarrow> B" shows "jointfun n f m g \<in> {l. l \<le> Suc (n + m)} \<rightarrow> A \<union> B"