hcju/isabelleps huggingface dataset:
gb_schema_aux_pmdl_init: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and rc :: "nat" assumes "struct_spec sel ap ab compl"   and "compl_pmdl compl"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ ab gs [] bs data) data"   and "rem_comps_spec (gs @ ab gs [] bs data) (rc, data)" shows "pmdl (fst ` set (gb_schema_aux sel ap ab compl gs (rc, data) (ab gs [] bs data) (ap gs [] [] bs data))) = pmdl (fst ` set (gs @ bs))"
simps: fixes R :: "'b sequent list \<times> 'b sequent \<Rightarrow> bool"   and a1 :: "'b sequent"   and a2 :: "nat" shows "derivablep R a1 a2 = ((\<exists>C. a1 = C \<and> a2 = 0 \<and> R ([], C)) \<or> (\<exists>r m. a1 = snd r \<and> a2 = m + 1 \<and> R r \<and> fst r \<noteq> [] \<and> (\<forall>p\<in>set (fst r). \<exists>n\<le>m. derivablep R p n)))"
ln_\<pi>_asymptotics: shows "(\<lambda>x. ln (\<pi> x)) \<sim>[at_top] ln"
d0_def: shows "st1.d0 = prod_list st1.dims0"
enat_0_iff_1: fixes x :: "nat" shows "(enat x = 0) = (x = 0)"
lossless_spmf_def: fixes p :: "'b spmf" shows "lossless_spmf p = (weight_spmf p = 1)"
induct: fixes P :: "'c list \<Rightarrow> bool"   and a0 :: "'c list" assumes "\<And>v w xs. P (v # w # xs)"   and "\<And>v. P [v]"   and "P []" shows "P a0"
feasible_boundedD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "feasible t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
cases: fixes a :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set" assumes "a \<in> grid b ds" obtains "a = b"   | p :: "(nat \<times> int) list" and d :: "nat" and dir :: "dir" where "a = child p dir d" and "p \<in> grid b ds" and "d \<in> ds"
D_mono': fixes e :: "expr"   and A :: "char list set option"   and A' :: "char list set option" assumes "\<D> e A"   and "A \<sqsubseteq> A'" shows "\<D> e A'"
Cons_3: shows "D \<in> \<C> - f \<turnstile> \<C>"
rCase: fixes Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and P :: "('a, 'b, 'c) psi"   and \<phi> :: "'c" assumes "cP = Cases Cs"   and "\<Psi> \<rhd> P \<longmapsto> cRs"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop"
elims_3: fixes x :: "'b kdt" assumes "\<not> KD_Tree.complete x" obtains uv_ :: "'b" and uw_ :: "real" and l :: "'b kdt" and r :: "'b kdt" where "x = Node uv_ uw_ l r" and "\<not> (KD_Tree.complete l \<and> KD_Tree.complete r \<and> height l = height r)"
not_getbase_shares: fixes D :: "char list"   and xs :: "base list" assumes "D \<notin> set (map getbase xs)" shows "Shares D \<notin> set xs"
good_item_simps_38: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "Abs xsa xa X $[y1 \<and> y2]_ys = Abs xsa (xa @xsa[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys)"
eigenvalue_det: fixes A :: "'b mat"   and n :: "nat"   and e :: "'b" assumes "A \<in> carrier_mat n n" shows "eigenvalue A e = (det (char_matrix A e) = (0::'b))"
bind_restrict_pmf: fixes p :: "'d spmf"   and A :: "'d set"   and g :: "'d option \<Rightarrow> 'c pmf" shows "p \<upharpoonleft> A \<bind> g = p \<bind> (\<lambda>x. if x \<in> Some ` A then g x else g None)"
TNil_eq_tmap_conv: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "('c, 'd) tllist" shows "(TNil y = tmap f g xs) = (\<exists>y'. xs = TNil y' \<and> g y' = y)"
eq4: shows "r2 (f x) (f y) = r1 x y"
meta_spec: fixes P :: "'d \<Rightarrow> prop"   and x :: "'d" assumes "\<And>x. PROP P x" shows "PROP P x"
intros: fixes q :: "'d"   and W :: "'d set"   and Q' :: "'d set"   and Q :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and W' :: "'d set" assumes "q \<in> W"   and "Q' = Q \<union> br_dsq \<delta> q (Q, W)"   and "W' = W - {q} \<union> (br_dsq \<delta> q (Q, W) - Q)" shows "((Q, W), Q', W') \<in> br_step \<delta>"
lowest_tops_lowest: fixes es :: "'a ifex list"   and a :: "'a"   and e :: "'a ifex"   and v :: "'a" assumes "lowest_tops es = Some a"   and "e \<in> set es"   and "ifex_ordered e"   and "v \<in> ifex_var_set e" shows "a \<le> v"
left_impl_times: fixes a :: "'a"   and b :: "'a" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
remove_min_def: fixes q :: "('c, 'd) pq" shows "remove_min q = (if is_empty q then PQ.empty else Abs_pq (tl (alist_of q)))"
least: fixes b :: "'a" assumes "\<And>a. a \<in> A \<Longrightarrow> a \<le> b" shows "x \<le> b"
invKey_image_eq: fixes x :: "nat"   and A :: "nat set" shows "(invKey x \<in> invKey ` A) = (x \<in> A)"
mod0_uint32_def: fixes x :: "uint32" shows "mod0_uint32 x = undefined (mod) x 0"
domintros: fixes x :: "'a" assumes "\<not> Q x \<Longrightarrow> recfun_aux_dom (h x)" shows "recfun_aux_dom x"
monoI: fixes f :: "'a word \<Rightarrow> 'b" assumes "\<And>x y. x \<le>s y \<Longrightarrow> f x \<le> f y" shows "signed.mono f"
inf_eq_top_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = (1::'a)) = (x = (1::'a) \<and> y = (1::'a))"
D1_def: shows "D1 = Sigma {0<..<1} (greaterThanLessThan 0)"
max2_greater_among: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(a, local.max2 a b) \<in> r \<and> (b, local.max2 a b) \<in> r \<and> local.max2 a b \<in> {a, b}"
continuous_map_into_subtopology: fixes X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_map X X' f"   and "f ` topspace X \<subseteq> T" shows "continuous_map X (subtopology X' T) f"
pasting_lemma: fixes I :: "'a set"   and X :: "'b topology"   and T :: "'a \<Rightarrow> 'b set"   and Y :: "'c topology"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x" shows "continuous_map X Y g"
const_map_is_Some: fixes y :: "'a"   and A :: "'b set"   and x :: "'b"   and z :: "'a" shows "(const_map y A x = Some z) = (z = y \<and> x \<in> A)"
x2aa: shows "\<forall>k<x1. x2a k * x4 * adjoint (x2a k) \<in> carrier_mat d d"
elims_2: fixes x :: "'d"   and xa :: "'e"   and xb :: "('f, 'g, 'd, 'e, 'h) qAbs" assumes "qFreshAbs x xa xb" obtains xs :: "'d" and x :: "'e" and ys :: "'d" and y :: "'e" and X :: "('f, 'g, 'd, 'e, 'h) qTerm" where "xa__ = xs" and "xa = x" and "xb = qAbs ys y X" and "xs = ys \<and> x = y \<or> qFresh xs x X"
edge_det: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a :: "'a"   and a' :: "'a" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a = targetnode a'" shows "a = a'"
smult_i_def: fixes ops :: "'b arith_ops_record"   and a :: "'b"   and pp :: "'b list" shows "smult_i ops a pp = (if a = arith_ops_record.zero ops then [] else strip_while ((=) (arith_ops_record.zero ops)) (map (arith_ops_record.times ops a) pp))"
residue_add_assoc: fixes k :: "nat"   and x :: "int"   and y :: "int"   and z :: "int" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> z = x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> (y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> z)"
intros: fixes C :: "('a, 'b) sequent"   and A :: "('a, 'b) form"   and Ps :: "('a, 'b) sequent list"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>)"   and "A \<noteq> ff"   and "(Ps, C) \<in> R" shows "leftPrincipal (Ps, C) A R"
transfer_bounded_bilinear_bounded_linearI: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and f :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>L 'd" assumes "g = (\<lambda>i. blinfun_apply (f i))" shows "bounded_bilinear g = bounded_linear f"
assms: shows "\<gamma> \<in> elts (\<omega> \<up> \<omega>)"
incrIndexList_help6: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "increasing ls"
abs_eq: fixes xa :: "'b"   and x :: "'c" shows "single xa x = Abs_poly_mapping (\<lambda>k'. x when xa = k')"
igSubstAbsIPresIGWlsAbsSTR_imp_igSubstAbsIPresIGWlsAbs: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" assumes "igSubstAbsIPresIGWlsAbsSTR MOD" shows "igSubstAbsIPresIGWlsAbs MOD"
iterate_size_correct: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"   and S0 :: "'a set" assumes "set_iterator it S0" shows "iterate_size it = card S0 \<and> finite S0"
less_h: shows "At = LessUni (a, b, c) \<longrightarrow> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
iso_in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<cong> b\<guillemotright> \<equiv> \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> local.iso f"
sign_changes_number_rootat_def: fixes ps :: "rat poly list" shows "sign_changes_number_rootat ps = length (remdups_adj (list_neq (map poly_number_rootat ps) 0)) - 1"
leftID: fixes a :: "'e"   and f :: "'e \<rightarrow> 'd Maybe" shows "Just\<cdot>a >>=\<^sub>M f = f\<cdot>a"
DOM: shows "atoms_ltlc \<phi> \<subseteq> dom f"
prefix_same_states: shows "\<forall>k<j. S t k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k"
is_mset_setI: fixes X :: "'d multiset" assumes "\<And>x. x \<in># X \<Longrightarrow> count X x = 1" shows "is_mset_set X"
domintros_8: fixes v :: "exp0"   and va :: "(pat \<times> exp0) list" shows "is_value_dom (Handle v va)"
simps_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Var x1 = Var y1) = (x1 = y1)"
merge_poly_merge_poly2: fixes a :: "(String.literal list \<times> int) list"   and b :: "(char list list \<times> int) list"   and a' :: "(String.literal list \<times> int) list"   and b' :: "(char list list \<times> int) list" assumes "(a, b) \<in> poly_rel"   and "(a', b') \<in> poly_rel" shows "(merge_poly a a', merge_poly b b') \<in> poly_rel"
axioms_1: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_class_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order get_attribute get_attribute_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name type_wf" shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order"
flip_fresh_fresh: fixes a :: "'b"   and x :: "'a"   and b :: "'b" assumes "atom a \<sharp> x"   and "atom b \<sharp> x" shows "(a \<leftrightarrow> b) \<bullet> x = x"
domain_subset_vimage: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> A \<rightarrow> B" shows "A \<subseteq> f -` B"
distinct_7: shows "cospan_shape.Null \<noteq> cospan_shape.AT"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A"
X: shows "X \<in> \<Omega> \<rightarrow> \<Omega>'"
mult_scalar_zero_left: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.mult_scalar 0 p = 0"
finprod_closed: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
endpoint_in_semifin: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set" assumes "[f[x ..]X]" shows "x \<in> X"
nemp_elem_setI: fixes u :: "'b list"   and S :: "'b list set" assumes "u \<in> S"   and "u \<noteq> \<epsilon>" shows "u \<in> S\<^sub>+"
distinct_59: fixes x2 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Minus x2 \<noteq> Power x151 x152"
unhindered_linkable_bounded: assumes "\<not> hindered (bipartite_web_of \<Gamma>)" shows "linkable (bipartite_web_of \<Gamma>)"
asm: shows "Re z \<le> 0 \<and> (cmod z)\<^sup>2 \<le> 4 * (Re z)\<^sup>2"
Jmp: fixes jump :: "jump"   and B :: "lname set"   and A :: "assigned"   and Env :: "env" assumes "jump = Ret \<longrightarrow> Result \<in> B"   and "nrm A = UNIV"   and "brk A = (case jump of Break l \<Rightarrow> \<lambda>k. if k = l then B else UNIV | _ \<Rightarrow> \<lambda>k. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>Jmp jump\<rangle>\<guillemotright> A"
lossless_gpv_inline1: fixes gpv :: "('b, 'call, 'ret) gpv"   and out :: "'call'"   and rpv :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and s :: "'s"   and input :: "'ret'" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "Inr (out, rpv, rpv') \<in> set_spmf (inline1 callee gpv s)"   and "input \<in> responses_\<I> \<I>' out"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_gpv \<I>' (callee s x)" shows "lossless_gpv \<I>' (rpv input)"
Nml_VcompNml_1: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t"   and "bicategorical_language.Nml V src trg u"   and "bicategorical_language.Dom V src trg t = bicategorical_language.Cod V src trg u" shows "bicategorical_language.Nml V src trg (bicategorical_language.VcompNml V t u)"
ocondition_def: fixes c :: "'c \<Rightarrow> bool"   and L :: "'c \<Rightarrow> 'd"   and R :: "'c \<Rightarrow> 'd" shows "ocondition c L R \<equiv> \<lambda>s. if c s then L s else R s"
zip_Nil: fixes ys :: "'h list" shows "zip [] ys = []"
direct_decomp_Cons_zeroI: fixes A :: "'a set"   and ss :: "'a set list" assumes "direct_decomp A ss" shows "direct_decomp A ({0::'a} # ss)"
has_vector_derivative_Pair: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. (f x, g x)) has_vector_derivative (f', g')) (at x within s)"
preserving_length: fixes n :: "'a"   and y :: "'a list" shows "length (local.times n y) = length y"
typing_swp: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and a :: "'a"   and \<sigma> :: "PreSimplyTyped.type"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and b :: "'a" assumes "\<Gamma>(a \<mapsto> \<sigma>) \<turnstile> M : \<tau>"   and "b \<notin> fvs M" shows "\<Gamma>(b \<mapsto> \<sigma>) \<turnstile> [a \<leftrightarrow> b] \<cdot> M : \<tau>"
f_mono_inj: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>x y. x < y \<longrightarrow> f x < f y" shows "\<forall>x y. f x = f y \<longrightarrow> x = y"
induced_function_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "induced_function A f = (\<lambda>x. \<Sum>i<return_time_function A x. f ((T ^^ i) x))"
fold_graph_finite: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and A :: "'b set"   and y :: "'c" assumes "fold_graph f z A y" shows "finite A"
interchange_rows_transpose': fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and b :: "'f" shows "interchange_rows A a b = Finite_Cartesian_Product.transpose (interchange_columns (Finite_Cartesian_Product.transpose A) a b)"
has_integral_Beta_real: fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 < b" shows "((\<lambda>t. t powr (a - 1) * (1 - t) powr (b - 1)) has_integral Beta a b) {0..1}"
space_lborel_add_dim: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and x :: "real" assumes "f \<in> space (lborel_f n)" shows "f(n := x) \<in> space (lborel_f (Suc n))"
sub1_divisor: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r', A')"   and "r \<noteq> (0::'a)"   and "q \<noteq> (0::'a)"   and "k + l < dim_row A"   and "l < dim_col A" shows "r' \<noteq> (0::'a)"
dom_cod: fixes f :: "'j" assumes "J.arr f" shows "J.dom (J.cod f) = J.cod f"
cpayloadp_cpayload_eq: shows "cpayloadp = (\<lambda>x. x \<in> cpayload)"
cases: fixes a :: "('d \<Rightarrow> 'e) set"   and \<R> :: "('d \<Rightarrow> 'e) set set"   and R :: "('d \<Rightarrow> 'e) set" assumes "a \<in> Succ \<R> R" obtains u :: "'d \<Rightarrow> 'e" and R' :: "('d \<Rightarrow> 'e) set" and t :: "'e" where "a = R'" and "u \<in> R" and "R \<in> \<R>" and "R' \<in> \<R>" and "(0::'e) \<le> t" and "R' = [u \<oplus> t]\<^sub>\<R>"
assms: shows "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f"
rep_eq: shows "apply_bcontfun 0 = (\<lambda>_. 0::'b)"
group: fixes S :: "'a set"   and T :: "'b set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'c" assumes "finite S"   and "finite T"   and "g ` S \<subseteq> T" shows "(\<Sum>y\<in>T. sum h {x \<in> S. g x = y}) = sum h S"
exhaust: fixes y :: "ty" obtains "y = One"   | x21 :: "ty" and x22 :: "ty" where "y = Fun x21 x22"   | x31 :: "ty" and x32 :: "ty" where "y = Syntax.Sum x31 x32"   | x41 :: "ty" and x42 :: "ty" where "y = Syntax.Prod x41 x42"   | x51 :: "Syntax.tvar" and x52 :: "ty" where "y = Mu x51 x52"   | x6 :: "Syntax.tvar" where "y = Alpha x6"   | x7 :: "ty" where "y = AuthT x7"
prems: shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t (a # A)"
linear_injective_left_inverse: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id"
commutative_squares_3: shows "C.commutative_square \<p>\<^sub>0[\<mu>.cod.leg0, assoc\<nu>\<pi>\<rho>.cod.leg1] assoc\<nu>\<pi>\<rho>.chine (chine_hcomp \<mu> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>) \<p>\<^sub>0[\<mu>.leg0, assoc\<nu>\<pi>\<rho>.dom.leg1]"
1: shows "C.cospan \<mu>.dom.leg0 f.dom.leg1"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool" shows "pred_ifex P Falseif"
geq1: shows "r'\<^sub>1 \<le> r1"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>"
m1_inv0_fin_def: shows "m1_inv0_fin \<equiv> {s. finite (dom (runs s))}"
matchesE: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and u :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.matches subst t u" obtains env :: "(name, 'b) fmap" where "subst t env = u"
comp_lunit_term_dup: fixes a :: "'a" assumes "ide a" shows "lunit a \<cdot> local.prod \<t>[a] a \<cdot> dup a = a"
thincomplexes: fixes A :: "'a set set" assumes "A \<in> \<A>" shows "ThinChamberComplex A"
lit_ueval: fixes x :: "'c"   and b :: "'d" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e b = x"
R23s_defs_2: fixes implem :: "chan \<Rightarrow> msg"   and t :: "l3_state" assumes "dhlvl3 implem" shows "dhlvl3.med23s implem t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t, chan = semivalid_implem.abs implem (ik t), bad = l3_state.bad t\<rparr>"
s: shows "i * b ^ n + sum_list (map ((^) b) ns) = j * b ^ m + sum_list (map ((^) b) ms)"
well_defined_Tensor: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and M1 :: "'a list list"   and M2 :: "'a list list" assumes "Matrix_Tensor.mult id f"   and "Matrix_Legacy.mat (mult.row_length M1) (length M1) M1 \<and> Matrix_Legacy.mat (mult.row_length M2) (length M2) M2" shows "Matrix_Legacy.mat (mult.row_length M1 * mult.row_length M2) (length M1 * length M2) (mult.Tensor f M1 M2)"
bf_ite_def: fixes i :: "'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd"   and e :: "'c \<Rightarrow> 'd" shows "bf_ite i t e \<equiv> \<lambda>l. if i l then t l else e l"
assms_3: assumes "x \<in>\<^sub>\<circ> A" shows "P"
prob_space_axioms: shows "prob_space (distr (uniform_measure M {\<omega> \<in> space M. X n \<omega> = s}) (count_space UNIV) (X (Suc n)))"
mprojTr3: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme"   and a :: "'e"   and a1 :: "'e" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "a \<in> carrier M"   and "a1 \<in> carrier M1"   and "a \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> a1 \<in> carrier M2" shows "(THE x1. x1 \<in> carrier M1 \<and> a \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> x1 \<in> carrier M2) = a1"
lm04: fixes X :: "'a set"   and x :: "'a" assumes "trivial X"   and "{x} \<subseteq> X" shows "{x} = X"