hcju/isabelleps huggingface dataset:
cod_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.cod (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
F_weight_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq" shows "d_IN f x \<le> weight \<Gamma> x"
decisive_coalitionD: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and C :: "'a set"   and x :: "'b"   and y :: "'b" assumes "decisive scf A Is C x y" shows "C \<subseteq> Is"
cases: fixes x :: "'a list" obtains "x = []"   | v :: "'a" and va :: "'a list" where "x = v # va"
QD_invar_def: fixes u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and Q :: "nat list" shows "QD_invar u f Q \<equiv> u \<in> V - {s, t} \<and> distinct Q \<and> set Q = {v \<in> V - {s, t, u}. excess f v \<noteq> (0::'capacity)}"
concat_map_maps: fixes f :: "'c \<Rightarrow> 'b list"   and xs :: "'c list" shows "concat (map f xs) = List.maps f xs"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>g \<cdot>\<^sub>C f : a \<Rightarrow>\<^sub>C c\<guillemotright>" obtains b :: "'a" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>"
OK_plus_OK_eq_Err_conv: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and fe :: "'b err \<Rightarrow> 'b err \<Rightarrow> 'b err" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, Err.le r, fe)" shows "(OK x \<squnion>\<^bsub>fe\<^esub> OK y = Err) = (\<not> (\<exists>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
a_rcos_sum: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "I +> x <+> (I +> y) = I +> x \<oplus> y"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "max (max a b) c = max a (max b c)"
U_LIMCONS_def: shows "U\<^bsub>LIM-CONS\<^esub> \<equiv> {vs @ [j] \<odot> p |vs j p. 2 \<le> j \<and> p \<in> \<R>\<^sub>0\<^sub>1 \<and> \<phi> j = vs @ [j] \<odot> p}"
substT_compose_eq_or: fixes s :: "'trm"   and t1 :: "'trm"   and t2 :: "'trm"   and x1 :: "'var"   and x2 :: "'var" assumes "s \<in> atrm"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<in> var"   and "x2 \<in> var"   and "x1 = x2 \<or> x2 \<notin> FvarsT s" shows "substT (substT s t1 x1) t2 x2 = substT s (substT t1 t2 x2) x1"
weight_space_dim_def: fixes rs :: "nat list"   and shared_weights :: "bool" assumes "deep_model_correct_params rs" shows "deep_model_correct_params.weight_space_dim shared_weights rs = count_weights shared_weights (deep_model_l rs)"
dim_le_card': fixes s :: "'b set" assumes "finite s" shows "dim s \<le> card s"
filter_i: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters"   and "a \<in> F"   and "b \<in> F" shows "a * b \<in> F"
l2_inv4I: fixes x :: "l2_state" assumes "\<And>Rb. \<lbrakk>test_ended x; guessed_frame test xsk = Some (NonceF (Rb $ sk))\<rbrakk> \<Longrightarrow> progress x Rb = Some {xpkE, xsk}" shows "x \<in> l2_inv4"
root_test_convergence': fixes f :: "nat \<Rightarrow> 'b" assumes "limsup (\<lambda>n. ereal (root n (norm (f n)))) < 1" shows "summable f"
bi_uniqueDr: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and z :: "'c" assumes "bi_unique A"   and "A x y"   and "A x z" shows "y = z"
idx_pqueue_push_correct: fixes xs :: "(nat \<times> 'b) list"   and m :: "nat option list"   and k :: "nat"   and r :: "(nat \<times> 'b) list \<times> nat option list"   and v :: "'b" assumes "index_of_pqueue (xs, m)"   and "k < length m"   and "\<not> has_key_alist xs k"   and "r = idx_pqueue_push_fun k v (xs, m)" shows "index_of_pqueue r \<and> fst r = xs @ [(k, v)] \<and> length (snd r) = length m"
log_exp2_ge: fixes n :: "nat" shows "n \<le> 2 * 2 ^ Discrete.log n"
rel_gpv''_refl_eq_on: fixes X :: "'c set"   and gpv :: "('a, 'b, 'c) gpv"   and A :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> A x x"   and "\<And>out. out \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> B out out" shows "rel_gpv'' A B (eq_on X) gpv gpv"
partial: fixes i :: "nat" shows "f i = g i 0 \<and> f (Suc i) = g i (n i) \<and> (\<forall>j<n i. (g i j, g i (Suc j)) \<in> SN_rel_ext_step P Pw R Rw (ty' i j)) \<and> (\<forall>j\<le>n i. M (g i j)) \<and> (ty i = top_s \<longrightarrow> (\<exists>ia<n i. ty' i ia \<in> {top_s})) \<and> (ty i = top_ns \<longrightarrow> (\<exists>ia<n i. ty' i ia \<in> {top_s, top_ns})) \<and> (ty i = normal_s \<longrightarrow> (\<exists>ia<n i. ty' i ia \<in> {top_s, normal_s}))"
basis_in_ex1_supset_circuit_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> B"   and "x \<in> \<E> - B" shows "\<exists>!C. indep_system.circuit_in indep \<E> C \<and> C \<subseteq> insert x B"
mset_nondetMs_6: fixes return :: "'a multiset \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a multiset \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind"   and "monad_reader return bind ask" shows "monad_reader (nondetM_base.return_nondet return (\<lambda>x. {#x#})) (nondetM_base.bind_nondet bind (mmerge return bind)) (nondetM_base.ask_nondet ask)"
subls_union: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set" shows "L\<^sub>1 \<union> L\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma> = (L\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>) \<union> (L\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>)"
permutations_of_set_aux_list_refine: fixes xs :: "'a list"   and acc :: "'a list" assumes "distinct xs" shows "set (permutations_of_set_aux_list acc xs) = permutations_of_set_aux acc (set xs)"
of_nat_1_eq_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
is_interval_closed_segment_1: fixes a :: "real"   and b :: "real" shows "is_interval {a--b}"
ShiftD: fixes kl :: "'a list"   and Kl :: "'a list set"   and k :: "'a" assumes "kl \<in> BNF_Greatest_Fixpoint.Shift Kl k" shows "k # kl \<in> Kl"
even_succ_div_exp: 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 bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"   and "0 < n" shows "divide (plus one a) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
B\<^sub>2_at_least_two_objects: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "u \<in> S"   and "real c < W B\<^sub>2 + w u" shows "2 \<le> card B\<^sub>2"
int_1: fixes m :: "'b"   and n :: "'b" assumes "{m, n} \<subseteq> N \<and> m \<noteq> n" shows "B1 m \<inter> B1 n = {}"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c stream" shows "sprefix (x # xs) ys = (x = shd ys \<and> sprefix xs (stl ys))"
nn_expl_cond_bounded: fixes X :: "'a \<Rightarrow> real"   and N :: "'c measure"   and Y :: "'a \<Rightarrow> 'c"   and w :: "'a" assumes "\<forall>w\<in>space M. 0 \<le> X w"   and "integrable M X"   and "point_measurable M (space N) Y"   and "w \<in> space M"   and "Y w \<in> space N" shows "\<integral>\<^sup>+ x. ennreal (expl_cond_expect M Y X x * indicat_real (Y -` {Y w} \<inter> space M) x) \<partial>M < \<infinity>"
abc_Hoare_plus_unhalt2: fixes Q :: "nat list \<Rightarrow> bool"   and B :: "abc_inst list"   and P :: "nat list \<Rightarrow> bool"   and A :: "abc_inst list" assumes "{Q} B \<up>"   and "{P} A {Q}" shows "{P} A [+] B \<up>"
assms: shows "(tsl, PR_CONST (it_to_sorted_list \<Omega>)) \<in> Rs\<^sup>k \<rightarrow>\<^sub>a list_assn Rk"
ex_eval_fm_iff_exists_tm': fixes e :: "(name, hf) finfun"   and k :: "name"   and A :: "fm" shows "eval_fm e (SyntaxN.Ex k A) = (\<exists>t. eval_fm e (A(k::=t)))"
Sigma_def: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" shows "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {(x, y)}"
assms_8: shows "min_lvar_not_in_bounds s = Some x\<^sub>i"
a: shows "a \<in> U \<rightarrow> {\<zero>}"
valid_path_compose: fixes g :: "real \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "valid_path g"   and "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable at x"   and "continuous_on (path_image g) (deriv f)" shows "valid_path (f \<circ> g)"
t_wf_trm: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
set_simps2: fixes x :: "'c"   and xs :: "'c list" shows "set (x # xs) = {x} \<union> set xs"
right_neutral: fixes a :: "'f" shows "a + (0::'f) = a"
sign_simps_34: fixes bm :: "'a"   and am :: "'a"   and ck :: "'a" shows "bm * (am * ck) = am * (bm * ck)"
or_lt_cong_gt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Lt C D \<or> A B Gt C D \<or> Cong A B C D"
tc: fixes x :: "'a ppath set" assumes "pp_prod x x \<subseteq> x" shows "pp_prod (pp_star x) x = x"
zero_order_3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
dynamic_mheadsD: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and S :: "qtname"   and statT :: "ref_ty"   and sig :: "sig"   and dynC :: "qtname" assumes "emh \<in> mheads G S statT sig"   and "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh"   and "isrtype G statT"   and "wf_prog G" shows "? m:dynlookup G statT dynC sig: is_static m = is_static emh \<and> G\<turnstile>resTy m\<preceq>resTy emh"
fun_upds_nth: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b" assumes "i < length xs"   and "length xs = length ys"   and "distinct xs" shows "f(xs [:=] ys) (xs ! i) = ys ! i"
lem2: fixes x :: "'a"   and y :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<noteq> y" shows "(x, M = y, N) = (y \<in># M \<and> N = x, M - (y, {#}))"
errMOD_igSubstAbsIPresIGWlsAbsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igSubstAbsIPresIGWlsAbs MOD"   and "igWlsAbsDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igSubstAbsIPresIGWlsAbsSTR (errMOD MOD)"
uMult_def: shows "uMult \<equiv> \<Lambda> x y. Discr (undiscr x * undiscr y)"
intro: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "semilat (A, r, f)" shows "Semilat A r f"
RG_sum_list_rddg_closed: fixes rgs :: "('a \<times> 'g) list" assumes "set (map snd rgs) \<subseteq> H" shows "(\<Sum>(r, g)\<leftarrow>rgs. r \<delta>\<delta> g) \<in> Supgroup.group_ring"
not_top_le_1_ennreal: shows "\<not> \<top> \<le> 1"
oalist_inv_alt: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a nat_term_order" shows "ko_ntm.oalist_inv (xs, ko) = ko.oalist_inv_raw (key_order_of_nat_term_order_inv ko) xs"
reads_restrict2: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "'a"   and \<sigma> :: "'b \<Rightarrow> 'e"   and \<sigma>' :: "'b \<Rightarrow> 'e" assumes "IFC prob"   and "\<forall>x\<in>IFC_def.reads prob n. \<sigma> x = \<sigma>' x" shows "\<sigma> \<restriction> IFC_def.reads prob n = \<sigma>' \<restriction> IFC_def.reads prob n"
simps_3: fixes K :: "nat"   and X :: "freemsg" shows "freenonces (CRYPT K X) = freenonces X"
elims_2: fixes x :: "atom set \<times> 'b"   and xa :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xb :: "'b \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom set \<times> 'b" assumes "x \<approx>set xa xb xc xd" obtains bs :: "atom set" and xe :: "'b" and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and f :: "'b \<Rightarrow> atom set" and p :: "perm" and cs :: "atom set" and y :: "'b" where "x = (bs, xe)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "f xe - bs = f y - cs \<and> (f xe - bs) \<sharp>* p \<and> R (p \<bullet> xe) y \<and> p \<bullet> bs = cs"
pnormal_degree: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times"   and "last p \<noteq> zero" shows "semiring_0.degree zero p = length p - 1"
contains_predE: fixes A :: "'a set"   and x :: "'a"   and y :: "unit" assumes "pred.eval (contains_pred A x) y" obtains "contains A x"
blinding_of_discrete_hash: shows "(=) \<le> vimage2p id id (=)"
f_plus_minus2: fixes f :: "'b \<Rightarrow> real" shows "f = (\<lambda>t. pp f t - np f t)"
last_take_nth_conv: fixes n :: "nat"   and l :: "'b list" assumes "n \<le> length l"   and "n \<noteq> 0" shows "last (take n l) = l ! (n - 1)"
trans: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" assumes "subseq a b"   and "subseq b c" shows "subseq a c"
nest_def: fixes r :: "('a \<times> 'a) set"   and s :: "((accent \<times> 'a) list \<times> (accent \<times> 'a) list) set" shows "nest r s = {(a, b). (ms_of_greek a, ms_of_greek b) \<in> mult (letter_less r <*lex*> s)}"
generate_valid_stateful_policy_IFSACS_2_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "generate_valid_stateful_policy_IFSACS_2 G M edgesList \<equiv> \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (filter_IFS_no_violations G M edgesList) \<inter> set (filter_compliant_stateful_ACS G M edgesList)\<rparr>"
finestpartSubset: fixes X :: "'c set"   and Y :: "'c set" assumes "X \<subseteq> Y" shows "finestpart X \<subseteq> finestpart Y"
norm_triangle_ineq2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (minus a b)"
path_append_conv: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "'b list"   and v :: "'b"   and w :: "'b" shows "path E u (p @ [v]) w = (path E u p v \<and> (v, w) \<in> E)"
irreducible_cases: fixes p :: "'a poly" shows "irreducible p = (degree p = 0 \<and> irreducible (coeff p 0) \<or> degree p \<noteq> 0 \<and> irreducible (map_poly to_fract p) \<and> content_ff (map_poly to_fract p) =dff 1)"
mcontI: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and luba :: "'f set \<Rightarrow> 'f"   and lubb :: "'g set \<Rightarrow> 'g" assumes "monotone orda ordb f"   and "cont luba orda lubb ordb f" shows "mcont luba orda lubb ordb f"
components_subset_ab: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'h"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'h"   and data' :: "nat \<times> 'g" assumes "ap_spec ap"   and "ab_spec ab"   and "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)" shows "component_of_term ` Keys (args_to_set (gs, ab gs bs hs data', ap gs bs (ps -- sps) hs data')) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps))"
pnormal_length: fixes p :: "'b list" assumes "p \<noteq> []" shows "pnormal p = (length (pnormalize p) = length p)"
semigroup_axioms: shows "Type_Semigroups.semigroup times"
not_infI: fixes cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "('d, 'f) xstate"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "\<And>f. \<lbrakk>f 0 = (cs, css, s); \<And>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)\<rbrakk> \<Longrightarrow> False" shows "\<not> AlternativeSmallStep.inf \<Gamma> cs css s"
fin: fixes s :: "'a rexp" shows "finite {fold Deriv w (lang s) |w. True}"
\<tau>move2BlockSome2: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and T :: "ty"   and v :: "'addr val"   and e :: "'addr expr1" shows "\<tau>move2 P h stk {V:T=\<lfloor>v\<rfloor>; e} (Suc 0) None"
trans: fixes x :: "'alt"   and y :: "'alt"   and z :: "'alt" assumes "Pareto R x y"   and "Pareto R y z" shows "Pareto R x z"
sets_uniform_count_measure: fixes A :: "'a set" shows "sets (uniform_count_measure A) = Pow A"
onode_disconnect_other: fixes i :: "nat"   and i' :: "nat"   and i'' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "i \<noteq> i'"   and "i \<noteq> i''"   and "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), disconnect(i', i''), \<sigma>', NodeS i s R) \<in> onode_sos S"
lens_indep_overrideI: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "vwb_lens X"   and "vwb_lens Y"   and "\<And>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>3 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2" shows "X \<bowtie> Y"
sel_1: fixes z :: "complex" shows "Re (cnj z) = Re z"
get_disconnected_nodes_locs_def: shows "get_disconnected_nodes_locs = (\<lambda>document_ptr. {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)})"
fps_zero_nth: fixes n :: "'d list" shows "0 $ n = (0::'c)"
simps_25: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case NT of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f4"
assms_2: shows "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form multiset" shows "MultiSequents.mset ( ant \<Rightarrow>* suc) = ant + suc"
get_M_Element_preserved8: fixes element_ptr :: "(_) element_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('az \<Rightarrow> 'az) \<Rightarrow> ('as, 'aq, 'ar, 'au, 'aw option) RElement_scheme \<Rightarrow> ('as, 'aq, 'ar, 'au, 'aw option) RElement_scheme"   and v :: "'az"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'ba" assumes "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
set_category_is_categorical: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<AA> :: "('b \<times> 'b) set"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<phi> :: "'a \<Rightarrow> 'c" assumes "set_category S \<AA>"   and "set_category S' \<AA>"   and "bij_betw \<phi> (Collect (category.terminal S)) (Collect (category.terminal S'))" shows "\<exists>\<Phi>. invertible_functor S S' \<Phi> \<and> (\<forall>m. set_category.incl S \<AA> m \<longrightarrow> set_category.incl S' \<AA> (\<Phi> m))"
rbt_union_def: shows "rbt_union = rbt_union_with_key (\<lambda>_ _ rv. rv)"
negr_def: fixes a :: "'a" shows "negr a = a r\<rightarrow> (0::'a)"
iteratei_alt_def: fixes m :: "('h, ('f, 'g) assoc_list) RBT.rbt" shows "HashMap_Impl.iteratei m = set_iterator_image snd (set_iterator_product (rm.iteratei m) (\<lambda>hclm. lm.iteratei (snd hclm)))"
poly_shift_eq_0: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "f n = (monom P (f 0) 0 \<oplus>\<^bsub>P\<^esub> X_poly R \<otimes>\<^bsub>P\<^esub> poly_shift f) n"
matrix_vector_column: fixes A :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s Finite_Cartesian_Product.transpose A $ i)"
pgalleryD_distinct: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.pmaxsimpchain X xs" shows "distinct xs"
SafeGlobMono_def: fixes c0 :: "('c, 'd, 'e) configuration_scheme"   and c1 :: "('c, 'd, 'f) configuration_scheme" shows "SafeGlobMono c0 c1 = (\<forall>p t. GlobVacantUpto c0 p t \<longrightarrow> GlobVacantUpto c1 p t)"
rref_condition4: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b))"
GuardFault: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<notin> g" shows "\<Gamma>\<turnstile> (Guard f g c # cs, css, Normal s) \<rightarrow> (cs, css, Fault f)"