hcju/isabelleps huggingface dataset:
\<A>1d_3: shows "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> \<A>1d"
axiom_impiff_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_impiff p q \<equiv> Thm (Imp (Imp p q) (Imp (Imp q p) (Iff p q)))"
simps_9: fixes N :: "nat"   and a :: "floatarith"   and n :: "nat" shows "plain_floatarith N (Power a n) = plain_floatarith N a"
cases: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "Rangep r a" obtains aa :: "'c" and b :: "'d" where "a = b" and "r aa b"
equiv_zmset_def: shows "equiv_zmset = (\<lambda>(Mp, Mn) (Np, Nn). Mp + Nn = Np + Mn)"
qAFresh_imp_qFresh: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "qAFresh xs x X" shows "qFresh xs x X"
wi_hom_mult: fixes a :: "int"   and b :: "int" shows "word_of_int a * word_of_int b = word_of_int (a * b)"
AbruptProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (c # cs, css, Abrupt s) \<rightarrow> (cs, css, Abrupt s)"
assms: shows "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f"
hyp: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f (0\<^sub>\<nat>) \<times>\<^sub>\<circ> f (1\<^sub>\<nat>)"
parts_insert_LtK: fixes K :: "ltkey"   and H :: "msg set" shows "parts (insert (LtK K) H) = insert (LtK K) (parts H)"
span_span: fixes A :: "'c set" shows "ideal (ideal A) = ideal A"
notEmptyFinestpart: fixes X :: "'a set" assumes "X \<noteq> {}" shows "finestpart X \<noteq> {}"
document_ptrs: shows "h \<turnstile> document_ptr_kinds_M \<rightarrow>\<^sub>r document_ptrs"
RA1: fixes P :: "('b, 'd) urel"   and Q :: "('d, 'e) urel"   and R :: "('e, 'c) urel" shows "P ;; Q ;; R = (P ;; Q) ;; R"
map_def: fixes Fg :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" shows "E.map Fg \<equiv> if A2_BxA2.arr Fg then A2_B.Map (fst Fg) (snd Fg) else B.null"
cmp_rfn5: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "\<forall>i\<le>r. cmp_rfn G r f s g (i * s) = f i"
tuple_in_hom: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<guillemotleft>\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> : local.dom h \<rightarrow> f \<down>\<down> g\<guillemotright>"
Sup_union_distrib: fixes A :: "('c \<times> 'd) set"   and B :: "('c \<times> 'd) set" shows "Sup_prod (A \<union> B) = sup_prod (Sup_prod A) (Sup_prod B)"
Transaction'_def: shows "transaction_merkle.Transaction' = (\<lambda>((cm, pm), views). Transaction cm pm views)"
reflp_fun1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "is_equality A"   and "reflp B" shows "reflp (rel_fun A B)"
eventually_principal: fixes P :: "'c \<Rightarrow> bool"   and S :: "'c set" shows "eventually P (principal S) = (\<forall>x\<in>S. P x)"
arr_char': fixes f :: "(bool, bool list) concrete_category.arr" shows "J.arr f = (f = J.MkIde False \<or> f = J.MkIde True \<or> f \<in> (\<lambda>x. J.MkArr False True [x]) ` {False, True})"
bind_eq_return_spmf: fixes p :: "'m spmf"   and f :: "'m \<Rightarrow> 'l spmf"   and x :: "'l" shows "(p \<bind> f = return_spmf x) = ((\<forall>y\<in>set_spmf p. f y = return_spmf x) \<and> lossless_spmf p)"
bind_advantage_def: fixes \<A> :: "nat \<times> nat \<Rightarrow> (nat \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "rivest_commit.bind_advantage \<A> \<equiv> spmf (rivest_commit.bind_game \<A>) True"
C_eq_All_untilSorted_withSimps: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = Cp p"
Conjs_def: shows "Conjs = {z. \<exists>A0 A1 Delta Gamma. z = (FConj Pos A0 A1 # Gamma @ Delta, {A0 # Gamma, A1 # Delta})}"
lrev_work3_lrev_work2_eq: shows "lrev_work3 = lrev_work2"
fds_zero_converges: fixes s :: "'a" shows "fds_converges 0 s"
ide_char: fixes a :: "'a" shows "B.ide a = (B.arr a \<and> B.dom a = a \<and> B.cod a = a)"
DisjointPaths_Q': shows "DisjointPaths H_x v0 v1 Q'"
counit_simps_1: shows "arr \<i>[trg \<mu>]"
trancl_into_trancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
goes_wrong_def: fixes e :: "exp" shows "goes_wrong e \<equiv> \<exists>e'. e \<longrightarrow>* e' \<and> stuck e' \<and> \<not> isval e'"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "list_all2 (R OO S) = list_all2 R OO list_all2 S"
C: shows "(x # xs @ ws) @ [w] \<cong> {ws @ [w], zs, P}"
iff_simps_5: fixes p :: "fm" shows "Ferrack.iff F p = not p"
yBij: shows "conjy \<in> Bij (subgroups_of_size p)"
h8: shows "0 < n \<longrightarrow> MPoly_Type.degree (sum f {0..n - 1}) x < n"
lcoset_refl: fixes a :: "'g" shows "a \<in> a +o G"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "ShadowRootClass.known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some shadow_root_ptr \<Rightarrow> is_shadow_root_ptr shadow_root_ptr))"
decide_universal_def: fixes fml :: "rat poly fml" shows "decide_universal fml = (let (fml_struct, polys) = convert fml in Let (find_consistent_signs polys) (list_all (lookup_sem fml_struct)))"
enabled_then_sas_applicable: fixes \<pi> :: "char list"   and s :: "nat \<Rightarrow> nat option" assumes "enabled \<pi> s" shows "is_operator_applicable_in s ((abs_ast_operator \<circ> the \<circ> lookup_operator) \<pi>)"
measure_spmf_spmf_of_pmf: fixes p :: "'a pmf" shows "measure_spmf (spmf_of_pmf p) = measure_pmf p"
exu_rename2: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "y = x \<or> y \<notin> Fvars \<phi>" shows "exu x \<phi> = exu y (subst \<phi> (Var y) x)"
mod_eq_0_iff_dvd: fixes a :: "'a"   and b :: "'a" shows "(a mod b = (0::'a)) = (b dvd a)"
"4_3": assumes "comp kx ky = Gt" shows "prod_ord_pair P ((kx, vx) # xs) ys = (lex_ord_pair (\<lambda>k x y. if P k x y then Some Eq else None) ((kx, vx) # xs) ys = Some Eq)"
partial_tauTE: fixes s :: "'c net_state"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, \<tau>, s') \<in> pnet_sos S T" obtains sa :: "'c net_state" and s'a :: "'c net_state" and t :: "'c net_state" where "s = SubnetS sa__ t__" and "s' = SubnetS s'a t__" and "(sa__, \<tau>\<^sub>n, s'a) \<in> S"   | t :: "'c net_state" and t' :: "'c net_state" and sa :: "'c net_state" where "s = SubnetS sa t" and "s' = SubnetS sa t'" and "(t, \<tau>\<^sub>n, t') \<in> T"
conv: fixes x :: "'n" assumes "x \<in> S" shows "(\<lambda>k. f k x) \<longlonglongrightarrow> g x"
reachable_extend_eq: fixes F :: "'a program" shows "reachable (extend h F) = extend_set h (reachable F)"
A: shows "map (\<lambda>p. sgn (poly p x)) ps = map (\<lambda>p. sgn (poly p y)) ps'"
sgnx_poly_at_top: fixes p :: "real poly" shows "sgnx (poly p) at_top = sgn_pos_inf p"
i_in_q: shows "\<iota> \<in> Red_I_\<G>_L_q q NL"
strategyproofness: shows "pmf (sds R35) b + pmf (sds R35) a \<le> pmf (sds R9) b + pmf (sds R9) a"
iter_le_lfp: fixes n :: "nat" shows "(f ^^ n) bot \<le> lfp f"
with_def: fixes zero :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "span.with zero plus \<equiv> \<lambda>scale b. {sum_with plus zero (\<lambda>a. scale (r a) a) t |t r. finite t \<and> t \<subseteq> b}"
comp_inv_arr: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.inverse_arrows f g" shows "g \<odot> f = Maps.dom f"
int_of_vint_def: shows "int_of_vint = inv vint_of_int"
HomObj: shows "Hom\<^bsub>C\<^esub>[\<midarrow>,X] \<in> obj\<^bsub>CatExp (Op C) SET\<^esub>"
caratheodory: fixes \<mu> :: "'a set \<Rightarrow> ennreal" assumes "positive (G ` I) \<mu>"   and "countably_additive (G ` I) \<mu>" shows "\<exists>\<mu>'. (\<forall>s\<in>G ` I. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G ` I)) \<mu>'"
linear: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "le a b \<or> le b a"
fresh_2: fixes a :: "name"   and x3 :: "'d"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) psi" shows "a \<sharp> x3\<langle>x2\<rangle>.x1 = (a \<sharp> x3 \<and> a \<sharp> x2 \<and> a \<sharp> x1)"
ring_hom_cring_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "ring_hom_cring R S h \<equiv> cring R \<and> cring S \<and> ring_hom_cring_axioms R S h"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat" assumes "\<And>i n. P (Bv i) n"   and "\<And>T body n. P body (n + 1) \<Longrightarrow> P (Abs T body) n"   and "\<And>f t n. \<lbrakk>P f n; P t n\<rbrakk> \<Longrightarrow> P (f $ t) n"   and "\<And>v va n. P (Ct v va) n"   and "\<And>v va n. P (Fv v va) n" shows "P a0 a1"
ideal_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.ring plus zero minus uminus times" shows "ring.ideal plus zero uminus times {zero}"
select_convs_2: fixes tag :: "obj_tag"   and "values" :: "vn \<Rightarrow> val option"   and more :: "'a" shows "values \<lparr>tag = tag, values = values, \<dots> = more\<rparr> = values"
nop_bij_vdomain: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_bij A n f" shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
dsf': fixes k :: "nat" assumes "k < n" shows "matrix_sum d f' k \<in> carrier_mat d d"
is_letter_def: fixes c :: "char" shows "is_letter c = (c \<in> set letters)"
sum2Chain: fixes Q :: "pi"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "Q \<noteq> Q'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> Q'"
ccomp_succs: fixes c :: "com"   and n :: "int" shows "succs (ccomp c) n \<subseteq> {n..n + size (ccomp c)}"
insert_is_Un: fixes a :: "'a"   and A :: "'a set" shows "insert a A = {a} \<union> A"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso (\<eta> a)"
homogeneous_hom_component: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "homogeneous (hom_component p n)"
R2: shows "R \<V>2 Tr\<^bsub>ES2\<^esub>"
awalk_verts_induce: fixes S :: "'a set" shows "pre_digraph.awalk_verts (G \<restriction> S) = awalk_verts"
sum_list_closed: fixes f :: "'c \<Rightarrow> 'g"   and as :: "'c list" assumes "set (map f as) \<subseteq> G" shows "sum_list (map f as) \<in> G"
generateI: fixes E :: "'a set"   and H :: "'a set" assumes "subgroup E B"   and "H \<subseteq> E"   and "\<And>K. \<lbrakk>subgroup K B; H \<subseteq> K\<rbrakk> \<Longrightarrow> E \<subseteq> K" shows "E = generate B H"
Smith_normal_form_min: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Smith_normal_form A = Smith_normal_form_upt_k A (min (nrows A) (ncols A))"
perron_frobenius_both: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "\<exists>v. eigen_vector A v (complex_of_real (HMA_Connect.spectral_radius A)) \<and> real_non_neg_vec v"
join_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
rel_prod2_def: fixes R :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'c" shows "rel_prod2 R a = (\<lambda>(c, b). R a b)"
mono_det_ref: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq> P'"   and "S \<sqsubseteq> S'" shows "(P [+] S) \<sqsubseteq> (P' [+] S')"
map_indets_inverseE_Polys: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "inj_on f X"   and "p \<in> P[X]" shows "map_indets (the_inv_into X f) (map_indets f p) = p"
ext_snoc_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_snoc ext \<equiv> ext ext \<and> ext_snoc_axioms ext"
finite_fourier_poly'_unique: fixes k :: "nat"   and p :: "complex poly"   and ws :: "nat \<Rightarrow> complex" assumes "0 < k"   and "degree p \<le> k - 1"   and "\<forall>m\<le>k - 1. ws m = poly p (unity_root k (int m))" shows "p = finite_fourier_poly' ws k"
reachable_mk_symmetricI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>* v" shows "u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v"
bal_substitute: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree"   and c :: "'b btree" assumes "bal (Node (ls @ (a, b) # rs) t)"   and "height t = height c"   and "bal c" shows "bal (Node (ls @ (c, b) # rs) t)"
prod: shows "mat_of_rows n fs = of_int_hom.mat_hom A * mat_of_rows n gs"
separating_of_bipartite_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "('a + 'a) set" assumes "web \<Gamma>" shows "web.separating_of_bipartite \<Gamma> S = (let A_S = Inl -` S; B_S = Inr -` S in A_S \<inter> B_S \<union> A \<Gamma> \<inter> A_S \<union> B \<Gamma> \<inter> B_S)"
nmp_eq: fixes x :: "i"   and y :: "i" assumes "M1" shows "PP x y = (P x y \<and> \<not> P y x)"
free_variable: fixes x :: "name"   and t :: "lam"   and xs :: "name list" assumes "\<not> x \<sharp> t"   and "x \<sharp> xs" shows "\<not> x \<sharp> bind xs t"
mono_nn_integral: fixes F :: "'b \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "mono F" shows "mono (\<lambda>x. integral\<^sup>N M (F x))"
no_messages_introduced_if_no_channel: fixes init :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and final :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and i :: "nat" assumes "trace init t final"   and "\<forall>i. channel i = None \<longrightarrow> msgs init i = []"   and "channel cid = None" shows "msgs (s init t i) cid = []"
in_homE: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>2 b\<guillemotright>" obtains "C2.arr f" and "C2.dom f = a" and "C2.cod f = b"
\<A>ss_OR_def: fixes X :: "'pub0 \<times> 'pub1"   and C1 :: "('msg0 \<times> 'msg1) \<times> 'bool \<times> ('bool \<times> 'response0) \<times> 'bool \<times> 'response1"   and C2 :: "('msg0 \<times> 'msg1) \<times> 'bool \<times> ('bool \<times> 'response0) \<times> 'bool \<times> 'response1" shows "\<A>ss_OR X C1 C2 = (if fst (fst (snd (snd C1))) \<noteq> fst (fst (snd (snd C2))) then \<A>ss0 (fst X) (fst (fst C1), fst (fst (snd (snd C1))), snd (fst (snd (snd C1)))) (fst (fst C2), fst (fst (snd (snd C2))), snd (fst (snd (snd C2)))) \<bind> (\<lambda>w0. return_spmf (Inl w0)) else \<A>ss1 (snd X) (snd (fst C1), fst (snd (snd (snd C1))), snd (snd (snd (snd C1)))) (snd (fst C2), fst (snd (snd (snd C2))), snd (snd (snd (snd C2)))) \<bind> (\<lambda>w1. return_spmf (Inr w1)))"
not_isAbrD: fixes s :: "('a, 'b) xstate" assumes "\<not> isAbr s" shows "(\<exists>s'. s = Normal s') \<or> s = Stuck \<or> (\<exists>f. s = Fault f)"
dag_T: shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>T\<rparr> \<in>\<^sub>\<circ> dg_Rel \<alpha>\<lparr>Arr\<rparr>"
components_are_iso: fixes a :: "'a" assumes "hom.ide a" shows "hom.iso (GF.map a)"
domain_very_strict: fixes x :: "'a" shows "(ad (ad x) = (zero_class.zero::'a)) = (x = (zero_class.zero::'a))"
Int_insert_right: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A \<inter> insert a B = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
lhss_preserves_cls: fixes ES :: "('b list set \<times> 'b trm set) set"   and Y :: "'b list set"   and yrhs :: "'b trm set" shows "lhss (Subst_all ES Y yrhs) = lhss ES"
dsum_def: fixes r :: "(('a, 'b) node set \<times> ('a, 'b) node set) set"   and s :: "(('a, 'b) node set \<times> ('a, 'b) node set) set" shows "dsum r s \<equiv> (\<Union>(x, x')\<in>r. {(In0 x, In0 x')}) \<union> (\<Union>(y, y')\<in>s. {(In1 y, In1 y')})"
that: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and m :: "nat" assumes "is_path \<pi>'"   and "\<pi> =\<^bsub>n\<^esub> \<pi>'"   and "k < m"   and "\<pi>' m = ipd (\<pi>' k)"   and "\<forall>l\<in>{k..<m}. \<pi>' l \<noteq> ipd (\<pi>' k)" shows "thesis"
is_strict_rewrite_ord_asym: fixes rel :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and x :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and y :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_strict_rewrite_ord rel"   and "rel x y" shows "\<not> rel y x"
prime_factors_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize x) = {p. normalization_semidom.prime zero times one normalize p \<and> dvd.dvd times p x}"
flow_usolves_ode: fixes x0 :: "'a" assumes "t\<^sub>0 \<in> T"   and "x0 \<in> S" shows "(flow t\<^sub>0 x0 usolves_ode f from t\<^sub>0) (ex_ivl x0) S"
reindex_bij_betw: fixes h :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "bij_betw h S T" shows "(\<Sum>x\<in>S. g (h x)) = sum g T"
int_of_real_mult: fixes a :: "real"   and b :: "real" assumes "real_is_int a"   and "real_is_int b" shows "int_of_real (a * b) = int_of_real a * int_of_real b"
rk_ac\<beta>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {a, c, \<beta>} = 2"
sim_submodel_aux: fixes s :: "'ss"   and n :: "nat" assumes "s \<in> worlds MCS\<^bsub>n\<^esub>" shows "gen_model MCS s = gen_model MCS\<^bsub>n\<^esub> s"
measurable_entropy_density: fixes b :: "real"   and M :: "'d measure"   and N :: "'d measure" shows "entropy_density b M N \<in> borel_measurable M"
abs_le_iff: fixes a :: "'b"   and b :: "'b" shows "(\<bar>a\<bar> \<le> b) = (a \<le> b \<and> - a \<le> b)"
eventually_inf: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and F' :: "'a filter" shows "eventually P (inf F F') = (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
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 "Cring_Poly.compose R (f \<otimes>\<^bsub>P\<^esub> h) g = Cring_Poly.compose R f g \<otimes>\<^bsub>P\<^esub> Cring_Poly.compose R h g"
indep_vars_subset: fixes M :: "'d measure"   and M' :: "'e \<Rightarrow> 'f measure"   and X :: "'e \<Rightarrow> 'd \<Rightarrow> 'f"   and I :: "'e set"   and J :: "'e set" assumes "prob_space M"   and "prob_space.indep_vars M M' X I"   and "J \<subseteq> I" shows "prob_space.indep_vars M M' X J"
SUP_sup_distrib: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'd" shows "sup (Sup (f ` A)) (Sup (g ` A)) = (SUP a\<in>A. sup (f a) (g a))"
Vfrom_succ: fixes A :: "V"   and i :: "V" shows "Vfrom A (ZFC_in_HOL.succ i) = A \<union>\<^sub>\<circ> VPow (Vfrom A i)"
responses_\<I>_mono: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('h, 'i) \<I>"   and x :: "'h" assumes "\<I> \<le> \<I>'"   and "x \<in> outs_\<I> \<I>" shows "responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x"
prefixes_length: fixes s :: "nat \<Rightarrow> nat option"   and z :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat" assumes "s \<in> \<R>" shows "length (prefixes s z i j x) = Suc x"
C\<pi>_def: shows "C\<pi> = classical_operator \<pi>"
right_idem: fixes a :: "'g"   and b :: "'g" shows "sup (sup a b) b = sup a b"
runit_in_hom_2: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>S.runit f : f \<star>\<^sub>S S.src f \<Rightarrow>\<^sub>S f\<guillemotright>"
assms_4: shows "card E * (r - 1) ^ k * fact k < card F"
omega_sum_aux_less: fixes k :: "nat"   and ms :: "nat list" shows "omega_sum_aux k ms < \<omega> \<up> ord_of_nat k"
vconn_implies_hpar_2: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C \<mu> = trg\<^sub>C f"
inner_simps_4: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x - y) \<bullet> z = x \<bullet> z - y \<bullet> z"
obs_split_det: fixes xs :: "'node list"   and x :: "'node"   and xs' :: "'node list"   and ys :: "'node list"   and y :: "'node"   and ys' :: "'node list"   and S :: "'node set" assumes "xs @ x # xs' = ys @ y # ys'"   and "obs_intra x S \<noteq> {}"   and "\<forall>x'\<in>set xs'. \<exists>x''. call_of_return_node x' x'' \<and> x'' \<in> S"   and "\<forall>zs z zs'. xs = zs @ z # zs' \<and> obs_intra z S \<noteq> {} \<longrightarrow> (\<exists>z''\<in>set (zs' @ [x]). \<exists>nx. call_of_return_node z'' nx \<and> nx \<notin> S)"   and "obs_intra y S \<noteq> {}"   and "\<forall>y'\<in>set ys'. \<exists>y''. call_of_return_node y' y'' \<and> y'' \<in> S"   and "\<forall>zs z zs'. ys = zs @ z # zs' \<and> obs_intra z S \<noteq> {} \<longrightarrow> (\<exists>z''\<in>set (zs' @ [y]). \<exists>ny. call_of_return_node z'' ny \<and> ny \<notin> S)" shows "xs = ys \<and> x = y \<and> xs' = ys'"
self_pref: fixes a :: "'b list" shows "a \<le>p a"
ide_dom: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and f :: "'ag" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.dom C f)"
nomatch_prx: shows "\<forall>pt\<in>set prx. \<not> repNodes_eq pt p low high repa"
vector_add_ldistrib: fixes c :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "c *s (x + y) = c *s x + c *s y"
simps_5: fixes b :: "cexpr"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "free_vars (IF\<^sub>c b THEN e1 ELSE e2) = free_vars b \<union> free_vars e1 \<union> free_vars e2"
simps_3: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "cl_inv (Node l B k v r) = (cl_inv l \<and> cl_inv r)"
pcr_cr_eq: shows "pcr_dual_ord (=) = cr_dual_ord"
abs_Gromov_completion_limit: fixes u :: "nat \<Rightarrow> 'b" assumes "Gromov_completion_rel u u" shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> abs_Gromov_completion u"
finite_Diff_insert: fixes A :: "'e set"   and a :: "'e"   and B :: "'e set" shows "finite (A - Set.insert a B) = finite (A - B)"
eq: shows "lucas_lehmer_to_real (a, b) = lucas_lehmer_to_real (c, d)"