hcju/isabelleps huggingface dataset:
vcomp_in_hhom: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>"   and "Span.src \<nu> = a"   and "Span.trg \<nu> = b" shows "Span.in_hhom (\<nu> \<bullet> \<mu>) a b"
list_all2_Cons2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
prefix_refl: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "prefix\<cdot>xs\<cdot>xs = TT"
ord_equiv_sym: fixes E :: "('c, 'd) Order_scheme" assumes "Order E"   and "ord_equiv D E" shows "ord_equiv E D"
dest_Num_fa_None: fixes fa :: "floatarith" assumes "dest_Num_fa fa = None" obtains v :: "floatarith" and va :: "floatarith" where "fa = Add v__ va__"   | v :: "floatarith" where "fa = Minus vb__"   | v :: "floatarith" and va :: "floatarith" where "fa = Mult vc__ vaa__"   | v :: "floatarith" where "fa = Inverse vd__"   | v :: "floatarith" where "fa = Cos ve__"   | v :: "floatarith" where "fa = Arctan vf__"   | v :: "floatarith" where "fa = Abs vg__"   | v :: "floatarith" and va :: "floatarith" where "fa = Max vh__ vab__"   | v :: "floatarith" and va :: "floatarith" where "fa = Min vi__ vac__"   | "fa = Pi"   | v :: "floatarith" where "fa = Sqrt vj__"   | v :: "floatarith" where "fa = Exp vk__"   | v :: "floatarith" and va :: "floatarith" where "fa = Powr vl__ vad__"   | v :: "floatarith" where "fa = Ln vm__"   | v :: "floatarith" and va :: "nat" where "fa = vn__ ^\<^sub>e va"   | v :: "floatarith" where "fa = Floor vo__"   | v :: "nat" where "fa = Var v"
assms_1: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"
simps_7: fixes \<sigma> :: "nat \<Rightarrow> val"   and dst :: "pdf_dist"   and e :: "expr" shows "expr_sem \<sigma> (Random dst e) = expr_sem \<sigma> e \<bind> dist_measure dst"
all_acquired_takeWhile_in: fixes x :: "nat"   and P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" assumes "x \<in> all_acquired (takeWhile P sb)" shows "x \<in> all_acquired sb"
times_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f * g = Mult f g"
focus_minus: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "focus X (p - q) = focus X p - focus X q"
simps_1: fixes T1 :: "'c trm"   and T2 :: "'c trm" shows "varsA (Eq T1 T2) = vars T1 \<union> vars T2"
fix_eq2: fixes f :: "'b"   and F :: "'b \<rightarrow> 'b" assumes "f \<equiv> fix\<cdot>F" shows "f = F\<cdot>f"
SourcesA92_L1: shows "Sources level1 sA92 = {sA6, sA71, sA81, sA91}"
map_poly_hom_monom: fixes a :: "'c"   and i :: "nat" shows "map_poly (\<lambda>a. [:a:]) (monom a i) = monom [:a:] i"
absolutely_integrable_componentwise: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A" shows "f absolutely_integrable_on A"
homotopy_eqv_rel_frontier_punctured_affine_hull: fixes S :: "'a set"   and a :: "'a" assumes "convex S"   and "bounded S"   and "a \<in> rel_interior S" shows "rel_frontier S homotopy_eqv affine hull S - {a}"
diag_mat_def: fixes A :: "'c mat" shows "diag_mat A = map (\<lambda>i. A $$ (i, i)) [0..<dim_row A]"
ifaceAny_def: shows "ifaceAny \<equiv> Iface ''+''"
PiE_def: fixes S :: "'b set"   and T :: "'b \<Rightarrow> 'c set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
dagger_unfoldl_eq: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
assms: shows "\<not> simple\<^sub>s\<^sub>t\<^sub>p x"
BuDuF_def: shows "BuDuF = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0..int q - 1}}"
graph_pred_it_axioms_def: fixes \<alpha> :: "'e \<Rightarrow> ('f, 'g) graph"   and invar :: "'e \<Rightarrow> bool"   and pred_list_it :: "'e \<Rightarrow> 'f \<Rightarrow> (('f \<times> 'g) list \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> ('f \<times> 'g) list \<Rightarrow> ('f \<times> 'g) list) \<Rightarrow> ('f \<times> 'g) list \<Rightarrow> ('f \<times> 'g) list" shows "graph_pred_it_axioms \<alpha> invar pred_list_it \<equiv> \<forall>g v. invar g \<longrightarrow> set_iterator (pred_list_it g v) (pred (\<alpha> g) v)"
hom: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "bounded_lattice_homomorphism phi"
atLeastAtMost_upt: fixes n :: "nat"   and m :: "nat" shows "{n..m} = set [n..<Suc m]"
simps_4: fixes r :: "real"   and x :: "octo" shows "Im3 (r *\<^sub>R x) = r * Im3 x"
landau_prop_equalsI: fixes L :: "real filter \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) set"   and L' :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) set"   and Lr :: "real filter \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) set"   and f1 :: "real \<Rightarrow> 'a"   and f2 :: "real \<Rightarrow> 'a"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "landau_symbol L L' Lr"   and "\<And>x. 1 < x \<Longrightarrow> f1 x = f2 x"   and "\<And>x. 1 < x \<Longrightarrow> g1 x = g2 x" shows "(f1 \<in> L at_top g1) = (f2 \<in> L at_top g2)"
e2stack_stack: fixes ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2stack (encode_config (ss, rv)) = Partial_Recursive.list_encode (map encode_frame ss)"
prime_dvd_mult_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "dvd.dvd times p (times a b) = (dvd.dvd times p a \<or> dvd.dvd times p b)"
n_rw: shows "nrows_iarray (IArray (map (\<lambda>x. vec_to_iarray (A $ mod_type_class.from_nat x)) [0..<CARD('c)])) = CARD('c)"
prems_1: shows "redT_updTs TS (TA # TAS) t = \<lfloor>(x, w)\<rfloor>"
path_strategies_good: fixes P :: "'a llist"   and n :: "nat" assumes "lset P \<subseteq> S - W"   and "enat n < llength P" shows "path_strategies P $ n \<in> good (P $ n)"
set_eq: shows "set (remdups (map (consistent_sign_vec qs) (characterize_root_list_p p))) = set signs \<inter> consistent_sign_vec qs ` set (characterize_root_list_p p)"
ring_hom_ring_axioms_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "ring_hom_ring_axioms R S h \<equiv> h \<in> ring_hom R S"
congruence_congruence_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "congruence M (congruence (mat_inv M) H) = H"
eSum_minus_x: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and l :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. x j \<in> carrier K"   and "\<forall>j\<le>n. b j \<in> carrier K"   and "l \<le> n"   and "\<forall>j\<in>{h. h \<le> n} - {l}. g j = x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a b j)"   and "g l = x l \<cdot>\<^sub>r (-\<^sub>a b l)" shows "\<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a b j)) n \<plusminus> -\<^sub>a x l = \<Sigma>\<^sub>e K g n"
fold2_ctor: shows "fold2 ctor1 ctor2 = id"
CON: fixes S :: "'a formula set"   and F :: "'a formula"   and G :: "'a formula"   and H :: "'a formula" assumes "\<And>s. \<lbrakk>s \<subseteq> S; finite s\<rbrakk> \<Longrightarrow> s \<in> C"   and "Con F G H"   and "F \<in> S" shows "G \<triangleright> H \<triangleright> S \<in> C \<union> {S. \<forall>s\<subseteq>S. finite s \<longrightarrow> s \<in> C}"
add_mono_thms_linordered_field_5: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i < j \<and> k < l" shows "i + k < j + l"
complementTA_correct_2: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "complete_tree_automaton TA A" shows "complete_tree_automaton (complete_tree_automaton.complementTA TA) A"
that: fixes \<sigma>t :: "heap option"   and rt :: "'b" assumes "run f \<sigma> \<sigma>t rt"   and "run g \<sigma>t \<sigma>'' res" shows "thesis"
trg_dom: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.trg (S.dom \<mu>) = S.trg \<mu>"
list_emb_prefix: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)"
Quotient_rel: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r :: "'d"   and s :: "'d" assumes "Quotient R Abs Rep T" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
fi_f_eq: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'c" assumes "g_rename_precond G f"   and "f x = y"   and "x \<in> g_V G" shows "g_rename_precond.fi G f y = x"
set_update_subsetI: fixes xs :: "'a list"   and A :: "'a set"   and x :: "'a"   and i :: "nat" assumes "set xs \<subseteq> A"   and "x \<in> A" shows "set (xs[i := x]) \<subseteq> A"
information_space_def: fixes M :: "'c measure"   and b :: "real" shows "information_space M b \<equiv> prob_space M \<and> information_space_axioms b"
phis_wf_inv: fixes n :: "'node"   and v :: "'val"   and args :: "'val list" assumes "phis' g (n, v) = Some args" shows "length (old.predecessors g n) = length args"
hyps: fixes Ds :: "nat list"   and Es :: "nat list" assumes "CT \<turnstile>+ tCs <: Ds"   and "CT \<turnstile>+ Ds <: Es" shows "CT \<turnstile>+ tCs <: Es"
hyps: fixes tab :: "'a \<Rightarrow> 'b option"   and ys :: "'b list"   and tab' :: "'a \<Rightarrow> 'b option" assumes "(tab(xs [\<mapsto>] ys)) vn = Some el"   and "(tab'(xs [\<mapsto>] ys)) vn = None" shows "tab vn = Some el"
psubseteq_pfun_upd1: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun"   and x :: "'a"   and v :: "'b" assumes "f \<subseteq>\<^sub>p g"   and "x \<notin> pdom g" shows "f \<subseteq>\<^sub>p g(x \<mapsto> v)\<^sub>p"
shows_prec_arctic_def: fixes p :: "nat"   and ai :: "arctic" shows "shows_prec p ai = shows_arctic ai"
lm031: shows "card \<circ> omega = card \<circ> snd"
phiUses_in_allUses: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> phiUses g n" shows "v \<in> allUses g n"
that_1: fixes y :: "Vars.var"   and e' :: "exp" assumes "z = Lam [y]. e'"   and "atom y \<sharp> c" shows "thesis"
thetaSDone_Sbis: shows "thetaSDone \<subseteq> Sbis"
mult_minus_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b Matrix.vec"   and w :: "'b Matrix.vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc"   and "w \<in> carrier_vec nc" shows "A *\<^sub>v (v - w) = A *\<^sub>v v - A *\<^sub>v w"
wf_syscls_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "wf_syscls P \<equiv> {Object} \<union> sys_xcpts \<subseteq> set (map fst P)"
Pg_def: shows "Pg = (\<lambda>n. {x \<in> space M. \<forall>l\<in>{1..n}. - delta l * real l < u n x - u (n - l) ((T ^^ l) x)})"
mk_eqcl_fst_nth: fixes xs :: "nat option list"   and T :: "bool list list"   and ii :: "nat"   and zs :: "nat list"   and i :: "nat"   and j :: "nat" assumes "\<And>i j k. \<lbrakk>i < length xs; j < length xs; xs ! i = Some k\<rbrakk> \<Longrightarrow> (xs ! j = Some k) = (\<not> tr_lookup T (i + ii) (j + ii))"   and "\<And>a b c. \<lbrakk>a \<le> length T; b \<le> length T; c \<le> length T; \<not> tr_lookup T a b; \<not> tr_lookup T b c\<rbrakk> \<Longrightarrow> \<not> tr_lookup T a c"   and "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < length zs"   and "length xs + ii = length T + 1"   and "i < length xs"   and "j < length xs" shows "(fst (mk_eqcl xs zs ii T) ! i = fst (mk_eqcl xs zs ii T) ! j) = (\<not> tr_lookup T (i + ii) (j + ii))"
monotoneE: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "PO cl"   and "Tarski.monotone f (pset cl) (order cl)"   and "x \<in> pset cl"   and "y \<in> pset cl"   and "(x, y) \<in> order cl" shows "(f x, f y) \<in> order cl"
delete2_delete: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and n' :: "nat"   and dl' :: "nat" shows "(local.delete2 x (t, size t, dl) = (t', n', dl')) = ((t', dl') = local.delete x (t, dl) \<and> n' = size t')"
App: fixes ts :: "dB list"   and x :: "nat" assumes "listall NF ts" shows "NF (dB.Var x \<degree>\<degree> ts)"
NatTransComp_def: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "\<eta>1 \<bullet> \<eta>2 \<equiv> MakeNT (\<eta>1 \<bullet>1 \<eta>2)"
weakenBisimE_4: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "weak substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> weak.weakenBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, Q, P) \<in> weak.weakenBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
interp_ins_monotonic: fixes pre :: "('a \<times> 'a option) list"   and suf :: "('a \<times> 'a option) list"   and oid :: "'a" assumes "insert_ops (pre @ suf)"   and "oid \<in> set (interp_ins pre)" shows "oid \<in> set (interp_ins (pre @ suf))"
app_def: shows "app = Cn 4 r_ifz [recf.Id 4 0, recf.Id 4 3, ch0]"
closed_orbit_recurrence_times_nonempty: fixes x :: "'a" assumes "closed_orbit x" shows "{T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x} \<noteq> {}"
ent_ex_postI: fixes P :: "assn"   and Q :: "'a \<Rightarrow> assn"   and x :: "'a" assumes "P \<Longrightarrow>\<^sub>A Q x" shows "P \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x"
disc_equity_market_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" shows "disc_equity_market M F \<equiv> triv_init_disc_filtr_prob_space M F"
induces_meta_adjunction: shows "meta_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi> \<psi>"
Singleton: fixes v :: "'a" assumes "v \<in> V" shows "walk [v]"
sublist_valid_plan: fixes as' :: "'a list"   and as :: "'a list"   and PROB :: "'a set" assumes "subseq as' as"   and "as \<in> valid_plans PROB" shows "as' \<in> valid_plans PROB"
L2_set_mono: fixes K :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"   and "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i" shows "L2_set f K \<le> L2_set g K"
pmat_extension_def: fixes m :: "'a mat" shows "stm1.pmat_extension m = stm1.ptensor_mat m (1\<^sub>m stm1.d2)"
sd_efficient_sds_axioms: shows "sd_efficient_sds agents alts sds"
in_degree_add_arc_iff: fixes aa :: "'a \<times> nat \<times> 'a"   and a :: "'a \<times> nat \<times> 'a"   and u :: "'a" shows "in_degree (H.add_arc aa a) u = (if head G a = u \<and> a \<notin> arcs (add_arc aa) then in_degree (add_arc aa) u + 1 else in_degree (add_arc aa) u)"
dom_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "B.dom (g \<cdot>\<^sub>B f) = B.dom f"
perm_restrict_def: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" shows "perm_restrict f S x \<equiv> if x \<in> S then f x else x"
intros_1: fixes M :: "'a trm" shows "M \<rightarrow>\<beta>\<^sup>* M"
subseqs_subformulas\<^sub>\<mu>_list: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln" shows "(X \<subseteq> subformulas\<^sub>\<mu> \<phi>) = (\<exists>xs. X = set xs \<and> xs \<in> set (subseqs (subformulas\<^sub>\<mu>_list \<phi>)))"
ring_tOp_rel: fixes x :: "'a"   and xa :: "'a"   and y :: "'a"   and ya :: "'a" assumes "x \<in> carrier R"   and "xa \<in> carrier R"   and "y \<in> carrier R"   and "ya \<in> carrier R" shows "x \<cdot>\<^sub>r xa \<cdot>\<^sub>r (y \<cdot>\<^sub>r ya) = x \<cdot>\<^sub>r y \<cdot>\<^sub>r (xa \<cdot>\<^sub>r ya)"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G F local.map"
close_world_def: fixes M :: "object atom formula set" shows "close_world M = M \<union> {\<^bold>\<not> (Atom (predAtm p as)) |p as. Atom (predAtm p as) \<notin> M} \<union> {Atom (Eq a a) |a. True} \<union> {\<^bold>\<not> (Atom (Eq a b)) |a b. a \<noteq> b}"
1: shows "mat (row_length (A1 \<circ> A2)) (length (A1 \<circ> A2)) (A1 \<circ> A2)"
ltl_convert_correct_4: fixes \<phi> :: "expr ltlc"   and APs :: "expr iarray"   and \<phi>\<^sub>i :: "nat ltlc" assumes "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)" shows "distinct (IArray.list_of APs)"
int_poly_factorization_algorithm_irreducible\<^sub>d: fixes alg :: "int_poly_factorization_algorithm"   and f :: "int poly"   and fs :: "int poly list" assumes "int_poly_factorization_algorithm alg f = fs"   and "square_free f"   and "0 < degree f" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
\<delta>complete_def: shows "\<delta>complete \<equiv> remap_rule Some ` \<delta> \<union> {None \<rightarrow> f qs |f qs. A f = Some (length qs) \<and> qs \<in> lists Qcomplete \<and> (\<nexists>qo qso. qo \<rightarrow> f qso \<in> \<delta> \<and> qs = map Some qso)}"
Union_mset_empty_conv: fixes M :: "'a multiset multiset" shows "(\<Sum>\<^sub># M = {#}) = (\<forall>i\<in>#M. i = {#})"
normal_eq_commuting: assumes "carrier G \<subseteq> A <#> B"   and "A \<inter> B \<subseteq> {\<one>}" shows "(A \<lhd> G \<and> B \<lhd> G) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x)"
distinct_1: fixes x2 :: "'g" shows "None \<noteq> Some x2"
IH_6: shows "((l, \<langle>c: t \<in> s\<rangle>) \<in> set A) = ((l, \<langle>c: t \<in> s\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
connected_space_subconnected: fixes X :: "'b topology" shows "connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>S. connectedin X S \<and> x \<in> S \<and> y \<in> S)"
accmodi_simp: fixes C :: "'a"   and m :: "'b decl_scheme" shows "accmodi (C, m) = access m"
var_negate: fixes l :: "lit" shows "var (negate l) = var l"
swapidseq_swap: fixes a :: "'b"   and b :: "'b" shows "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
rep_eq: fixes x :: "char list" shows "explode (implode x) = map ascii_of x"
A: shows "(evsFR5, S, A, U) \<in> protocol"
csq_2: shows "Maps.commutative_square SPN_fgh.\<mu>.leg0 (SPN_fgh.\<nu>.leg1 \<odot> SPN_fgh.\<nu>\<pi>.prj\<^sub>1) SPN_fgh.Prj\<^sub>1\<^sub>1 tuple_BC"
pr_algo_maintains_hb_labeling: fixes a :: "pr_operation"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), a, f', l') \<in> pr_algo_lts" shows "Height_Bounded_Labeling c s t f' l'"
relpow_E2: fixes x :: "'b"   and z :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" assumes "(x, z) \<in> R ^^ n" obtains "n = 0" and "x = z"   | y :: "'b" and m :: "nat" where "n = Suc m" and "(x, y) \<in> R" and "(y, z) \<in> R ^^ m"
M_def: shows "M = middle (rep_Bonk_Schramm_extension x) (rep_Bonk_Schramm_extension y)"
sqnf_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip)" shows "sqnf (update rt dip (dsn, dsk, flg, hops, sip)) dip = dsk"
simps_3: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" shows "ccomp (c\<^sub>1;; c\<^sub>2) = ccomp c\<^sub>1 @ ccomp c\<^sub>2"
starset_n_def: fixes As :: "nat \<Rightarrow> 'c set" shows "*sn* As = Iset (star_n As)"
induct: fixes x1 :: "expr"   and x2 :: "Unary_PCF.type"   and P :: "expr \<Rightarrow> Unary_PCF.type \<Rightarrow> bool" assumes "x1 ::: x2"   and "\<And>x T. P \<langle>(x, T)\<rangle> T"   and "\<And>b. P b \<B>"   and "\<And>e1 e2. \<lbrakk>e1 ::: \<B>; P e1 \<B>; e2 ::: \<B>; P e2 \<B>\<rbrakk> \<Longrightarrow> P (e1 ? e2) \<B>"   and "\<And>e1 T U e2. \<lbrakk>e1 ::: T \<rightarrow> U; P e1 (T \<rightarrow> U); e2 ::: T; P e2 T\<rbrakk> \<Longrightarrow> P (e1 \<cdot> e2) U"   and "\<And>T X e U. \<forall>x. (x, T) |\<notin>| X \<longrightarrow> open0_Var (x, T) e ::: U \<and> P (open0_Var (x, T) e) U \<Longrightarrow> P (\<Lambda>\<langle>T\<rangle> e) (T \<rightarrow> U)" shows "P x1 x2"
inverse_unique: fixes a :: "'d"   and b :: "'d" assumes "a + b = (0::'d)" shows "- a = b"
All_None_def: fixes xs :: "'b com option list" shows "All_None xs \<equiv> \<forall>c\<in>set xs. c = None"
Sup_bot_conv_2: 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 A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup A) = (\<forall>x\<in>A. x = bot)"
circ_sup_one_left_unfold: fixes x :: "'a" assumes "top \<le> x" shows "x \<sqinter> top = top"
preserves_cod: fixes f :: "'a" assumes "\<phi>.A.arr f" shows "\<phi>.B.cod (\<phi> f) = G (\<phi>.A.cod f)"
i_aggregate_hd: fixes f :: "nat \<Rightarrow> 'd"   and k :: "nat"   and ag :: "'d list \<Rightarrow> 'd" shows "i_aggregate f k ag 0 = ag (f \<Down> k)"
norm_of_int: fixes z :: "int" shows "norm (of_int z) = \<bar>real_of_int z\<bar>"
nres_of_THE_DRES_le: fixes x :: "'a option"   and y :: "'a nres" assumes "\<And>v. x = Some v \<Longrightarrow> RETURN v \<le> y" shows "nres_of (THE_DRES x) \<le> y"
S_def: shows "S \<equiv> \<lambda>A t. {x. (\<forall>i\<in>A. 0 \<le> x i) \<and> sum x A \<le> t}"
Max_wrt_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Max_wrt R = Max_wrt_among R UNIV"
rep_eq: fixes x :: "'c poly"   and xa :: "'c poly" shows "coeff (x + xa) = (\<lambda>n. coeff x n + coeff xa n)"
advantage_single_def: fixes \<A> :: "'c_a \<Rightarrow> ('guess, 'call, 'ret) gpv" shows "advantage_single \<A> = spmf (game_single \<A>) True"
lunit'_simps_3: fixes f :: "'a" assumes "S.ide f" shows "S.trg (S.lunit' f) = S.trg f"
2: shows "coreflexive ((w \<sqinter> - prim_EP w v e) * prim_P w v e\<^sup>T\<^sup>T)"
not_eq_order_implies_strict: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<noteq> b"   and "b \<subseteq># a" shows "b \<subset># a"
sumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<sim>\<^sub>l Q \<oplus> P"
induced_map_iterates_stabilize_A: fixes x :: "'a"   and A :: "'a set"   and n :: "nat" assumes "x \<in> A" shows "(induced_map A ^^ n) x \<in> A"
WCFG_SeqConnect: fixes c\<^sub>1 :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and c\<^sub>2 :: "cmd" assumes "c\<^sub>1 \<turnstile> n -et\<rightarrow> (_Exit_)"   and "n \<noteq> (_Entry_)" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> n -et\<rightarrow> (_ 0 _) \<oplus> #:c\<^sub>1"
Sup_subset_mono: fixes B :: "V set"   and A :: "V set" assumes "small B"   and "A \<subseteq> B" shows "\<Squnion> A \<le> \<Squnion> B"
conj_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x AND y) AND z = x AND y AND z"