hcju/isabelleps huggingface dataset:
add_cancel_right_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
field_divide_inverse: fixes a :: "'b"   and b :: "'b" shows "a / b = a * inverse b"
def_lfp_unfold: fixes h :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "h \<equiv> lfp f"   and "mono f" shows "h = f h"
add_closed: fixes V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x + y \<in> V"
sup_least_basic_type: fixes \<tau> :: "'a basic_type"   and \<rho> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<le> \<rho>"   and "\<sigma> \<le> \<rho>" shows "\<tau> \<squnion> \<sigma> \<le> \<rho>"
f_def: shows "f = (\<lambda>\<sigma>. signof \<sigma> * (\<Prod>i = 0..<m + n. M $$ (i, \<sigma> i)))"
hrelation_hunion: fixes f :: "hf"   and g :: "hf" shows "hrelation (f \<squnion> g) = (hrelation f \<and> hrelation g)"
b_assn_def: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and P :: "'c \<Rightarrow> bool" shows "b_assn A P \<equiv> \<lambda>x y. A x y * \<up> (P x)"
isFDERIV_lengthD_1: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list" assumes "isFDERIV n xs fas vs" shows "length fas = n"
assms_2: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"
mod_minus1_right: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo a (uminus one) = zero"
IH: assumes "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop"   and "\<forall>t1\<le>t. req t1 = []" shows "ack t = [init_state]"
Fourier_sum_limit_Dirichlet_kernel_part: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "0 < d"   and "d \<le> pi" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l = (\<lambda>n. LINT x|lebesgue_on {0..d}. Dirichlet_kernel n x * (f (t + x) + f (t - x) - 2 * l)) \<longlonglongrightarrow> 0"
fun_lub_def: fixes L :: "'e set \<Rightarrow> 'd"   and A :: "('c \<Rightarrow> 'e) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
card_of_Times_same_infinite: fixes A :: "'a set" assumes "infinite A" shows "|A \<times> A| =o |A|"
assertion_isol: fixes p :: "'a"   and y :: "'a"   and x :: "'a" assumes "test p" shows "(y \<le> p\<^sup>o \<cdot> x) = (p \<cdot> y \<le> x)"
ep_pair_sfun: shows "ep_pair sfun_emb sfun_prj"
treap_unique: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "treap t1"   and "treap t2"   and "set_tree t1 = set_tree t2"   and "inj_on snd (set_tree t1)" shows "t1 = t2"
assms_1: shows "find_sub_False xs ns = Some ((rs, w, ws), ys, z, zs)"
is_character_data_ptr_kind_obtains: fixes node_ptr :: "(_) node_ptr" assumes "is_character_data_ptr_kind node_ptr" obtains character_data_ptr :: "(_) character_data_ptr" where "cast character_data_ptr = node_ptr"
reachable_finsert: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and aa :: "nat"   and ba :: "nat"   and b :: "transition" assumes "reachable s e" shows "reachable s (finsert ((aa, ba), b) e)"
a: shows "R[x\<turnstile>n>y] = NotL <c>.N u"
is_monomial_setD: fixes A :: "('d \<Rightarrow>\<^sub>0 'e) set"   and p :: "'d \<Rightarrow>\<^sub>0 'e" assumes "is_monomial_set A"   and "p \<in> A" shows "is_monomial p"
cart_prod_fst: fixes x :: "'e \<times> 'f"   and A :: "'e set"   and B :: "'f set" assumes "x \<in> A \<times> B" shows "fst x \<in> A"
ring_subset_carrier: fixes x :: "'e"   and A :: "'e set"   and R :: "('e, 'f) partial_object_scheme" assumes "x \<in> A"   and "A \<subseteq> carrier R" shows "x \<in> carrier R"
has_subprob_densityD_4: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "subprob_space M"
indexed_var_def: fixes i :: "'d" shows "L.indexed_var i = L.indexed_pmult (L.indexed_const \<one>\<^bsub>L\<^esub>) i"
check_def: fixes h :: "'a"   and a :: "'a"   and e :: "nat"   and z :: "nat \<times> nat" shows "check h a e z = (\<^bold>g [^] fst z \<otimes> g' [^] snd z = a \<otimes> h [^] e \<and> a \<in> carrier \<G>)"
wf_sig_iff_exe_wf_sig_pre: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "alist_conds cto"   and "alist_conds tao"   and "exe_osig_conds sa" shows "wf_sig (map_of cto, map_of tao, translate_osig sa) = (exe_wf_osig sa \<and> fst ` set (snd sa) = fst ` set tao \<and> (\<forall>type\<in>fst ` set (snd sa). \<forall>ars\<in>snd ` set (the (lookup (\<lambda>k. k = type) (snd sa))). the (lookup (\<lambda>k. k = type) tao) = length ars) \<and> (\<forall>ty\<in>snd ` set cto. typ_ok_sig (map_of cto, map_of tao, translate_osig sa) ty))"
eval\<^sub>1_preserves_unmod: fixes P :: "nat exp prog"   and e :: "nat exp"   and h :: "heap"   and ls :: "val list"   and sh :: "sheap"   and e' :: "nat exp"   and h' :: "heap"   and ls' :: "val list"   and sh' :: "sheap"   and i :: "nat" assumes "P \<turnstile>\<^sub>1 \<langle>e,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>e',(h', ls', sh')\<rangle>"   and "unmod e i"   and "i < length ls" shows "ls ! i = ls' ! i"
split: fixes P :: "'k \<Rightarrow> bool"   and f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and prod :: "'l \<times> 'm" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
encode_state_variable_def: fixes t :: "nat"   and k :: "nat"   and v :: "bool option" shows "encode_state_variable t k v \<equiv> case v of Some True \<Rightarrow> Atom (State t k) | Some False \<Rightarrow> \<^bold>\<not> (Atom (State t k))"
wf_word_append: fixes n :: "nat"   and ws :: "('a \<times> bool list) list"   and vs :: "('a \<times> bool list) list" shows "wf_word n (ws @ vs) = (wf_word n ws \<and> wf_word n vs)"
inv_comp_right_2: fixes g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" assumes "S.iso (g \<cdot>\<^sub>S f)"   and "S.iso f" shows "S.iso g"
pos_quad_def: shows "pos_quad = {p. - snd p \<le> 0} \<inter> {p. - fst p \<le> 0}"
add_sheaf_spec_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and s' :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.add_sheaf_spec R addition multiplication zero U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.add_rel (R\<setminus>\<pp>) R addition multiplication zero (s \<pp>) (s' \<pp>)"
axioms_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexRetraction X f" shows "ChamberComplexEndomorphism X f"
wMinU: fixes \<theta>\<^sub>1 :: "trm"   and \<theta>\<^sub>2 :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([trm.Min \<theta>\<^sub>1 \<theta>\<^sub>2]<>\<nu>) = (let (l1, u1) = [\<theta>\<^sub>1]<>\<nu>; (l2, u2) = [\<theta>\<^sub>2]<>\<nu> in (wmin l1 l2, wmin u1 u2))"
T_ins3_Same: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree" assumes "ins3 n d x t = Same2" shows "T_ins3 n d x t \<le> 2 * height t + 1"
monic_normalize: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "monic (normalize p)"
prod_list_split: fixes xs :: "real list"   and m :: "real" shows "prod_list xs = prod_list (list_neq xs m) * prod_list (list_eq xs m)"
qtableI: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and P :: "'b option list \<Rightarrow> bool"   and Q :: "'b option list \<Rightarrow> bool" assumes "table n A X"   and "\<And>x. \<lbrakk>x \<in> X; wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> Q x"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x; Q x\<rbrakk> \<Longrightarrow> x \<in> X" shows "qtable n A P Q X"
assms_4: shows "\<phi> \<in> set \<phi>s"
span_alt: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'d set" assumes "module scale" shows "module.span scale B = {\<Sum>x | f x \<noteq> (0::'c). scale (f x) x |f. {x. f x \<noteq> (0::'c)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'c)}}"
T_def_3: shows "\<forall>x\<in>T. f x \<in> S"
mu_sfis: fixes u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "\<And>m n. \<exists>a. a \<in> sfis (u m n) M"   and "measure_space M" shows "\<exists>c. \<forall>m. c m \<in> sfis (mu u m) M"
FGModuleHom_fscalar_mul: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and a :: "'b" assumes "FGModuleHom G smult V smult' T" shows "FGModuleHom G smult V smult' (\<lambda>v. aezfun_scalar_mult.fsmult smult' a (T v))"
nonzero_memE_2: fixes a :: "'a" assumes "a \<in> nonzero R" shows "a \<noteq> \<zero>"
MD_rows: shows "dim_row (mat_delete (make_mat n k B) 0 0) = n - 1"
indep_vars_PiM_components': fixes A :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'a \<Rightarrow> 'c measure" assumes "\<And>i. i \<in> A \<Longrightarrow> prob_space (M i)"   and "\<And>i. i \<in> A \<Longrightarrow> g i \<in> M i \<rightarrow>\<^sub>M N i" shows "prob_space.indep_vars (Pi\<^sub>M A M) N (\<lambda>i f. g i (f i)) A"
map_G_id0: shows "map_G id id id id id id = id"
gbm_succ_list_it_def: fixes g :: "'m1"   and v :: "'V" shows "gbm_succ_list_it g v \<equiv> case m1.lookup v g of None \<Rightarrow> set_iterator_emp | Some m2 \<Rightarrow> set_iterator_image (\<lambda>((v', m2), w). (w, v')) (set_iterator_product (m2.iteratei m2) (\<lambda>(v', y). s3.iteratei y))"
antisym: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" assumes "less_eq_prod b a"   and "less_eq_prod a b" shows "a = b"
hyps_5: shows "P \<turnstile> \<langle>RI (D,throw a) ; Cs \<leftarrow> e',(h', l', sh'')\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
appr_rel_def: shows "appr_rel = aforms_rel O \<langle>lv_rel\<rangle>set_rel"
assms_7: shows "x' \<notin> vars (p - Var x')"
sum_bounds_lt_plus1: fixes f :: "nat \<Rightarrow> 'b"   and mm :: "nat" shows "(\<Sum>k<mm. f (Suc k)) = sum f {1..mm}"
nfv_simps_2: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "nfv (Or \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"
ibounded_by_def: shows "local.ind_cpa.ibounded_by = (\<lambda>(\<A>1, \<A>2) q. \<exists>q1 q2. interaction_any_bounded_by \<A>1 q1 \<and> (\<forall>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2) \<and> q1 + q2 \<le> q)"
ball_eq_cball_iff: fixes x :: "'b"   and d :: "real"   and y :: "'b"   and e :: "real" shows "(ball x d = cball y e) = (d \<le> 0 \<and> e < 0)"
ereal_limsup_lim_mult: fixes u :: "nat \<Rightarrow> ereal"   and a :: "ereal"   and v :: "nat \<Rightarrow> ereal" assumes "u \<longlonglongrightarrow> a"   and "0 < a"   and "a \<noteq> \<infinity>" shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
null_space_eq_ker: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "linear (*s) (*s) f" shows "null_space (matrix f) = {x. f x = 0}"
assms_4: shows "(\<sigma>, p) \<in> oreachable A S U"
queue_invar_simps_2: fixes t :: "('c, 'd) BinomialTree"   and q :: "('c, 'd) BinomialTree list" shows "queue_invar (t # q) = (tree_invar t \<and> queue_invar q)"
Cons_path: fixes x :: "'a"   and xs :: "'a list"   and t :: "'a ttree" shows "(x # xs \<in> paths t) = (possible t x \<and> xs \<in> paths (nxt t x))"
pick_sum_def: fixes s0 :: "nat"   and l :: "nat set list" shows "pick_sum s0 l \<equiv> rfoldl (\<lambda>s x. ASSERT (x \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>y. y \<in> x) \<bind> (\<lambda>y. RETURN (s + y)))) s0 l"
trans'_def: fixes u :: "'c \<Rightarrow> real"   and u' :: "'c \<Rightarrow> real" shows "trans' u u' \<equiv> ((\<forall>c\<in>\<X>. real (k c) < u c \<and> real (k c) < u' c \<and> u \<noteq> u') \<longrightarrow> u' = u \<oplus> 5 / 10) \<and> ((\<exists>c\<in>\<X>. u c = 0 \<and> 0 < u' c \<and> (\<forall>c\<in>\<X>. \<not> (\<exists>d\<le>k c. u' c = real d))) \<longrightarrow> u' = delayedR ([u']\<^sub>\<R>) u)"
of_real_def: fixes r :: "real" shows "of_real r = r *\<^sub>R (1::'a)"
C_def: shows "C \<equiv> - ((ln (d * a + b) - 1 / (c + 1)) * (d * a + b) powr (c + 1) / (d * (c + 1)))"
ind_set_rows: shows "gs.lin_indpt (set (rows B))"
intro: fixes cl :: "'b potype" assumes "cl \<in> CompleteLattice" shows "CL cl"
P2_case_l_new_1_gt_e0_rewrite: fixes s :: "nat"   and r :: "nat"   and t :: "nat"   and x :: "nat"   and xa :: "nat"   and \<alpha> :: "nat" assumes "s < order \<G>" shows "(\<^bold>g [^] (r * ((order \<G> * order \<G> - s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)) + x) mod order \<G>) + (r * s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)) + xa) mod order \<G>), \<^bold>g [^] (r * \<alpha> * ((order \<G> * order \<G> - s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)) + x) mod order \<G>) + (r * s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)) + xa) mod order \<G> * \<alpha>) \<otimes> \<^bold>g [^] (t * ((order \<G> * order \<G> - s * nat (fst (bezw t (order \<G>)) mod int (order \<G>)) + x) mod order \<G> + s * nat (fst (bezw t (order \<G>)) mod int (order \<G>))))) = (\<^bold>g [^] (r * x + xa), \<^bold>g [^] (r * \<alpha> * x + xa * \<alpha>) \<otimes> \<^bold>g [^] (t * x))"
min_less_iff_disj: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.linorder less_eq less" shows "less (ord.min less_eq x y) z = (less x z \<or> less y z)"
X_rv: fixes n :: "nat" shows "random_variable borel (X n)"
valid_refs_invD_8: fixes m :: "'e"   and x :: "'f"   and s :: "'e process_name \<Rightarrow> ('g, 'e, 'h, 'f) Model.local_state" assumes "mut_m.reachable m x s"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) x s"
SeqStTermP_lemma: fixes m :: "name"   and v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name" assumes "atom m \<sharp> (v, i, t, u, s, k, n, sm, sm', sn, sn')"   and "atom n \<sharp> (v, i, t, u, s, k, sm, sm', sn, sn')"   and "atom sm \<sharp> (v, i, t, u, s, k, sm', sn, sn')"   and "atom sm' \<sharp> (v, i, t, u, s, k, sn, sn')"   and "atom sn \<sharp> (v, i, t, u, s, k, sn')"   and "atom sn' \<sharp> (v, i, t, u, s, k)" shows "{SeqStTermP v i t u s k} \<turnstile> (t EQ v AND u EQ i OR (IndP t OR t NEQ v) AND u EQ t) OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN k AND Var n IN k AND SeqStTermP v i (Var sm) (Var sm') s (Var m) AND SeqStTermP v i (Var sn) (Var sn') s (Var n) AND t EQ Q_Eats (Var sm) (Var sn) AND u EQ Q_Eats (Var sm') (Var sn')))))))"
bs_val_is_val: fixes w :: "bval"   and v :: "exp" assumes "bs_val w v" shows "is_val v"
minf_4: fixes t :: "'a" shows "\<exists>z. \<forall>x<z. (x \<noteq> t) = True"
simps_1: fixes x :: "'a"   and y :: "'a" shows "srcNet (DenyAllFromTo x y) = x"
singular_relboundary_def: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set" shows "singular_relboundary p X S \<equiv> \<lambda>c. \<exists>d. singular_chain (Suc p) X d \<and> (chain_boundary (Suc p) d, c) \<in> mod_subset p (subtopology X S)"
comp_galcon_def: fixes G :: "('f, 'e, unit, unit) galcon"   and F :: "('d, 'f, unit, unit) galcon" shows "G \<circ>\<^sub>g F = \<lparr>orderA = \<X>\<^bsub>F\<^esub>, orderB = \<Y>\<^bsub>G\<^esub>, lower = \<pi>\<^sup>*\<^bsub>G\<^esub> \<circ> \<pi>\<^sup>*\<^bsub>F\<^esub>, upper = \<pi>\<^sub>*\<^bsub>F\<^esub> \<circ> \<pi>\<^sub>*\<^bsub>G\<^esub>\<rparr>"
scaleR_adj: fixes a :: "real"   and A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(a *\<^sub>R A)* = a *\<^sub>R A*"
f3: shows "\<forall>i ia. \<not> 0 \<le> i \<or> 0 \<le> i + - 1 * ia \<or> i mod ia = i"
assms_2: shows "dgrad_p_set_le d {f} G"
lconcat_eq_LCons_conv: fixes xss :: "'c llist llist"   and x :: "'c"   and xs :: "'c llist" shows "(lconcat xss = LCons x xs) = (\<exists>xs' xss' xss''. xss = lappend (llist_of xss') (LCons (LCons x xs') xss'') \<and> xs = lappend xs' (lconcat xss'') \<and> set xss' \<subseteq> {xs. lnull xs})"
ask_env_def: fixes f :: "'e \<Rightarrow> ('e, 'f) envT" shows "ask_env f = EnvT (\<lambda>r. run_env (f r) r)"
shift_append: fixes xs :: "'a list"   and ys :: "'a list"   and s :: "'a stream" shows "(xs @ ys) @- s = xs @- ys @- s"
le_iff_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (inf x y = x)"
firstOccurrence_def: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and msg :: "('a, 'c) message"   and n :: "nat" assumes "execution trans sends start exec trace" shows "execution.firstOccurrence exec trace msg n \<equiv> (\<exists>p. isReceiverOf p msg) \<and> enabled (last exec) msg \<and> n < length exec \<and> enabled (exec ! n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> trace ! n') \<and> (n \<noteq> 0 \<longrightarrow> \<not> enabled (exec ! (n - 1)) msg \<or> msg = trace ! (n - 1))"
JN_preserves: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program"   and v :: "'c \<Rightarrow> 'e" shows "(JOIN I F \<in> preserves v) = (\<forall>i\<in>I. F i \<in> preserves v)"
vors_2: shows "(f12, f21) = split_face f ram1 ram2 []"
sum1: shows "gs.M.sumlist (map (\<lambda>j. fs.gs.\<mu> (i - 1) j \<cdot>\<^sub>v fs.gs.gso j) [0..<i - 1]) \<in> Rn"
data_prev'_def: shows "data_prev' \<equiv> dropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_prev"
bintr_cat: fixes m :: "nat"   and n :: "nat"   and b :: "int"   and a :: "int" shows "take_bit m (concat_bit n b a) = concat_bit n (take_bit (min m n) b) (take_bit (m - n) a)"
assms_3: shows "lr_of_tran rt fw ifs = Inr oft"
epsclo_def: fixes M :: "'b nfa"   and Q :: "hf set" assumes "nfa M" shows "nfa.epsclo M Q \<equiv> nfa.states M \<inter> (\<Union>q\<in>Q. {q'. (q, q') \<in> (eps M)\<^sup>*})"
comp_permute: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and k :: "'c \<times> 'c"   and l :: "'c \<times> 'c"   and h :: "'c \<times> 'c" assumes "f \<odot> g = k \<odot> l"   and "CopxC.seq f g"   and "CopxC.seq g h" shows "f \<odot> g \<odot> h = k \<odot> l \<odot> h"
run_raise_iff: fixes s :: "String.literal"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b" shows "run (raise s) \<sigma> \<sigma>' r = (\<sigma>' = None)"
supp_3: fixes x1 :: "pi" shows "supp (\<tau>.x1) = supp x1"
fun_upd_same: fixes f :: "'g \<Rightarrow> 'f"   and x :: "'g"   and y :: "'f" shows "(f(x := y)) x = y"
multi_member_last: fixes x :: "'a" shows "x \<in># \<LM> x  \<RM>"
tl_hd_not_empty: fixes l :: "'a list"   and a :: "'a" assumes "tl (l @ [a]) \<noteq> []" shows "l \<noteq> []"
le_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_eq z (ord.max less_eq x y) = (less_eq z x \<or> less_eq z y)"
valid_stepE: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s \<leadsto> s'" shows "\<exists>r. revision_step r s s'"
equal_pair_count_maximality1a: fixes ps :: "('b \<times> 'b) list" shows "equal_pair_count ps = length ps \<or> \<not> equal_pair (ps ! equal_pair_count ps)"
list_of_eucl_nth: fixes n :: "nat"   and x :: "'b" assumes "n < DIM('b)" shows "list_of_eucl x ! n = x \<bullet> Basis_list ! n"
exec_gpv_simps: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and gpv :: "('b, 'e, 'f) gpv"   and s :: "'d" shows "exec_gpv callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out rpv. callee s out \<bind> (\<lambda>(x, y). exec_gpv callee (rpv x) y))"
ab: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> T" shows "b + a \<bullet> y < a \<bullet> x"
assms_2: shows "t \<in> carrier R\<^bsup>\<omega>\<^esup>"
nth_bl_of_nth: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m < n" shows "rev (bl_of_nth n f) ! m = f m"
vequalityD1: fixes A :: "V"   and B :: "V" assumes "A = B" shows "A \<subseteq>\<^sub>\<circ> B"
set_disconnected_nodes_def: shows "set_disconnected_nodes \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes"
fst_set_ExcessTable_subset: fixes g :: "graph" shows "fst ` set (ExcessTable g (vertices g)) \<subseteq> \<V> g"
lspace_g: fixes n :: "nat" shows "g n \<in> lspace (lebesgue_on S) (ennreal p)"
mon_pl_def: fixes w :: "('d set \<times> 'd set) list" shows "mon_pl w \<equiv> foldl (\<union>) {} (map (\<lambda>e. fst e \<union> snd e) w)"
mono_ndet_F_left: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>F Q" shows "(P |-| S) \<sqsubseteq>\<^sub>F Q"
in_vars_is_Fun_not_unifiable: fixes x :: "'c"   and t :: "('d, 'c) Term.term" assumes "x \<in> vars_term t"   and "is_Fun t" shows "\<not> unifiable {(Var x, t)}"
cfifi: shows "f\<inverse> O f\<inverse> \<subseteq> f\<inverse>"
hom_induced3: shows "\<exists>hom_relmap. ((\<forall>p X S Y T f c. \<not> (continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> c \<in> carrier (relative_homology_group p X S)) \<longrightarrow> hom_relmap p X S Y T f c = \<one>\<^bsub>relative_homology_group p Y T\<^esub>) \<and> (\<forall>p X S Y T f. hom_relmap p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)) \<and> (\<forall>p X S Y T f c. continuous_map X Y f \<and> f ` (topspace X \<inter> S) \<subseteq> T \<and> singular_relcycle p X S c \<longrightarrow> hom_relmap (int p) X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)) \<and> (\<forall>p X S Y T. hom_relmap p X S Y T = hom_relmap p X (topspace X \<inter> S) Y (topspace Y \<inter> T))) \<and> (\<forall>p X S Y f T c. hom_relmap p X S Y T f c \<in> carrier (relative_homology_group p Y T)) \<and> (\<forall>p<0. hom_relmap p = (\<lambda>X S Y T f c. undefined))"