hcju/isabelleps huggingface dataset:
oguardT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and g :: "'e \<Rightarrow> 'e set"   and \<sigma> :: "nat \<Rightarrow> 'e"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'g, 'f, 'h) seqp"   and l :: "'h"   and p :: "('e, 'g, 'f, 'h) seqp" assumes "\<sigma>' i \<in> g (\<sigma> i)" shows "oseqp_sosp \<Gamma> i \<sigma> ({l}\<langle>g\<rangle> p) \<tau> \<sigma>' p"
the_inv_f_o_f_id: fixes f :: "'a \<Rightarrow> 'b"   and z :: "'a" assumes "inj f" shows "(the_inv f \<circ> f) z = id z"
C: fixes x :: "'b" shows "norm (g' x) \<le> C * norm x"
assms_1: shows "ccApprox t \<sqsubseteq> G"
wf_interp_for_formula_FOr: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi>1 :: "'b formula"   and \<phi>2 :: "'b formula" assumes "formula \<Sigma>" shows "formula.wf_interp_for_formula \<Sigma> (w, I) (FOr \<phi>1 \<phi>2) = (formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>1 \<and> formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>2)"
cases: fixes x :: "'c DBMEntry" obtains t :: "'c" where "x = Le t__"   | t :: "'c" where "x = Lt t"   | "x = \<infinity>"
bdd: fixes x :: "V" assumes "x \<in> A" shows "small (f ` B x)"
refl: fixes t :: "('a, 'b) tm" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b t"
subset_Times_ap: fixes ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "punit.ap_spec ap"   and "ab_spec ab"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)" shows "set (ap gs bs (ps -- sps) hs data) \<subseteq> set (ab gs bs hs data) \<times> (set gs \<union> set (ab gs bs hs data))"
cell_complex_subdivision_exists: fixes e :: "real"   and \<F> :: "'a set set"   and d :: "int" assumes "0 < e"   and "finite \<F>"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X" obtains \<F>' :: "'a set set" where "finite \<F>'" and "\<Union> \<F>' = \<Union> \<F>" and "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e" and "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X" and "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d" and "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X" and "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D" and "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
CR_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "CR_on r A"   and "a \<in> A" obtains "(b, c) \<in> r\<^sup>\<down>"   | "(a, b) \<notin> r\<^sup>*"   | "(a, c) \<notin> r\<^sup>*"
assms_2: shows "s \<in> carrier R\<^bsup>\<omega>\<^esup>"
ctr_simps_mem_Collect_eq: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
assms_1: shows "rs = rs\<^sub>1 @ rs\<^sub>2"
supt_ctxtE: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<rhd> t" obtains C :: "('d, 'e) ctxt" where "C \<noteq> \<box>" and "s = C\<langle>t\<rangle>"
dbm_entry_val_mono_3: fixes u :: "'b \<Rightarrow> 'c"   and c :: "'b"   and b :: "'c DBMEntry"   and b' :: "'c DBMEntry" assumes "dbm_entry_val u (Some c) None b"   and "b \<preceq> b'" shows "dbm_entry_val u (Some c) None b'"
Union_plus: fixes f :: "'b + 'c \<Rightarrow> 'a set"   and A :: "'b set"   and B :: "'c set" shows "\<Union> (f ` (A <+> B)) = (\<Union>a\<in>A. f (Inl a)) \<union> (\<Union>b\<in>B. f (Inr b))"
surj_diff_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "surj (\<lambda>x. minus x a)"
eq_filter_fset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b fset"   and Q :: "'b \<Rightarrow> bool" shows "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
d4_def: fixes a :: "'a"   and b :: "'a" shows "d4 a b = d2 b a"
index_uminus_mat_2: fixes A :: "'c mat" shows "dim_row (- A) = dim_row A"
ereal_add_strict_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal"   and d :: "ereal" assumes "a \<le> b"   and "0 \<le> a"   and "a \<noteq> \<infinity>"   and "c < d" shows "a + c < b + d"
le_sup_iff: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
rel_funE: fixes A :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and B :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and f :: "'n \<Rightarrow> 'p"   and g :: "'o \<Rightarrow> 'q"   and x :: "'n"   and y :: "'o" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
f_le_g: fixes x :: "'a" assumes "x \<in> S" shows "f x \<bullet> k \<le> g x \<bullet> k"
splice_Nil2: fixes xs :: "'c list" shows "splice xs [] = xs"
nn_integral_fst: fixes M :: "'a measure"   and f :: "'b \<times> 'a \<Rightarrow> ennreal"   and M1 :: "'b measure" assumes "sigma_finite_measure M"   and "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f"
fineD: fixes d :: "'b \<Rightarrow> 'c set"   and s :: "('b \<times> 'c set) set"   and x :: "'b"   and k :: "'c set" assumes "d fine s"   and "(x, k) \<in> s" shows "k \<subseteq> d x"
simps: fixes a1 :: "'a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a"   and a2 :: "'a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" shows "a1 \<lessapprox> a2 = (\<exists>ve ve' ds ds' proc b. a1 = (proc, ds, ve, b) \<and> a2 = (proc, ds', ve', b) \<and> ve \<lessapprox> ve' \<and> ds \<lessapprox> ds')"
\<tau>move1BlockThrow: fixes P :: "'b prog"   and h :: "'heap"   and V :: "'c"   and T :: "ty"   and a :: "'addr" shows "\<tau>move1 P h {V:T=None; Throw a}"
inv_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
f: fixes K :: "'b set" assumes "compactin Y K" shows "compactin X {x \<in> topspace X. f x \<in> K}"
normalize_replace_primitive_matchexpr_preserves_normalized_n_primitive: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr"   and disc2 :: "'b \<Rightarrow> bool"   and sel2 :: "'b \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> bool"   and m' :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<And>a m'. m' \<in> set (normalize_match (replace_f a)) \<Longrightarrow> normalized_n_primitive (disc2, sel2) f m'"   and "normalized_n_primitive (disc2, sel2) f m"   and "m' \<in> set (normalize_match (replace_primitive_matchexpr (disc, sel) replace_f m))" shows "normalized_n_primitive (disc2, sel2) f m'"
no_Empty_ge0: fixes h :: "'b heap" assumes "no_Empty h" shows "0 < size_hp h"
hb_broadcast_exists1: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and m2 :: "'c" assumes "network history msg_id"   and "network.hb history m1 m2" shows "\<exists>i. Broadcast m1 \<in> set (history i)"
add_commute: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "ab_semigroup_add_ow S pls"   and "a \<in> S"   and "b \<in> S" shows "pls a b = pls b a"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "Group G"
\<pi>'_def: fixes x :: "'b" shows "\<pi>' x = (if x \<in> A then ((!) xs \<circ> \<pi> \<circ> index xs) x else x)"
assms_1: shows "monotone (\<le>) ord (\<lambda>x. f x (eSuc x))"
dedekind_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (z \<sqinter> id y) \<sqinter> (y \<sqinter> (id x \<sqinter> z)) \<sqinter> z"
dg_is_arrD_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
neg_less_iff_less: fixes b :: "'a"   and a :: "'a" shows "(- b < - a) = (a < b)"
igSubstInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) model_scheme"   and ys :: "'e"   and Y :: "'b"   and y :: "'h"   and inp :: "'a \<Rightarrow> 'b option" shows "igSubstInp MOD ys Y y inp \<equiv> lift (igSubst MOD ys Y y) inp"
block_set_nempty_imp_block_ex: assumes "\<B>' \<noteq> {#}" shows "\<exists>bl. bl \<in># \<B>'"
fimage_finsert: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e"   and B :: "'e fset" shows "f |`| finsert a B = finsert (f a) (f |`| B)"
ReflectL_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.ReflectL Bet Cong P' P A B \<equiv> (\<exists>X. Tarski_neutral_dimensionless.Midpoint Bet Cong X P P' \<and> Tarski_neutral_dimensionless.Col Bet A B X) \<and> (Tarski_neutral_dimensionless.Perp Bet Cong A B P P' \<or> P = P')"
A: shows "((s, c), ee, s', csp + c) \<in> ntrs fg"
nnfis_dom_conv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "f \<in> rv M"   and "b \<in> nnfis g M"   and "f \<le> g"   and "nonnegative f" shows "\<exists>a. a \<in> nnfis f M \<and> a \<le> b"
antisymp_onI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; P a b; P b a\<rbrakk> \<Longrightarrow> a = b" shows "antisymp_on P A"
implements_oreach: fixes pi :: "'e \<Rightarrow> 'f"   and Sa :: "('g, 'f) spec"   and Sc :: "('h, 'e) spec" assumes "implements pi Sa Sc" shows "pi ` oreach Sc \<subseteq> oreach Sa"
L: shows "L = (K - vSF) / (1 - vSF)"
valid_history: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "history_consistent \<theta> (hd_prog p sb) sb"
create_name_gba_alt: fixes \<phi> :: "'b ltlr" shows "create_name_gba \<phi> = create_graph\<^sub>T \<phi> \<bind> (\<lambda>nds. ASSERT (nds_invars nds) \<bind> (\<lambda>_. RETURN (gba_rename name (create_gba_from_nodes \<phi> nds))))"
object_ptr_kinds_eq_h2: shows "object_ptr_kinds h2 = object_ptr_kinds h3"
osafe: shows "osafe (OSing vid1 (f1 fid1 vid1))"
node_total_order_antisym: fixes e1 :: "'a event"   and i :: "nat"   and e2 :: "'a event" assumes "e1 \<sqsubset>\<^sup>i e2"   and "e2 \<sqsubset>\<^sup>i e1" shows "False"
inPlus_def: fixes r :: "'c rexp"   and s :: "'c rexp" shows "inPlus r s = (case Pluss (Plus r s) of None \<Rightarrow> Full | Some x \<Rightarrow> PLUS x)"
canonical_subs_len: fixes n :: "nat"   and I :: "nat set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "canonical_subs n I M"   and "i \<le> n"   and "j \<le> n"   and "set xs \<subseteq> I"   and "I \<subseteq> {0..n}" shows "M i j \<le> len M i j xs"
minus_minus: fixes x :: "'c" assumes "x \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> x) = x"
sum_in_Reals: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "sum f s \<in> \<real>"
simps: fixes a1 :: "'c"   and a2 :: "'d list"   and a3 :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" shows "((a1, a2, a3) \<in> trcl t) = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c a c' w c''. a1 = c \<and> a2 = a # w \<and> a3 = c'' \<and> (c, a, c') \<in> t \<and> (c', w, c'') \<in> trcl t))"
interior_of_interior_of: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of X interior_of S = X interior_of S"
sb2: fixes t :: "real"   and b :: "complex" assumes "1 / 12 < t" shows "ball (C * complex_of_real r * b) r' \<subseteq> (*) (C * complex_of_real r) ` ball b t"
find_first_insert_nth_eq: fixes n :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "n \<le> length xs"   and "x \<notin> set (take n xs)" shows "find_first x (insert_nth n x xs) = Some n"
mod_div_decomp: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" obtains q :: "'c" and r :: "'c" where "q = divide a b" and "r = modulo a b" and "a = plus (times q b) r"
simps_9: fixes f :: "'c \<Rightarrow> 'b"   and x21 :: "'c Tree"   and x22 :: "'c"   and x23 :: "'c Tree" shows "map_Tree f (T x21 x22 x23) = T (map_Tree f x21) (f x22) (map_Tree f x23)"
c: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A' \<turnstile> e' : B'"
set_eqI: fixes A :: "'d set"   and B :: "'d set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
shows_list_monom_def: fixes ts :: "'a monom list" shows "shows_list ts = showsp_list shows_prec 0 ts"
dual_star_def: fixes x :: "'a" shows "x ^ \<otimes> = ((x ^ o) ^ *) ^ o"
dom_char: fixes f :: "'a" shows "L.dom f = (if L.arr f then local.dom f else null)"
is_dual_ccpo: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.ccpo Inf (\<lambda>x y. less_eq y x) (\<lambda>x y. less y x)"
distinct_5: fixes x1 :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option" shows "\<Up>x1 \<noteq> x41\<hookleftarrow>\<^bsub>x42\<^esub>x43"
unat_eq_0: fixes x :: "'b word" shows "(unat x = 0) = (x = 0)"
starfun_exp_gt_one: fixes x :: "real star" assumes "0 < x" shows "1 < (*f* exp) x"
no_await_trancl_impl: fixes ctx :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and ctx' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "ctx \<leadsto>\<^sup>+ ctx'" shows "no_await (fst (fst ctx)) \<longrightarrow> no_await (fst (fst ctx'))"
listset_closed_map: fixes ys :: "'b list"   and xs :: "'b set list"   and f :: "'b \<Rightarrow> 'b" assumes "ys \<in> listset xs"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> x\<rbrakk> \<Longrightarrow> f y \<in> x" shows "map f ys \<in> listset xs"
4: shows "bprv (imp (cnj (PP \<langle>PP \<langle>\<phi>L \<phi>\<rangle>\<rangle>) (PP \<langle>imp (PP \<langle>\<phi>L \<phi>\<rangle>) \<phi>\<rangle>)) (PP \<langle>\<phi>\<rangle>))"
finprod_mono_neutral_cong_right: fixes B :: "'d set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "finite B"   and "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> g i = \<one>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "g \<in> B \<rightarrow> carrier G" shows "finprod G g B = finprod G h A"
list_all2_conv_all_nth: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
type_less_x_Set: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> < Set \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Set \<rho>" and "\<rho> < \<sigma>"
prefix_distinct: fixes xs :: "'msg event list"   and i :: "nat" assumes "xs prefix of i" shows "distinct xs"
path_connectedin_Un: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "path_connectedin X S"   and "path_connectedin X T"   and "S \<inter> T \<noteq> {}" shows "path_connectedin X (S \<union> T)"
atLeast_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b" shows "ord.atLeast less_eq l \<equiv> {x. less_eq l x}"
dominant_term_bigo: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis" shows "eval F \<in> O(eval_monom (1, snd (dominant_term F)) basis)"
cont_nosp2: shows "continuous_on (cball (0::'a) 1 \<inter> affine hull S) (\<lambda>x. norm x *\<^sub>R (surf \<circ> proj) x)"
cancel_div_mod_rules_2: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "plus (plus (times b (divide a b)) (modulo a b)) c = plus a c"
trms\<^sub>e\<^sub>s\<^sub>t_ik_subtermsI: fixes t :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t A" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)"
LPDs_aux1: fixes r :: "mregex" shows "lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> LPD)) \<subseteq> LPDs r"
tObsC_ii_absI: fixes t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and v :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" assumes "t' \<in> SPR.jkbpC"   and "tObsC t' = tObsC t"   and "v = tLast t'" shows "v \<in> tObsC_ii_abs t"
eval_wlp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wlp (Embed t) = t"
trancl_image: fixes i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" assumes "(i, j) \<in> R\<^sup>+" shows "(f i, f j) \<in> (map_prod f f ` R)\<^sup>+"
assms: fixes l :: "'a list" assumes "distinct l"   and "sorted_wrt R l" shows "foldli l c f \<sigma> = result (set l)"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = (n < min LENGTH('c) m)"
coeff_all_0: fixes p :: "'b mpoly" assumes "\<And>m. MPoly_Type.coeff p m = (0::'b)" shows "p = 0"
vector_cart: fixes f :: "(real, 'a) vec \<Rightarrow> real" shows "(\<chi>i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
run_alt_option: fixes m1 :: "('a, 'm) optionT"   and m2 :: "('a, 'm) optionT" shows "run_option (local.alt_option m1 m2) = alt (run_option m1) (run_option m2)"
is_iso_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<and> is_iso_dghm_axioms \<BB> \<FF>"
join_cong_r: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and y' :: "'c" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "y' \<in> carrier L"   and "y .=\<^bsub>L\<^esub> y'" shows "x \<squnion>\<^bsub>L\<^esub> y .=\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> y'"
simps: fixes T :: "'a itself"   and nrandom :: "natural"   and sz :: "natural"   and seed :: "natural \<times> natural" shows "iter' T nrandom sz seed = (if nrandom = 0 then \<bottom> else let ((x, uu_), seed') = random_class.random sz seed in Predicate.Seq (\<lambda>u. seq.Insert x (iter' T (nrandom - 1) sz seed')))"
ssort_def: fixes l :: "'a list" shows "ssort l = ssort' (of_list l) []"
leg_simps_4: shows "C.dom f.dom.leg1 = f.dom.apex"
pders_changes_c: fixes r :: "real poly"   and q :: "real poly" assumes "poly r c \<noteq> 0"   and "q \<noteq> 0" shows "changes_poly_at (r # pders q) c = (if sign_r_pos q c = (0 < poly r c) then changes_poly_at (pders q) c else 1 + changes_poly_at (pders q) c)"
label_incr_start_Node_smaller: fixes l :: "nat"   and n :: "label"   and i :: "nat" assumes "Label l = n \<oplus> i" shows "n = Label (l - i)"
dvd_ideal_generated_singleton': fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "ideal_generated {a} \<subseteq> ideal_generated {b}"
npeadd_correct: fixes xs :: "'a list"   and e1 :: "pexpr"   and e2 :: "pexpr" assumes "in_carrier xs" shows "peval xs (npeadd e1 e2) = peval xs (PExpr1 (PAdd e1 e2))"
distinct_313: fixes x61 :: "tf"   and x62 :: "b_e list"   and x23 :: "v" shows "Loop x61 x62 \<noteq> C x23"
AG_def: fixes p :: "'b set" shows "\<^bold>A\<^bold>G p = - \<^bold>E\<^bold>F - p"
rctxt_def: shows "rctxt \<equiv> {x. rctxtp x}"
i0: shows "local.adverse 0 i j \<triangleright> Suc xmin = e_snoc (local.adverse z i j \<triangleright> xmin) 0"