hcju/isabelleps huggingface dataset:
tendsto_Im7: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im7 (g x)) \<longlongrightarrow> Im7 a) F"
one_aux: fixes f :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f" shows "f (Suc 0) = (1::'b)"
cont: shows "\<forall>\<^sub>F n in F. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
x_1: shows "x \<in> subst_domain (\<theta>1 \<circ>\<^sub>s \<theta>2)"
stc_part_Ex: fixes x :: "complex star" assumes "x \<in> HFinite" shows "\<exists>t\<in>SComplex. x \<approx> t"
ipresIGVar_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'e, 'j, 'k) model_scheme" shows "ipresIGVar h termMOD MOD = ipresVar h MOD"
inv_gHom: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "gbij\<^bsub>F,G\<^esub> f" shows "f\<inverse>\<^bsub>carrier G,carrier F\<^esub> \<in> gHom G F"
RedTry: fixes v :: "val"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>try Val v catch(C V) e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
assms_1: shows "has_as_product J0 D0 a0"
\<Z>_def: fixes \<alpha> :: "V" shows "\<Z> \<alpha> \<equiv> Limit \<alpha> \<and> \<omega> \<in>\<^sub>\<circ> \<alpha>"
indep_var_eq: fixes S :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'a \<Rightarrow> 'c" shows "indep_var S X T Y = ((random_variable S X \<and> random_variable T Y) \<and> indep_set (sigma_sets (space M) {X -` A \<inter> space M |A. A \<in> sets S}) (sigma_sets (space M) {Y -` A \<inter> space M |A. A \<in> sets T}))"
uminus_carrier_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A \<in> carrier_mat nr nc"
nnfis_integral_2: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> nnfis f M"   and "measure_space M" shows "\<integral> f \<partial>M = a"
uminus_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Sup A) = Inf (uminus ` A)"
span_clauses_4: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and S :: "'c set"   and c :: "'b" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
sup2I1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "A x y" shows "sup A B x y"
eq: shows "pm_of_idx_pm fs (vectorize_poly r) = pm_of_idx_pm (take j fs) (vectorize_poly r)"
map_sel_1: fixes a :: "'b tree"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> Leaf" shows "l (map_tree f a) = l a"
ltake_lappend2: fixes xs :: "'b llist"   and n :: "enat"   and ys :: "'b llist" assumes "llength xs \<le> n" shows "ltake n (lappend xs ys) = lappend xs (ltake (n - llength xs) ys)"
refl: fixes x :: "'o"   and a :: "'a" shows "le x a a"
im_sheaf_def: fixes V :: "'a set" shows "local.im_sheaf V \<equiv> \<FF> (identity S \<^sup>\<inverse> S V)"
to_ordered_collection_det: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "to_ordered_collection \<tau> \<sigma>\<^sub>1"   and "to_ordered_collection \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
list_all2_eq: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs = ys) = list_all2 (=) xs ys"
field_differentiable_rGamma: fixes z :: "'a"   and A :: "'a set" shows "rGamma field_differentiable at z within A"
cos_aux_2: fixes x :: "float"   and n :: "nat"   and prec :: "nat" shows "(\<Sum>i = 0..<n. (- 1) ^ i * (1 / fact (2 * i)) * real_of_float x ^ (2 * i)) \<le> real_of_float (ub_sin_cos_aux prec n 1 1 (x * x))"
elims: fixes x :: "('c, 'd) atm"   and y :: "Preliminaries.var set" assumes "varsA x = y" obtains T1 :: "'c trm" and T2 :: "'c trm" where "x = Eq T1 T2" and "y = vars T1 \<union> vars T2"   | p :: "'d" and Tl :: "'c trm list" where "x = Pr p Tl" and "y = \<Union> (set (map vars Tl))"
multi_psub_self: fixes A :: "'c multiset" shows "(A \<subset># A) = False"
rsquarefree_roots: fixes p :: "'a poly" shows "rsquarefree p = (\<forall>a. \<not> (poly p a = (0::'a) \<and> poly (pderiv p) a = (0::'a)))"
neutral: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>A. g x = (0::'c)" shows "sum g A = (0::'c)"
is_strongly_scdom_component_safe_def: shows "Shadow_DOM_SC_DOM_Components.is_strongly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_scdom_component_safe Shadow_DOM.get_owner_document get_disconnected_nodes Shadow_DOM.to_tree_order"
greatest_plus_one_not_zero: shows "Greatest_plus_one \<noteq> (0::'rows)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_begin1 n (l, r) = ((l, r) = ([], Oc \<up> n))"
assms_2: fixes P :: "'state"   and Q :: "'state" assumes "P \<in> S"   and "P \<approx>\<cdot> Q" shows "Q \<in> S"
plain_nres_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>plain_nres_rel \<equiv> {(c, a). RETURN c \<le> \<Down> R a}"
vector_scalar_mult_def: fixes c :: "'d"   and x :: "('d, 'e) Finite_Cartesian_Product.vec" shows "c *s x = (\<chi>i. c * x $h i)"
continuous_on_Dirichlet_L: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and A :: "complex set" assumes "1 < n"   and "\<chi> \<noteq> principal_dchar n \<and> dcharacter n \<chi> \<or> \<chi> = principal_dchar n \<and> 1 \<notin> A" shows "continuous_on A (Dirichlet_L n \<chi>)"
expected_ipl_random_bst_eq: fixes A :: "'a set" assumes "finite A" shows "measure_pmf.expectation (random_bst A) (\<lambda>x. real (ipl x)) = rqs_cost_exp (card A)"
INF_mono': fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` A))"
sel_2: fixes x1 :: "('b \<times> 'a \<times> 'a) set"   and x2 :: "'a set" shows "vertices (LG x1 x2) = x2"
dictatesx_def: fixes F :: "(indi \<Rightarrow> alt \<Rightarrow> real) \<Rightarrow> alt \<Rightarrow> real"   and i :: "indi"   and c :: "alt" assumes "arrow F" shows "arrow.dictatesx F i c \<equiv> \<forall>a b. c \<notin> {a, b} \<longrightarrow> arrow.dictates F i a b"
i_Exec_Stream_Init_take: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream_Init trans_fun input c \<Down> Suc n = f_Exec_Comp_Stream_Init trans_fun (input \<Down> n) c"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (\<I>\<^sub>1 \<otimes>\<^sub>D a)"
len_downshift_helper: fixes x :: "nat"   and P :: "nat fset"   and xa :: "nat" assumes "x |\<in>| P"   and "Suc (fMax ((\<lambda>x. x - Suc 0) |`| (P |-| {|0|}))) \<noteq> fMax P"   and "xa |\<in>| P" shows "xa = 0"
non_trivial_field_hom_is_inj: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S"   and "field R"   and "field S" shows "inj_on h (carrier R)"
poly_circlepath_tan_eq: fixes t :: "real"   and p :: "complex poly"   and z0 :: "complex"   and r :: "real" assumes "0 \<le> t"   and "t \<le> 1"   and "t \<noteq> 1 / 2" shows "poly p (circlepath z0 r t) = poly (fcompose p [:(z0 + complex_of_real r) * \<i>, z0 - complex_of_real r:] [:\<i>, 1:]) (complex_of_real (tan (pi * t))) / poly ([:\<i>, 1:] ^ degree p) (complex_of_real (tan (pi * t)))"
frontier_halfspace_ge: assumes "normal sctn \<noteq> (0::'a) \<or> pstn sctn \<noteq> 0" shows "frontier {x. pstn sctn \<le> normal sctn \<bullet> x} = {x. normal sctn \<bullet> x = pstn sctn}"
SN_rel_on_alt_r_empty: fixes S :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_rel_on_alt {} S T"
simps_6: fixes p :: "'a poly"   and n :: "nat" shows "polynate (Pw p n) = polynate p ^\<^sub>p n"
length_equal: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs = ys" shows "length xs = length ys"
lcoeff_llist_of_stream: fixes xs :: "'a msstream"   and n :: "nat" shows "lcoeff (msllist_of_msstream xs) n = mssnth xs n"
assms: fixes w :: "complex" assumes "w \<in> ball z r" shows "(\<lambda>n. a n * (w - z) ^ n) sums f w"
Inf_from_def: fixes M :: "'a literal multiset set"   and N :: "'a literal multiset set" shows "G.Inf_from M N = {\<iota> \<in> G_Inf M. set (prems_of \<iota>) \<subseteq> N}"
IH: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qGood Y \<longrightarrow> (\<forall>X'. Y #= X' \<longrightarrow> qGood X')"
endo_comp: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f"   and "ChamberComplexEndomorphism X g" shows "ChamberComplexEndomorphism X (g \<circ> f)"
Range_insert: fixes a :: "'d"   and b :: "'c"   and r :: "('d \<times> 'c) set" shows "Range (insert (a, b) r) = insert b (Range r)"
lF_simps_3: fixes f :: "'c" assumes "C.ide f" shows "trg\<^sub>D (lF f) = map\<^sub>0 (trg\<^sub>C f)"
frame_strong_hoare_r: fixes a :: "'a \<Longrightarrow> 'b"   and r :: "'b upred"   and q :: "'b upred"   and p :: "'b upred"   and S :: "'b hrel" assumes "vwb_lens a"   and "a \<sharp> r"   and "a \<natural> q"   and "\<lbrace>p \<and> r\<rbrace> S \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> r\<rbrace> {a}:[S] \<lbrace>q \<and> r\<rbrace>\<^sub>u"
correct_6: fixes ma :: "('c, 'd) RBT.rbt"   and k :: "'c"   and v :: "'d" assumes "rm.invar ma" shows "rm.\<alpha> (rm.update k v ma) = rm.\<alpha> ma(k \<mapsto> v)"
transD: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "trans r"   and "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<preceq> z" shows "x \<^bsub>r\<^esub>\<preceq> z"
le_infI2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and x :: "'a"   and a :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq b x" shows "less_eq (inf a b) x"
order_refl: fixes x :: "'b list" shows "x \<le>p x"
reach_inj: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and a :: "'b list"   and q :: "'a"   and q' :: "'a" assumes "reach tr p a q"   and "reach tr p a q'" shows "q = q'"
finite_PlusD_2: fixes A :: "'a set"   and B :: "'b set" assumes "finite (A <+> B)" shows "finite B"
indRelRT_modulo_pred_impl_indRelRTPO_modulo_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>P. Pred (P, P)"   and "\<forall>P Q R. Pred (P, Q) \<and> Pred (Q, R) \<longrightarrow> Pred (P, R)" shows "(\<forall>(P, Q)\<in>indRelRT TRel. Pred (P, Q)) = (\<forall>(P, Q)\<in>indRelRTPO TRel. Pred (P, Q))"
partial_span': fixes i :: "nat" assumes "i \<le> length (map of_int_hom.vec_hom fs')" shows "gs.span (gs2.gso ` {0..<i}) = gs.span ((!) (map of_int_hom.vec_hom fs') ` {0..<i})"
IO: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<Gamma> :: "('e, 'f) \<I>"   and x :: "'d" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out"   and "resultsp_gpv \<Gamma> x (c input)" shows "resultsp_gpv \<Gamma> x gpv"
primerec_rec_right_1: shows "primerec rec_right (Suc 0)"
sep_false_def: shows "sep_false \<equiv> \<lambda>s. False"
lim_ereal: fixes f :: "'c \<Rightarrow> real"   and x :: "real"   and net :: "'c filter" shows "((\<lambda>n. ereal (f n)) \<longlongrightarrow> ereal x) net = (f \<longlongrightarrow> x) net"
finpref_suff: fixes r :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "r \<in> finpref A t"   and "t \<in> A\<^sup>\<infinity>" shows "t \<in> suff A r"
form_of_lit_is_lit: fixes x :: "'b literal" shows "is_lit_plus (form_of_lit x)"
continuous_on_tendsto_compose: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and l :: "'b"   and F :: "'d filter" assumes "continuous_on s f"   and "(g \<longlongrightarrow> l) F"   and "l \<in> s"   and "\<forall>\<^sub>F x in F. g x \<in> s" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
bidirected_digraph_rev_conv_pair: fixes G :: "'a pair_pre_digraph"   and rev_G :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a" assumes "bidirected_digraph (with_proj G) rev_G" shows "rev_G = swap_in (parcs G)"
has_bochner_integral_cong_AE: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "has_bochner_integral M f x = has_bochner_integral M g x"
effective_scalar_product_times: fixes w1 :: "'a list"   and w2 :: "'a list"   and x :: "'a"   and y :: "'a" assumes "length w1 = length w2" shows "x * y * scalar_product w1 w2 = scalar_product (local.times x w1) (local.times y w2)"
le_infI: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x a"   and "less_eq x b" shows "less_eq x (inf a b)"
emeasure_pmf_UNIV: fixes M :: "'c pmf" shows "emeasure (measure_pmf M) UNIV = 1"
IH: fixes x2a :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "x2a \<in> set ss"   and "NS x2a t" shows "(S x2a t \<longrightarrow> S (x2a \<cdot> \<sigma>) (t \<cdot> \<sigma>)) \<and> NS (x2a \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
vec_def: fixes x :: "'c" shows "vec x = (\<chi>i. x)"
asIMOD_igFreshAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom"   and ys :: "'varSort"   and y :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> ('varSort, 'a) sAbs" shows "igFreshAbs (asIMOD SEM) ys y A = (\<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> A val = A val')"
HFail_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
abs_ccache_union: fixes c1 :: "((nat \<times> (nat \<Rightarrow> contour option)) \<times> d) set"   and c2 :: "((nat \<times> (nat \<Rightarrow> contour option)) \<times> d) set" shows "|c1 \<union> c2| \<lessapprox> |c1| \<union> |c2|"
cs_def_1: shows "\<S>\<bullet>\<C> \<turnstile> cs1 : [] _> ts"
set_map0_2: fixes f1 :: "'o \<Rightarrow> 'r"   and f2 :: "'p \<Rightarrow> 'q" shows "F'set4 \<circ> Fmap f1 f2 = (`) f2 \<circ> F'set4"
iso_in_homE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "category C"   and "category.iso_in_hom C f a b" obtains "partial_magma.in_hom C f a b" and "category.iso C f"
bnd: shows "0 \<le> Max (f ` space M) \<and> Max (f ` space M) < \<infinity>"
digraph_dg_Set: shows "digraph \<alpha> (dg_Set \<alpha>)"
holomorphic_cong: fixes s :: "complex set"   and t :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "s = t"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(f holomorphic_on s) = (g holomorphic_on t)"
lemma_2_6_20: fixes a :: "'a"   and A :: "'a set"   and b :: "'a" assumes "a \<in> infimum A" shows "b r\<rightarrow> a \<in> infimum ((r\<rightarrow>) b ` A)"
finalNonConflictState_2: fixes state :: "State"   and decisionVars :: "nat set" assumes "getConflictFlag state = False"   and "\<not> applicableDecide state decisionVars"   and "\<not> applicableConflict state" shows "decisionVars \<subseteq> vars (elements (getM state))"
m1_inv4_rfreshD: fixes x :: "'a m1_state_scheme"   and Rb1 :: "fid_t"   and A1 :: "agent"   and B1 :: "agent"   and Kab :: "key"   and Ts1 :: "nat"   and Ta1 :: "nat"   and Rb2 :: "fid_t"   and A2 :: "agent"   and B2 :: "agent"   and Ts2 :: "nat"   and Ta2 :: "nat" assumes "runs x Rb1 = Some (Resp, [A1, B1], [aKey Kab, aNum Ts1, aNum Ta1])"   and "runs x Rb2 = Some (Resp, [A2, B2], [aKey Kab, aNum Ts2, aNum Ta2])"   and "B1 \<notin> bad"   and "A1 \<notin> bad"   and "Kab \<notin> Domain (m1_state.leak x)"   and "x \<in> m1_inv4_rfresh" shows "Rb1 = Rb2"
base: fixes N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "inputChain [] N P = \<rparr>N.P"
presheaf_of_rings_axioms_def: fixes is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" shows "presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> ((\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> V \<subseteq> U \<longrightarrow> ring_homomorphism (\<rho> U V) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF> V) (add_str V) (mult_str V) (zero_str V) (one_str V)) \<and> \<FF> {} = {b}) \<and> (\<forall>U. is_open U \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> \<rho> U U x = x)) \<and> (\<forall>U V W. is_open U \<longrightarrow> is_open V \<longrightarrow> is_open W \<longrightarrow> V \<subseteq> U \<longrightarrow> W \<subseteq> V \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> \<rho> U W x = (\<rho> V W \<circ> \<rho> U V) x))"
right_diff_distrib': 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 a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times a (minus b c) = minus (times a b) (times a c)"
inf_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
closed_csubspace_INF: fixes \<A> :: "'d set set" assumes "\<forall>A\<in>\<A>. closed_csubspace A" shows "closed_csubspace (\<Inter> \<A>)"
elims_2: fixes x :: "('c, 'd) tm"   and xa :: "dir list" assumes "position_of x xa" obtains uu_ :: "('c, 'd) tm" where "x = uu_" and "xa = []"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t__ sa__" and "xa = dir.Left # ds__" and "position_of t__ ds__"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t s" and "xa = dir.Right # ds" and "position_of s ds"
simps_1: shows "Tree2.inorder \<langle>\<rangle> = []"
supt_subset_LPO_S: shows "{\<rhd>} \<subseteq> LPO_S"
rev_is_rev_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "(rev xs = rev ys) = (xs = ys)"
op_type_alt_simps_10: fixes op :: "string_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type" shows "string_unop_type op \<tau> \<sigma> = (op = SizeOp \<and> \<tau> = String[1] \<and> \<sigma> = Integer[1] \<or> op = CharactersOp \<and> \<tau> = String[1] \<and> \<sigma> = Sequence String[1] \<or> op = ToUpperCaseOp \<and> \<tau> = String[1] \<and> \<sigma> = String[1] \<or> op = ToLowerCaseOp \<and> \<tau> = String[1] \<and> \<sigma> = String[1] \<or> op = ToBooleanOp \<and> \<tau> = String[1] \<and> \<sigma> = Boolean[1] \<or> op = string_unop.ToIntegerOp \<and> \<tau> = String[1] \<and> \<sigma> = Integer[1] \<or> op = ToRealOp \<and> \<tau> = String[1] \<and> \<sigma> = Real[1])"
fundamental_theorem_of_calculus_interior: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)" shows "(f' has_integral f b - f a) {a..b}"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>"
stable_def: fixes A :: "'d set" shows "stable A \<equiv> A co A"
Sup_state_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "Sup_state Sts = (Sup_llist (lmap N_of_state Sts), Sup_llist (lmap P_of_state Sts), Sup_llist (lmap Q_of_state Sts))"
sfirst_finite: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "(sfirst P \<omega> < \<infinity>) = ev P \<omega>"
ge: shows "(\<Union>c. {\<lambda>x. if c \<le> x then f x else 0}) equiintegrable_on {a..b}"
msteps0_snoc: fixes \<pi> :: "((String.literal \<times> event_data list) set \<times> nat) list"   and tdb :: "(String.literal \<times> event_data list) set \<times> nat"   and st :: "('msaux, 'muaux) mstate" shows "msteps0 (\<pi> @ [tdb]) st = (let (V', st') = msteps0 \<pi> st; (V'', st'') = mstep (map_prod mk_db id tdb) st' in (V' @ V'', st''))"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> B.VVV.arr f" shows "\<a>\<^sub>B[fst f, fst (snd f), snd (snd f)] = B.null"
AxSound: fixes \<phi> :: "('t, 'f) Language" assumes "\<phi> \<in> aAxioms Ax" shows "L\<lbrakk>\<phi>\<rbrakk> \<rightarrow> IBool True"
pmult: shows "poly_mult (p @ [\<zero>]) q = r @ [\<zero>]"
Nonce_notin_used_empty: fixes N :: "nat" shows "Nonce N \<notin> used []"
Ret_def: fixes x1 :: "trm" shows "[x1] \<equiv> Abs_trm (Ret_Rep (Rep_trm x1))"
assms_3: shows "p \<in> ideal (set (take j fs))"
semiring_normalization_rules_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and m :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus (times a m) (times b m) = times (plus a b) m"
isnpolyh_zero_iff: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "\<forall>bs. wf_bs bs p \<longrightarrow> Ipoly bs p = (0::'b)" shows "p = 0\<^sub>p"
HF_ax_def: shows "HF_ax \<equiv> {\<lbrakk>\<guillemotleft>HF1\<guillemotright>\<rbrakk>e0, \<lbrakk>\<guillemotleft>HF2\<guillemotright>\<rbrakk>e0}"
from_hma_to_hma\<^sub>m: fixes A :: "'b mat" assumes "A \<in> carrier_mat CARD('c) CARD('d)" shows "from_hma\<^sub>m (to_hma\<^sub>m A) = A"
set_valued_transformation_axioms: shows "set_valued_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) Cones.map \<Phi>"
assms: shows "\<Union> (f ` A) \<subseteq> carrier G"
prems: shows "y < output_size (shallow_model' Z M (Suc N))"