hcju/isabelleps huggingface dataset:
direct_decomp_appendD_2: fixes A :: "'b set"   and ss1 :: "'b set list"   and ss2 :: "'b set list" assumes "direct_decomp A (ss1 @ ss2)"   and "{} \<notin> set ss1" shows "direct_decomp (sum_list ` listset ss2) ss2"
H: fixes F :: "'a set set" assumes "F \<in> Filters"   and "F \<longlongrightarrow> x"   and "A \<in> F" shows "R"
le_fininf_def: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2 \<equiv> \<exists>v\<^sub>2. v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2 \<and> w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
finite_measure_eq_sum_singleton: fixes S :: "'h set"   and M :: "'h pmf" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> measure_pmf.events M" shows "measure_pmf.prob M S = (\<Sum>x\<in>S. measure_pmf.prob M {x})"
composable_char\<^sub>P\<^sub>B: fixes \<tau> :: "'a"   and \<sigma> :: "'a" shows "(\<tau> \<star> \<sigma> \<noteq> null) = (arr \<sigma> \<and> arr \<tau> \<and> some_src \<tau> = some_trg \<sigma>)"
boundary_of_region_is_subset_of_partition_boundary: shows "\<gamma> \<subseteq> two_chain_boundary two_chain"
ap_def: fixes f :: "('c, 'e \<Rightarrow> 'd) state"   and x :: "('c, 'e) state" shows "f \<diamondop> x = State (\<lambda>s. case run_state f s of (g, s') \<Rightarrow> case run_state x s' of (y, s'') \<Rightarrow> (g y, s''))"
path_nth_conv: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "'b list"   and v :: "'b" shows "path E u p v = (let p' = p @ [v] in u = p' ! 0 \<and> (\<forall>i<length p' - 1. (p' ! i, p' ! Suc i) \<in> E))"
red_rtrancl_plus_higher: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* p q"   and "\<And>u v. \<lbrakk>u \<in> keys p; v \<in> keys r\<rbrakk> \<Longrightarrow> u \<prec>\<^sub>t v" shows "(red F)\<^sup>*\<^sup>* (p + r) (q + r)"
dist_triangle3: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and a :: "'b" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist a x + dist a y"
pcompose_mult: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "(p * q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r * q \<circ>\<^sub>p r"
mapping_of_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b" shows "mapping_of (monom m a) = Poly_Mapping.single m a"
S: fixes r :: "'d rexp"   and r' :: "'d rexp" assumes "r ~ r'" shows "Star r ~ Star r'"
SNF_preserved_multiples_identity: fixes S :: "'b mat"   and m :: "nat"   and n :: "nat"   and k :: "'b" assumes "S \<in> carrier_mat m n"   and "Smith_normal_form_mat S" shows "Smith_normal_form_mat (S * (k \<cdot>\<^sub>m 1\<^sub>m n))"
stepfun_imp_tranclp: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat"   and z :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "f 0 = x"   and "f (Suc n) = z"   and "\<forall>i\<le>n. P (f i) (f (Suc i))" shows "P\<^sup>+\<^sup>+ x z"
eint_ord_number_1: fixes m :: "num"   and n :: "num" shows "(eint (numeral m) \<le> eint (numeral n)) = (numeral m \<le> numeral n)"
assms: shows "(a, b) \<in> r\<^sup>*"
contlub_cfun_fun: fixes F :: "nat \<Rightarrow> 'b \<rightarrow> 'c"   and x :: "'b" assumes "chain F" shows "(\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
to_list_repr: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)" shows "list_repr i f (map ((!) fs) [0..<m])"
path_prefix_io_targets: fixes M :: "('d, 'e, 'f) FSM"   and xs :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f"   and i :: "nat" assumes "path M (xs || tr) q"   and "length xs = length tr"   and "0 < length xs" shows "last (take (Suc i) tr) \<in> io_targets M q (take (Suc i) xs)"
funpow_swap1: fixes f :: "'k \<Rightarrow> 'k"   and n :: "nat"   and x :: "'k" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
algebraic_int_power_iff: fixes x :: "'a"   and n :: "nat" shows "algebraic_int (x ^ n) = (n = 0 \<or> algebraic_int x)"
f_Exec_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun (xs @ ys) c = f_Exec_Comp trans_fun ys (f_Exec_Comp trans_fun xs c)"
complex_Re_le_cmod: fixes x :: "complex" shows "Re x \<le> cmod x"
riesz_frechet_representation_unique: fixes f :: "'a \<Rightarrow> complex"   and t :: "'a"   and u :: "'a" assumes "\<And>x. f x = t \<bullet>\<^sub>C x"   and "\<And>x. f x = u \<bullet>\<^sub>C x" shows "t = u"
stake_add: fixes m :: "nat"   and s :: "'a stream"   and n :: "nat" shows "stake m s @ stake n (sdrop m s) = stake (m + n) s"
IH: assumes "Subobjs P C (rev Cs')" shows "P,C \<turnstile> [C] \<sqsubseteq> rev Cs'"
psubstAbs_compose: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodAbs A"   and "goodEnv rho"   and "goodEnv rho'" shows "A $[rho] $[rho'] = A $[rho &[rho']]"
diff_fresh_vsubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y \<noteq> y1" shows "fresh ys y (X #[y1 // y]_ys)"
smult_l_null: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and x :: "'j" assumes "module R M"   and "x \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
leof_trans_nofail: fixes a :: "'c nres"   and b :: "'c nres"   and c :: "'c nres" assumes "a \<le>\<^sub>n b"   and "nofail b"   and "b \<le>\<^sub>n c" shows "a \<le>\<^sub>n c"
subdprodl_def: fixes x :: "nat list"   and y :: "nat list" shows "subdprodl x y = (\<forall>k\<le>m. take k a \<bullet> take k x \<le> b \<bullet> y)"
eq: shows "vec (length ts) (\<lambda>i. c * poly_to_row ts p $ i) = vec (length ts) (\<lambda>i. c * lookup p (ts ! i))"
a_d_one: fixes x :: "'a" shows "d x \<oplus> ad x = (1::'a)"
Cs_def: shows "Cs = (ARG_MIN length Cs. domain.gallery (C # Cs @ [D]))"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma>\<^sub>\<sigma> :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and state_update :: "'b \<Rightarrow> final_decision \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma>\<^sub>0 :: "'b"   and x1 :: "char list \<times> action"   and x2 :: "'c list"   and x3 :: "('c \<times> final_decision) list"   and x4 :: "'b"   and P :: "char list \<times> action \<Rightarrow> 'c list \<Rightarrow> ('c \<times> final_decision) list \<Rightarrow> 'b \<Rightarrow> bool" assumes "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_update \<sigma>\<^sub>0 x1 x2 x3 x4"   and "\<And>built_in_chain default_policy ps. P (built_in_chain, default_policy) ps [] \<sigma>\<^sub>0"   and "\<And>built_in_chain default_policy p ps ps_processed \<sigma>' X. \<lbrakk>semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) (p # ps) ps_processed \<sigma>'; P (built_in_chain, default_policy) (p # ps) ps_processed \<sigma>'; \<Gamma>,\<gamma>\<^sub>\<sigma> \<sigma>',p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> Decision X\<rbrakk> \<Longrightarrow> P (built_in_chain, default_policy) ps (ps_processed @ [(p, X)]) (state_update \<sigma>' X p)" shows "P x1 x2 x3 x4"
2: shows "((ereal \<circ> (\<lambda>x. log q (1 - x)) \<circ> real_of_ereal) \<longlongrightarrow> \<infinity>) (at_left 1)"
exec_unit_SE': fixes \<sigma>\<^sub>0 :: "'d"   and f :: "'d \<Rightarrow> bool" shows "(\<sigma>\<^sub>0 \<Turnstile> (\<lambda>\<sigma>. Some (f \<sigma>, \<sigma>))) = f \<sigma>\<^sub>0"
Limit_vpair_in_VsetI: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Limit \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> Vset \<alpha>"
diag_PAQ_dvd_Gcd_minors: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "invertible_mat P"   and "invertible_mat Q"   and "Smith_normal_form_mat (P * A * Q)" shows "(\<Prod>i = 0..<k. (P * A * Q) $$ (i, i)) dvd Gcd (minors A k)"
finalE: fixes e :: "'a exp" assumes "final e" obtains v :: "val" where "e = Val v"   | a :: "nat" where "e = Throw a"
wf_bs_I: fixes bs :: "'b list"   and p :: "'b poly"   and bs' :: "'b list" assumes "wf_bs bs p" shows "Ipoly (bs @ bs') p = Ipoly bs p"
lpred_conj_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P land Q \<equiv> \<lambda>x. P x \<and> Q x"
nonunit_principal_id: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<not> Unit R a" shows "R \<diamondsuit>\<^sub>p a \<noteq> carrier R"
meets_asym: fixes p :: "'b"   and q :: "'b" assumes "p \<parallel> q" shows "\<not> q \<parallel> p"
S_AND_def: fixes X :: "'g \<times> 'a"   and e :: "'e" shows "S_AND X e = (let (x0, x1) = X in S0_raw x0 e \<bind> (\<lambda>(a0, z0). S1_raw x1 e \<bind> (\<lambda>(a1, z1). return_spmf ((a0, a1), z0, z1))))"
pwd_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_pwd \<union> bad_shakey \<inter> bad_prikey" shows "Pwd n \<notin> spied s"
S_intersections_defs_3: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection1_rep p q \<equiv> S_intersection_coeff1 p q *\<^sub>R p + q"
rat_of_vrat_def: shows "rat_of_vrat = inv vrat_of_rat"
has_derivative_def: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (Lim F (\<lambda>x. x)) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> (0::'c)) F)"
unit_div_commute: 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 b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "times (divide a b) c = divide (times a c) b"
while_upt_while_direct: fixes a :: "nat"   and b :: "nat"   and c :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a nres"   and \<sigma> :: "'a" assumes "a \<le> b" shows "WHILE\<^sub>T (FOREACH_cond c) (\<lambda>x. ASSERT (FOREACH_cond c x) \<bind> (\<lambda>_. FOREACH_body f x)) ([a..<b], \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>) = WHILE\<^sub>T (\<lambda>(i, x). i < b \<and> c x) (\<lambda>(i, x). ASSERT (i < b) \<bind> (\<lambda>_. f i x \<bind> (\<lambda>\<sigma>'. RETURN (i + 1, \<sigma>')))) (a, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)"
rename_rename_inv: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'d program" assumes "bij h" shows "rename h (rename (inv h) F) = F"
zcount_a: shows "0 < zcount (c_pts c loc1) a"
tau_steps_eqvt_raw: fixes p :: "perm" shows "p \<bullet> tau_steps = tau_steps"
zero_hcoords_def: shows "0\<^sub>h\<^sub>c \<equiv> Abs_complex_homo_coords 0\<^sub>v"
freeze_def: fixes a :: "'a array" shows "Array.freeze a = tap (\<lambda>h. Array.get h a)"
f_b: shows "Id_on (vertices B) O (f_a \<union> f_b) = f_b"
g2: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "\<nexists>G. geodesic_segment_between G x y \<and> G \<subseteq> S" shows "g x S y = {x, y}"
that: fixes a :: "'a"   and M0 :: "'a multiset"   and K :: "'a multiset" assumes "M = add_mset a M0"   and "N = M0 + K"   and "a \<notin># K"   and "\<And>b. b \<in># K \<Longrightarrow> b < a" shows "thesis"
Rep_dataspace: fixes x :: "'b dataspace" shows "Rep_dataspace x \<in> dataspace"
SN_imp_SN_trancl: fixes R :: "('a \<times> 'a) set" assumes "SN R" shows "SN (R\<^sup>+)"
poly_map_poly: fixes p :: "'a mod_ring poly poly"   and x :: "'a mod_ring poly" shows "poly (map_poly (map_poly rebase) p) (map_poly rebase x) = map_poly rebase (poly p x)"
BINDINGn_decreasing: fixes X :: "ctrm set"   and Y :: "ctrm set"   and B :: "ty" assumes "X \<subseteq> Y" shows "BINDINGn B Y \<subseteq> BINDINGn B X"
less_separate: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less x y" shows "\<exists>a b. x \<in> ord.lessThan less a \<and> y \<in> ord.greaterThan less b \<and> ord.lessThan less a \<inter> ord.greaterThan less b = {}"
hyps_1: shows "\<Gamma> \<turnstile> (c\<^sub>1, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
designI: assumes "\<And>b. b \<in># add_block bl \<Longrightarrow> b \<noteq> {}"   and "add_block bl \<noteq> {#}"   and "\<V> \<union> bl \<noteq> {}" shows "design (\<V> \<union> bl) (add_block bl)"
LI_soundness: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and S\<^sub>1 :: "('a, 'b) strand_step list"   and \<theta>\<^sub>1 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and S\<^sub>2 :: "('a, 'b) strand_step list"   and \<theta>\<^sub>2 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and public :: "'a \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "((S\<^sub>1, \<theta>\<^sub>1), S\<^sub>2, \<theta>\<^sub>2) \<in> (intruder_model.LI_rel arity public)\<^sup>*"   and "intruder_model.constr_sem_c arity public \<I> S\<^sub>2 \<theta>\<^sub>2" shows "intruder_model.constr_sem_c arity public \<I> S\<^sub>1 \<theta>\<^sub>1"
intvs_decr_l: fixes l :: "int"   and h :: "int" assumes "l \<le> h" shows "{l - 1..<h} = insert (l - 1) {l..<h}"
supplement2_Legendre: fixes p :: "int" assumes "2 < p"   and "prime p" shows "Legendre 2 p = (- 1) ^ (((nat p)\<^sup>2 - 1) div 8)"
assms_3: shows "bn \<alpha> \<sharp>* (\<langle>f\<rangle>Q, F, f)"
src_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src (local.dom \<mu>) = src \<mu>"
min_elems_charn: fixes t :: "'a list"   and A :: "'a list set" assumes "t \<in> A" shows "\<exists>t' r. t = t' @ r \<and> t' \<in> min_elems A"
EX_def: fixes p :: "'b set" shows "\<^bold>E\<^bold>X p = {s. \<exists>s'. (s, s') \<in> \<M> \<and> s' \<in> p}"
dual_upset_set: shows "\<Up> = ord.downset_set (\<lambda>x y. y \<le> x)"
triangle_num_altdef: fixes n :: "nat" shows "triangle_num n = \<Sum> {..n}"
zero_when: fixes zero :: "'a"   and P :: "bool" shows "zero.when zero zero P = zero"
bindI: fixes P :: "'a Predicate.pred"   and x :: "'a"   and Q :: "'a \<Rightarrow> 'b Predicate.pred"   and y :: "'b" assumes "pred.eval P x"   and "pred.eval (Q x) y" shows "pred.eval (P \<bind> Q) y"
strict_trans1: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le> a"   and "c < b" shows "c < a"
wlog_moebius_01inf: fixes P :: "complex_homo \<Rightarrow> complex_homo \<Rightarrow> complex_homo \<Rightarrow> bool"   and z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo" assumes "P 0\<^sub>h 1\<^sub>h \<infinity>\<^sub>h"   and "z1 \<noteq> z2"   and "z2 \<noteq> z3"   and "z1 \<noteq> z3"   and "\<And>M a b c. P a b c \<Longrightarrow> P (moebius_pt M a) (moebius_pt M b) (moebius_pt M c)" shows "P z1 z2 z3"
power_decreasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "nat"   and N :: "nat"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a N) (power.power one times a n)"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list" assumes "\<And>uu_. P uu_ []"   and "\<And>R x xs. \<lbrakk>P R (filter (\<lambda>y. R y x) xs); P R (filter (\<lambda>y. \<not> R y x) xs)\<rbrakk> \<Longrightarrow> P R (x # xs)" shows "P a0 a1"
finite_cartesian_product_iff: fixes A :: "'a set"   and B :: "'b set" shows "finite (A \<times> B) = (A = {} \<or> B = {} \<or> finite A \<and> finite B)"
merge_pure_star: fixes a :: "bool"   and b :: "bool" shows "\<up> a * \<up> b = \<up> (a \<and> b)"
map_of_zip_distinct_inj: fixes B :: "'b list"   and A :: "'c list" assumes "distinct B"   and "length A = length B" shows "inj_on (the \<circ> map_of (zip A B)) (set A)"
def_scheme: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "f x \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>"
orthogonal_gso: shows "orthogonal (map fs.gs.gso [0..<length (map of_int_hom.vec_hom fs)])"
SUP_eq_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Sup (f ` I) = x"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (F.map\<^sub>0 a) = F.map\<^sub>0 a"
str_del_point_blocks_alt: fixes p :: "'a" shows "str_del_point_blocks p = \<B> - filter_mset ((\<in>) p) \<B>"
length_eq_card_rows: fixes A :: "(('b, 'c) vec, 'd) vec" shows "IArray.length (matrix_to_iarray A) = CARD('d)"
rev_eq_Cons_iff: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
ENF_refl_not_offedning: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a"   and f :: "('v \<times> 'v) set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<not> sinvar G nP"   and "f \<in> set_offending_flows G nP"   and "ENF_refl P" shows "\<forall>(e1, e2)\<in>f. e1 \<noteq> e2"
reduce_prog_to_body: fixes p :: "prog"   and x :: "nat"   and y :: "nat" assumes "para p = [''x'', ''y'']" shows "wfp p = wfe \<Gamma>\<^sub>x\<^sub>y (body p) \<and> exec p [x, y] = eval (\<E>\<^sub>x\<^sub>y x y) (body p)"
quasinorm_intersection_def: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm" shows "N1 \<inter>\<^sub>N N2 = quasinorm_of (max (defect N1) (defect N2), \<lambda>f. eNorm N1 f + eNorm N2 f)"
inf_sup_aci_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
nat_pow_closed: fixes x :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier (function_ring (carrier Zp) Zp)" shows "x [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n \<in> carrier (function_ring (carrier Zp) Zp)"
Abs_ainj_ex: fixes X :: "('c, 'd, 'b, 'a, 'e) qTerm set"   and X' :: "('c, 'd, 'b, 'a, 'e) qTerm set"   and xs :: "'b"   and x :: "'a"   and xs' :: "'b"   and x' :: "'a" assumes "FixVars TYPE('a) TYPE('b)"   and "FixVars.good X"   and "FixVars.good X'" shows "(FixVars.Abs xs x X = FixVars.Abs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> fresh xs y X \<and> fresh xs y X' \<and> FixVars.swap xs y x X = FixVars.swap xs y x' X'))"
independent_card_le_aff_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "\<not> affine_dependent B" shows "int (card B) \<le> aff_dim V + 1"
intros_1: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P Stop \<Gamma> \<S> P"
separated_by_def: fixes w :: "'b set set"   and x :: "'b"   and y :: "'b" shows "separated_by w x y \<equiv> \<exists>A B. w = {A, B} \<and> x \<in> A \<and> y \<in> B"
j_3: shows "\<forall>n\<ge>m\<^sub>0. s ((\<lambda>x. if x = Suc n\<^sub>0 then Some 1 else Some 0) \<triangleright> n) \<down>= Suc j"
fis_singleton_fthe_elem: fixes A :: "'b fset" shows "fis_singleton A = (A = {|fthe_elem A|})"
cfs_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "f \<in> carrier (UP Zp)" shows "f n \<in> carrier Zp"
not_fin_crename: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and y :: "name" assumes "\<not> fin (M[a\<turnstile>c>b]) y" shows "\<not> fin M y"
comp_def: fixes f :: "'p \<Rightarrow> 'o"   and g :: "'n \<Rightarrow> 'p" shows "f \<circ> g = (\<lambda>x. f (g x))"
uminus_not_id: fixes x :: "'a literal" shows "x \<noteq> - x"
Sup_distl: fixes x :: "'a"   and Y :: "'a set" shows "x \<cdot> Sup Y = Sup {z. \<exists>y\<in>Y. z = x \<cdot> y}"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, t\<rangle> \<Rightarrow> t'"
assms: shows "\<langle>\<langle>r, A\<rangle>, s\<rangle> \<in>\<^sub>\<circ> vrrestriction D"
out_arcs_single_per_vert: fixes v :: "'a"   and u :: "'a"   and e1 :: "'b"   and e2 :: "'b" assumes "v \<in> verts G"   and "u \<in> verts G"   and "e1 \<in> out_arcs G v"   and "e2 \<in> out_arcs G v"   and "head G e1 = u"   and "head G e2 = u" shows "e1 = e2"
filt_equiv_slt: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 \<le> q"   and "q \<le> 1"   and "0 < p"   and "p < 1"   and "filt_equiv nat_filtration M N" shows "q < 1"
Iod_segment_segment: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier (Iod D (segment D b))" shows "Iod (Iod D (segment D b)) (segment (Iod D (segment D b)) a) = Iod D (segment D a)"
notf: shows "\<forall>e\<in>{a, b, c}. r e \<noteq> f"
if_False: fixes x :: "'a"   and y :: "'a" shows "(if False then x else y) = y"
well_related_extend: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "well_related_set A r"   and "well_related_set B r"   and "\<forall>a\<in>A. \<forall>b\<in>B. r a b" shows "well_related_set (A \<union> B) r"
assms_4: shows "set (bvars\<^sub>s\<^sub>t\<^sub>p x) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
TESL_interp_unfold_stepwise_strictly_precedes: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
assms_3: shows "Q_I_pos = projectQuery I (filterQuery I Q)"