hcju/isabelleps huggingface dataset:
rel_G_neg_distr_cond_eq: fixes tytok :: "('r \<times> 's \<times> 't \<times> 'u \<times> 'v \<times> 'w \<times> 'x) itself" shows "rel_G_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
simps_2: fixes g :: "graph"   and f :: "face"   and u :: "nat"   and n :: "nat"   and vo :: "nat option"   and vos :: "nat option list" shows "subdivFace' g f u n (vo # vos) = (case vo of None \<Rightarrow> subdivFace' g f u (Suc n) vos | Some v \<Rightarrow> if f \<bullet> u = v \<and> n = 0 then subdivFace' g f v 0 vos else let ws = [countVertices g..<countVertices g + n]; (f\<^sub>1, f\<^sub>2, g') = splitFace g u v f ws in subdivFace' g' f\<^sub>2 v 0 vos)"
inputs2_def: shows "inputs2 = drop (length (input_sizes m1)) inputs"
indicator_leI: fixes x :: "'b"   and A :: "'b set"   and y :: "'c"   and B :: "'c set" assumes "x \<in> A \<Longrightarrow> y \<in> B" shows "indicator A x \<le> indicator B y"
continuous_map_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "continuous_map X Y f \<equiv> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
intros_3: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs2 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" assumes "small_step \<pi> cs = Some cs1"   and "small_steps \<pi> cs1 cs2" shows "small_steps \<pi> cs cs2"
the_x: fixes x :: "nat" assumes "x < m" shows "(THE k. k < m \<and> row B x = row B k) = x"
is_map: shows "is_left_adjoint Tgh_Hgh.chine"
distinct_1: fixes x2 :: "'a" shows "\<bottom> \<noteq> \<lfloor>x2\<rfloor>"
admD: fixes P :: "'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c" assumes "adm P"   and "chain Y"   and "\<And>i. P (Y i)" shows "P (\<Squnion>i. Y i)"
UnderS_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "UnderS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (b, a) \<in> r}"
prob_space_pair: fixes M :: "'b measure"   and N :: "'c measure" assumes "prob_space M"   and "prob_space N" shows "prob_space (M \<Otimes>\<^sub>M N)"
sp\<^sub>\<mu>_rel_self: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('c, 'a, 'b) sp\<^sub>\<mu>" assumes "(=) \<le> R1"   and "(=) \<le> R2" shows "rel_sp\<^sub>\<mu> R1 R2 x x"
image_Suc_lessThan: fixes n :: "nat" shows "Suc ` {..<n} = {1..n}"
that: fixes z :: "'b" assumes "(x, z) \<in> R"   and "(y, z) \<in> R" shows "thesis"
fps_sin_even: fixes c :: "'a" shows "fps_sin (- c) = - fps_sin c"
simps_3: fixes va :: "nat" shows "len (Suc (Suc va)) = Suc (len (Suc (Suc va) div 2))"
cod_char: fixes f :: "'d" shows "D'.cod f = (if D'.arr f then D.cod f else D.null)"
s2: fixes s :: "'i word" shows "runFw s ip1 c rs = runFw s ip2 c rs"
good_item_simps_42: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "skelAbs (Abs xsa xa X) = Branch (\<lambda>i. Some (skel X)) Map.empty"
"apply": fixes x :: "'b perm" shows "(\<langle>$\<rangle>) x \<in> {f. bij f \<and> finite {a. f a \<noteq> a}}"
in_set_impl_in_set_zip2: fixes xs :: "'d list"   and ys :: "'e list"   and y :: "'e" assumes "length xs = length ys"   and "y \<in> set ys" obtains x :: "'d" where "(x, y) \<in> set (zip xs ys)"
clause_processing: fixes N :: "('a literal multiset \<times> nat) multiset"   and C :: "'a literal multiset"   and i :: "nat"   and P :: "('a literal multiset \<times> nat) multiset"   and Q :: "('a literal multiset \<times> nat) multiset"   and n :: "nat" shows "(N + {#(C, i)#}, P, Q, n) \<leadsto>\<^sub>w (N, P + {#(C, i)#}, Q, n)"
connected_simple_path_endless: fixes c :: "real \<Rightarrow> 'a" assumes "simple_path c" shows "connected (path_image c - {pathstart c, pathfinish c})"
abs_eq: fixes x :: "nat" shows "integer_of_nat x = integer_of_int (int x)"
maximal: fixes w :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "current \<Gamma> w"   and "wave \<Gamma> w"   and "f \<le> w" shows "f = w"
unchanged: shows "\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>"
trace_parallel_plan_step_effect_is: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "trace_parallel_plan_strips I \<pi> ! Suc k = execute_parallel_operator (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k)"
n_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x \<oplus> !y \<cdot> !z = (!x \<oplus> !y) \<cdot> (!x \<oplus> !z)"
dom_eqI': fixes a :: "'c"   and f :: "'c" assumes "a \<in> C.domains f" shows "a = C.dom f"
lunit'_simps_3: fixes f :: "'a" assumes "ide f" shows "trg \<l>\<^sup>-\<^sup>1[f] = trg f"
mod_0_imp_dvd: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo"   and "modulo a b = zero" shows "dvd.dvd times b a"
\<eta>_in_hom: shows "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>"
update_def: fixes x :: "'c"   and y :: "'d"   and t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "update x y t = paint Black (upd x y t)"
right_cont_F: fixes a :: "real" shows "continuous (at_right a) F"
insert_commute: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" shows "{x, y} \<union> A = {y, x} \<union> A"
MakeCatObj: fixes C :: "('d, 'e, 'f) Category_scheme" shows "obj\<^bsub>MakeCat C\<^esub> = obj\<^bsub>C\<^esub>"
Lt_Lt_dbm_lt_D: fixes a :: "'c"   and b :: "'c" assumes "Lt a \<prec> Lt b" shows "a < b"
norm_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "norm one = 1"
ptrm_prm_fvs: fixes \<pi> :: "'b prm"   and X :: "'b ptrm" shows "ptrm_fvs (\<pi> \<bullet> X) = \<pi> {$} ptrm_fvs X"
assms: fixes k :: "'b"   and v :: "'c" assumes "(k, v) \<in> set (fst xs)" shows "f k v = g k v"
until_not_relesased_now: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "(\<phi> until \<psi>) \<omega>"   and "\<not> \<psi> \<omega>" shows "\<phi> \<omega>"
trunc_simps_1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)" shows "f = Cring_Poly.truncate Zp f \<oplus>\<^bsub>UP Zp\<^esub> monom (UP Zp) (f (deg Zp f)) (deg Zp f)"
bounded_norm_le_SUP_norm: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "bounded (range f)" shows "norm (f x) \<le> (SUP x. norm (f x))"
wlp_whileI: fixes I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and \<ss>\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "I \<ss>\<^sub>0"   and "\<And>\<ss>. \<lbrakk>I \<ss>; bval b \<ss>\<rbrakk> \<Longrightarrow> wlp \<pi> c I \<ss>"   and "\<And>\<ss>. \<lbrakk>I \<ss>; \<not> bval b \<ss>\<rbrakk> \<Longrightarrow> Q \<ss>" shows "wlp \<pi> (WHILE b DO c) Q \<ss>\<^sub>0"
obj_def: fixes a :: "'c" shows "C.obj a \<equiv> C.arr a \<and> src\<^sub>C a = a"
that: fixes q :: "nat \<Rightarrow> nat" assumes "\<forall>i<n. q i < p"   and "\<forall>i<n. \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and> (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and> label r i \<noteq> label s i" shows "thesis"
wellformed_union: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "wf_digraph H"   and "compatible G H" shows "wf_digraph (Digraph_Component.union G H)"
split_asm: fixes P :: "'k \<Rightarrow> bool"   and f1 :: "'l \<Rightarrow> 'k"   and f2 :: "'m \<Rightarrow> 'k"   and sum :: "'l + 'm" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Inr x2 \<and> \<not> P (f2 x2))))"
ipresWlsAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'b, 'c, 'h) model_scheme" shows "ipresWlsAll h hA MOD \<equiv> ipresWls h MOD \<and> ipresWlsAbs hA MOD"
onorm_le: fixes f :: "'a \<Rightarrow> 'b"   and b :: "real" assumes "\<And>x. norm (f x) \<le> b * norm x" shows "onorm f \<le> b"
cases: fixes x :: "'a list" obtains "xa__ = []"   | x :: "'a" where "xa__ = [xb__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "xa__ = x # y # xs"
E_sup_def: fixes s :: "'s"   and f :: "'s stream \<Rightarrow> ennreal" shows "E_sup s f = (\<Squnion>cfg\<in>cfg_on s. integral\<^sup>N (T cfg) f)"
e'_2: shows "(\<lambda>x. g x - eval C x * b x powr e) \<in> o(\<lambda>x. b x powr e')"
assms: shows "p \<cdot> x \<cdot> q = x"
util_def_conf: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes" shows "(measure_pmf.expectation y u \<le> measure_pmf.expectation x u) = x \<succeq> y"
hlog_mult: fixes a :: "real star"   and x :: "real star"   and y :: "real star" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x"   and "0 < y" shows "hlog a (x * y) = hlog a x + hlog a y"
quasi_commute_def: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "quasi_commute r s = (s O r \<subseteq> r O (r \<union> s)\<^sup>*)"
minus_rat_cancel: fixes a :: "int"   and b :: "int" shows "Fract (- a) (- b) = Fract a b"
w_simps_3: shows "src \<omega>.chine = src r\<^sub>0"
f_def: fixes i :: "nat" shows "f i = isolate_variable_sparse p var i * sq ^ (i div 2) * Const (real (i mod 2)) * Var var"
decl_acc_modi_simp: fixes d :: "'a decl_scheme" shows "accmodi d = access d"
cong_iff_lin: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and m :: "'c" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = (\<exists>k. b = plus a (times m k))"
computation_pull_pos_neg: shows "rat_poly_plus (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A)) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A) = 0"
bi_unique_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
swap_in_def: fixes S :: "('a \<times> 'a) set"   and x :: "'a \<times> 'a" shows "swap_in S x = (if x \<in> S then prod.swap x else x)"
cong2_lt__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B Lt C D"   and "Cong A B A' B'"   and "Cong C D C' D'" shows "A' B' Lt C' D'"
span_explicit': fixes b :: "'d set" shows "cspan b = {\<Sum>v | f v \<noteq> 0. f v *\<^sub>C v |f. finite {v. f v \<noteq> 0} \<and> (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)}"
that: fixes d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(d, t) \<rightarrow>*c (d', t')"   and "s' \<approx> t'"   and "c' \<approx>T d'" shows "thesis"
RSD_pareto_eqclass_rsd_winners: fixes agents' :: "'agent list" assumes "finite alts"   and "alts \<noteq> {}"   and "set agents' = agents" shows "RSD_pareto_eqclass agents alts R (rsd_winners R alts agents')"
sum_bounded_below: fixes A :: "'a set"   and K :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> A \<Longrightarrow> K \<le> f i" shows "of_nat (card A) * K \<le> sum f A"
proj_spherI: fixes x :: "'a" assumes "x \<in> affine hull S"   and "x \<noteq> (0::'a)" shows "proj x \<in> sphere (0::'a) 1 \<inter> affine hull S"
mult_sup_associative_one: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y * (1::'a)) * z \<le> x * (y * z)"
ex_conj_contr_right: fixes x :: "'a \<Longrightarrow> 'b"   and Q :: "'b upred"   and P :: "'b upred" assumes "x \<sharp> Q" shows "(\<exists> x \<bullet> P \<and> Q) = ((\<exists> x \<bullet> P) \<and> Q)"
hyps: fixes x2a :: "('f, 'a) Term.term" assumes "x2a \<in> set ss" shows "NS x2a x2a"
w_scale: fixes b :: "'a \<Rightarrow> real"   and r :: "real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> w x \<in> UNIV"   and "b \<in> diff_fun_space" shows "w (r *\<^sub>R b) = r *\<^sub>R w b"
less_eq_prod_def: fixes p1 :: "'b \<times> 'c"   and p2 :: "'b \<times> 'c" shows "(p1 \<le> p2) = (fst p1 \<le> fst p2 \<and> snd p1 \<le> snd p2)"
closed_sum_comm: fixes A :: "'b set"   and B :: "'b set" shows "A +\<^sub>M B = B +\<^sub>M A"
poly_add_normalize_2: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_add p1 p2 = poly_add p1 (local.normalize p2)"
no_marker_if_no_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and cid :: "nat"   and p :: "'d"   and q :: "'d"   and i :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "channel cid = Some (p, q)"   and "\<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None" shows "Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t i) cid)"
P4: shows "Bet E F F' \<and> Cong F F' B C"
distinct_swap: fixes i :: "nat"   and xs :: "'b list" shows "distinct (swap i xs) = distinct xs"
U0_def: shows "U0 = ($0-\<Sigma>\<acute> =\<^sub>u $\<^bold>v)"
taylor_expansion_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "c \<in> carrier R" shows "taylor_expansion R c (p \<oplus>\<^bsub>UP R\<^esub> q) = taylor_expansion R c p \<oplus>\<^bsub>UP R\<^esub> taylor_expansion R c q"
dvd_minus_mod: 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 b :: "'b"   and a :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times b (minus a (modulo a b))"
NSLIM_mult: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and l :: "'b"   and g :: "'a \<Rightarrow> 'b"   and m :: "'b" assumes "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"   and "g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m" shows "(\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l * m"
valid_as_submap_init_submap_exec_i: fixes s :: "('d, 'e) fmap"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" shows "fmdom' s \<subseteq> fmdom' (state_succ s a)"
col_image_spec__eq: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and P' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B P"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong P P' A B" shows "P = P'"
Skip_is_CSP4: shows "action.relation_of Skip is CSP4 healthy"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 's stream set"   and s :: "'s" assumes "range A \<subseteq> T.events s"   and "decseq A" shows "(\<lambda>n. T.prob s (A n)) \<longlonglongrightarrow> T.prob s (\<Inter> (range A))"
M: shows "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
conf_Null: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and T :: "ty" shows "(P,h \<turnstile> Null :\<le> T) = P \<turnstile> NT \<le> T"
state_upd_in_atomicsR: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and c' :: "('e, 'd, 'f) com"   and t :: "'e"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a :: "('e, 'd, 'f) ann" assumes "\<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c', Normal t)"   and "s \<noteq> t"   and "ann_matches \<Gamma> \<Theta> a c"   and "s \<in> pre a" shows "\<exists>p cm x. atomicsR \<Gamma> \<Theta> a c (p, cm) \<and> s \<in> p \<and> \<Gamma>\<turnstile> (cm, Normal s) \<rightarrow> (x, Normal t) \<and> final (x, Normal t)"
0: fixes f :: "'a" assumes "ide f"   and "b \<in> targets f"   and "b' \<in> targets f" shows "b \<cong> b'"
holomorphic_on_const: fixes c :: "complex"   and s :: "complex set" shows "(\<lambda>z. c) holomorphic_on s"
distrib_left: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semiring_ow U plus times"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "times a (plus b c) = plus (times a b) (times a c)"
chain_imp_rtrancl: fixes S :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. (S i, S (Suc i)) \<in> r"   and "i \<le> j" shows "(S i, S j) \<in> r\<^sup>*"
simps_1: fixes c :: "'b" shows "get_const (Le c) = c"
MPair_used: fixes X :: "msg"   and Y :: "msg"   and H :: "event list" assumes "\<lbrace>X, Y\<rbrace> \<in> used H" obtains "X \<in> used H" and "Y \<in> used H"
simps_73: fixes x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com"   and x41 :: "('d, 'e, 'f) com"   and x42 :: "('d, 'e, 'f) com" shows "Guard x91 x92 x93 \<noteq> Seq x41 x42"
norm_signed_Cons: fixes b :: "bit"   and bs :: "bit list" shows "norm_signed (b # bs) = (case b of \<zero> \<Rightarrow> if norm_unsigned bs = [] then [] else b # norm_unsigned bs | \<one> \<Rightarrow> b # rem_initial b bs)"
IH: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "dsafe \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'" shows "dterm_sem I (Tsubst \<theta> \<sigma>) \<nu> = dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> \<nu>"
square_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "square R S T U = (\<forall>x y. R x y \<longrightarrow> (\<forall>z. S x z \<longrightarrow> (\<exists>u. T y u \<and> U z u)))"
row_echelon_form_def: fixes A :: "'b mat" shows "row_echelon_form A \<equiv> \<exists>f. pivot_fun A f (dim_col A)"
state_rank_oldest_token_le: fixes p :: "'b"   and n :: "nat"   and i :: "nat"   and q :: "'b"   and j :: "nat"   and x :: "nat"   and y :: "nat" assumes "state_rank p n = Some i"   and "state_rank q n = Some j"   and "oldest_token p n = Some x"   and "oldest_token q n = Some y" shows "(i \<le> j) = (x \<le> y)"
mes0: fixes i :: "nat" shows "(\<lambda>x. f ((T ^^ i) x)) \<in> borel_measurable M"
Lower_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "Lower L A \<subseteq> carrier L"
space_subgroup_props_2: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "R.dimension n K E" shows "\<zero> \<in> E"
triangle: fixes F :: "'a set" assumes "finite F" shows "L2_set (cmod \<circ> (\<lambda>i. x i + y i)) F \<le> L2_set (cmod \<circ> x) F + L2_set (cmod \<circ> y) F"
ag_inv_inv: fixes x :: "'a" assumes "x \<in> carrier M" shows "-\<^sub>a (-\<^sub>a x) = x"
simps: fixes ops :: "'b arith_ops_record"   and f :: "'b list"   and n :: "nat"   and g :: "'b list"   and m :: "nat" shows "karatsuba_main_i ops f n g m = (if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then let ff = poly_of_list_i ops f in foldr (\<lambda>a p. plus_poly_i ops (smult_i ops a ff) (cCons_i ops (arith_ops_record.zero ops) p)) g [] else let n2 = n div 2 in if n2 < m then case split_at n2 f of (f0, f1) \<Rightarrow> case split_at n2 g of (g0, g1) \<Rightarrow> let p1 = karatsuba_main_i ops f1 (n - n2) g1 (m - n2); p2 = karatsuba_main_i ops (coeffs_minus_i ops f1 f0) n2 (coeffs_minus_i ops g1 g0) n2; p3 = karatsuba_main_i ops f0 n2 g0 n2 in plus_poly_i ops (monom_mult_i ops (n2 + n2) p1) (plus_poly_i ops (monom_mult_i ops n2 (plus_poly_i ops (minus_poly_i ops p1 p2) p3)) p3) else case split_at n2 f of (f0, f1) \<Rightarrow> let p1 = karatsuba_main_i ops f1 (n - n2) g m in Let (karatsuba_main_i ops f0 n2 g m) (plus_poly_i ops (monom_mult_i ops n2 p1)))"
s_max: fixes x :: "real" shows "ego.s x \<le> ego.s_stop"
linear_imp_differentiable_on: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f" shows "f differentiable_on S"
supp_set: fixes xs :: "'a list" shows "supp (set xs) = supp xs"
concat_map_filter_filter: fixes xs :: "'a list"   and Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b list" assumes "\<And>x. \<lbrakk>x \<in> set xs; \<not> Q x\<rbrakk> \<Longrightarrow> filter P (f x) = []" shows "concat (map (filter P \<circ> f) (filter Q xs)) = concat (map (filter P \<circ> f) xs)"
new_character_data_ptr_not_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "new_character_data_ptr |\<notin>| character_data_ptr_kinds h"
keys_map_key: fixes p :: "'b \<Rightarrow>\<^sub>0 'd" shows "keys (map_key f p) = f -` keys p"
asm: shows "v = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), 0, 0, \<alpha> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), 0, 0, \<beta> / complex_of_real (sqrt 2)]]"
refine_list_2: fixes R :: "('g \<times> 'h) set" shows "((#), (#)) \<in> R \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
lead_coeff_monom: fixes c :: "'a"   and n :: "nat" shows "lead_coeff (monom c n) = c"
SN_on_iff_wf_below: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "SN_on r A = wf_below (r\<inverse>) A"