hcju/isabelleps huggingface dataset:
skeleton_code_def: fixes node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'd) gen_g_impl_scheme" shows "skeleton_code node_hash_impl node_def_hash_size node_eq_impl G_impl \<equiv> stat_start_nres \<bind> (\<lambda>x. let y' = Impl_Array_Hash_Map.ahm_empty node_def_hash_size in LIST_FOREACH' (RETURN (id (gi_V0 G_impl))) (\<lambda>xa. True) (\<lambda>xa xb. if \<not> (case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xa xb of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then False else True) then let y'a = (as_singleton xa, as_singleton 0, Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl xa (int 0) xb, if is_Nil (gi_E G_impl xa) then as_empty () else as_singleton (0, gi_E G_impl xa)) in WHILE\<^sub>T (\<lambda>xc. \<not> as_is_empty (case xc of (xd, xe, xf, xg) \<Rightarrow> xd)) (\<lambda>xc. select_edge_code node_eq_impl xc \<bind> (\<lambda>(xd, xe). case xd of None \<Rightarrow> pop_code node_eq_impl node_hash_impl xe | Some xf \<Rightarrow> if case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xf (case xe of (xg, xh, xi, xj) \<Rightarrow> xi) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then True else False then collapse_code node_eq_impl node_hash_impl xf xe else if \<not> (case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xf (case xe of (xg, xh, xi, xj) \<Rightarrow> xi) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then False else True) then RETURN (push_code node_eq_impl node_hash_impl G_impl xf xe) else RETURN xe)) y'a \<bind> (\<lambda>(xc, xd, xe, xf). RETURN xe) else RETURN xb) y' \<bind> (\<lambda>xa. stat_stop_nres \<bind> (\<lambda>xb. RETURN xa)))"
wf_graph_remove_edges: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E - X\<rparr>"
inv_comp_left_2: fixes g :: "'d"   and f :: "'d" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso g" shows "D.iso f"
assms: shows "false\<^sub>r \<notin> old (fst n_ns) \<and> (\<forall>nd\<in>snd n_ns. false\<^sub>r \<notin> old nd)"
equivp_aeseq_eq: shows "equivp aeseq_eq"
open_delete: fixes s :: "'b set"   and x :: "'b" assumes "open s" shows "open (s - {x})"
fr_nGet2a: fixes nReturn :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and k :: "'b"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "'b \<Rightarrow> Config"   and nStore :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "'b \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat" assumes "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)"   and "\<not> t mod cycleLength (nC k) mem schedule (nC k)" shows "nGet k t = []"
countable_atoms: shows "countable {x. 0 < M2.prob {x}}"
cases: fixes x :: "'a factor_tree" obtains i :: "'a" and xa :: "int poly" where "x = Factor_Leaf i__ xa"   | i :: "'a" and l :: "'a factor_tree" and r :: "'a factor_tree" where "x = Factor_Node i l r"
neg_mask_add_mask: fixes x :: "'b word"   and n :: "nat" shows "(x AND NOT (mask n)) + (2 ^ n - 1) = x OR mask n"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b" assumes "category C"   and "binary_product_diagram_axioms C a0 a1" shows "binary_product_diagram C a0 a1"
WT_gpv_full: fixes gpv :: "('c, 'a, 'b) gpv" shows "\<I>_full \<turnstile>g gpv \<surd>"
expandTauFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C" obtains \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P' :: "'b" where "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>" and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'" and "A\<^sub>P' \<sharp>* C" and "A\<^sub>P' \<sharp>* P'" and "distinct A\<^sub>P'"
fr1: shows "(q \<bullet> (as \<inter> supp x)) \<sharp>* (x, c, f (as \<inter> supp x) x c, f (bs \<inter> supp y) y c)"
enum_idx_correct: fixes i :: "'b" shows "enum_class.enum ! enum_idx i = i"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<sim>\<^sub>l P"
att_and_empty: fixes s' :: "'a set"   and s'' :: "'a set" assumes "\<turnstile>[] \<oplus>\<^sub>\<and>\<^bsup>(s', s'')\<^esup>" shows "s' \<subseteq> s''"
assms_2: shows "B\<^sub>2 \<subseteq> S"
row_copy_first_matrix: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat" assumes "i < nr" shows "row (copy_first_matrix nr nc) i = unit_vec nc 0"
1: shows "(\<epsilon> (C.cod f) \<cdot>\<^sub>C F (G f)) \<cdot>\<^sub>C C.inv (\<epsilon> (C.dom f)) = (f \<cdot>\<^sub>C \<epsilon> (C.dom f)) \<cdot>\<^sub>C C.inv (\<epsilon> (C.dom f))"
dom_char: shows "C'.dom = (\<lambda>i. if i \<in> S then \<psi> (local.dom (\<phi> i)) else n)"
hom_component_monom_mult: fixes c :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "MPoly_Type_Class.punit.monom_mult c t (hom_component p n) = hom_component (MPoly_Type_Class.punit.monom_mult c t p) (deg_pm t + n)"
nat_mirror_less: fixes a :: "nat"   and b :: "nat"   and l :: "nat"   and r :: "nat" assumes "a < b"   and "a < l + r" shows "nat_mirror b l r < nat_mirror a l r"
simps_1: shows "product_lists \<epsilon> = [\<epsilon>]"
prod_into_DSTEP1: fixes S :: "'c"   and A :: "'c"   and B :: "'c"   and G :: "('c \<times> ('c, 'd) RHS) list"   and L :: "('c + 'd) list"   and R :: "('c + 'd) list" assumes "(S, Branch A B) \<in> set G" shows "L \<cdot> [Inl S] \<cdot> R -G\<rightarrow> L \<cdot> [Inl A, Inl B] \<cdot> R"
hyps_3: fixes ss :: "dB list" assumes "ys => ss" shows "xs [\<rightarrow>\<^sub>s] ss"
is_local_def: fixes U :: "'a set" shows "stfx.is_local U \<equiv> local_ring stfx.carrier_stalk stfx.add_stalk stfx.mult_stalk (stfx.zero_stalk U) (stfx.one_stalk U)"
Domainp_forall_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A" shows "((A ===> (=)) ===> (=)) (transfer_bforall (Domainp A)) transfer_forall"
ih: fixes C :: "'a literal list"   and C' :: "'a literal list" assumes "mset C = mset C'" shows "reduce Ds C E = reduce Ds C' E"
well_order_consistent: fixes i :: "example_peers \<times> nat"   and j :: "example_peers \<times> nat" assumes "example.happened_immediately_before i j" shows "witness_acyclic_events i < witness_acyclic_events j"
red_external_hext: fixes t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>" shows "h \<unlhd> h'"
distinct_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
reflP_def: fixes P :: "'b \<Rightarrow> bool"   and \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "reflP P \<phi> \<equiv> (\<forall>x y. \<phi> x y \<or> \<phi> y x \<longrightarrow> P x) \<and> (\<forall>x. P x \<longrightarrow> \<phi> x x)"
assms: shows "x \<in> results_gpv \<I> (restrict_gpv \<I>' gpv)"
comp_null_1: fixes f :: "'a" shows "R.comp R.null f = R.null"
simps_1: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and f :: "'d"   and F :: "'d fs_set"   and P :: "'a"   and \<alpha>P' :: "(('c, 'd) L_action, ('a, 'd) L_state) residual" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.L_transition transition effect_apply L (AC (f, F, P)) \<alpha>P' = (\<exists>\<alpha> P'. transition P \<langle>\<alpha>,P'\<rangle> \<and> \<alpha>P' = \<langle>L_action.Act \<alpha>,EF (L (\<alpha>, F, f), P')\<rangle> \<and> bn \<alpha> \<sharp>* (F, f))"
comp: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and h :: "'c \<Rightarrow> 'g"   and i :: "'d \<Rightarrow> 'h" shows "map_prod f g \<circ> map_prod h i = map_prod (f \<circ> h) (g \<circ> i)"
integral_ln_powr_over_x: fixes y :: "real"   and a :: "real"   and b :: "real" assumes "y \<noteq> - 1"   and "a \<le> b"   and "1 < a" shows "integral {a..b} (\<lambda>x. ln x powr y / x) = inverse (y + 1) * (ln b powr (y + 1) - ln a powr (y + 1))"
DERIV_g: fixes z :: "real" assumes "a < z"   and "z < b" shows "(g has_real_derivative g' z) (at z)"
rep_eq: fixes x :: "('c, 'd) lifted_pair"   and xa :: "('c, 'd) lifted_pair" shows "Rep_lifted_pair (x \<sqinter> xa) = (\<lambda>f. triple.pairs_inf (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
index_swaps_size: fixes s :: "'b list"   and q :: "'b"   and sws :: "nat list" assumes "distinct s" shows "index s q \<le> index (swaps sws s) q + length sws"
wffp_imp_appendR: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl1 :: "('a, 'b, 'c) state list"   and sl2 :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do (sl1 @ sl2)"   and "sl2 \<noteq> []" shows "GM_sec_model.wffp do sl2"
rev_ordsetmap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "a \<in> P"   and "b \<in> P"   and "less_eq' (f a) (f b)" shows "less_eq a b"
iso_char: fixes f :: "'b" shows "hhom\<^sub>D.iso f = (D.iso f \<and> hhom\<^sub>D.arr f \<and> hhom\<^sub>D.arr (D.inv f))"
bot_meet: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<sqinter>\<^bsub>L\<^esub> x = \<bottom>\<^bsub>L\<^esub>"
Skip_eq: fixes G :: "prog"   and s :: "state"   and s' :: "state" shows "G\<turnstile>s \<midarrow>Skip\<rightarrow> s' = (s = s')"
assms_2: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
user_reg_val_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and win :: "'b word"   and ur :: "word5" assumes "low_equal s1 s2" shows "user_reg_val win ur s1 = user_reg_val win ur s2"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a) Term.term" assumes "e \<in> fv a" obtains "a = Var e"   | z1 :: "'b" and z2 :: "('b, 'a) Term.term list" and x :: "('b, 'a) Term.term" where "a = Fun z1 z2" and "x \<in> set z2" and "e \<in> fv x"
MGF_lemma: fixes A :: "triple set"   and c :: "stmt"   and e :: "expr" assumes "\<forall>M Z. A |\<turnstile> {MGT (Impl M) Z}" shows "(\<forall>Z. A |\<turnstile> {MGT c Z}) \<and> (\<forall>Z. A |\<turnstile>\<^sub>e MGT\<^sub>e e Z)"
path_length_is_upper_bound: fixes p :: "(symbol \<times> character list) list"   and \<alpha> :: "symbol list"   and u :: "symbol list" assumes "wellformed_tokens p"   and "is_word \<alpha>"   and "derives (\<alpha> @ u) (terminals p)" shows "length \<alpha> \<le> length p"
R: fixes x :: "'b"   and \<sigma> :: "'a" shows "nres_of (fi x \<sigma>) \<le> f x \<sigma>"
is_derivation_derives: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "derives \<alpha> \<beta>"   and "is_derivation (u @ \<alpha> @ v)" shows "is_derivation (u @ \<beta> @ v)"
dg_is_arrD_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
rev_sub_mult: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "h \<in> carrier P" shows "rev_compose R g (f \<otimes>\<^bsub>P\<^esub> h) = rev_compose R g f \<otimes>\<^bsub>P\<^esub> rev_compose R g h"
induct: fixes P :: "'a list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "nat" assumes "\<And>n. P [] n"   and "\<And>x xs. P (x # xs) 0"   and "\<And>x xs n. P xs n \<Longrightarrow> P (x # xs) (Suc n)" shows "P a0 a1"
hyps: fixes ys :: "'a list"   and x :: "'a"   and thesis :: "bool" assumes "\<And>y. \<lbrakk>y \<in> set (local.filter_min_aux xs ys); rel y x\<rbrakk> \<Longrightarrow> thesis"   and "x \<in> set xs"   and "x \<notin> set (local.filter_min_aux xs ys)" shows "thesis"
D_def: fixes P :: "'c process" shows "D P = DIVERGENCES (Rep_process P)"
case_prodI2: fixes p :: "'k \<times> 'l"   and c :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
inj_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f A"   and "order.greater_eq A B" shows "inj_on f B"
Quotient3_interface: shows "Quotient3 (\<simeq>) abs_interface rep_interface"
HComplex_mult_hcomplex_of_hypreal: fixes x :: "real star"   and y :: "real star"   and r :: "real star" shows "HComplex x y * hcomplex_of_hypreal r = HComplex (x * r) (y * r)"
methdMembr_simp: fixes c :: "qtname"   and m :: "mdecl" shows "methdMembr (c, m) = (c, mdecl m)"
nat_pow_Suc: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
PrfP_inference: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and \<alpha>1 :: "tm"   and \<alpha>2 :: "tm"   and \<beta> :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, \<alpha>1, \<alpha>2, \<beta>)"   and "atom k \<sharp> (s1, s2, k1, k2, \<alpha>1, \<alpha>2, \<beta>)" shows "{PrfP s1 k1 \<alpha>1, PrfP s2 k2 \<alpha>2, ModPonP \<alpha>1 \<alpha>2 \<beta> OR ExistsP \<alpha>1 \<beta> OR SubstP \<alpha>1 \<beta>} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (PrfP (Var s) (Var k) \<beta>))"
find_remove_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "find_remove P xs = (case find P xs of None \<Rightarrow> (None, xs) | Some x \<Rightarrow> (Some x, remove1 x xs))"
is_ddi_def: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and v :: "'var"   and m :: "nat" shows "(n dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m) = (is_path \<pi> \<and> m < n \<and> v \<in> local.reads (\<pi> n) \<inter> local.writes (\<pi> m) \<and> (\<forall>l\<in>{m<..<n}. v \<notin> local.writes (\<pi> l)))"
WT_callee: fixes s :: "'b" shows "\<I> \<turnstile>c callee s \<surd>"
refl_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
arr_dom_iff_arr: fixes f :: "'a arrow_of_spans_data" shows "V.arr (V.dom f) = V.arr f"
Pluss_Some: fixes r :: "'c rexp"   and xs :: "'c rexp list" shows "(Pluss r = Some xs) = (Full \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_summands r - {Zero}))"
sw_def: shows "sw = (\<lambda>A xs. fold (\<lambda>(i, j). swaprows i j) xs A)"
not_0_cCons_eq: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "a ## coeffs p = a # coeffs p"
delete_notin_dom: fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "k \<notin> fst ` set (AList.delete k al)"
iAll_False: fixes I :: "nat set" shows "(\<box> t I. False) = (I = {})"
simps_9: fixes bs :: "real list"   and c :: "int"   and a :: "MIR.num"   and b :: "MIR.num" shows "Inum bs (CF c a b) = real_of_int c * real_of_int \<lfloor>Inum bs a\<rfloor> + Inum bs b"
wtSB_tpOf: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and \<pi> :: "Preliminaries.var \<Rightarrow> 'f trm"   and x :: "Preliminaries.var" assumes "Signature TYPE('e) wtFsym wtPsym"   and "Signature.wtSB wtFsym arOf resOf \<pi>" shows "Signature.tpOf resOf (\<pi> x) = Signature.tpOfV x"
cases: fixes a1 :: "'v node"   and a2 :: "'v node" assumes "edge' a1 a2" obtains "a1 = SOURCE" and "a2 = Inner (source \<Delta>)"   | x :: "'v" and y :: "'v" where "a1 = Inner x" and "a2 = Inner y" and "edge \<Delta> x y"
T_vrestr_composed_1: fixes A :: "'a set"   and n :: "nat"   and m :: "nat" assumes "A \<in> sets M" shows "(T ^^ n) --` (T ^^ m) --` A = (T ^^ (n + m)) --` A"
assms_4: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "g \<in> Y" shows "g e \<le> f e"
inL4': fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and I :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, G, H, I, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "H, I, F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
fresh_fun_upd: fixes a :: "atom"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f(x := y)"
Lower_memD: fixes l :: "'a"   and L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and x :: "'a" assumes "l \<in> Lower L A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "l \<sqsubseteq>\<^bsub>L\<^esub> x \<and> l \<in> carrier L"
intsquare: shows "((\<lambda>x. (1 - x)\<^sup>2) has_integral 1 ^ 3 / 3 - 1\<^sup>2 + 1 - (0 ^ 3 / 3 - 0\<^sup>2 + 0)) (cbox 0 1)"
dvd_abs_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and m :: "'d"   and k :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "dvd.dvd times m (abs k) = dvd.dvd times m k"
eSuc_pred_enat: fixes n :: "enat" assumes "enat 0 < n" shows "eSuc (n - eSuc (enat 0)) = n"
i_UN_set_min_def: shows "i_UN_set_min \<equiv> iFROM_UN_set \<union> iIN_UN_set \<union> iMOD2_UN_set \<union> iMODb2_UN_set"
assms: shows "\<lless>\<pi>\<^sub>1, eV\<ggreater> V\<rightarrow> \<lless>\<pi>\<^sub>2, eV'\<ggreater>"
unch_def: fixes v :: "'b \<Rightarrow> 'c" shows "Unchanged v \<equiv> v$ = $v"
prob_Collect_eq_1: fixes s :: "'s"   and P :: "'s stream \<Rightarrow> bool" assumes "{x \<in> space (T s). P x} \<in> T.events s" shows "(T.prob s {x \<in> space (T s). P x} = 1) = almost_everywhere (T s) P"
gen_degen_step: fixes cs :: "('b \<Rightarrow> bool) list"   and w :: "'b stream"   and k :: "nat" assumes "gen infs cs w" obtains u :: "'b list" and a :: "'b" and v :: "'b stream" where "w = u @- a ## v" and "degen cs (a, fold (count cs) u k)"
simps_3: fixes f :: "'a"   and u :: "'a local.term" shows "Diag (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u) = (C.arr f \<and> Diag u \<and> u \<noteq> \<^bold>\<I>)"
fg: shows "f = g \<or> (\<exists>a b. (a, b) \<in> c \<and> f = Abs a \<and> g = Abs b)"
cfun_eqvtI: fixes p :: "perm"   and f :: "'a \<rightarrow> 'b"   and f' :: "'a \<rightarrow> 'b" assumes "\<And>x. p \<bullet> f\<cdot>x = f'\<cdot>(p \<bullet> x)" shows "p \<bullet> f = f'"
def_scheme: fixes x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "f x \<equiv> \<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>"
path_split_single2: fixes \<beta> :: "'e list"   and x :: "'e"   and s :: "'s" assumes "s0\<^bsub>SES\<^esub> (\<beta> @ [x])\<Longrightarrow>\<^bsub>SES\<^esub> s" shows "\<exists>s'\<in>S\<^bsub>SES\<^esub>. s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' x\<longrightarrow>\<^bsub>SES\<^esub> s \<and> reachable SES s'"
emeasure_embed_measure_image': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and X :: "'c set" assumes "inj_on f (space M)"   and "X \<in> sets M" shows "emeasure (embed_measure M f) (f ` X) = emeasure M X"
SC_sound_complete: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "\<Gamma> \<Rightarrow> \<Delta> = \<Turnstile> \<Gamma> \<Rightarrow> \<Delta>"
cf_h_len2: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S c" shows "fst (cf_h f c) = fst c"
ocircline_set_opposite_ocircline: fixes H :: "ocircline" shows "ocircline_set (opposite_ocircline H) = ocircline_set H"
cases: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com \<times> 'b"   and a2 :: "'b" assumes "PL.MtransT tval aval a1 a2" obtains cf :: "('a, 'c) com \<times> 'b" and cf' :: "('a, 'c) com \<times> 'b" and s'' :: "'b" where "a1 = cf" and "a2 = s''" and "PL.MtransC tval aval cf cf'" and "PL.transT tval aval cf' s''"
UN_E: fixes b :: "'e"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'f" where "x \<in> A" and "b \<in> B x"
hyps: assumes "\<And>v. v \<in> set V \<Longrightarrow> f v \<le> g v" shows "ListSum V f \<le> ListSum V g"
inverse_le_imp_le: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less zero a" shows "less_eq b a"
eventually_isFDERIV: fixes fas :: "floatarith list"   and x :: "'a"   and params :: "real list" assumes "isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x @ params)"   and "max_Var_floatariths fas \<le> DIM('a) + length params" shows "\<forall>\<^sub>F x in at x. isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x @ params)"