hcju/isabelleps huggingface dataset:
silent_move_obs_slice: fixes S :: "'node SDG_node set"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and msx :: "'node list" assumes "S,kind \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"   and "msx \<in> obs ms' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "Ball (set (tl ms')) return_node" shows "msx \<in> obs ms \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
const_type_inv_wf: fixes c :: "'fun"   and X :: "('fun, 'var) Term.term list"   and a :: "'atom" assumes "\<Gamma> (Fun c X) = Var a"   and "wf\<^sub>t\<^sub>r\<^sub>m (Fun c X)" shows "X = []"
matches_casesE: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" assumes "Matching_Ternary.matches (\<beta>, \<alpha>) m a p" obtains "ternary_ternary_eval (map_match_tac \<beta> p m) = TernaryUnknown" and "\<alpha> a p"   | "ternary_ternary_eval (map_match_tac \<beta> p m) = TernaryTrue"
stutter_equiv_def: fixes \<sigma> :: "nat \<Rightarrow> 'f"   and \<tau> :: "nat \<Rightarrow> 'f" shows "\<sigma> \<approx> \<tau> \<equiv> \<natural>\<sigma> = \<natural>\<tau>"
\<C>_Low: fixes mem :: "'Var\<^sub>A \<Rightarrow> 'Val" shows "\<forall>x\<in>\<C>\<^sub>A. dma\<^sub>A mem x = Low"
distinct_7: fixes x1 :: "agent"   and x5 :: "msg" shows "Agent x1 \<noteq> Hash x5"
mult_g_square: shows "multiplicative_function (\<lambda>n. g (n\<^sup>2))"
abs_summable_on_union: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "f abs_summable_on A"   and "f abs_summable_on B" shows "f abs_summable_on A \<union> B"
comp1: fixes g :: "'e \<Rightarrow> 'd" assumes "bounded_linear g" shows "bounded_bilinear (\<lambda>x. (**) (g (g x)))"
get_nondet_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) nondetT" shows "get_nondet get f = NondetT (get (\<lambda>s. run_nondet (f s)))"
select_convs_1: fixes verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and more :: "'d" shows "V\<^bsub>\<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr>\<^esub> = verts"
binary_endofunctor_axioms: shows "binary_endofunctor (\<cdot>) local.Prod"
elims_2: fixes x :: "'b mat \<times> 'b mat \<times> 'b mat"   and xa :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "HMA_M3 x xa" obtains P :: "'b mat" and A :: "'b mat" and Q :: "'b mat" and P' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" and A' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" and Q' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" where "x = (P, A, Q)" and "xa = (P', A', Q')" and "Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q'"
card_eq: shows "card {a \<in> f ` {0..<n}. a < f i} = i"
distinct_3: fixes ty1' :: "ty"   and ty2' :: "ty" shows "TBase \<noteq> ty1'\<rightarrow>ty2'"
exproj_psi_minus_1_tensor: shows "2 \<cdot>\<^sub>m tensor_P proj_psi (1\<^sub>m K) - 1\<^sub>m d = tensor_P (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) (1\<^sub>m K)"
\<nu>\<upsilon>_surj: shows "surj \<nu>\<upsilon>"
arg_cong2: fixes a :: "'c"   and b :: "'c"   and c :: "'d"   and d :: "'d"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "a = b"   and "c = d" shows "f a c = f b d"
factorial_monoid: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ufd times one plus zero minus uminus" shows "factorial_monoid \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>"
size_mult_mono: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "euclidean_size a \<le> euclidean_size (a * b)"
bad_worst_chain_0: shows "local.bad (local.worst_chain 0)"
dst_found: shows "dst \<in> dom PRED \<inter> Vd d"
summable_of_real_iff: fixes f :: "nat \<Rightarrow> real" shows "summable (\<lambda>x. of_real (f x)) = summable f"
a3: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> F"   and "B \<in> F" shows "A \<inter> B \<in> F"
prime_multiplicity_gt_zero_iff: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "(0 < factorial_semiring.multiplicity times one p x) = dvd.dvd times p x"
surjective: fixes r :: "('d, 'e, 'f) category_scheme" shows "r = \<lparr>ob = Ob\<^bsub>r\<^esub>, ar = Ar\<^bsub>r\<^esub>, dom = category.dom r, cod = cod r, id = category.id r, comp = (\<bullet>\<^bsub>r\<^esub>), \<dots> = more r\<rparr>"
simps_5: fixes x :: "nat"   and e :: "cexpr"   and b :: "cexpr"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_subst x e (IF\<^sub>c b THEN e1 ELSE e2) = (IF\<^sub>c cexpr_subst x e b THEN cexpr_subst x e e1 ELSE cexpr_subst x e e2)"
dest_fun_mk_fun: fixes Ts :: "Unary_PCF.type list"   and T :: "Unary_PCF.type" shows "dest_fun (Ts \<rightarrow>\<rightarrow> T) = Ts @ dest_fun T"
forest_def: fixes G :: "('a, 'b) graph" shows "forest G \<equiv> valid_graph G \<and> forest_axioms G"
pmat_extension_carrier: fixes m :: "'a mat" shows "st2.pmat_extension m \<in> carrier_mat st2.d0 st2.d0"
set_bind_conv_fold: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'b set" shows "set xs \<bind> f = fold ((\<union>) \<circ> f) xs {}"
filterlim_compose: fixes g :: "'a \<Rightarrow> 'b"   and F3 :: "'b filter"   and F2 :: "'a filter"   and f :: "'c \<Rightarrow> 'a"   and F1 :: "'c filter" assumes "filterlim g F3 F2"   and "filterlim f F2 F1" shows "LIM x F1. g (f x) :> F3"
imirror_bounds_imirror_bounds_ident: fixes I :: "nat set"   and l :: "nat"   and r :: "nat" assumes "finite I"   and "Max I \<le> l + r" shows "imirror_bounds (imirror_bounds I l r) l r = I"
fps_X_neq_zero: shows "fps_X \<noteq> 0"
that: fixes h' :: "heap"   and r :: "'a" assumes "effect c h h' r" shows "thesis"
DBM_reset_len_mono: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and xs :: "nat list"   and i :: "nat"   and j :: "nat" assumes "DBM_reset M n k d M'"   and "k \<notin> set xs"   and "i \<noteq> k"   and "j \<noteq> k"   and "set (i # j # xs) \<subseteq> {0..n}" shows "len M' i j xs \<le> len M i j xs"
ladj_radj_dual: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c" shows "ladj f x = \<partial> (radj (\<partial>\<^sub>F f) (\<partial> x))"
scaleR_cong_right: fixes x :: "'b"   and r :: "real"   and p :: "real" assumes "x \<noteq> (0::'b) \<Longrightarrow> r = p" shows "r *\<^sub>R x = p *\<^sub>R x"
HC_consistent: shows "pcp {\<Gamma> |\<Gamma>::'b formula set. \<not> (\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<bottom>)}"
prime_coprime: fixes x :: "nat"   and y :: "nat" assumes "Prime x"   and "Prime y"   and "x \<noteq> y" shows "coprime x y"
comm_monoid_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "comm_monoid_add_hom hom \<equiv> monoid_add_hom hom"
mult_left_le_imp_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq (times c a) (times c b)"   and "less zero c" shows "less_eq a b"
linear_with_transfer: fixes B\<^sub>1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B\<^sub>2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique B\<^sub>1"   and "right_total B\<^sub>1"   and "bi_unique B\<^sub>2"   and "right_total B\<^sub>2" shows "rel_fun (rel_fun B\<^sub>1 (rel_fun B\<^sub>1 B\<^sub>1)) (rel_fun B\<^sub>1 (rel_fun (rel_fun B\<^sub>1 (rel_fun B\<^sub>1 B\<^sub>1)) (rel_fun (rel_fun B\<^sub>1 B\<^sub>1) (rel_fun (rel_fun (=) (rel_fun B\<^sub>1 B\<^sub>1)) (rel_fun (rel_fun B\<^sub>2 (rel_fun B\<^sub>2 B\<^sub>2)) (rel_fun B\<^sub>2 (rel_fun (rel_fun B\<^sub>2 (rel_fun B\<^sub>2 B\<^sub>2)) (rel_fun (rel_fun B\<^sub>2 B\<^sub>2) (rel_fun (rel_fun (=) (rel_fun B\<^sub>2 B\<^sub>2)) (rel_fun (rel_fun B\<^sub>1 B\<^sub>2) (=))))))))))) (\<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1. linear_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B\<^sub>2))) linear_with"
distr_cong_AE: fixes M :: "'b measure"   and K :: "'b measure"   and N :: "'c measure"   and L :: "'c measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = K"   and "sets N = sets L"   and "AE x in M. f x = g x"   and "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> K \<rightarrow>\<^sub>M L" shows "distr M N f = distr K L g"
set_mult_finite: fixes H :: "'a set"   and J :: "'a set" assumes "finite H"   and "finite J"   and "H \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "finite (H <#> J)"
that: assumes "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>" shows "thesis"
monotoneI: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
Ass: fixes C :: "'a literal set"   and S :: "'a literal set set" assumes "C \<in> S" shows "S \<turnstile> C"
env_delete_below_arg: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "env_delete x m \<sqsubseteq> m"
vconn_implies_hpar_3: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "src g = src f"
tendsto_ereal: fixes f :: "'b \<Rightarrow> real"   and x :: "real"   and F :: "'b filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
trigonometry: shows "(cos \<alpha> - cos \<beta>)\<^sup>2 + (sin \<alpha> - sin \<beta>)\<^sup>2 = (2 * sin ((\<beta> - \<alpha>) / 2))\<^sup>2"
PrimRec1p_PrimRec1_eq: shows "PrimRec1p = (\<lambda>x. x \<in> PrimRec1)"
decomp_propa: fixes qs :: "(nat \<times> nat list) list"   and ss :: "nat list list"   and w :: "nat list" assumes "\<forall>(q, t)\<in>set qs. q < length ss \<and> t \<in> A"   and "Sorted_Less.sorted w"   and "set ss \<subseteq> A" shows "propa f qs ss w = (merges f qs ss, sorted_list_of_set ({q. \<exists>t. (q, t) \<in> set qs \<and> t \<squnion>\<^bsub>f\<^esub> ss ! q \<noteq> ss ! q} \<union> set w))"
real_euclid_B_def: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec" shows "B\<^sub>\<real> a b c \<equiv> \<exists>l\<ge>0. l \<le> 1 \<and> b - a = l *\<^sub>R (c - a)"
intro: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom (add_monoid G) (add_monoid H) h" shows "abelian_group_hom_axioms G H h"
valid_weak_tau_modality: fixes P :: "'state"   and x :: "('idx, 'pred, 'act) formula" shows "P \<Turnstile> weak_tau_modality x = (\<exists>P'. P \<Rightarrow> P' \<and> P' \<Turnstile> x)"
ge_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A BGe C D" shows "B AGe C D"
bounded_by_mono: fixes a :: "real"   and P :: "'b \<Rightarrow> real"   and b :: "real" assumes "bounded_by a P"   and "a \<le> b" shows "bounded_by b P"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "noDenyAll x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "\<not> member DenyAll xa \<and> noDenyAll xs"   | "x = []"
add2_comm: fixes a :: "'a option"   and b :: "'a option" shows "a ++ b = b ++ a"
lm075: fixes XX :: "'b set set" shows "finestpart (\<Union> XX) \<subseteq> \<Union> (finestpart ` XX)"
comm_semiring_0_ow_axioms: shows "comm_semiring_0_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (*\<^sub>o\<^sub>w)"
dim_affine_hull: fixes S :: "'a set" shows "dim (affine hull S) = dim S"
eq_id_iff: fixes f :: "'h \<Rightarrow> 'h" shows "(\<forall>x. f x = x) = (f = id)"
etran_or_ctran2_disjI2: fixes x :: "('b com option \<times> 'b) list"   and i :: "nat" assumes "x \<in> cptn"   and "Suc i < length x"   and "x ! i -e\<rightarrow> x ! Suc i" shows "(x ! i, x ! Suc i) \<notin> ctran"
dag_Null: fixes l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref" shows "dag Null l r = Tip"
simps_3: fixes x :: "('a, 'b) aformula"   and v :: "('a, 'b) aformula"   and va :: "('a, 'b) aformula list" shows "nFOR (x # v # va) = FOr x (nFOR (v # va))"
one_ne_\<omega>: shows "1 \<noteq> \<omega>"
set_Collect_arr_Rel: fixes x :: "V"   and \<alpha> :: "V" shows "(x \<in>\<^sub>\<circ> ZFC_in_HOL.set (Collect (arr_Rel \<alpha>))) = arr_Rel \<alpha> x"
pm_of_idx_pm_uminus: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd" shows "pm_of_idx_pm xs (- f) = - pm_of_idx_pm xs f"
sub1: shows "(\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> - U"
swap_cols_rows_carrier_2: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_col (swap_cols_rows k l A) = dim_col A"
rewrite_iter_fast_sound: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n rewrite_iter_fast \<phi> = w \<Turnstile>\<^sub>n \<phi>"
inject_1: fixes x1 :: "trm"   and y1 :: "trm" shows "(Base x1 = Base y1) = (x1 = y1)"
of_complex_of_nat_eq: fixes n :: "nat" shows "of_complex (of_nat n) = of_nat n"
observable_productF: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM"   and FAIL :: "'g \<times> 'f"   and PM :: "('d, 'e, 'g \<times> 'f) FSM" assumes "observable M1"   and "observable M2"   and "productF M2 M1 FAIL PM" shows "observable PM"
rank_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = row_rank A"
wf_iff_wfp_on: fixes p :: "('b \<times> 'b) set" shows "wf p = wfp_on (\<lambda>a b. (a, b) \<in> p) UNIV"
is_subprob_density_imp_has_density: fixes N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "is_subprob_density N f"   and "M = density N f" shows "has_density M N f"
has_vderiv_on_eq_rhs: fixes f :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and T :: "real set"   and f' :: "real \<Rightarrow> 'b" assumes " D f = g' on T"   and "\<And>x. x \<in> T \<Longrightarrow> g' x = f' x" shows " D f = f' on T"
det_zero_row_1: fixes i :: "'b"   and A :: "(('a, 'b) vec, 'b) vec" assumes "row i A = 0" shows "det A = (0::'a)"
D: fixes X :: "'b set" assumes "open X"   and "X \<noteq> {}" shows "\<exists>d\<in>D. d \<in> X"
eval_fm_Equality_axP: fixes e :: "(name, hf) finfun"   and x :: "tm" shows "eval_fm e (Equality_axP x) = (\<lbrakk>x\<rbrakk>e \<in> Equality_ax)"
F2in_mono23: fixes A2 :: "'d set"   and A2a :: "'d set"   and A3 :: "'e set"   and A3a :: "'e set"   and A1 :: "'f set" assumes "A2 \<subseteq> A2a"   and "A3 \<subseteq> A3a" shows "F2in A1 A2 A3 \<subseteq> F2in A1 A2a A3a"
card_AF: fixes k :: "nat"   and i :: "nat" shows "card (AF k i) = \<Sqinter> (DF k i)"
wf_inv_image: fixes r :: "('f \<times> 'f) set"   and f :: "'g \<Rightarrow> 'f" assumes "wf r" shows "wf (inv_image r f)"
quot_to_ratfps_quot_of_ratfps: fixes x :: "'a ratfps" shows "local.quot_to_ratfps (local.quot_of_ratfps x) = x"
binomial_int_ineq_3: fixes y :: "int"   and x :: "int"   and k :: "nat" assumes "0 \<le> y"   and "y \<le> 2 * x" shows "2 * (x gchoose k) \<le> y gchoose k + (2 * x - y gchoose k)"
either_belowI: fixes t :: "'a ttree"   and t'' :: "'a ttree"   and t' :: "'a ttree" assumes "t \<sqsubseteq> t''"   and "t' \<sqsubseteq> t''" shows "t \<oplus>\<oplus> t' \<sqsubseteq> t''"
tendsto_discrete: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and F :: "'c filter" assumes "class.discrete_topology open" shows "filterlim f (topological_space.nhds open y) F = (\<forall>\<^sub>F x in F. f x = y)"
Subst_all_preserves_validity: fixes ES :: "('b list set \<times> 'b trm set) set"   and Y :: "'b list set"   and yrhs :: "'b trm set" assumes "validity (ES \<union> {(Y, yrhs)})" shows "validity (Subst_all ES Y (Arden Y yrhs))"
lookup_default_fold_combine_plus: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('d, real) mapping"   and x :: "'d" assumes "finite A" shows "Mapping.lookup_default 0 (fold_combine_plus f A) x = (\<Sum>y\<in>A. Mapping.lookup_default 0 (f y) x)"
irreflexive_complement_reflexive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.irreflexive x = matrix_idempotent_semiring.reflexive (\<ominus> x)"
select_convs_4: fixes ok :: "bool"   and wait :: "bool"   and tr :: "'d list"   and ref :: "'d set"   and more :: "'c" shows "alpha_rp.more \<lparr>ok = ok, wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr> = more"
preserves_ide: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "functor A B F"   and "partial_magma.ide A a" shows "partial_magma.ide B (F a)"
in_homE: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr"   and b :: "('o, 'a) arr" assumes "in_hom f a b" obtains "arr f" and "local.dom f = a" and "cod f = b"
length_asm: shows "length (map snd (pivot_positions A)) = length (pivot_positions A)"
mod_starD: fixes A :: "assn"   and B :: "assn"   and h :: "Heap.heap \<times> nat set" assumes "h \<Turnstile> A * B" shows "\<exists>h1 h2. h1 \<Turnstile> A \<and> h2 \<Turnstile> B"
product_topology_countable_basis: shows "\<exists>K. topological_basis K \<and> countable K \<and> (\<forall>k\<in>K. \<exists>X. k = Pi\<^sub>E UNIV X \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV})"
downset_directed_downset: shows "directed \<circ> \<Down> = directed"
simps_6: fixes t :: "'a local.term" shows "\<^bold>\<lfloor>\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
span_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly"   and S :: "'b poly set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> poly_vs.span S" shows "sum f A \<in> poly_vs.span S"
assms: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gxz :: "'a set"   and z :: "'a"   and t :: "real" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "t \<in> {0..Gromov_product_at x y z}" shows "dist (geodesic_segment_param Gxy x t) (geodesic_segment_param Gxz x t) \<le> delta"
fmeasurable_eq_sets: shows "fmeasurable M = M.events"
dom_kildall_def: shows "dom_kildall = kildall (fst (snd nodes_semi)) (snd (snd nodes_semi)) exec"
set_mult_union: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A <#> B \<union> C = (A <#> B) \<union> (A <#> C)"
h_outside_E: fixes x :: "'v"   and y :: "'v"   and n :: "nat" assumes "(x, y) \<notin> \<^bold>E" shows "h n (x, y) = 0"
VPow_iff: fixes y :: "V"   and x :: "V" shows "(y \<in> elts (VPow x)) = (y \<le> x)"
heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: shows "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed Shadow_DOM.get_child_nodes get_disconnected_nodes"
C1: shows "C1 \<in> UNIV // K1.communicating"
norm_sq_mtx_diag: fixes f :: "'a \<Rightarrow> real" shows "\<parallel>sq_mtx_diag f\<parallel> = Max {\<bar>f i\<bar> |i. i \<in> UNIV}"
in_renamed_in_unlabelled_global_2: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and l :: "'b"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<beta> l \<in> LID\<^sub>G (\<R>\<^sub>G \<alpha> \<beta> s)) = (l \<in> LID\<^sub>G s)"
assms_3: fixes x :: "\<nu>"   and y :: "\<nu>"   and z :: "\<nu>" shows "[\<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) in v]"
sifum_lang_no_dma_axioms: shows "sifum_lang_no_dma ev\<^sub>A\<^sub>C ev\<^sub>B\<^sub>C aexp_vars\<^sub>C bexp_vars\<^sub>C"
1: shows "regular (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_irrefl ext" shows "ext ext"
assms_3: shows "next_change_multiplicity' c0 c1 loc t n"
induct: fixes x1 :: "value"   and x2 :: "value"   and P :: "value \<Rightarrow> value \<Rightarrow> bool" assumes "veq_structure x1 x2"   and "\<And>uu_ uv_ uw_ ux_. P (Vabs uu_ uv_) (Vabs uw_ ux_)"   and "\<And>uy_ uz_ va_ vb_ vc_ vd_. P (Vrecabs uy_ uz_ va_) (Vrecabs vb_ vc_ vd_)"   and "\<And>ts us name. list_all2 (\<lambda>x1 x2. veq_structure x1 x2 \<and> P x1 x2) ts us \<Longrightarrow> P (Vconstr name ts) (Vconstr name us)" shows "P x1 x2"
borel_sigma_sets_subset: fixes A :: "'b set set" assumes "A \<subseteq> sets borel" shows "sigma_sets UNIV A \<subseteq> sets borel"
F3: shows "\<forall>x\<^sub>1 x\<^sub>3. x\<^sub>3 \<le> \<bar>h x\<^sub>1\<bar> \<longrightarrow> x\<^sub>3 \<le> c * \<bar>f x\<^sub>1\<bar>"
simps: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and a1 :: "'a itrm"   and a2 :: "'a itrm" shows "itrm_cong R a1 a2 = ((\<exists>x y. a1 = x \<and> a2 = y \<and> R x y) \<or> (\<exists>x y. a1 = Pure x \<and> a2 = Pure y \<and> x \<leftrightarrow> y) \<or> (\<exists>f f' x x'. a1 = f \<diamondop> x \<and> a2 = f' \<diamondop> x' \<and> itrm_cong R f f' \<and> itrm_cong R x x') \<or> (\<exists>x. a1 = x \<and> a2 = x) \<or> (\<exists>x y. a1 = y \<and> a2 = x \<and> itrm_cong R x y) \<or> (\<exists>x y z. a1 = x \<and> a2 = z \<and> itrm_cong R x y \<and> itrm_cong R y z))"
Suc: shows "nths (replicate n x) A = replicate (card {i. i < n \<and> i \<in> A}) x"
fic_substc_crename: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and P :: "trm"   and y :: "name" assumes "fic M a"   and "a \<noteq> b"   and "a \<sharp> P" shows "M[a\<turnstile>c>b]{b:=y.P} = Cut <a>.M{b:=y.P} y.P"
subgroup_incl: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I (G\<lparr>carrier := conj.stabilizer P\<rparr>)"   and "subgroup J (G\<lparr>carrier := conj.stabilizer P\<rparr>)"   and "I \<subseteq> J" shows "subgroup I (G\<lparr>carrier := conj.stabilizer P, carrier := J\<rparr>)"
zerosign_def: fixes s :: "nat"   and a :: "('a, 'b) IEEE.float" shows "zerosign s a = (if IEEE.is_zero a then if s = 0 then 0 else - 0 else a)"