hcju/isabelleps huggingface dataset:
split_mult_pos_le: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero" shows "less_eq zero (times a b)"
scalar_product_eq0: fixes ns :: "nat list"   and "is" :: "int list" assumes "\<forall>c\<in>set ns. c = 0" shows "scalar_product ns is = 0"
c: shows "lookup q1 (t2 \<oplus> lt f2) = lookup p (t2 \<oplus> lt f2)"
SPLIT_reflection: fixes x :: "'c"   and y :: "'c" assumes "SPLIT x y" shows "x \<equiv> y"
finite: shows "(\<Sum>n. ennreal (1 / R n)) = \<top>"
lindep_span: fixes U :: "'a vec set" assumes "U \<subseteq> carrier_vec n"   and "finite U" shows "lin_dep U = (\<exists>u\<in>U. u \<in> local.span (U - {u}))"
vs_def: shows "vs = remdups (map fst (snd G_list))"
g_sm_p: shows "\<forall>i. i < g_sm i \<and> f (g_sm i) < f i"
rpo_axioms: shows "rpo ground_heads_var (>\<^sub>s) extf arity_sym arity_var"
components_are_equivalences: fixes a :: "'a" assumes "C.obj a" shows "D.equivalence_map (\<tau>\<^sub>0 a)"
urel_integer_def: fixes pp :: "integer"   and x :: "integer"   and y :: "int" assumes "p = int_of_integer pp" shows "urel_integer x y = (y = int_of_integer x \<and> 0 \<le> y \<and> y < p)"
p'_def: fixes a :: "real"   and v :: "real"   and t :: "real" shows "movement.p' a v t = v + a * t"
irreflexive_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i. f (i, i) = bot" shows "matrix_stone_relation_algebra.irreflexive f"
linorder_le_of_nat_term_order: fixes to :: "'b nat_term_order" shows "class.linorder (le_of_nat_term_order to) (lt_of_nat_term_order to)"
ext_inject: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d"   and counter' :: "nat"   and discovered' :: "'c \<Rightarrow> nat option"   and finished' :: "'c \<Rightarrow> nat option"   and pending' :: "('c \<times> 'c) set"   and stack' :: "'c list"   and tree_edges' :: "('c \<times> 'c) set"   and back_edges' :: "('c \<times> 'c) set"   and cross_edges' :: "('c \<times> 'c) set"   and more' :: "'d" shows "(\<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = \<lparr>counter = counter', discovered = discovered', finished = finished', pending = pending', stack = stack', tree_edges = tree_edges', back_edges = back_edges', cross_edges = cross_edges', \<dots> = more'\<rparr>) = (counter = counter' \<and> discovered = discovered' \<and> finished = finished' \<and> pending = pending' \<and> stack = stack' \<and> tree_edges = tree_edges' \<and> back_edges = back_edges' \<and> cross_edges = cross_edges' \<and> more = more')"
lhd_cons_def: fixes s :: "'s" shows "local.lhd_cons s = lhd (lunstream g s)"
equiv_trans: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.equiv eqv"   and "eqv x y"   and "eqv y z" shows "eqv x z"
mdecls_visible: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list" assumes "wf_prog wf_md P"   and "is_class P C"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" shows "\<exists>Mm. P \<turnstile> C sees_methods Mm \<and> (\<forall>(M, b, Ts, T, m)\<in>set ms. Mm M = \<lfloor>((b, Ts, T, m), C)\<rfloor>)"
magicl_applied_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and F :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.magicl_applied plus sep_disj h1 h2 h0 F \<equiv> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> F h2"
IFind_3: shows "ro_ifex (IF xv xb1 xb2)"
from_nat_to_nat_id: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (mod_type.to_nat Rep x) = x"
hyp: shows "(\<Sum>i\<le>b. monom (g i) i) mod f = (\<Sum>i\<le>b. monom (g i) i)"
simps_1: fixes uu :: "String.literal"   and uv :: "typ" shows "fv (Ct uu uv) = {}"
context_free_Cut: fixes \<Gamma> :: "'a formula multiset"   and A :: "'a formula"   and \<Delta> :: "'a formula multiset"   and \<Gamma>' :: "'a formula multiset"   and \<Delta>' :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> A, \<Delta>"   and "A, \<Gamma>' \<Rightarrow> \<Delta>'" shows "\<Gamma> + \<Gamma>' \<Rightarrow> \<Delta> + \<Delta>'"
size_big_union_sum: fixes M :: "'b multiset multiset" shows "size (\<Sum>\<^sub># M) = \<Sum>\<^sub># (image_mset size M)"
filter_map_splitE: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'b list"   and zs :: "'b list" assumes "filter P (map f xs) = ys @ zs" obtains us :: "'c list" and vs :: "'c list" where "xs = us @ vs" and "filter P (map f us) = ys" and "filter P (map f vs) = zs"
loc_srj_lm_2: fixes x :: "nat" assumes "1 < x" shows "nat_to_sch x = (let u = mod7 (c_fst x); v = c_snd x; v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
swap_restrict: fixes A :: "'c set"   and B :: "'d set"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "finite A"   and "finite B" shows "(\<Sum>x\<in>A. sum (g x) {y \<in> B. R x y}) = (\<Sum>y\<in>B. \<Sum>x | x \<in> A \<and> R x y. g x y)"
gen_dfs_post_imp_outer: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and V0 :: "'a set"   and it :: "'a set"   and Vis0 :: "'a set"   and v0 :: "'a"   and Vis :: "'a set" assumes "gen_dfs_outer E U V0 it Vis0 False"   and "gen_dfs_post E U {} Vis0 v0 Vis False"   and "v0 \<in> it"   and "it \<subseteq> V0"   and "v0 \<notin> Vis0" shows "gen_dfs_outer E U V0 (it - {v0}) Vis False"
affine_hull_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "bounded_linear f" shows "f ` (affine hull s) = affine hull f ` s"
assms_1: shows "(\<^bold>\<diamond> p) at a in (ps, a) # branch"
dghm_dghm_comp_dghm_id_left: shows "dghm_id \<DD> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> = \<GG>"
simps_4: fixes T :: "typ"   and t :: "Core.term" shows "tvs (Abs T t) = tvsT T \<union> tvs t"
nts_snoc: shows "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi>'' j) (Monitor.progress \<sigma> \<psi>'' j)..<Suc j] (nts @ [\<tau> \<sigma> j])"
deleteMin_def: fixes q :: "('c, 'd) SkewBinomialHeap" shows "deleteMin q \<equiv> if q = local.empty then local.empty else Abs_SkewBinomialHeap (BsSkewBinomialHeapStruc.bs_deleteMin (Rep_SkewBinomialHeap q))"
simps_7: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_nat (Rec_op t1 t2) = c_pair 6 (c_pair (sch_to_nat t1) (sch_to_nat t2))"
disj_com_Cn: fixes p :: "'a"   and q :: "'a" shows "Cn {p .\<or>. q} = Cn {q .\<or>. p}"
fun_evaluate_matchE: fixes s :: "'b state"   and env :: "v sem_env"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and s' :: "'b state"   and vs :: "v list" assumes "fun_evaluate_match s env v pes err_v = (s', Rval vs)" obtains v :: "v" where "vs = [v]"
assms_3: shows "(G ===> (=)) \<alpha>n1 \<alpha>n2"
assert_bound_nolhs_tableau_id: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and a :: "'a \<times> 'b atom" assumes "AssertBoundNoLhs assert_bound"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s" shows "\<T> (assert_bound a s) = \<T> s"
D_hiding: fixes P :: "'a process"   and A :: "'a set" shows "D (P \ A) = div_hide P A"
indoc_single: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "traces ind" shows "traces.indoc ind a [b] = (a = b)"
FD_F: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>F Q"
inner_complex_def: fixes x :: "complex"   and y :: "complex" shows "x \<bullet> y = Re x * Re y + Im x * Im y"
red_diff_rtrancl': fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and r :: "'a \<Rightarrow>\<^sub>0 'd" assumes "(punit.red F)\<^sup>*\<^sup>* (p - q) r" obtains p' :: "'a \<Rightarrow>\<^sub>0 'd" and q' :: "'a \<Rightarrow>\<^sub>0 'd" where "(punit.red F)\<^sup>*\<^sup>* p p'" and "(punit.red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"
Poly_snoc: fixes xs :: "'b list"   and x :: "'b" shows "Poly (xs @ [x]) = Poly xs + monom x (length xs)"
cg: shows "continuous_on UNIV (\<lambda>(x, y). g x y)"
prefix_of_appendD: fixes xs :: "'msg event list"   and ys :: "'msg event list"   and i :: "nat" assumes "xs @ ys prefix of i" shows "xs prefix of i"
wp1_alternativesep: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (separate p)"
ide_mkarr_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "\<F>C.ide (\<F>C.mkarr a)"
complete_if_size1_height: fixes t :: "'b tree" assumes "size1 t = 2 ^ height t" shows "complete t"
filter_compliant_stateful_ACS_subseteq_input: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "set (filter_compliant_stateful_ACS G M Es) \<subseteq> set Es"
refl: fixes x :: "'a" shows "(x, x) \<in> r"
valid_Cn_equiv: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(equiv_PL p q \<in> Cn {}) = (Cn {p} = Cn {q})"
is_rewrite_ordD4': fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_rewrite_ord rword"   and "rword (spp_of a) (spp_of b)"   and "rword (spp_of b) (spp_of a)" shows "lt a = lt b"
rewrite_iter_slow_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (rewrite_iter_slow \<phi>) \<subseteq> atoms_ltln \<phi>"
naturality: fixes f :: "'b" assumes "Adj.D.arr f" shows "Adj'.\<eta> (Adj.D.cod f) \<cdot>\<^sub>D Adj.D.map f = Adj'.GF.map f \<cdot>\<^sub>D Adj'.\<eta> (Adj.D.dom f)"
CX4h_singular: fixes h :: "H1"   and X :: "X4 set" shows "inj_on X4d (CX4h h X)"
sub_trans: fixes T1 :: "funty"   and T2 :: "funty"   and T3 :: "funty" assumes "T1 <:: T2"   and "T2 <:: T3" shows "T1 <:: T3"
set_image_mset: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e multiset" shows "set_mset (image_mset f M) = f ` set_mset M"
record_simps_28: fixes f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "ffi (update_defined_types f x) = ffi x"
reduce'_semantics: fixes q :: "'a pfm"   and M :: "('a, 'b) kripke"   and w :: "'b"   and p :: "'a pfm" assumes "static q" shows "(M, w \<Turnstile>\<^sub>! [p]\<^sub>! q) = (M, w \<Turnstile>\<^sub>! reduce' p q)"
spmf_rel_conversep: fixes R :: "'i \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_spmf R\<inverse>\<inverse> = (rel_spmf R)\<inverse>\<inverse>"
no_step: fixes A :: "('a \<times> 'a) set"   and a :: "'a" assumes "A `` {a} = {}" shows "a \<in> NF A"
un_non_empty: shows "i \<squnion> j \<noteq> bot"
iwlsFSwSb_termFSwSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSwSb MOD" shows "termFSwSbImorph h hA MOD = ipresCons h hA MOD"
elims_2: fixes x :: "'a"   and xa :: "'b"   and xb :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "qAFreshAbs x xa xb" obtains xs :: "'a" and xc :: "'b" and ys :: "'a" and y :: "'b" and X :: "('c, 'd, 'a, 'b, 'e) qTerm" where "x = xs" and "xa = xc" and "xb = qAbs ys y X" and "(xs \<noteq> ys \<or> xc \<noteq> y) \<and> qAFresh xs xc X"
boundedE: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" assumes "less_eq_prod a (inf_prod b c)" obtains "less_eq_prod a b" and "less_eq_prod a c"
push_bit_word_beyond: fixes n :: "nat"   and w :: "'a word" assumes "LENGTH('a) \<le> n" shows "push_bit n w = 0"
of_real_poly_eq_iff: fixes p :: "real poly"   and q :: "real poly" shows "(map_poly of_real p = map_poly of_real q) = (p = q)"
isecure_def: shows "int.isecure \<equiv> int.NI_indirect_sources \<and> int.NI_unrelated"
orbit_sym_aux: fixes g :: "'a"   and x :: "'c"   and y :: "'c" assumes "g \<in> carrier G"   and "x \<in> E"   and "\<phi> g x = y" shows "\<phi> (inv g) y = x"
perfect_hiding: fixes \<eta> :: "nat"   and \<A> :: "(('grp \<times> 'grp) \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "chaum_ped_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<eta> \<A>"
asdf: shows "snd (mult_aform X Y) \<le> Radius' p (fst X) * Radius' p (fst Y) + truncate_up p (\<bar>snd X\<bar> * truncate_up p (\<bar>fst (fst Y)\<bar> + Radius' p (fst Y))) + truncate_up p (\<bar>snd Y\<bar> * truncate_up p (\<bar>fst (fst X)\<bar> + Radius' p (fst X))) + truncate_up p (\<bar>snd X\<bar> * \<bar>snd Y\<bar>)"
assms_3: shows "\<forall>n>0. degree (F $ n) < degree f"
\<T>e_ide_2: fixes b :: "'c" assumes "Cop.ide b" shows "\<T> e b = S.mkArr (Hom.set (b, a)) (Ya'.SET b) (\<lambda>x. Ya'.FUN (\<psi> (b, a) x) e)"
onll_step_invariant_sterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('i \<times> ('f, 'g, 'e, 'h) seqp, 'j) automaton"   and I :: "'j \<Rightarrow> bool"   and P :: "('i \<times> 'h) \<times> 'j \<times> 'i \<times> 'h \<Rightarrow> bool"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and a :: "'j"   and \<xi>' :: "'i"   and q :: "('f, 'g, 'e, 'h) seqp"   and l' :: "'h"   and p' :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p), a, \<xi>', q) \<in> automaton.trans A"   and "I a"   and "l' \<in> labels \<Gamma> q"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P ((\<xi>, l), a, \<xi>', l')"
cont_eff_Par_UNlift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and II :: "nat set"   and P :: "nat \<Rightarrow> nat set set"   and ii :: "nat"   and jj :: "nat"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "SbisL cl dl"   and "II \<in> UNpart cl P"   and "ii \<in> II"   and "jj \<in> UNlift cl dl P F II"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)"   and "\<And>n I i j. \<lbrakk>n < length cl; I \<in> P n; i \<in> I; j \<in> F n I\<rbrakk> \<Longrightarrow> eff (cl ! n) s i \<approx> eff (dl ! n) t j \<and> local.cont (cl ! n) s i \<approx>s local.cont (dl ! n) t j"   and "s \<approx> t" shows "eff (Par cl) s ii \<approx> eff (Par dl) t jj \<and> (local.cont (Par cl) s ii, local.cont (Par dl) t jj) \<in> thetaSPar"
lower_comp: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "\<pi>\<^sup>* (\<pi>\<^sub>* (\<pi>\<^sup>* x)) = \<pi>\<^sup>* x"
units_of_carrier: fixes G :: "('b, 'c) monoid_scheme" shows "carrier (units_of G) = Units G"
polyhedron_def: fixes n :: "nat"   and A :: "'b mat"   and b :: "'b vec" shows "gram_schmidt.polyhedron n A b = {x \<in> carrier_vec n. A *\<^sub>v x \<le> b}"
comp_null_1: fixes f :: "'c" shows "C.null \<cdot>\<^sub>C f = C.null"
deterministic_stepI: fixes M :: "('a \<times> ('a, 'b) Op) set" assumes "deterministic M" shows "deterministic (step M)"
smult_normalize_field_eq: fixes p :: "'b poly" shows "p = smult (lead_coeff p) (normalize p)"
lookup_one: fixes k :: "'c" shows "lookup 1 k = ((1::'b) when k = (0::'c))"
sec_has_right_codom: fixes s :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"   and U :: "('a \<times> 'a) set set set"   and \<pp> :: "('a \<times> 'a) set set" assumes "s \<in> local.local.sheaf_spec U"   and "\<pp> \<in> U" shows "s \<pp> \<in> local.carrier_quotient_ring \<^bsub>\<pp> local.add_rel local.mult_rel local.zero_rel\<^esub>"
min_algs2: fixes i :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set"   and s1 :: "('f, 'e, 'd) F1 \<Rightarrow> 'e"   and s2 :: "('f, 'e, 'd) F2 \<Rightarrow> 'd" assumes "i \<in> Field SucFbd" shows "snd (min_algs s1 s2 i) = min_G2 (min_algs s1 s2) i \<union> s2 ` F2in UNIV (min_G1 (min_algs s1 s2) i) (min_G2 (min_algs s1 s2) i)"
Suc_0_xor_eq: fixes n :: "nat" shows "Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)"
del_unEdge_def: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "del_unEdge v e v' g \<equiv> \<lparr>nodes = nodes g, edges = edges g - {(v, e, v'), (v', e, v)}\<rparr>"
Lcm_fin_dvd_iff: fixes A :: "'a set"   and b :: "'a" assumes "finite A" shows "(Lcm\<^sub>f\<^sub>i\<^sub>n A dvd b) = (\<forall>a\<in>A. a dvd b)"
nsum_eq: fixes A :: "('e, 'f) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e"   and g :: "nat \<Rightarrow> 'e" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A"   and "\<forall>j\<le>n. f j = g j" shows "\<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A g n"
semilat_le_err_OK2: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b err \<Rightarrow> 'b err \<Rightarrow> 'b err"   and z :: "'b" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, Err.le r, f)"   and "OK x \<squnion>\<^bsub>f\<^esub> OK y = OK z" shows "y \<sqsubseteq>\<^bsub>r\<^esub> z"
hyps_4: fixes k' :: "'a" assumes "k' \<in> fst ` set xs" shows "lt k k'"
purgeIdle_Cons_iff: fixes sl :: "('a, 'b, 'c) state list"   and s :: "('a, 'b, 'c) state"   and sll :: "('a, 'b, 'c) state list" shows "(GM_sec_model.purgeIdle sl = s # sll) = (\<exists>sl1 sl2. sl = sl1 @ s # sl2 \<and> (\<forall>s1\<in>set sl1. \<not> isState s1) \<and> isState s \<and> GM_sec_model.purgeIdle sl2 = sll)"
simps_3: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and q :: "'b" shows "\<xi> \<Turnstile>\<^sub>c prop\<^sub>c(q) = (q \<in> \<xi> 0)"
singleton: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "semilattice_set f" shows "semilattice_set.F f {x} = x"
simps_1: shows "slast\<cdot>[::] = \<bottom>"
is_path_split: fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_path v (p1 @ p2) v' = (\<exists>u. is_path v p1 u \<and> is_path u p2 v')"
monoid_axioms: shows "Group_Theory.monoid A addA zeroA"
fmupd_same: fixes x1 :: "'a"   and x1aa :: "('a, 'b) fmap" assumes "x1 \<in># dom_m x1aa" shows "fmupd x1 (the (fmlookup x1aa x1)) x1aa = x1aa"
typing_alt_simps_28: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E CollectNestedIteratorCall src k its its_ty body : \<tau>) = (k = ArrowCall \<and> (\<exists>\<tau>' \<sigma> \<rho>. \<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>) \<and> length its \<le> Suc 0 \<and> (\<exists>\<upsilon>. to_nonunique_collection \<tau>' \<upsilon> \<and> update_element_type \<upsilon> \<rho> \<tau>)))"
elims: fixes x :: "'b"   and y :: "'b list" assumes "wrap x = y" obtains xa :: "'b" where "x = xa" and "y = [xa]"
assms: fixes \<tau> :: "'a"   and \<tau>' :: "'a" assumes "P \<tau> = P \<tau>'"   and "Q \<tau> = Q \<tau>'" shows "R \<tau> = R \<tau>'"
P1: shows "Bet B A A0 \<and> Cong A A0 B' A' \<and> Bet B C C0 \<and> Cong C C0 B' C' \<and> Bet B' A' A1 \<and> Cong A' A1 B A \<and> Bet B' C' C1 \<and> Cong C' C1 B C \<and> Cong A0 C0 A1 C1"
snd_proj_def: fixes M :: "('d \<times> 'c) set" shows "snd_proj M \<equiv> {B. \<exists>A. (A, B) \<in> M}"
pfn_1: fixes x :: "'a set" assumes "x \<in> I" shows "pfn x tagged_division_of x"
forget: fixes x :: "name"   and t :: "lam"   and s :: "lam" assumes "x \<sharp> t" shows "t[x::=s] = t"
mod_add_cong: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (plus a b) c = modulo (plus a' b') c"
mult_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d" shows "c * a \<le> d * b"
ordertype_empty: fixes r :: "('a \<times> 'a) set" shows "ordertype {} r = 0"
smult_minus_left: fixes a :: "'a"   and p :: "'a poly" shows "smult (- a) p = - smult a p"
reindex: fixes h :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'f" assumes "inj_on h A" shows "sum g (h ` A) = sum (g \<circ> h) A"
continuous_on_frac: fixes t :: "'a set" assumes "t \<subseteq> {0::'a..<1::'a}" shows "continuous_on t frac"
lset_llist_of_stream: fixes xs :: "'c stream" shows "lset (llist_of_stream xs) = sset xs"
smallstepsV_consumes_proofstream: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and eV :: "Syntax.term"   and i :: "nat"   and \<pi>\<^sub>2 :: "Syntax.term list"   and eV' :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>\<^sub>2, eV'\<ggreater>" obtains \<pi> :: "Syntax.term list" where "\<pi>\<^sub>1 = \<pi> @ \<pi>\<^sub>2"
fds_eqI_truncate: fixes f :: "'a fds"   and g :: "'a fds" assumes "\<And>m. 0 < m \<Longrightarrow> fds_truncate m f = fds_truncate m g" shows "f = g"
EM_axioms_def: shows "EM_axioms \<equiv> \<forall>x y. \<not> P x y \<longrightarrow> (\<exists>z. P z x \<and> \<not> O z y)"
existence_ivl_empty_iff: fixes t0 :: "real"   and x0 :: "'a" shows "(existence_ivl t0 x0 = {}) = (t0 \<notin> T \<or> x0 \<notin> X)"
singleton: fixes a :: "nat" shows "\<not> (\<forall>\<^sub>F x in \<U>. x = a)"
distinct_3: fixes x1 :: "nat" shows "Label x1 \<noteq> Exit"