hcju/isabelleps huggingface dataset:
cs_done_implies_has_snapshotted: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = Done"   and "channel cid = Some (p, q)" shows "ps (distributed_system.s channel trans send recv init t i) q \<noteq> None"
as_proj_eq_as: fixes as :: "(('c, 'd) fmap \<times> ('c, 'e) fmap) list"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'e) fmap) set"   and vs :: "'c set" assumes "no_effectless_act as"   and "as \<in> valid_plans PROB"   and "prob_dom PROB \<subseteq> vs" shows "as_proj as vs = as"
xs_def: shows "(xs\<^sub>L, xs\<^sub>R) = split_at (n div 2) xs"
sig_inv_setD: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "r \<in> sig_inv_set' j" shows "keys (vectorize_poly r) \<subseteq> {0..<j}"
all_zero_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(\<forall>u. inner x u = 0) = (x = zero)"
hgsub: shows "h ` g ` (S - {b}) \<subseteq> S"
sub: fixes k :: "nat" shows "integral S (\<lambda>x. f k x - f 0 x) = integral S (f k) - integral S (f 0)"
pcr_stream_def: fixes T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_stream T \<equiv> llist_all2 T OO cr_stream"
swap_Recv_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
natural_extension_measure_def: shows "natural_extension_measure = projective_family.lim UNIV (\<lambda>I. distr M (Pi\<^sub>M I (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>I. (T ^^ nat (i - Min I)) x)) (\<lambda>i. borel)"
free_generator_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme"   and H :: "'j set" shows "free_generator R M H = (generator R M H \<and> (\<forall>n s f. s \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> f \<in> {j. j \<le> n} \<rightarrow> H \<and> inj_on f {j. j \<le> n} \<and> l_comb R M n s f = \<zero>\<^bsub>M\<^esub> \<longrightarrow> s \<in> {j. j \<le> n} \<rightarrow> {\<zero>\<^bsub>R\<^esub>}))"
ax_thin: fixes G :: "prog"   and A' :: "'b triple set"   and ts :: "'b triple set"   and A :: "'b triple set" assumes "G,A'|\<turnstile>ts"   and "A' \<subseteq> A" shows "G,A|\<turnstile>ts"
lnull_cons_code: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and s :: "'c" shows "lnull_cons g s = (case g s of Done \<Rightarrow> True | Skip s' \<Rightarrow> lnull_cons g s' | Yield x xa \<Rightarrow> False)"
the_eq_trivial: fixes a :: "'c" shows "(THE x. x = a) = a"
ov: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> q \<and> q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)"
produces_imp_Max_atom: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a" assumes "production N C = {A}" shows "A = Max (atms_of C)"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat" assumes "\<And>i k. P (Bv i) k"   and "\<And>t u k. \<lbrakk>P t k; P u k\<rbrakk> \<Longrightarrow> P (t $ u) k"   and "\<And>uu_ t k. P t (k + 1) \<Longrightarrow> P (Abs uu_ t) k"   and "\<And>v va uw_. P (Ct v va) uw_"   and "\<And>v va uw_. P (Fv v va) uw_" shows "P a0 a1"
arrE: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "functor_category A B"   and "partial_magma.arr (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>)) f" obtains "f \<noteq> partial_magma.null (concrete_category.COMP (Collect (functor A B)) (\<lambda>F G. Collect (natural_transformation A B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A B \<sigma> \<tau>))" and "natural_transformation A B (concrete_category.Dom f) (concrete_category.Cod f) (concrete_category.Map f)"
strongly_fair_def2: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "strongly_fair enabled taken = \<box>(\<lambda>s. \<not> (\<box>(\<lambda>s. (\<diamond>enabled) s \<and> \<not> taken s)) s)"
antisymPD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
simps: fixes A :: "'a mat" shows "square_mat A = (dim_col A = dim_row A)"
gext_cong2: fixes s1 :: "st"   and l :: "lname \<Rightarrow> val option"   and s2 :: "st" shows "s1\<le>|set_locals l s2 = s1\<le>|s2"
bounded_def: fixes P :: "'a \<Rightarrow> real" shows "Expectations.bounded P \<equiv> \<exists>b. bounded_by b P"
quotient_is_field: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "maximalideal I R"   and "cring R" shows "field (R Quot I)"
ucast_add: fixes a :: "'b word"   and b :: "'b word" shows "UCAST('b \<rightarrow> 'b signed) (a + b) = UCAST('b \<rightarrow> 'b signed) a + UCAST('b \<rightarrow> 'b signed) b"
monom_1_dvd_iff: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0" shows "(monom (1::'a) n dvd p) = (n \<le> Polynomial.order (0::'a) p)"
div_int_pos_iff: fixes k :: "int"   and l :: "int" shows "(0 \<le> k div l) = (k = 0 \<or> l = 0 \<or> 0 \<le> k \<and> 0 \<le> l \<or> k < 0 \<and> l < 0)"
t1_space_derived_set_of_finite: fixes X :: "'b topology" shows "t1_space X = (\<forall>S. finite S \<longrightarrow> X derived_set_of S = {})"
uminus_bit0_def: fixes x :: "'a bit0" shows "- x = Abs_bit0' (- Rep_bit0 x)"
Units_l_inv_ex: fixes x :: "'a mat" assumes "x \<in> Units (ring_mat TYPE('a) n ())" shows "\<exists>y\<in>carrier (ring_mat TYPE('a) n ()). y \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>"
arr_Par_ArrCod_in_Vset: shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
extract_matrix_def: fixes a :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "extract_matrix a m n = mat m n (\<lambda>(i, j). a (i * n + j))"
intro: fixes R :: "('c, 'd) Ring_scheme" assumes "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>" shows "Idomain_axioms R"
p_squarefree: fixes x :: "real" shows "\<not> (poly p x = 0 \<and> poly (ps ! 1) x = 0)"
rangeI: fixes f :: "'b::type \<Rightarrow> 'a::type"   and x :: "'b::type" shows "(f::'b::type \<Rightarrow> 'a::type) (x::'b::type) \<in> range f"
unitary_adjoint: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "unitary (adjoint A)"
step_less: fixes r :: "'a"   and r' :: "'a" assumes "I r"   and "C r"   and "B r = Some r'" shows "(r', r) \<in> M"
simps_1: fixes s :: "'b sparc_state" shows "SEQ 0 s = Some s"
binary_meet_exists': fixes x :: "'a"   and y :: "'a" shows "\<exists>l. {x, y} >>| l"
length_replicate: fixes n :: "nat"   and x :: "'a" shows "|replicate n x| = n"
1: shows "tab = (rep r \<star> tab\<^sub>0 r) \<cdot> tab\<^sub>0.trnr\<^sub>\<eta> (tab\<^sub>1 r) (tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*)"
power_minus: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) n = times (power.power one times (uminus one) n) (power.power one times a n)"
to_nat_on_def: fixes S :: "'c set" shows "to_nat_on S = (SOME f. if finite S then bij_betw f S {..<card S} else bij_betw f S UNIV)"
B_def: shows "B = {c * arctan (- b / a), c * pi / 2}"
minus_eq: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" shows "x \<ominus>\<^bsub>F\<^esub> y = x \<oplus>\<^bsub>F\<^esub> \<ominus>\<^bsub>F\<^esub> y"
are_all_operators_applicable_set: fixes s :: "'c \<Rightarrow> bool option"   and ops :: "'c strips_operator list" shows "are_all_operators_applicable s ops = (\<forall>op\<in>set ops. \<forall>v\<in>set (strips_operator.precondition_of op). s v = Some True)"
simps_2: fixes ts :: "ct list"   and v :: "t" shows "select_return_top ts TAny (TSome v) = TopType (take (length ts - 3) ts @ [TSome v])"
has_subprob_density_cong: fixes N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_subprob_density M N f = has_subprob_density M N g"
mult_less_cancel_left1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times c b) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
Nil_in_star: fixes A :: "'a list set" shows "[] \<in> star A"
subcls_notin_has_fields: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and F :: "String.literal"   and D :: "String.literal"   and Tm :: "ty \<times> fmod" assumes "P \<turnstile> C has_fields FDTs"   and "((F, D), Tm) \<in> set FDTs" shows "\<not> (subcls1 P)\<^sup>+\<^sup>+ D C"
sup_left_commute: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
assms_5: fixes S :: "('a, 'b) strand_step list"   and t :: "('a, 'b) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "simple S"   and "P S"   and "fv t \<subseteq> V \<union> \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S))" shows "P (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
cong_dvd_modulus_poly: fixes x :: "'b poly"   and y :: "'b poly"   and m :: "'b poly"   and n :: "'b poly" assumes "[x = y] (mod m)"   and "n dvd m" shows "[x = y] (mod n)"
rem: fixes A :: "'a set"   and \<phi> :: "'a" shows "A .\<bottom>. \<phi> \<equiv> {B. B \<subseteq> A \<and> \<not> B \<turnstile> \<phi> \<and> (\<forall>B'\<subseteq>A. B \<subset> B' \<longrightarrow> B' \<turnstile> \<phi>)}"
one_closed: shows "\<one>\<^bsub>G i\<^esub> \<in> carrier (G i)"
mat_of_cblinfun_inverse: fixes B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "cblinfun_of_mat (mat_of_cblinfun B) = B"
lt_in_keys: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "punit.lt p \<in> keys p"
induct: fixes P3 :: "assertion_gadget \<Rightarrow> bool"   and P1 :: "diagram \<Rightarrow> interface \<Rightarrow> interface \<Rightarrow> bool"   and P2 :: "command_gadget \<Rightarrow> interface \<Rightarrow> interface \<Rightarrow> bool"   and x1 :: "diagram"   and x2 :: "interface"   and x3 :: "interface"   and x4 :: "command_gadget"   and x5 :: "interface"   and x6 :: "interface"   and x7 :: "assertion_gadget" assumes "\<And>p. P3 (Rib p)"   and "\<And>G uu_ uv_ x. \<lbrakk>prov_dia G uu_ uv_; P1 G uu_ uv_\<rbrakk> \<Longrightarrow> P3 (Exists_dia x G)"   and "\<And>P c Q. prov_triple (asn P, c, asn Q) \<Longrightarrow> P2 (Com c) P Q"   and "\<And>G P Q H. \<lbrakk>prov_dia G P Q; P1 G P Q; prov_dia H P Q; P1 H P Q\<rbrakk> \<Longrightarrow> P2 (Choose_dia G H) P Q"   and "\<And>G P. \<lbrakk>prov_dia G P P; P1 G P P\<rbrakk> \<Longrightarrow> P2 (Loop_dia G) P P"   and "\<And>G. \<lbrakk>wf_dia G; \<And>v. v \<in> fset G^V \<Longrightarrow> prov_ass (G^\<Lambda> v); \<And>v. v \<in> fset G^V \<Longrightarrow> P3 (G^\<Lambda> v); \<And>e. e \<in> set G^E \<Longrightarrow> prov_com (snd3 e) (\<Otimes>v|\<in>|fst3 e. bot_ass (G^\<Lambda> v)) (\<Otimes>v|\<in>|thd3 e. top_ass (G^\<Lambda> v)); \<And>e. e \<in> set G^E \<Longrightarrow> P2 (snd3 e) (\<Otimes>v|\<in>|fst3 e. bot_ass (G^\<Lambda> v)) (\<Otimes>v|\<in>|thd3 e. top_ass (G^\<Lambda> v))\<rbrakk> \<Longrightarrow> P1 G (top_dia G) (bot_dia G)" shows "(prov_dia x1 x2 x3 \<longrightarrow> P1 x1 x2 x3) \<and> (prov_com x4 x5 x6 \<longrightarrow> P2 x4 x5 x6) \<and> (prov_ass x7 \<longrightarrow> P3 x7)"
uniformly_continuous_on_compose: fixes s :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "uniformly_continuous_on s g"   and "uniformly_continuous_on (g ` s) f" shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
as_needed_asses_submap_exec_vii: fixes f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v" shows "f \<subseteq>\<^sub>f g"
A: fixes R :: "real" assumes "0 \<le> R" shows "\<exists>F\<subseteq>cball (0::'a) R. (\<forall>x\<in>F. \<forall>y\<in>F. x = y \<or> D__ \<le> dist x y) \<and> finite F \<and> Ca * R ^ DIM('a) \<le> real (card F)"
dot_ortho: fixes x :: "'a" shows "x \<bullet> rot x = 0"
distinct_205: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x13 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Roll x13"
d'_1: fixes d' :: "real" assumes "0 < d'"   and "d' \<le> d" shows "sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) + d')) = sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) + d)) \<and> sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) - d')) = sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) - d))"
random_oracle_def: fixes \<sigma> :: "'d \<Rightarrow> 'a option"   and x :: "'d" shows "random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> p \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
routing_def: shows "CIPHER.routing \<equiv> (1\<^sub>C |\<^sub>= lassocr\<^sub>C) \<odot> swap_lassocr \<odot> (1\<^sub>C |\<^sub>= (1\<^sub>C |\<^sub>= swap_lassocr) \<odot> swap_lassocr) \<odot> rassocl\<^sub>C"
frac_less2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and w :: "'b"   and z :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less_eq x y"   and "less zero w"   and "less w z" shows "less (divide x z) (divide y w)"
cEnum_fun_def: shows "CENUM('a \<Rightarrow> 'b) = (case ID CENUM('a) of None \<Rightarrow> None | Some (enum_a, enum_all_a, enum_ex_a) \<Rightarrow> case ID CENUM('b) of None \<Rightarrow> None | Some (enum_b, enum_all_b, enum_ex_b) \<Rightarrow> Some (map (\<lambda>ys. the \<circ> map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum_b), \<lambda>P. all_n_lists enum_all_b (\<lambda>bs. P (the \<circ> map_of (zip enum_a bs))) (length enum_a), \<lambda>P. ex_n_lists enum_ex_b (\<lambda>bs. P (the \<circ> map_of (zip enum_a bs))) (length enum_a)))"
embed_ge_0: fixes G :: "'a \<Rightarrow> bool"   and s :: "'a" shows "0 \<le> \<guillemotleft> G \<guillemotright> s"
unit_factor_power: 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 a :: "'a"   and n :: "nat" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "unit_factor (power.power one times a n) = power.power one times (unit_factor a) n"
pq_coprime_int: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "coprime (int p) (int q)"
eval_fps_mult: fixes z :: "'a"   and f :: "'a fps"   and g :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f"   and "ereal (norm z) < fps_conv_radius g" shows "eval_fps (f * g) z = eval_fps f z * eval_fps g z"
oalist_inv_raw_ConsD3: fixes k :: "'a"   and v :: "'c"   and xs :: "('a \<times> 'c) list"   and k' :: "'a" assumes "oalist_inv_raw ((k, v) # xs)"   and "k' \<in> fst ` set xs" shows "lt k k'"
select_convs_5: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "mmu \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = mmu"
wary_cases_apps: fixes t :: "('s, 'v) tm" assumes "wary t" obtains \<zeta> :: "('s, 'v) hd" and ss :: "('s, 'v) tm list" where "t = apps (Hd \<zeta>) ss" and "\<And>sa. sa \<in> set ss \<Longrightarrow> wary sa" and "enat (length ss) \<le> arity_hd \<zeta>"
update_convs_8: fixes dests' :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> nat option"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "dests_update dests' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests' dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr>"
lprefix_llength_le: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys" shows "llength xs \<le> llength ys"
YMap1: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and \<eta> :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) NatTrans" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "NT \<eta> : YFtor C @@ X \<Longrightarrow> F" shows "YMapInv C X F (YMap C X \<eta>) = \<eta>"
equals0I: fixes A :: "'e set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
RG_el_decomp_aezdeltafun: fixes c :: "('a, 'g) aezfun" assumes "c \<in> Supgroup.group_ring" shows "\<exists>ras. set (map snd ras) \<subseteq> H \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
Plus_Times_pre_cong_mono: fixes L' :: "'a language"   and L :: "'a language"   and K :: "'a language"   and K' :: "'a language"   and R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "L' \<le> L"   and "K \<le> K'"   and "Plus_Times_pre_cong R L K" shows "Plus_Times_pre_cong R L' K'"
arr_cod: fixes f :: "'d" assumes "D.arr f" shows "D.arr (D.cod f)"
rep_eq: fixes x :: "'c ltl_prop_equiv_quotient"   and xa :: "'c ltl_prop_equiv_quotient" shows "x \<up>\<longrightarrow>\<^sub>P xa = Rep x \<longrightarrow>\<^sub>P Rep xa"
hyps_3: shows "e \<turnstile>\<^sub>R t : T"
abc_in_Q: shows "a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
left_commute: fixes b :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" shows "(b::'a::refinement_lattice) \<parallel> ((a::'a::refinement_lattice) \<parallel> (c::'a::refinement_lattice)) = a \<parallel> (b \<parallel> c)"
orderI: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "a = inf_prod a b" shows "less_eq_prod a b"
assms_2: shows "G\<^sub>2 \<sqsubseteq> G\<^sub>3"
seqp_unicastTE: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and r :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), a, \<xi>', r) \<in> seqp_sos \<Gamma>" obtains "a = unicast (s\<^sub>i\<^sub>p \<xi>) (s\<^sub>m\<^sub>s\<^sub>g \<xi>)" and "\<xi>' = \<xi>" and "r = p"   | "a = \<not>unicast (s\<^sub>i\<^sub>p \<xi>)" and "\<xi>' = \<xi>" and "r = q"
inject_3: fixes x4 :: "'b"   and y4 :: "'b" shows "(Inca.instr.IStore x4 = Inca.instr.IStore y4) = (x4 = y4)"
take_take: fixes m :: "nat"   and n :: "nat"   and x :: "Value" shows "\<psi>\<^sup>D\<^bsub>m\<^esub>\<cdot>(\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x) = \<psi>\<^sup>D\<^bsub>min m n\<^esub>\<cdot>x"
le: shows "\<guillemotleft> G \<guillemotright> && (wp a Q \<circ> \<phi>) \<tturnstile> \<guillemotleft> G \<guillemotright> && wp b (Q \<circ> \<phi>)"
results'_gpv_Done: fixes x :: "'d" shows "results'_gpv (Generative_Probabilistic_Value.Done x) = {x}"
intros_8: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<or>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<or>\<^sub>! [r]\<^sub>! q)"