hcju/isabelleps huggingface dataset:
subst_cls_union: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and \<sigma> :: "'s" shows "(C + D) \<cdot> \<sigma> = C \<cdot> \<sigma> + D \<cdot> \<sigma>"
cone_mono: fixes Vs :: "'b vec set"   and Ws :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> Ws" shows "gram_schmidt.cone n Vs \<subseteq> gram_schmidt.cone n Ws"
not_empty: fixes M :: "'c measure" assumes "prob_space M" shows "space M \<noteq> {}"
IH_1: fixes u :: "'a set list"   and w :: "nat \<Rightarrow> 'a set" assumes "suspendable \<xi>" shows "(u \<frown> w) \<Turnstile>\<^sub>n \<xi> = w \<Turnstile>\<^sub>n \<xi>"
rel_conversep: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "lrel R\<inverse>\<inverse> = (lrel R)\<inverse>\<inverse>"
AE_impI: fixes P :: "bool"   and Q :: "'a \<Rightarrow> bool"   and M :: "'a measure" assumes "P \<Longrightarrow> almost_everywhere M Q" shows "AE x in M. P \<longrightarrow> Q x"
comp_ipresIGSwapAbs: fixes hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'ae, 'w, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ab, 'ai, 'ad, 'aj, 'x, 'ak) model_scheme"   and hA' :: "'x \<Rightarrow> 'al"   and MOD'' :: "('am, 'an, 'aa, 'ao, 'ap, 'ad, 'aq, 'al, 'ar) model_scheme" assumes "ipresIGWlsAbs hA MOD MOD'"   and "ipresIGSwapAbs hA MOD MOD'"   and "ipresIGSwapAbs hA' MOD' MOD''" shows "ipresIGSwapAbs (hA' \<circ> hA) MOD MOD''"
IH: assumes "insert_ops (xs @ ys)"   and "oid \<in> set (interp_ins (xs @ ys))"   and "oid \<in> set (map fst xs)" shows "oid \<in> set (interp_ins xs)"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Eq of Eq \<Rightarrow> f1 | Add \<Rightarrow> f2) = f1"
update_conv: fixes k :: "'d"   and v :: "'c"   and al :: "('d \<times> 'c) list"   and k' :: "'d" shows "map_of (update k v al) k' = (map_of al(k \<mapsto> v)) k'"
psubst_in: fixes x :: "W.var"   and S :: "tyS"   and \<Gamma> :: "(W.var \<times> tyS) list"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "(x, S) \<in> set \<Gamma>" shows "(x, \<theta><S>) \<in> set (\<theta><\<Gamma>>)"
\<theta>_twopow_less: fixes r :: "nat" shows "\<theta> (2 ^ r) < 2 ^ (r + 1) * ln 2"
sym_symcl: fixes Rel :: "('c \<times> 'c) set" shows "sym (symcl Rel)"
hd_exp_inverse: fixes xs :: "('a \<times> real) msllist" assumes "xs \<noteq> MSLNil" shows "ms_aux_hd_exp (inverse_ms_aux xs) = map_option uminus (ms_aux_hd_exp xs)"
hd_conv_nth: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "hd xs = xs ! 0"
S_edge_reachable: fixes u :: "'v"   and v :: "'v" assumes "u \<in> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges rg"
rank_def: fixes A :: "'a mat" shows "rank A = vectorspace.dim class_ring (vs (local.span (set (cols A))))"
residue_primroot_odd_prime_power_exists: fixes p :: "nat" assumes "prime p"   and "odd p" obtains g :: "nat" where "\<forall>k>0. residue_primroot (p ^ k) g"
intros_2: fixes t1 :: "('c, 'd) tm"   and t2 :: "('c, 'd) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t2"
elliptic_isometry_stable_translation_length: fixes f :: "'b \<Rightarrow> 'b" assumes "elliptic_isometry f" shows "stable_translation_length f = 0"
C_def: shows "C = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. B x \<inter> B y)"
ads_d_def: fixes x :: "'a" shows "antidomain_semiringl.ads_d a x = a (a x)"
yun_monic_factorization_def: fixes p :: "'b poly" shows "yun_monic_factorization p = (let pp = pderiv p; u = gcd p pp; b0 = p div u; c0 = pp div u in filter (\<lambda>(a, i). a \<noteq> 1) (yun_factorization_main b0 c0 0 []))"
hyps_1: fixes fstate :: "d \<times> d list \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour"   and fstate_a :: "'c \<aproc> \<times> 'c \<aproc> set list \<times> ((nat \<times> char list) \<times> 'c \<Rightarrow> 'c \<aproc> set) \<times> 'c" assumes "|fstate| \<lessapprox> fstate_a" shows "|evalF\<cdot>(Discr fstate)| \<lessapprox> \<aF>\<cdot>(Discr fstate_a)"
unique_convergence: fixes F :: "'a set set"   and x :: "'a"   and y :: "'a" assumes "Topology.filter F T"   and "x \<in> carrier"   and "y \<in> carrier"   and "F \<longlongrightarrow> x"   and "F \<longlongrightarrow> y" shows "x = y"
cut_of_sep_def: fixes S :: "('v \<times> 'v) set" shows "local.cut_of_sep S = RF\<^sup>N\<^bsub>\<Delta>\<^esub> (fst ` \<E> S)"
le_cases: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
dom_in_domains: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C"   and "partial_magma.domains C f \<noteq> {}" shows "partial_magma.dom C f \<in> partial_magma.domains C f"
permutes_not_in: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "f permutes S"   and "x \<notin> S" shows "f x = x"
HEndPhase2_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4c s q" shows "HInv4c s' q"
vector_matrix_mul_rid: fixes v :: "('a, 'b) vec" shows "v v* mat (1::'a) = v"
exhaust: fixes y :: "const" obtains x1 :: "int" where "y = Const x1"   | "y = Any"
wt_step_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list" shows "wt_step r T step \<tau>s = (\<forall>p<length \<tau>s. \<tau>s ! p \<noteq> T \<and> stable r step \<tau>s p)"
transitive_rule: fixes G :: "('b, 'c) labeled_graph"   and l :: "'b" assumes "graph G"   and "maintained (transitive_rule l) G" shows "trans (getRel l G)"
domintros: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and C :: "acom" assumes "bval b s \<Longrightarrow> postQs_dom (C, b, postQ C s)" shows "postQs_dom (C, b, s)"
le_cSUP_iff: 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 A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
homeomorphic_scaling: fixes c :: "real"   and S :: "'c set" assumes "c \<noteq> 0" shows "S homeomorphic (*\<^sub>R) c ` S"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q \<alpha> Q' A\<^sub>Q \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) \<alpha> (P \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
mono_on_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
no_neighbors_relabel_precond: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and u :: "nat" assumes "neighbor_invar c s t f l n"   and "n u = {}"   and "u \<noteq> t"   and "u \<noteq> s"   and "u \<in> Graph.V c"   and "Network.excess c f u \<noteq> (0::'a)" shows "Network.relabel_precond c t f l u"
seqp_guardTE: fixes \<xi> :: "'e"   and l :: "'h"   and g :: "'e \<Rightarrow> 'e set"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}\<langle>g\<rangle> p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = \<tau>" and "\<xi>' \<in> g \<xi>" and "q = p"
power_mono_iff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b"   and "0 < n" shows "(a ^ n \<le> b ^ n) = (a \<le> b)"
select_convs_2: fixes NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "IntMapK \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = IntMapK"
2: fixes xs :: "'a llist"   and x' :: "'a" assumes "x \<in> lset xs"   and "P xs" shows "P (LCons x' xs)"
subset_singleton_iff: fixes X :: "'c set"   and a :: "'c" shows "(X \<subseteq> {a}) = (X = {} \<or> X = {a})"
SFAccInitRed: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh),False\<rangle> \<rightarrow> \<langle>INIT D ([D],False) \<leftarrow> C\<bullet>\<^sub>sF{D},(h, l, sh),False\<rangle>"
REC_rule: fixes body :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and pre :: "'b \<Rightarrow> bool"   and x :: "'b"   and M :: "'b \<Rightarrow> 'c" assumes "trimono body"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x. pre x \<Longrightarrow> f x \<le> M x; pre x; f \<le> REC body\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC body x \<le> M x"
list_pred_cases: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "list_all P xs" obtains "xs = []"   | y :: "'b" and ys :: "'b list" where "xs = y # ys" and "P y" and "list_all P ys"
assms_2: shows "n \<approx>\<^sub>\<circ> A"
inject_2: fixes x2 :: "'b equation"   and y2 :: "'b equation" shows "(Neg x2 = Neg y2) = (x2 = y2)"
expand: fixes "term" :: "('c, 'd) Term.term"   and term' :: "('c, 'd) Term.term" assumes "is_Var term = is_Var term'"   and "\<lbrakk>is_Var term; is_Var term'\<rbrakk> \<Longrightarrow> the_Var term = the_Var term'"   and "\<lbrakk>is_Fun term; is_Fun term'\<rbrakk> \<Longrightarrow> the_Fun term = the_Fun term' \<and> args term = args term'" shows "term = term'"
Bang_2: shows "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> cRs"
mono_aval': fixes S1 :: "'a st"   and S2 :: "'a st"   and e :: "aexp" assumes "S1 \<le> S2" shows "aval' e S1 \<le> aval' e S2"
that: assumes "ok r = ok r'"   and "tr r = tr r'"   and "wait r = wait r'"   and "ref r = ref r'"   and "alpha_rp.more r = alpha_rp.more r'" shows "thesis"
induct: fixes x :: "lam"   and P :: "lam \<Rightarrow> bool" assumes "SN x"   and "\<And>t. \<lbrakk>\<And>t'.  t \<longrightarrow>\<^sub>\<beta> t' \<Longrightarrow> SN t'; \<And>t'.  t \<longrightarrow>\<^sub>\<beta> t' \<Longrightarrow> P t'\<rbrakk> \<Longrightarrow> P t" shows "P x"
irrefl: fixes a :: "'a multiset" shows "\<not> a \<subset># a"
carrier_ttree_restr: fixes S :: "'b set"   and t :: "'b ttree" shows "carrier (ttree_restr S t) = S \<inter> carrier t"
ur_closure: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "p at i in' H"   and "\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A" shows "\<forall>a\<in>assign A H i. p at a in' H"
proj2_no_3_Col_span: fixes S :: "proj2 set"   and p :: "proj2" assumes "proj2_no_3_Col S"   and "p \<in> S" shows "span (proj2_rep ` (S - {p})) = UNIV"
fps_power_first: fixes a :: "'a fps"   and n :: "nat" shows "fps_nth (a ^ n) 1 = of_nat n * fps_nth a 0 ^ (n - 1) * fps_nth a 1"
subcomplex_chamber: fixes Y :: "'b set set"   and C :: "'b set" assumes "codomain.ChamberSubcomplex Y"   and "SimplicialComplex.maxsimp Y C" shows "codomain.chamber C"
has_bochner_integral_cong: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x"   and "x = y" shows "has_bochner_integral M f x = has_bochner_integral N g y"
Skorohod: shows "\<exists>\<Omega> Y_seq Y. prob_space \<Omega> \<and> (\<forall>n. Y_seq n \<in> borel_measurable \<Omega>) \<and> (\<forall>n. distr \<Omega> borel (Y_seq n) = \<mu> n) \<and> Y \<in> \<Omega> \<rightarrow>\<^sub>M lborel \<and> distr \<Omega> borel Y = M \<and> (\<forall>x\<in>space \<Omega>. (\<lambda>n. Y_seq n x) \<longlonglongrightarrow> Y x)"
2: shows "fmlookup (action_needed_asses a s2) v = fmlookup (fst a) v"
Rep_filter_iff_eventually: fixes F :: "'c filter"   and P :: "'c \<Rightarrow> bool" shows "Rep_filter F P = eventually P F"
abs_eq: fixes x :: "rat" shows "of_rat_3 x = Real_Alg_Invariant (of_rat_2 x)"
store_sub2_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (store_sub2 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
funToCone_coneToFun: fixes \<chi> :: "'j \<Rightarrow> 's" assumes "\<chi> \<in> cones S.unity" shows "funToCone (coneToFun \<chi>) = \<chi>"
inclusion_morphism: fixes X :: "'b set set"   and Y :: "'b set set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y" shows "ChamberComplexMorphism Y X id"
set_pps_to_list: fixes S :: "('x \<Rightarrow>\<^sub>0 nat) set" assumes "finite S" shows "set (punit.pps_to_list S) = S"
indaut_eq_foldch_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "fun_eq_on (OpposedThinChamberComplexFoldings.induced_automorphism X f g) g C"
f1: shows "4 \<le> rk {A, B, C, a, b, c, \<alpha>, \<beta>, \<gamma>}"
less_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
strict_implies_order: fixes a :: "nat list"   and b :: "nat list" assumes "a <\<^sub>v b" shows "a \<le>\<^sub>v b"
assms_1: shows "g \<turnstile> m\<^sub>1 \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"
rb_aux_inv_D1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, ps)" shows "rb_aux_inv1 bs"
sign_simps_3: fixes x :: "'a" assumes "x < (0::'a)" shows "Sturm_Tarski.sign x = - 1"
BAD_iff: fixes f :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "(f \<in> mbs.BAD A P) = ((\<forall>i. f i \<in> A) \<and> bad P f)"
insert_remove: fixes a :: "'a"   and A :: "'a set" shows "Gcd\<^sub>f\<^sub>i\<^sub>n (insert a A) = gcd a (Gcd\<^sub>f\<^sub>i\<^sub>n (A - {a}))"
spanE: fixes p :: "'b"   and B :: "'b set" assumes "p \<in> span B" obtains A :: "'b set" and q :: "'b \<Rightarrow> 'a" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b *s b)"
PA_QA: fixes a :: "'a" assumes "QA a" shows "PA a"
smc_Cod_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
INFM_inj: fixes P :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<exists>\<^sub>\<infinity>x. P (f x)"   and "inj f" shows "\<exists>\<^sub>\<infinity>x. P x"
product_topology_open_contains_basis: fixes T :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and U :: "('c \<Rightarrow> 'd) set"   and x :: "'c \<Rightarrow> 'd" assumes "openin (product_topology T I) U"   and "x \<in> U" shows "\<exists>X. x \<in> Pi\<^sub>E I X \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> Pi\<^sub>E I X \<subseteq> U"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {0..<Suc n} = prod g {0..<n} * g n"
necess_cond_1_rhs: fixes x :: "'a" assumes "x \<in> \<V>" shows "int (size {#p \<in># mset_set (\<V> - {x}) \<times># filter_mset ((\<in>) x) \<B>. fst p \<in> snd p#}) = \<Lambda> * (\<v> - 1)"
AR_imp_absolute_extensor: fixes S :: "'a set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and U :: "'b set" assumes "AR S"   and "continuous_on T f"   and "f ` T \<subseteq> S"   and "closedin (top_of_set U) T" obtains g :: "'b \<Rightarrow> 'a" where "continuous_on U g" and "g ` U \<subseteq> S" and "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
of_nat_eq_1_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero n = one) = (n = 1)"
dirichlet_prod_commutes: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g = dirichlet_prod g f"
shift_operator_ergodic_def: fixes M :: "'a measure" shows "shift_operator_ergodic M \<equiv> prob_space M"
embP3_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "embP3 P = (\<lambda>(ps, n). dom ps = UNIV \<and> (\<forall>s. P (Partial_Evaluation.emb ps s)) \<and> n = (0::'a))"
infsetsum_Sigma: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "countable A"   and "\<And>i. countable (B i)"   and "f abs_summable_on Sigma A B" shows "infsetsum f (Sigma A B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B x. f (x, y))"
assms_2: shows "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> prob_with_noop \<Psi>  t"
rep_eq: fixes x :: "('b, 'a) comp_fun_idem" shows "set_fold_cfi x = Finite_Set.fold (comp_fun_idem_apply x)"
leadsTo_Union: fixes S :: "'c set set"   and F :: "'c program"   and B :: "'c set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A \<longmapsto> B" shows "F \<in> \<Union> S \<longmapsto> B"
rel_fun_Sup_distr: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and G :: "('c \<Rightarrow> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "rel_magma.times_rel_fun \<rho> (\<Squnion>G) f = \<Squnion>{rel_magma.times_rel_fun \<rho> g f |g. g \<in> G}"
order_trans_rules_27: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < b"   and "b = c" shows "a < c"
dgrad_p_set_leI_insert: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set"   and f :: "'t \<Rightarrow>\<^sub>0 'd" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d {f} G" shows "dgrad_p_set_le d (insert f F) G"
not_NoMsg_eq: fixes m :: "'b message_af" shows "(m \<noteq> NoMsg) = (\<exists>x. m = Msg x)"
f12: fixes ss :: "symbol list"   and ssa :: "symbol list"   and n :: "nat" shows "take (length ss) ssa = take n (take (length ss) ssa) \<or> length (take n ss) = n"
power_Sum_any: fixes f :: "'a \<Rightarrow> nat"   and c :: "'b" assumes "finite {a. f a \<noteq> 0}" shows "c ^ Sum_any f = (\<Prod>a. c ^ f a)"
glueing: fixes U :: "'a set"   and I :: "real set"   and V :: "real \<Rightarrow> 'a set"   and s :: "real \<Rightarrow> 'b" assumes "open_cover_of_open_subset S is_open U I V"   and "\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> \<FF> (V i)"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<rho> (V i) (V i \<inter> V j) (s i) = \<rho> (V j) (V i \<inter> V j) (s j)" shows "\<exists>t. t \<in> \<FF> U \<and> (\<forall>i. i \<in> I \<longrightarrow> \<rho> U (V i) t = s i)"
adopt_node_def: shows "Shadow_DOM.adopt_node \<equiv> l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.remove_child get_disconnected_nodes set_disconnected_nodes get_ancestors_di"
inv: shows "invariant ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b m\<^sub>s\<^sub>b"
lspasl_starl_inv: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Gamma :: "bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a"   and Delta :: "bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "Gamma \<and> sep_algebra.sep_conj plus sep_disj A B h0 \<longrightarrow> Delta" shows "(\<exists>h1 h2. Gamma \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> B h2) \<longrightarrow> Delta"
is_merge_status_3: fixes a :: "status"   and st' :: "status" shows "is_success (merge_status a st') = (is_success a \<and> is_success st')"
Un_insert_left: fixes a :: "'x"   and B :: "'x set"   and C :: "'x set" shows "insert a B \<union> C = insert a (B \<union> C)"
is_unit_gcd: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = algebraic_semidom.coprime times one a b"
geodesic_segment_param_in_geodesic_spaces_6: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and t :: "real"   and x :: "'a"   and y :: "'a" assumes "class.geodesic_space dist uniformity open"   and "t \<in> {0..dist x y}" shows "dist x (geodesic_segment_param {x--y} x t) = t"
scount_infinite_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(scount P \<omega> = \<infinity>) = alw (ev P) \<omega>"
seq_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>\<^sub>SC.seq g f = (\<F>\<^sub>SC.arr f \<and> \<F>\<^sub>SC.arr g \<and> seq g f)"
less_max_iff_disj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "less z (ord.max less_eq x y) = (less z x \<or> less z y)"
simps_2: fixes x31 :: "acom"   and x32 :: "acom"   and y31 :: "acom"   and y32 :: "acom" shows "(x31;; x32 = y31;; y32) = (x31 = y31 \<and> x32 = y32)"
elims: fixes x :: "'b infty"   and xa :: "'b infty"   and y :: "'b infty" assumes "x + xa = y" obtains uu_ :: "'b infty" where "x = uu_" and "xa = Infty" and "y = Infty"   | v :: "'b" where "x = Infty" and "xa = Num v" and "y = Infty"   | a :: "'b" and b :: "'b" where "x = Num a" and "xa = Num b" and "y = Num (a + b)"