hcju/isabelleps huggingface dataset:
intro: fixes X :: "'b set set"   and C0 :: "'b set" assumes "SimplicialComplex.maxsimp X C0"   and "\<And>C D. \<lbrakk>SimplicialComplex.maxsimp X C; SimplicialComplex.maxsimp X D; C \<sim> D; C \<noteq> D\<rbrakk> \<Longrightarrow> \<exists>f g. OpposedThinChamberComplexFoldings X f g C \<and> D = g ` C" shows "ThinChamberComplexManyFoldings_axioms X C0"
assms_2: shows "RModule FG (\<cdot>) U \<and> U \<subseteq> V"
subsets_smash_def: fixes n :: "nat"   and s1 :: "nat list list"   and s2 :: "nat list list" shows "subsets_smash n s1 s2 = concat (map (\<lambda>l1. map (\<lambda>l2. l1 @ map ((+) n) l2) s2) s1)"
Cons_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set (x__ # xs)"   and "y \<in> set (x__ # xs)" shows "rel x y \<or> rel y x"
runit_in_hom_1: fixes f :: "'a" assumes "ide f" shows "arr \<r>[f] \<and> src f \<in> sources \<r>[f] \<and> trg f \<in> targets \<r>[f]"
StepRecWhile: fixes cx :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat"   and b :: "expr" assumes "cx \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "while (b) cx \<turnstile> \<langle>c;; while (b) cx,s,l + 2\<rangle> \<leadsto> \<langle>c';; while (b) cx,s',l' + 2\<rangle>"
S2MClkFwd_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> ($S2 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> <MClkFwd memCh crCh cst p>_(c p) \<longrightarrow> S3 rmhist p$"
arc_imp_inj_on: fixes g :: "real \<Rightarrow> 'c" assumes "arc g" shows "inj_on g {0..1}"
real_alg_2_1: fixes rai :: "int poly \<times> rat \<times> rat" assumes "invariant_1 rai" shows "invariant_2 (real_alg_2 rai)"
res_compose: fixes c :: "cars" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (res ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (res ts) c = lan u \<and> |lan u| = 1) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
EF_lem2b: fixes x :: "'a"   and f :: "'a set" assumes "x \<in> EX' (EF f)" shows "x \<in> EF f"
table_project: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set" assumes "table (Suc n) A X" shows "table n ((\<lambda>x. x - Suc 0) ` (A - {0})) (tl ` X)"
assms: shows "d \<cdot>\<^sub>p \<delta> \<cdot>\<^sub>p \<theta> \<noteq> d' \<cdot>\<^sub>p \<I>"
simps_8: fixes x2 :: "'b" shows "\<infinity> \<noteq> Lt x2"
set_bit_nonnegative_int_iff: fixes n :: "nat"   and k :: "int" shows "(0 \<le> set_bit n k) = (0 \<le> k)"
simps_1: fixes d :: "'a" shows "the_default d None = d"
R1: fixes u :: "nat"   and ns :: "nat list"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list" assumes "(u, ns, xs) \<in> gcsort_set index key p t" shows "gcsort_round index key p ns xs \<in> gcsort_set index key p t"
r_psi_recfn: shows "recfn 2 r_psi"
word_le_add: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> y" shows "\<exists>n. y = x + word_of_nat n"
distr_PiM_restrict_finite: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "distr (Pi\<^sub>M I M) (Pi\<^sub>M J M) (\<lambda>x. restrict x J) = Pi\<^sub>M J M"
nonneg_bounded: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
bit_shiftr1_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (shiftr1 w) n = bit w (Suc n)"
C: shows "U \<in> key_sets (Pwd n) (crypts (Log -` spied s)) \<longrightarrow> (\<exists>SK. U = {SesKey SK} \<and> ((Owner n, Crypt (SesK SK) (Pwd n)) \<in> s \<or> (Asset n, Crypt (SesK SK) (Num 0)) \<in> s))"
intros_1: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and n :: "nat"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and n' :: "nat"   and prob :: "('c, 'a, 'b, 'd) ifc_problem" assumes "((\<sigma>, n), \<sigma>', n') \<in> IFC_def.cp prob"   and "IFC_def.path prob \<sigma> n \<in> dom (IFC_def.att prob)" shows "((\<sigma>, n), \<sigma>', n') \<in> IFC_def.cop prob"
P2_def: shows "P2 = {p. p \<in># prime_factorization n \<and> [p = 1] (mod 4)}"
simps: fixes f :: "nat \<Rightarrow> bool" shows "wf_set_bits_int f = ((\<exists>n. \<forall>n'\<ge>n. \<not> f n') \<or> (\<exists>n. \<forall>n'\<ge>n. f n'))"
exhaust: fixes y :: "'b extended" obtains "y = \<turnstile>"   | x2 :: "'b" where "y = \<lbrakk>x2\<rbrakk>"   | "y = \<stileturn>"
fls_lr_inverse_shift_nonzero_1: fixes f :: "'a fls"   and m :: "int"   and x :: "'a" assumes "f \<noteq> 0" shows "fls_left_inverse (fls_shift m f) x = fls_shift (- m) (fls_left_inverse f x)"
not_Carmichael_number_imp_card_fermat_witness_bound_2: fixes n :: "nat" assumes "\<not> prime n"   and "\<not> Carmichael_number n"   and "odd n"   and "1 < n" shows "real (card {a. 2 \<le> a \<and> a < n \<and> strong_fermat_liar a n}) < real (n - 2) / 2"
rtrancl_into_rtrancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
sum_subtractff: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and B :: "nat set"   and A :: "nat set" shows "(\<Sum>x\<in>A. \<Sum>y\<in>B. h x y - g x y) = (\<Sum>x\<in>A. sum (h x) B) - (\<Sum>x\<in>A. sum (g x) B)"
singleE': fixes x :: "'a"   and y :: "'a" assumes "eval (single x) y" obtains "x = y"
relative_to_def: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" shows "P relative_to S \<equiv> \<lambda>T. \<exists>U. P U \<and> S \<inter> U = T"
lunit_def: fixes f :: "'a" shows "PB.lunit f \<equiv> left_hom_with_unit.lunit (\<cdot>) (\<star>) \<a> (\<i>' (trg f)) (trg f) f"
fds_nth_const: fixes c :: "'a" shows "fds_nth (fds_const c) = (\<lambda>n. if n = 1 then c else (0::'a))"
rel_simps_18: fixes m :: "num" shows "(num.Bit0 m = num.One) = False"
hom_sum: fixes f :: "'b \<Rightarrow> 'a poly"   and X :: "'b set"   and i :: "nat" shows "coeff (sum f X) i = (\<Sum>x\<in>X. coeff (f x) i)"
merge_guards_res_Guard: fixes c :: "('d, 'e, 'f) com"   and f :: "'f"   and g :: "'d set"   and c' :: "('d, 'e, 'f) com" assumes "merge_guards c = Guard f g c'" shows "\<exists>c'' f' g'. c = Guard f' g' c''"
aux_2: shows "\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High"
monomial_uminus: fixes s :: "'b"   and c :: "'c" shows "- monomial c s = monomial (- c) s"
simps_2: fixes n :: "'a"   and l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0" shows "is_bal (MKT\<^sub>0 n l r) = ((height l = height r \<or> height l = 1 + height r \<or> height r = 1 + height l) \<and> is_bal l \<and> is_bal r)"
csmall_nstep_exists: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state" assumes "\<sigma>' \<in> big P \<sigma>" shows "\<exists>n coll. (\<sigma>', coll) \<in> csmall_nstep P \<sigma> n \<and> \<sigma>' \<in> endset"
prod_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "local.prod a b = hf_to_ide (ide_to_hf a * ide_to_hf b)"
linord_of_list_append: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "((x, y) \<in> linord_of_list (xs @ ys)) = ((x, y) \<in> linord_of_list xs \<or> (x, y) \<in> linord_of_list ys \<or> y \<in> set xs \<and> x \<in> set ys)"
lem: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and P :: "'d \<Rightarrow> bool"   and b :: "'d" assumes "\<forall>y. r\<^sup>+\<^sup>+ a y \<longrightarrow> P a \<longrightarrow> P y" shows "((b, a) \<in> {(y, x). P x \<and> r x y}\<^sup>+) = ((b, a) \<in> {(y, x). P x \<and> r\<^sup>+\<^sup>+ x y})"
compat_S_NS_point: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(x, y) \<in> S"   and "(y, z) \<in> NS" shows "(x, z) \<in> S"
assms_2: fixes x :: "'a" shows "0 \<le> u x"
image_mset_empty: fixes f :: "'b \<Rightarrow> 'a" shows "image_mset f (\<Empt>) = \<Empt>"
vec_iff: fixes x :: "'a" shows "(d x \<cdot> U = x) = (x \<cdot> 1\<^sub>\<pi> \<parallel> U = x)"
choose_pivot3_choose_pivot: fixes lo :: "nat"   and xs :: "'c list"   and hi :: "nat"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "lo < length xs"   and "hi < length xs"   and "lo \<le> hi" shows "choose_pivot3 R h xs lo hi \<le> \<Down> nat_rel (choose_pivot R h xs lo hi)"
atom_name_def: fixes a :: "name" shows "atom a = Rep_name a"
quot_of_fract_uminus: fixes x :: "'b fract" shows "quot_of_fract (- x) = (let (a, b) = quot_of_fract x in (- a, b))"
Collect_inj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "Collect P = Collect Q" shows "P = Q"
mult_le_cancel_right1_hmset: fixes y :: "hmultiset"   and x :: "hmultiset" assumes "0 < y" shows "x \<le> y * x"
sgn_octo_def: fixes x :: "octo" shows "sgn x = x /\<^sub>R norm x"
T_on_rand'_append: fixes A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and s :: "('state \<times> 'b) pmf"   and xs :: "'request list"   and ys :: "'request list" shows "T_on_rand' A s (xs @ ys) = T_on_rand' A s xs + T_on_rand' A (config'_rand A s xs) ys"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('d \<times> ('e, 'f) stateful_strand_step) list"   and N :: "('d \<times> ('e, 'f) stateful_strand_step) list" assumes "set M \<subseteq> set N" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t M \<subseteq> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t N"
\<psi>\<^sub>3_is_state: shows "state 2 (Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) + (real (f 1) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2)), real (1 - f 0) / (2 * sqrt 2) - real (f 0) / (2 * sqrt 2) - real (1 - f 1) / (2 * sqrt 2) + real (f 1) / (2 * sqrt 2), real (f 0) / (2 * sqrt 2) - real (1 - f 0) / (2 * sqrt 2) - real (f 1) / (2 * sqrt 2) + real (1 - f 1) / (2 * sqrt 2)]]))"
counit_simps_3: fixes a :: "'d" assumes "D.obj a" shows "src\<^sub>D (\<eta>'.counit a) = GF.map\<^sub>0 a"
cup_pmatch_list_nomatch: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace" assumes "length vs = length ps" shows "Matching.fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> cupcake_pmatch cenv p v env) Match_type_error ps vs No_match = No_match"
prv_dsj_cnj_distrib2: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (cnj (dsj \<phi> \<chi>1) (dsj \<phi> \<chi>2)) (dsj \<phi> (cnj \<chi>1 \<chi>2)))"
le_type_scheme_def2: fixes sch' :: "type_scheme"   and sch :: "type_scheme" shows "(sch' \<le> sch) = (\<exists>B. sch' = bound_scheme_inst B sch)"
tiTable_i1_4: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop"   and "\<forall>t1\<le>t. req t1 = []"   and "req (Suc t) = [init]"   and "\<forall>m<k + 3. req (t + m) \<noteq> [send]"   and "\<forall>m\<le>k. ack (Suc (Suc (t + m))) = [connection_ok]"   and "\<forall>j\<le>k + 3. lose (t + j) = [False]" shows "\<forall>t2<t + 3 + k. i1 t2 = []"
dom_delete_subset: fixes k :: "'d"   and al :: "('d \<times> 'e) list" shows "fst ` set (delete k al) \<subseteq> fst ` set al"
Abs_inverse: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'b" assumes "type_definition Rep Abs A"   and "y \<in> A" shows "Rep (Abs y) = y"
theory_full_exhaust: fixes \<Theta> :: "('a \<times> 'b \<times> 'c) \<times> 'd" obtains cto :: "'a" and tao :: "'b" and sorts :: "'c" and axioms :: "'d" where "\<Theta> = ((cto, tao, sorts), axioms)"
spmf_eq_0_set_spmf: fixes p :: "'d spmf"   and x :: "'d" shows "(spmf p x = 0) = (x \<notin> set_spmf p)"
subnetAux: fixes Dd :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "Dd \<inter> A \<noteq> {}"   and "A \<subseteq> B" shows "Dd \<inter> B \<noteq> {}"
distinct_append: fixes xs :: "'a list"   and ys :: "'a list" shows "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
infsetsum'_tendsto: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "infsetsum'_converges f S" shows "(sum f \<longlongrightarrow> infsetsum' f S) (finite_subsets_at_top S)"
effect_to_assignments_iii_b: fixes v :: "'a"   and op :: "'a strips_operator"   and as :: "('a \<times> bool) list" assumes "v \<in> set (delete_effects_of op)"   and "as = effect_to_assignments op" obtains a :: "'a \<times> bool" where "a \<in> set as" and "a = (v, False)"
seq_sim_mono: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> bool) set"   and \<tau> :: "nat \<Rightarrow> 'b"   and B :: "('b \<Rightarrow> bool) set" assumes "\<sigma> \<simeq>A\<simeq> \<tau>"   and "B \<subseteq> A" shows "\<sigma> \<simeq>B\<simeq> \<tau>"
neighborhoods_def: fixes x :: "'a set" shows "pr.neighborhoods x \<equiv> {U. is_zariski_open U \<and> x \<in> U}"
BotLe: fixes \<Gamma> :: "'b formula list"   and \<Delta> :: "'b formula list" assumes "\<bottom> \<in> set \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta>"
nn_integral_snd: fixes M1 :: "'a measure"   and M2 :: "'b measure"   and f :: "'a \<times> 'b \<Rightarrow> ennreal" assumes "pair_sigma_finite M1 M2"   and "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M1 \<partial>M2 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
Quotient_dlist: shows "Quotient dlist_eq dlist rep_dlist cr_dlist"
division_ofD_5: fixes s :: "'c set set"   and i :: "'c set"   and K1 :: "'c set"   and K2 :: "'c set" assumes "s division_of i"   and "K1 \<in> s"   and "K2 \<in> s"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
pred_inject: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and aa :: "'b stream" shows "pred_stream P (a ## aa) = (P a \<and> pred_stream P aa)"
Some_those_length: fixes ys :: "'a option list"   and xs :: "'a list" assumes "those ys = Some xs" shows "length xs = length ys"
span_card_ge_dim: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set"   and V :: "'e set" assumes "vector_space scale"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "finite B" shows "vector_space.dim scale V \<le> card B"
ring_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "ring_ow U plus zero minus uminus times \<equiv> semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
index_not_in_insert_Some_None: fixes i :: "'c"   and xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" assumes "i \<notin> fst ` set xs" shows "insert xs e (Some i) = None"
int_pow_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int"   and m :: "int" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "(x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> n) [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> m = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> (n * m)"
r_step_Cn_prim: shows "prim_recfn 1 r_step_Cn"
fcb2: shows "(bs \<inter> supp y) \<sharp>* f (bs \<inter> supp y) y c"
trss_stack_comp_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and r :: "'f list" assumes "((s, c), e, s', c') \<in> trss fg" shows "((s @ r, c), e, s' @ r, c') \<in> trss fg"
dvd_left_imp_subdegree_le: fixes g :: "'b fps"   and f :: "'b fps" assumes "\<exists>k. g = f * k"   and "g \<noteq> 0" shows "subdegree f \<le> subdegree g"
fg: fixes z :: "'b" assumes "z \<in> T" shows "f (g z) = z"
unrest_lit: fixes x :: "'a \<Longrightarrow> 'b"   and v :: "'c" shows "x \<sharp> \<guillemotleft>v\<guillemotright>"
Int_UN_distrib: fixes B :: "'d set"   and A :: "'e \<Rightarrow> 'd set"   and I :: "'e set" shows "B \<inter> \<Union> (A ` I) = (\<Union>i\<in>I. B \<inter> A i)"
fold_reduce_add: fixes m :: "nat"   and n :: "nat" shows "fold_reduce 0 (m + n) = fold_reduce 0 m \<circ> fold_reduce m n"
Q3''_a_def: shows "Q3''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . p \<noteq> nil \<and> (\<exists>i. g mrk lbl (lnk i) p \<and> p' = lnk i p \<and> t' = p \<and> lnk' = lnk(i := (lnk i)(p := t)) \<and> lbl' = lbl(p := i) \<and> mrk' = mrk \<union> {lnk i p}) :]"
hext_upd_obj: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option" assumes "h a = \<lfloor>(C, fs)\<rfloor>" shows "h \<unlhd> h(a \<mapsto> (C, fs'))"
\<delta>: shows "0 < (MIN i. SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)))"
fsmult_assoc: fixes r :: "'f"   and s :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "s \<in> UNIV"   and "m \<in> V" shows "r \<sharp>\<cdot> s \<sharp>\<cdot> m = (r * s) \<sharp>\<cdot> m"
RG_mult_closed: fixes x :: "('a, 'g) aezfun"   and y :: "('a, 'g) aezfun" assumes "x \<in> ActingGroup.group_ring"   and "y \<in> ActingGroup.group_ring" shows "x * y \<in> ActingGroup.group_ring"
setdist_closure_1: fixes S :: "'b set"   and T :: "'b set" shows "setdist (closure S) T = setdist S T"
succ_diff: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "succ r i \<noteq> i"
ring_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.ring plus zero minus uminus times \<equiv> class.ab_group_add plus zero minus uminus \<and> class.semiring plus times"
abs_of_nonpos: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a zero" shows "abs a = uminus a"
asymp_equiv_imp_bigo: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "f \<in> O[F](g)"
cod_runit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.cod C (monoidal_category.runit C T \<alpha> \<iota> a) = a"
fromIMorAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'b" shows "fromIMorAbs hA \<equiv> snd \<circ> hA"
simps_2: fixes x :: "'a"   and n :: "nat" shows "power_rd x (Suc n) = 1\<^sub>\<sigma> + x \<cdot> power_rd x n"
simps_5: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com" shows "well_com (While M S) = (measurement d 2 M \<and> well_com S)"
dual_filters: shows "ord.filters (\<lambda>x y. y \<le> x) = ideals"
map_def: fixes f :: "'j" shows "A.map f = (if J.arr f then a else C.null)"
lengthClosed: fixes p :: "(name \<times> name) list"   and xs :: "'a list" shows "length (p \<bullet> xs) = length xs"
r: fixes i :: "'a \<Rightarrow> 'b"   and j :: "'c \<Rightarrow> 'd" assumes "p i"   and "q j" shows "r (\<lambda>(x, y). (i x, j y))"
sup_upper: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> A" shows "x \<sqsubseteq>\<^bsub>L\<^esub> \<Squnion>\<^bsub>L\<^esub>A"
w'\<theta>'\<nu>': shows "ide w' \<and> \<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : g' \<Rightarrow> g \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> \<rho>' = ((r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w')) \<cdot> \<nu>'"
3: shows "(w \<sqinter> - d)\<^sup>\<star> * e * top \<le> w\<^sup>T\<^sup>\<star> * d\<^sup>T * top"
T_mtf_ub: fixes n :: "nat" assumes "\<forall>i<n. rs ! i \<in> set s0" shows "T_mtf n \<le> int (n * length s0)"
simps: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and nods :: "'a set" shows "completely_specified_rel_over rel nods = (\<forall>s1\<in>nods. \<forall>x\<in>(fst \<circ> fst \<circ> snd) ` rel. \<exists>y\<in>(snd \<circ> fst \<circ> snd) ` rel. \<exists>s2. (s1, (x, y), s2) \<in> rel)"
spectral_radius_def: fixes x :: "'b" shows "spectral_radius x = Inf {root n (norm (x ^ n)) |n. 0 < n}"
utp_theory_mono_axioms: shows "utp_theory_mono \<H>"
measurable_envelopeD1: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set"   and F :: "'b set" assumes "measurable_envelope M A E"   and "F \<in> sets M"   and "F \<subseteq> E - A" shows "emeasure M F = 0"
is_glb_above_iff: fixes S :: "'a set"   and x :: "'a"   and u :: "'a" assumes "S >>| x" shows "(u \<sqsubseteq> x) = S >| u"
simps_17: fixes v :: "'a local.term" shows "Nml \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[v\<^bold>] = False"
choicefun_def: fixes E :: "'a set" shows "choicefun E = (SOME x. \<exists>X\<in>Pow E - {{}}. x \<in> X)"
aS: shows "adjuster n w us \<in> local.span (insert (adjuster n w us + w) (set us))"
662: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> y \<squnion> (x \<sqinter> (z \<squnion> y) \<squnion> u) = x \<sqinter> (z \<squnion> y) \<squnion> u"
finite_measure_axioms: shows "finite_measure (Pi\<^sub>M I (\<lambda>i. M (f i)))"
in_set_conv_decomp_first: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in>. xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
lincomb_list_as_mat_mult: fixes ws :: "rat vec list"   and c :: "nat \<Rightarrow> rat" assumes "\<forall>w\<in>set ws. dim_vec w = n" shows "gs.lincomb_list c ws = mat_of_cols n ws *\<^sub>v vec (length ws) c"
spr_simAbs_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agent :: "'b" assumes "FiniteSingleAgentEnvironment jkbp agent" shows "FiniteSingleAgentEnvironment.spr_simAbs \<equiv> \<lambda>ss. rel_ext (\<lambda>uu_. \<exists>s. uu_ = (toSet ss, s) \<and> s \<in> toSet ss)"
divide_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "A div B = A * inverse B"