hcju/isabelleps huggingface dataset:
indifferent_pref_def: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c" shows "x \<^bsub>r\<^esub>\<approx> y \<equiv> x \<^bsub>r\<^esub>\<preceq> y \<and> y \<^bsub>r\<^esub>\<preceq> x"
Units_inv_closed: fixes x :: "cltn2" assumes "x \<in> Units \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "inv\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
klift_prop1: fixes f :: "'g \<Rightarrow> 'f set"   and g :: "'e \<Rightarrow> 'g set" shows "(f\<^sup>\<dagger> \<circ> g)\<^sup>\<dagger> = f\<^sup>\<dagger> \<circ> g\<^sup>\<dagger>"
cases: fixes a :: "ind" assumes "Nat a" obtains "a = Zero_Rep"   | i :: "ind" where "a = Suc_Rep i" and "Nat i"
valuesum_rec_Mapping_2: fixes x :: "'d \<times> 'c"   and xs :: "('d \<times> 'c) list" shows "valuesum (AList_Mapping.Mapping (x # xs)) = (let l = SOME l. l \<in> Mapping.keys (AList_Mapping.Mapping (x # xs)) in the (Mapping.lookup (AList_Mapping.Mapping (x # xs)) l) + valuesum (Mapping.delete l (AList_Mapping.Mapping (x # xs))))"
execute_bind_success: fixes f :: "'a Heap"   and h :: "heap"   and g :: "'a \<Rightarrow> 'b Heap" assumes "success f h" shows "execute (f \<bind> g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
pseudominimal_ne_bottom: fixes P :: "'b set"   and x :: "'b" assumes "order.pseudominimal_in P x" shows "x \<noteq> order.bottom P"
hyps: assumes "p \<in> set ps" shows "\<turnstile> imply ps p"
map_cong_pred: fixes x :: "'c dual"   and ya :: "'c dual"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "pred_dual (\<lambda>z. f z = g z) ya" shows "map_dual f x = map_dual g ya"
AG_in_lem: fixes x :: "'a"   and s :: "'a set" assumes "x \<in> AG s" shows "x \<in> s"
rtrancl_fold_product1: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "{(r, s). \<exists>a\<in>A. s = f a r}\<^sup>* = {(r, s). \<exists>a\<in>lists A. s = fold f a r}"
Mem_Zero_E: fixes x :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (x IN Zero) H \<turnstile> A"
bij_Rep: shows "bij_betw Rep UNIV {0..<int CARD('a)}"
min_gallery_pgallery: fixes xs :: "'a set list" assumes "min_gallery xs" shows "pgallery xs"
subprob_space_axioms: fixes p :: "'c spmf" shows "subprob_space (measure_spmf p)"
merge_winning_strategies: fixes p :: "Player" shows "\<exists>\<sigma>. G'.strategy p \<sigma> \<and> (\<forall>v\<in>G'.winning_region p. G'.winning_strategy p \<sigma> v)"
distinct_31: fixes x31 :: "nat"   and x32 :: "'e"   and x6 :: "nat" shows "unicast x31 x32 \<noteq> deliver x6"
f_def: shows "f = (\<lambda>(q, m). m(k \<mapsto> q))"
B: shows "padic_val p (\<ominus>\<^bsub>padic_int p\<^esub> a) \<le> padic_val p a"
simps_1: fixes ts :: "t list" shows "type_update_select (Type ts) = (if 3 \<le> length ts \<and> ts ! (length ts - 2) = ts ! (length ts - 3) then consume (Type ts) [TAny, TSome T_i32] else Bot)"
hyps: shows "length (filter p xs) = card {i. i < length xs \<and> p (xs ! i)}"
ptok0: shows "\<forall>i\<in>{p..k}. int (n mod p choose i) * (1 + fps_X ^ p) ^ (n div p) $ (k - i) = 0"
character_data_ptr_casts_commute2: fixes character_data_ptr :: "(_) character_data_ptr" shows "cast (cast character_data_ptr) = Some character_data_ptr"
no_volatile_Write\<^sub>s\<^sub>b_sb: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"
TVarB_def: fixes x2 :: "tyvrs"   and x1 :: "ty" shows "TVarB x2 x1 \<equiv> Abs_binding (TVarB_Rep x2 x1)"
sel_2: fixes x1 :: "'f"   and x2 :: "'e" shows "snd (x1, x2) = x2"
E_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "Graph.E c \<equiv> {(u, v). c (u, v) \<noteq> (0::'a)}"
assms_4: shows "\<forall>x\<in>cbox a b - s. norm (f x) \<le> B"
enqueue_correct_1: fixes s :: "'queue"   and x :: "'t" assumes "queue_invar s" shows "queue_\<alpha> (queue_enqueue x s) = queue_\<alpha> s @ [x]"
assms_1: shows "Q (\<sigma> |\<^sub>s i)"
inj_semiring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_semiring_hom hom \<equiv> semiring_hom hom \<and> injective hom"
prems_3: shows "fdisjnt (frees f) (freess (x # xs))"
is_lub_unique: fixes T1 :: "ty"   and T2 :: "ty"   and T3 :: "ty"   and T4 :: "ty" assumes "\<turnstile> lub(T1, T2) = T3"   and "\<turnstile> lub(T1, T2) = T4" shows "T3 = T4"
pdg_dghm_proj_is_dghm': fixes i :: "V"   and \<CC> :: "V"   and \<DD> :: "V" assumes "i \<in>\<^sub>\<circ> I"   and "\<CC> = dg_prod I \<AA>"   and "\<DD> = \<AA> i" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
simps_17: fixes v :: "'a exp" shows "val_of (throw v) = None"
simps_16: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x1 :: "'b" shows "(case Normal x1 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f1 x1"
SN_relto_SN_rel_conv: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" shows "SN (S\<^sup>* O R O S\<^sup>*) = SN_rel R S"
tauPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "\<tau>.P \<simeq> \<tau>.Q"
factgroup_subgroup_union_subgroup: fixes A :: "'a set set" assumes "subgroup A (G Mod H)" shows "subgroup (\<Union> A) G"
pred_sets1: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and N :: "'c measure" assumes "{x \<in> space M. P x} \<in> sets M"   and "f \<in> N \<rightarrow>\<^sub>M M" shows "Measurable.pred N (\<lambda>x. P (f x))"
assms_2: shows "(y, \<sigma>') \<in> set_spmf (exec_gpv random_permutation \<A> \<sigma>)"
prems_2: shows "n \<notin> set (tl (ns @ [m']))"
RedNew: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h' = h(a \<mapsto> (C, init_fields FDTs))" shows "P \<turnstile> \<langle>new C,(h, l)\<rangle> \<rightarrow> \<langle>addr a,(h', l)\<rangle>"
subsFAtom_def: fixes atms :: "(signs \<times> predicate \<times> vbl list) list"   and nAs :: "(nat \<times> formula) list"   and z :: "signs"   and P :: "predicate"   and vs :: "vbl list" shows "subsFAtom atms nAs z P vs = {((z, P, vs) # atms, nAs)}"
higher_deriv_add: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
MStar_1: fixes ms :: "mregex" assumes "ok m mr"   and "ms \<in> LPD mr" shows "ok m ms"
weight_less_def: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "weight_fun.weight_less w w0 scf s t \<equiv> vars_term_ms (scf_term scf s) \<subseteq># vars_term_ms (scf_term scf t) \<and> weight_fun.weight w w0 scf s < weight_fun.weight w w0 scf t"
DirProds_order: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "finite I" shows "order (DirProds G I) = prod (order \<circ> G) I"
cases: fixes x :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" obtains p :: "'t \<Rightarrow>\<^sub>0 'b" and q :: "'t \<Rightarrow>\<^sub>0 'b" where "x = Inl (p, q)"   | j :: "nat" where "x = Inr j"
scalar_nop: fixes A :: "V"   and f :: "V" assumes "scalar A f" shows "nop A []\<^sub>\<circ> f"
assms_3: shows "(\<lambda>(x, l, v, r). n x l v r) \<in> B \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M M \<Otimes>\<^sub>M tree_sigma M \<rightarrow>\<^sub>M A"
sel_15: fixes x21 :: "nat"   and x22 :: "'b"   and x23 :: "'b"   and x24 :: "'a"   and x25 :: "'a"   and x26 :: "'c" shows "un_Send5 (Send x21 x22 x23 x24 x25 x26) = x25"
enttI_true: fixes P :: "assn"   and Q :: "assn" assumes "P * true \<Longrightarrow>\<^sub>A Q * true" shows "P \<Longrightarrow>\<^sub>t Q"
quasinorm_sum_subset_2: fixes N2 :: "'c quasinorm"   and N1 :: "'c quasinorm" shows "N2 \<subseteq>\<^sub>N N1 +\<^sub>N N2"
adjoint_col: fixes i :: "nat"   and A :: "'c mat" assumes "i < dim_row A" shows "col (adjoint A) i = conjugate (row A i)"
redT_updLs_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" shows "redT_updLs ls t las \<equiv> (\<lambda>(l, la). upd_locks l t la) \<circ>$ ($ls, las$)"
has_dist_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat" shows "has_dist G m n = (\<exists>p. is_shortest_path G m n p)"
assms: shows "\<forall>\<^sub>F x in sequentially. f x = g x"
nth_Cons_Suc: fixes x :: "'d"   and xs :: "'d list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
\<phi>_def: shows "\<phi> \<equiv> (1 + sqrt 5) / 2"
rinit_ext: fixes P :: "(char list list \<times> char list exp) prog"   and e'' :: "char list exp"   and s'' :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "\<And>e' s'. P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> \<Longrightarrow> P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "P \<turnstile> \<langle>RI (C,e'') ; Cs \<leftarrow> e\<^sub>0,s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; Cs \<leftarrow> e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
abs_eq: fixes xa :: "'a"   and x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "smult xa (MPoly x) = MPoly (Poly_Mapping.map ((*) xa) x)"
dens: shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure (PRODUCT t t)) f"
induct: fixes P :: "(char list, char list, nat \<times> tid_or_exn) namespace \<Rightarrow> v store_v list \<Rightarrow> pat \<Rightarrow> v \<Rightarrow> (char list \<times> v) list \<Rightarrow> bool"   and a0 :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and a1 :: "v store_v list"   and a2 :: "pat"   and a3 :: "v"   and a4 :: "(char list \<times> v) list" assumes "\<And>envC s v' env. P envC s Pany v' env"   and "\<And>envC s x v' env. P envC s (Pvar x) v' env"   and "\<And>envC s l l' env. P envC s (Plit l) (Litv l') env"   and "\<And>envC s n ps n' t' vs env. (\<And>x2 x y init xa ya x3. \<lbrakk>nsLookup envC n = Some x2; (x, y) = x2; same_tid y t' \<and> length ps = x; same_ctor (id_to_n n, y) (n', t'); xa \<in> set ps; ya \<in> set vs; init = Match x3\<rbrakk> \<Longrightarrow> P envC s xa ya x3) \<Longrightarrow> P envC s (Pcon (Some n) ps) (Conv (Some (n', t')) vs) env"   and "\<And>envC s ps vs env. (\<And>init x y x3. \<lbrakk>length ps = length vs; x \<in> set ps; y \<in> set vs; init = Match x3\<rbrakk> \<Longrightarrow> P envC s x y x3) \<Longrightarrow> P envC s (Pcon None ps) (Conv None vs) env"   and "\<And>envC s p lnum env. (\<And>x2 x1. \<lbrakk>store_lookup lnum s = Some x2; x2 = Refv x1\<rbrakk> \<Longrightarrow> P envC s p x1 env) \<Longrightarrow> P envC s (Pref p) (Loc lnum) env"   and "\<And>envC s p t1 v2 env. P envC s p v2 env \<Longrightarrow> P envC s (Ptannot p t1) v2 env"   and "\<And>envC uu_ v va vb env. P envC uu_ (Plit v) (Conv va vb) env"   and "\<And>envC uu_ v va vb vc env. P envC uu_ (Plit v) (Closure va vb vc) env"   and "\<And>envC uu_ v va vb vc env. P envC uu_ (Plit v) (Recclosure va vb vc) env"   and "\<And>envC uu_ v va env. P envC uu_ (Plit v) (Loc va) env"   and "\<And>envC uu_ v va env. P envC uu_ (Plit v) (Vectorv va) env"   and "\<And>envC uu_ va v env. P envC uu_ (Pcon None va) (Litv v) env"   and "\<And>envC uu_ va vc vb env. P envC uu_ (Pcon None va) (Conv (Some vc) vb) env"   and "\<And>envC uu_ va v vb vc env. P envC uu_ (Pcon None va) (Closure v vb vc) env"   and "\<And>envC uu_ va v vb vc env. P envC uu_ (Pcon None va) (Recclosure v vb vc) env"   and "\<And>envC uu_ va v env. P envC uu_ (Pcon None va) (Loc v) env"   and "\<And>envC uu_ va v env. P envC uu_ (Pcon None va) (Vectorv v) env"   and "\<And>envC uu_ v va vb env. P envC uu_ (Pcon v va) (Litv vb) env"   and "\<And>envC uu_ vb va vc env. P envC uu_ (Pcon (Some vb) va) (Conv None vc) env"   and "\<And>envC uu_ v va vb vc vd env. P envC uu_ (Pcon v va) (Closure vb vc vd) env"   and "\<And>envC uu_ v va vb vc vd env. P envC uu_ (Pcon v va) (Recclosure vb vc vd) env"   and "\<And>envC uu_ v va vb env. P envC uu_ (Pcon v va) (Loc vb) env"   and "\<And>envC uu_ v va vb env. P envC uu_ (Pcon v va) (Vectorv vb) env"   and "\<And>envC uu_ v va env. P envC uu_ (Pref v) (Litv va) env"   and "\<And>envC uu_ v va vb env. P envC uu_ (Pref v) (Conv va vb) env"   and "\<And>envC uu_ v va vb vc env. P envC uu_ (Pref v) (Closure va vb vc) env"   and "\<And>envC uu_ v va vb vc env. P envC uu_ (Pref v) (Recclosure va vb vc) env"   and "\<And>envC uu_ v va env. P envC uu_ (Pref v) (Vectorv va) env" shows "P a0 a1 a2 a3 a4"
ocrit_specD: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and m :: "nat"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'e" assumes "ocrit_spec crit"   and "dickson_grading d"   and "set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "unique_idx (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps) data"   and "\<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "p \<in> B"   and "q \<in> B"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data hs ps p q" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
sort_key_def: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "sort_key f xs = foldr (insort_key f) xs []"
simps: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "rquicksort R xs = (if xs = [] then return_pmf ([], 0) else pmf_of_set {..<length xs} \<bind> (\<lambda>i. let x = xs ! i in case partition (\<lambda>y. (y, x) \<in> R) (delete_index i xs) of (ls, rs) \<Rightarrow> rquicksort R ls \<bind> (\<lambda>(ls, n1). rquicksort R rs \<bind> (\<lambda>(rs, n2). return_pmf (ls @ [x] @ rs, length xs - 1 + n1 + n2)))))"
scaleR_scaleR: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "a *\<^sub>R b *\<^sub>R x = (a * b) *\<^sub>R x"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'b" shows "s1 (a + b) x = s1 a x + s1 b x"
addNonSync: fixes sa :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list"   and y1 :: "'c event" shows "sa setinterleaves ((t, u), A) \<and> y1 \<notin> A \<longrightarrow> sa @ [y1] setinterleaves ((t @ [y1], u), A) \<and> sa @ [y1] setinterleaves ((t, u @ [y1]), A)"
Tree_def: fixes G :: "('a, 'b) Graph_scheme" shows "Tree G \<equiv> Graph G \<and> Tree_axioms G"
g_def: fixes w :: "complex" shows "g w = (\<Sum>i. (deriv ^^ (i + n)) f \<xi> / fact (i + n) * (w - \<xi>) ^ i)"
state_succ_pair: fixes s :: "('c, 'd) fmap"   and p :: "('c, 'd) fmap"   and e :: "('c, 'd) fmap" shows "state_succ s (p, e) = (if p \<subseteq>\<^sub>f s then e ++ s else s)"
T8_\<nu>: fixes w :: "i"   and \<psi> :: "\<nu> \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<^sub>\<nu>x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
\<theta>_4: shows "fv s \<union> fv t - subst_domain \<theta> \<subseteq> X"
mC_C_mono: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set"   and theta' :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "mC_C theta c d s t P F"   and "theta \<subseteq> theta'" shows "mC_C theta' c d s t P F"
mult_local_ring_at_def: shows "pi.mult_local_ring_at \<equiv> local.mult_rel"
getEnv_updEnv_idEnv: fixes x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and y :: "'var" shows "(idEnv [x \<leftarrow> X]_xs) ys y = (if ys = xs \<and> y = x then Some X else None)"
oproj_def: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "oproj r s f \<equiv> Field s \<subseteq> f ` Field r \<and> compat r s f"
zip_commute: fixes xs :: "'b list"   and ys :: "'c list" shows "zip xs ys = Map2.map2 (\<lambda>x y. (y, x)) ys xs"
intros: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. R a b \<Longrightarrow> \<not> R b a" shows "asymp R"
map_def: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c" assumes "identity_functor C" shows "identity_functor.map C f = (if partial_magma.arr C f then f else partial_magma.null C)"
starfun_ln_HInfinite_gt_zero: fixes x :: "real star" assumes "x \<in> HInfinite"   and "0 < x" shows "0 < (*f* real_ln) x"
case_1: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x11 :: "'e"   and x12 :: "'f"   and x13 :: "'f" shows "(case Trans x11 x12 x13 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13"
tkt_thread_def: shows "tkt_thread \<equiv> \<lambda>ts k. THE t. t < N \<and> has_ticket (ts t) k"
assms_5: fixes n :: "'a" assumes "n \<in> A" shows "\<exists>m\<in>B. f n \<le>\<^sub>o\<^sub>w g m"
le_bot: fixes a :: "'a" assumes "a \<le> 0" shows "a = 0"
assms: shows "c = ((Cn n f gs, xs, ls) # fs, rv)"
eq_smooth_gen: fixes f :: "real \<Rightarrow> 'a"   and s :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f C1_differentiable_on s"   and "\<forall>x. f x = g x" shows "g C1_differentiable_on s"
linear_homeomorphism_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "inj f" obtains g :: "'c \<Rightarrow> 'b" where "homeomorphism (f ` S) S g f"
omega_coinduct_var2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x \<cdot> y" shows "y \<le> x\<^sup>\<omega>"
leg_simps_1: shows "C.arr \<mu>\<nu>_\<pi>.leg0"
bernoulli_pmf_half_conv_pmf_of_set: shows "bernoulli_pmf (1 / 2) = pmf_of_set UNIV"
elims_3: fixes x :: "'a \<times> 'a"   and xa :: "'a \<times> 'a" assumes "\<not> my_int_rel x xa" obtains xb :: "'a" and y :: "'a" and u :: "'a" and v :: "'a" where "x = (xb, y)" and "xa = (u, v)" and "xb + v \<noteq> u + y"
foundation11: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> y" shows "(\<tau> \<Turnstile> x or y) = (\<tau> \<Turnstile> x \<or> \<tau> \<Turnstile> y)"
exesig_def: fixes exetheory :: "exetheory" shows "exesig exetheory = (case exetheory of ExeTheory x1 x2 \<Rightarrow> x1)"
normalize_snd_quot_of_fract: fixes x :: "'a fract" shows "normalize (snd (quot_of_fract x)) = snd (quot_of_fract x)"
get_ancestors_si_is_l_get_ancestors: shows "l_get_ancestors get_ancestors_si"
rel_S_pos_distr_imp: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok_G :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself"   and tytok_S :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_S_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_S"
lower_or_eq_def: fixes t :: "'a trm"   and s :: "'a trm" shows "lower_or_eq t s = (t = s \<or> (t, s) \<in> trm_ord)"
fps_of_poly_const: fixes c :: "'b" shows "fps_of_poly [:c:] = fps_const c"
kat_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<cdot> y \<le> y \<cdot> d z" shows "y \<cdot> ad z \<le> ad x \<cdot> y"
assms_4: shows "(u, v) \<notin> uconnected F"
generate_PS_def: fixes emBits :: "nat"   and hLen :: "nat" shows "generate_PS emBits hLen = bv_prepend (roundup emBits 8 * 8 - sLen - hLen - 16) \<zero> []"
S_index: fixes s :: "'b" assumes "s \<in> \<Union> (S ` I)" shows "s \<in> S (index s)"
alist_and_NegPos_map_getNeg_getPos_matches: fixes spts :: "'a negation_type list"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and C :: "'a \<Rightarrow> 'b"   and a :: "action"   and p :: "'c" shows "((\<forall>m\<in>set (getNeg spts). matches \<gamma> (MatchNot (Match (C m))) a p) \<and> (\<forall>m\<in>set (getPos spts). matches \<gamma> (Match (C m)) a p)) = matches \<gamma> (alist_and (NegPos_map C spts)) a p"
reciprocal_preserve_orientation: fixes H :: "ocircline" assumes "0\<^sub>h \<in> disc_compl H" shows "pos_oriented (moebius_ocircline moebius_reciprocal H)"
less_eq_dres_split_conv: fixes a :: "'c dres"   and b :: "'c dres" shows "(a \<le> b) = (case (a, b) of (dSUCCEEDi, xa) \<Rightarrow> True | (dFAILi, dFAILi) \<Rightarrow> True | (dFAILi, _) \<Rightarrow> False | (dRETURN aa, dSUCCEEDi) \<Rightarrow> False | (dRETURN aa, dFAILi) \<Rightarrow> True | (dRETURN aa, dRETURN b) \<Rightarrow> aa = b)"
weak_consensus_def: fixes vals :: "'f \<Rightarrow> 'g"   and dec :: "'h \<Rightarrow> 'g option"   and rho :: "'i \<Rightarrow> 'j \<Rightarrow> 'h" shows "weak_consensus vals dec rho \<equiv> (\<forall>v. (\<forall>p. vals p = v) \<longrightarrow> (\<forall>n p w. dec (rho n p) = Some w \<longrightarrow> w = v)) \<and> (\<forall>m n p q v w. dec (rho m p) = Some v \<and> dec (rho n q) = Some w \<longrightarrow> v = w) \<and> (\<forall>p. \<exists>n. dec (rho n p) \<noteq> None)"