hcju/isabelleps huggingface dataset:
skl1_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "skl1_step4 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> (can_signal s A B \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xgnx = Some gnx \<and> sklvl1.guessed_frame Ra xni = Some Ni \<and> sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)) \<and> sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))))) \<and> (Rb = test \<longrightarrow> Exp gnx (NonceF (Rb $ ny)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp2 s\<rparr>}"
len: shows "n + max_varss (Val v # es') \<le> length xs"
cSup_least: 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 X :: "'a set"   and z :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "less_eq (Sup X) z"
irreducible_imp_maximalideal: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c" assumes "principal_domain R"   and "p \<in> carrier R"   and "ring_irreducible\<^bsub>R\<^esub> p" shows "maximalideal (PIdl\<^bsub>R\<^esub> p) R"
if_wasm: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es1 :: "b_e list"   and es2 :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es1 : tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es2 : tn _> tm" shows "\<C> \<turnstile> [b_e.If tf es1 es2] : tn @ [T_i32] _> tm"
subst_typ_irrelevant_order: fixes pairs :: "((variable \<times> String.literal set) \<times> typ) list"   and pairs' :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" assumes "distinct (map fst pairs)"   and "distinct (map fst pairs')"   and "set pairs = set pairs'" shows "subst_typ pairs T = subst_typ pairs' T"
dependent_live_vars_def: shows "dependent_live_vars \<equiv> \<lambda>n'. {(xb, xa, x). dependent_live_varsp n' xb xa x}"
real_is_int_def2: fixes x :: "real" shows "real_is_int x = (x = real_of_int (int_of_real x))"
try_gpv_Fail2: fixes gpv :: "('d, 'e, 'f) gpv" shows "TRY gpv ELSE Fail = gpv"
wgraph_from_list_algo_axioms: shows "wgraph_from_list_algo \<alpha> invar succ empty_graph add_edge"
map_fun_apply: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'b"   and h :: "'d \<Rightarrow> 'e"   and x :: "'c" shows "map_fun f g h x = g (h (f x))"
finsum_reindex: fixes f :: "'e \<Rightarrow> 'a"   and h :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "f \<in> h ` A \<rightarrow> carrier R"   and "inj_on h A" shows "finsum R f (h ` A) = (\<Oplus>x\<in>A. f (h x))"
SUP1_I: fixes a :: "'b"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" assumes "a \<in> A"   and "B a b" shows "Sup (B ` A) b"
cases: fixes a :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "a \<in> oreachable A S U" obtains s :: "'a \<times> 'b" where "a = s__" and "s__ \<in> init A"   | s :: "'a \<times> 'b" and aa :: "'c" and s' :: "'a \<times> 'b" where "a = s'" and "sa__ \<in> oreachable A S U" and "(sa__, aa, s') \<in> automaton.trans A" and "S (fst sa__) (fst s') aa"   | s :: "'a \<times> 'b" and \<sigma>' :: "'a" where "a = (\<sigma>', snd s)" and "s \<in> oreachable A S U" and "U (fst s) \<sigma>'"
cexpr_sem_integrate_var: fixes \<sigma> :: "nat \<Rightarrow> val"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "nat"   and e :: "cexpr" shows "cexpr_sem \<sigma> (integrate_var \<Gamma> v e) = RealVal (LINT x|stock_measure (\<Gamma> v). extract_real (cexpr_sem (\<sigma>(v := x)) e))"
sum_clauses_2: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite S" shows "sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
that: assumes "a \<^bold>\<in> A"   and "b \<^bold>\<in> B a" shows "thesis"
inverse_transformations_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" shows "inverse_transformations A B F G \<tau> \<tau>' \<equiv> (category A \<and> category B \<and> functor A B F) \<and> (functor A B G \<and> natural_transformation A B F G \<tau>) \<and> natural_transformation A B G F \<tau>' \<and> inverse_transformations_axioms A B \<tau> \<tau>'"
disjK_def: shows "disjK \<equiv> \<Lambda> g h s f. g\<cdot>s\<cdot>(h\<cdot>s\<cdot>f)"
fMax_ge: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" shows "x \<le> fMax A"
set_mset_single: fixes b :: "'d" shows "set_mset {#b#} = {b}"
universal: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "\<exists>!l. \<pp>\<^sub>1[cod f, cod g] \<cdot> l = f \<and> \<pp>\<^sub>0[cod f, cod g] \<cdot> l = g"
A_rw: shows "(\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) = (\<Prod>i = 0..<n. A $$ (i, f i))"
strict_evaluation_functor_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd" shows "strict_evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V \<equiv> (strict_monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D \<and> evaluation_map C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V) \<and> free_monoidal_category C \<and> FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V \<and> free_strict_monoidal_category C"
subspace_UNIV: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module scale" shows "module.subspace scale UNIV"
nat_minus_as_int: shows "(-) = (\<lambda>a b. nat (int a - int b))"
IH: assumes "ref = None \<or> ref = Some r \<and> r \<in> set list" shows "\<exists>xs ys. list = xs @ ys \<and> insert_spec list (oid, ref) = xs @ oid # ys"
nat_even_power_sums_complex: fixes n' :: "nat" assumes "0 < n'" shows "(\<lambda>k. 1 / of_nat (Suc k) ^ (2 * n')) sums complex_of_real ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))"
subset_memb_def: fixes qs :: "bool list" shows "subset_memb qs = (\<lambda>(bdd, qss). bdd_lookup bdd qs \<noteq> None)"
m_strictly_prefers_def: fixes m :: "M"   and \<mu> :: "(M \<times> W) set" shows "Non_Strict.m_strictly_prefers m \<mu> = {w' \<in> Field (Pm m). \<forall>w\<in>{w. (m, w) \<in> \<mu>}. (w, w') \<in> Pm m \<and> (w', w) \<notin> Pm m}"
inc_never_one: fixes x :: "nat" shows "inc x \<noteq> 1"
pair_measure_def: fixes A :: "'d measure"   and B :: "'e measure" shows "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B} (\<lambda>X. \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>B \<partial>A)"
opt_mru_guard_imp_mru_guard: fixes s :: "v_state"   and Q :: "process set"   and v :: "val" assumes "s \<in> Vinv1"   and "s \<in> SV_inv3"   and "opt_mru_guard (process_mru (votes s)) Q v" shows "mru_guard s Q v"
reducecoeff_numbound0: fixes t :: "MIR.num" assumes "numbound0 t" shows "numbound0 (reducecoeff t)"
finite_has_minimal: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq b m \<longrightarrow> m = b"
exp_inj_iff: fixes x :: "real"   and y :: "real" shows "(exp x = exp y) = (x = y)"
simps_1: fixes A :: "'c \<Rightarrow> 's"   and uu :: "'c \<Rightarrow> 's"   and uv :: "'c \<Rightarrow> 's" shows "Cop_S.Dom (Cop_S.MkArr A uu uv) = A"
inner_verts_def: fixes p :: "'b list" shows "inner_verts p \<equiv> tl (map (tail G) p)"
elims_2: fixes x :: "'b formula" assumes "is_cnf x" obtains F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<and> G" and "is_cnf F \<and> is_cnf G"   | v :: "'b" where "x = Atom v__" and "is_disj (Atom v__)"   | "x = \<bottom>" and "is_disj \<bottom>"   | v :: "'b formula" where "x = \<^bold>\<not> va__" and "is_disj (\<^bold>\<not> va__)"   | v :: "'b formula" and va :: "'b formula" where "x = vb__ \<^bold>\<or> vaa__" and "is_disj (vb__ \<^bold>\<or> vaa__)"   | v :: "'b formula" and va :: "'b formula" where "x = v \<^bold>\<rightarrow> va" and "is_disj (v \<^bold>\<rightarrow> va)"
elims_1: fixes x :: "'c btree"   and y :: "bool" assumes "nonempty_lasttreebal x = y" obtains "x = Leaf" and "y"   | ts :: "('c btree \<times> 'c) list" and t :: "'c btree" where "x = Node ts t" and "y = ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
suffix_singleton_suffix: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "w i ## w [Suc i..] = w [i..]"
res: shows "local.tighten_poly_bounds_for_x l r sr = (l', r', sr')"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>)). \<CC>\<lparr>Dom\<rparr>\<lparr>a\<rparr> = b"
le_fininfI: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v\<^sub>2 :: "'a list"   and w\<^sub>2 :: "'a stream"   and w\<^sub>1 :: "'a list" assumes "traces ind"   and "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "traces.le_fin ind w\<^sub>1 v\<^sub>2" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>2"
that: fixes X1 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and X2 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set" assumes "X0 = X1 \<union> X2"   and "flowsto X1 {0<..t} (CX \<inter> {x. x \<bullet> n < c} \<times> UNIV) (CX \<inter> {x \<in> R. x \<bullet> n = c} \<times> UNIV)"   and "flowsto X2 {hl..t} (CX \<inter> {x. x \<bullet> n < c} \<times> UNIV) (Y \<inter> {x. x \<bullet> n < c} \<times> UNIV)"   and "poincare_mapsto {x \<in> R. x \<bullet> n = c} X1 UNIV (fst ` CX \<inter> {x. x \<bullet> n < c}) PDP" shows "thesis"
midpoint_sym: fixes a :: "'c"   and b :: "'c" shows "midpoint a b = midpoint b a"
label: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and k :: "nat"   and lholed :: "Lholed"   and les :: "e list"   and les' :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "Lfilled k lholed es les"   and "Lfilled k lholed es' les'" shows "\<lparr>s;vs;les\<rparr> \<leadsto>_ i \<lparr>s';vs';les'\<rparr>"
grid: fixes i :: "nat" shows "t i \<le> t (Suc i)"
sum_list_list_update_Suc: fixes i :: "nat"   and u :: "nat list" assumes "i < length u" shows "sum_list (u[i := Suc (u ! i)]) = Suc (sum_list u)"
has_fps_expansion_divide: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and g :: "'a \<Rightarrow> 'a"   and G :: "'a fps"   and c :: "'a" assumes "f has_fps_expansion F"   and "g has_fps_expansion G"   and "subdegree G \<le> subdegree F"   and "G \<noteq> 0"   and "c = fps_nth F (subdegree G) / fps_nth G (subdegree G)" shows "(\<lambda>x. if x = (0::'a) then c else f x / g x) has_fps_expansion F / G"
\<sigma>_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sym :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<times> 'b" assumes "elementary_symmetric_monoidal_category C tensor unity lunit runit assoc sym" shows "elementary_symmetric_monoidal_category.\<sigma> C tensor sym f \<equiv> if partial_magma.arr (product_category.comp C C) f then C (sym (partial_magma.cod C (fst f)) (partial_magma.cod C (snd f))) (tensor (fst f) (snd f)) else partial_magma.null C"
diff_union_swap2: fixes y :: "'k"   and M :: "'k multiset"   and x :: "'k" assumes "y \<in># M" shows "add_mset x M - {#y#} = add_mset x (M - {#y#})"
induced_modulus_dvd: fixes d :: "nat" assumes "\<Phi>.induced_modulus d" shows "d dvd d__"
append_Cons: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "(x # xs) @ ys = x # xs @ ys"
alpha_Tree_eqvt': fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
bin_last_numeral_simps_5: fixes w :: "num" shows "\<not> odd (numeral (num.Bit0 w))"
normalize_unit_factor: 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 a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero" shows "normalize (unit_factor a) = one"
1: shows "x mod p ^ k = ((x - y) mod p ^ k + y) mod p ^ k"
wf_graph_remove_edges_union: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
equivalence_respects_iso: fixes f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b"   and \<phi> :: "'b"   and f' :: "'b"   and \<psi> :: "'b"   and g' :: "'b" assumes "D.internal_equivalence f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : f \<Rightarrow>\<^sub>D f'\<guillemotright>"   and "D.iso \<phi>"   and "\<guillemotleft>\<psi> : g \<Rightarrow>\<^sub>D g'\<guillemotright>"   and "D.iso \<psi>" shows "D.internal_equivalence f' g' ((g' \<star>\<^sub>D \<phi>) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D f) \<cdot>\<^sub>D \<eta>) (\<epsilon> \<cdot>\<^sub>D (D.inv \<phi> \<star>\<^sub>D g) \<cdot>\<^sub>D (f' \<star>\<^sub>D D.inv \<psi>))"
P: shows "P \<in> subgroups_of_size (p ^ a)"
aprogress: fixes \<Gamma> :: "char list \<Rightarrow> ty"   and a :: "aexp"   and \<tau> :: "ty"   and s :: "char list \<Rightarrow> val" assumes "\<Gamma> \<turnstile> a : \<tau>"   and "\<Gamma> \<turnstile> s" shows "\<exists>v. taval a s v"
sel_1: fixes x1 :: "'a" shows "atm_of (Pos x1) = x1"
Constrains_eq_constrains: fixes A :: "'c set"   and B :: "'c set" shows "A Co B = {F. F \<in> reachable F \<inter> A co reachable F \<inter> B}"
prop2: fixes a :: "letter"   and b :: "letter"   and xs :: "letter list list"   and ys :: "letter list list"   and zs :: "letter list list" assumes "a \<noteq> b"   and "bar xs"   and "bar ys"   and "T a xs zs"   and "T b ys zs" shows "bar zs"
idinfo_sep: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and n :: "nat"   and Z :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "s' = insert (Spy, X) (insert (Spy, Y) s) \<and> (Spy, \<lbrace>X, Y\<rbrace>) \<in> s"   and "\<langle>n, Z\<rangle> = X \<or> \<langle>n, Z\<rangle> = Y" shows "\<langle>n, Z\<rangle> \<in> spied s"
tau_refl: fixes P :: "'state" shows "P \<Rightarrow> P"
thetaParW_Wretr: shows "thetaParW \<subseteq> Wretr (thetaParW \<union> Wbis)"
sint_signed_drop_bit_eq: fixes n :: "nat"   and w :: "'b word" shows "sint (signed_drop_bit n w) = drop_bit n (sint w)"
rep_eq: fixes x :: "('f, 'e) generator"   and xa :: "'f" shows "generator (list_update_trans x xa) = list_update_raw (generator x) xa"
le_Inf_iff: fixes b :: "'b"   and A :: "'b set" shows "(b \<le> \<Sqinter>A) = (\<forall>a\<in>A. b \<le> a)"
NormalAction: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x' :: "'x2"   and m' :: "'m2" assumes "r2.r_syntax t x m ta x' m'" shows "r2.init_fin t ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
leg_simps_3: shows "C.arr \<mu>\<nu>.leg1"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>a as b bs accs. \<lbrakk>a < b \<Longrightarrow> P as (b # bs) (accs @ [a]); \<lbrakk>\<not> a < b; a = b\<rbrakk> \<Longrightarrow> P as bs (accs @ [a]); \<lbrakk>\<not> a < b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a # as) bs (accs @ [b])\<rbrakk> \<Longrightarrow> P (a # as) (b # bs) accs"   and "\<And>bs accs. P [] bs accs"   and "\<And>v va accs. P (v # va) [] accs" shows "P a0 a1 a2"
HARoot_ran_CompFun2: fixes S :: "('a, 'b, 'c) seqauto set"   and HA :: "('a, 'b, 'c) hierauto" shows "S \<in> ran (CompFun HA) \<longrightarrow> HARoot HA \<notin> S"
extract_splitres_def: fixes r :: "('a, 'b) splitres" shows "extract_splitres r \<equiv> (extract_splitres_l r, extract_splitres_a r, extract_splitres_r r)"
outside_reduces_domain: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "Domain (Outside' X P) = Domain P - X"
fid_on_small: fixes A :: "V" shows "small {[a, a]\<^sub>\<circ> |a. a \<in>\<^sub>\<circ> A}"
ennreal_diff_add_assoc: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<le> b" shows "c + b - a = c + (b - a)"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "B.ide b"   and "B.seq b f" shows "b \<cdot>\<^sub>B f = f"
row_conjugate: fixes i :: "nat"   and m :: "'b mat" assumes "i < dim_row m" shows "row (conjugate m) i = conjugate (row m i)"
c: shows "set_mset K \<subseteq> r \<down>m J"
5: shows "r = (\<lambda>u s w. \<exists>x. \<nu>\<upsilon> x = u \<and> Some x = Some (\<omega>\<nu> o\<^sub>1))"
lessThan_Suc: fixes k :: "nat" shows "{..<Suc k} = insert k {..<k}"
document: shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r document"
N1_2: shows "\<forall>n'\<ge>n1. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg"
SumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<equiv>\<^sub>e P \<oplus> Q \<oplus> R"
fref_cons: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set"   and R' :: "('a \<times> 'c) set"   and Q :: "'c \<Rightarrow> bool"   and S' :: "('b \<times> 'd) set" assumes "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"   and "\<And>c a. \<lbrakk>(c, a) \<in> R'; Q a\<rbrakk> \<Longrightarrow> P a"   and "R' \<subseteq> R"   and "S \<subseteq> S'" shows "(f, g) \<in> [Q]\<^sub>f R' \<rightarrow> S'"
def_fold: fixes l :: "('a state \<times> 'a state \<Rightarrow> 'c) list" assumes "list_all (\<lambda>f. \<tau> \<Turnstile> \<upsilon> f) l" shows "\<tau> \<Turnstile> \<delta> foldl UML_Set.OclIncluding Set{} l"
cxt_inv_assign: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and E :: "('a, 'c, 'd) Stmt list"   and c :: "('a, 'c, 'd) Stmt"   and x :: "'a"   and e :: "'c" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "sifum_lang_no_dma.cxt_to_stmt E c = x \<leftarrow> e" shows "c = x \<leftarrow> e \<and> E = []"
dghm_ArrMap_app_in_HomCod_Arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
prognosis_Var_thunk: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and u :: "Arity"   and as :: "Arity list"   and S :: "stack_elem list"   and a :: "Arity" assumes "CardinalityPrognosisVar prognosis"   and "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u"   and "\<not> isVal e" shows "prognosis ae as u (delete x \<Gamma>, e, Upd x # S) \<sqsubseteq> record_call x\<cdot>(prognosis ae as a (\<Gamma>, Var x, S))"
cosafe_lpd: fixes r :: "'a regex"   and s :: "'a regex"   and test :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'d" assumes "local.safe_regex Futu Lax r"   and "s \<in> lpd test i r" shows "local.safe_regex Futu Lax s"
add_diff_cancel_right': fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
eq_vector_fraction_iff: fixes x :: "'d"   and u :: "real"   and v :: "real"   and a :: "'d" shows "(x = (u / v) *\<^sub>R a) = (if v = 0 then x = (0::'d) else v *\<^sub>R x = u *\<^sub>R a)"
IF_PARALLEL_tp: fixes G :: "Var \<Rightarrow> L"   and b :: "BExpr"   and p :: "L"   and c1 :: "IMP"   and H :: "Var \<Rightarrow> L"   and c2 :: "IMP"   and q :: "L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(G, b, p) \<in> HS_B"   and "(p, G, c1, H) \<in> HS"   and "(p, G, c2, H) \<in> HS"   and "\<exists>x. LEQ p (H x) \<and> LEQ (H x) q"   and "X \<rhd> c1 : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>"   and "X \<rhd> c2 : Sec (Q p H) (EQ G q) (EQ H q) \<Psi>" shows "X \<rhd> Iff b c1 c2 : Sec (Q p H) (EQ G q) (EQ H q) (\<lambda>(r, u). (evalB b u \<longrightarrow> \<Phi> (r, u)) \<and> (\<not> evalB b u \<longrightarrow> \<Psi> (r, u)))"
nat_pow_Suc2: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier P" shows "[Suc n] \<cdot>\<^bsub>P\<^esub> x = x \<oplus>\<^bsub>P\<^esub> [n] \<cdot>\<^bsub>P\<^esub> x"
r_enumdom_empty_domain: fixes i :: "nat"   and y :: "nat" assumes "\<And>x. eval r_phi [i, x] \<up>" shows "eval r_enumdom [i, y] \<up>"
finter_funion_distrib: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |\<inter>| (B |\<union>| C) = A |\<inter>| B |\<union>| (A |\<inter>| C)"
distributed_transform_integral: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real"   and P :: "'f measure"   and Y :: "'d \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and T :: "'e \<Rightarrow> 'f"   and f :: "'f \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> Px x"   and "distributed M P Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space P \<Longrightarrow> 0 \<le> Py x"   and "Y = T \<circ> X"   and "T \<in> N \<rightarrow>\<^sub>M P"   and "f \<in> borel_measurable P" shows "LINT x|P. Py x * f x = LINT x|N. Px x * f (T x)"
thetaParW_Wbis: shows "thetaParW \<subseteq> Wbis"
1: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> y = y)"