hcju/isabelleps huggingface dataset:
limsup_subseq_mono: fixes r :: "nat \<Rightarrow> nat"   and X :: "nat \<Rightarrow> 'a" assumes "strict_mono r" shows "limsup (X \<circ> r) \<le> limsup X"
spanning_subset_independent: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set"   and A :: "'e set" assumes "module scale"   and "B \<subseteq> A"   and "\<not> module.dependent scale A"   and "A \<subseteq> module.span scale B" shows "A = B"
diamond_bot_edges_def: shows "diamond_bot_edges = (1, \<lambda>x. (diamond_x x, - diamond_y (diamond_x x)))"
a2: shows "(\<lambda>i. map_option (fst \<circ> sr) (netmap s2 i)) = fst (netgmap sr s2)"
sum_mult_mem: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R"   and "\<forall>j\<le>n. f j \<in> set_mult R A B" shows "\<Sigma>\<^sub>e R f n \<in> carrier R"
lotteries_on_def: fixes Oc :: "'a set" shows "\<P> Oc = {p. set_pmf p \<subseteq> Oc}"
homotopic_paths_imp_pathstart: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q" shows "pathstart p = pathstart q"
M9: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> F \<and> \<circle>\<box>F"
arr_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "arr (local.dom f)"
B: fixes Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "P Y \<tau>' = P (\<lambda>_. Y \<tau>') \<tau>'"
bound2a: shows "real_of_int (sint w2) < real_of_int (sint POS_INF)"
igSwapInp_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'c, 'j, 'k) model_scheme"   and zs :: "'f"   and z1 :: "'i"   and z2 :: "'i"   and inp :: "'b \<Rightarrow> 'c option" shows "igSwapInp MOD zs z1 z2 inp \<equiv> lift (igSwap MOD zs z1 z2) inp"
axioms_1: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "module_pair_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "module_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"
extension_conforms: fixes v' :: "'a" assumes "v' \<in> VV p \<Longrightarrow> \<sigma> v' = v0" shows "path_conforms_with_strategy p (LCons v' P) \<sigma>"
ipurge_tr_aux_single_dom: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list" shows "ipurge_tr_aux I D {u} xs = ipurge_tr I D u xs"
ord_equiv_def: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" shows "ord_equiv D E = (\<exists>f. ord_isom D E f)"
fun_upd_eqD: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d"   and g :: "'c \<Rightarrow> 'd"   and z :: "'d" assumes "f(x := y) = g(x := z)" shows "y = z"
vars_term_ms_subst: fixes t :: "('d, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "vars_term_ms (t \<cdot> \<sigma>) = (\<Sum>x\<in>#vars_term_ms t. vars_term_ms (\<sigma> x))"
gen_iapath_triv_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "H.gen_iapath (pverts H) u p v" shows "p = [(u, v)]"
solvable_seq_condition: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c"   and K :: "('e, 'f) monoid_scheme"   and g :: "'c \<Rightarrow> 'e"   and I :: "'a set"   and J :: "'c set" assumes "group_hom G H f"   and "group_hom H K g"   and "f ` I \<subseteq> J"   and "kernel H K g \<subseteq> f ` I"   and "subgroup J H"   and "solvable_seq G I"   and "solvable_seq K (g ` J)" shows "solvable_seq H J"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "ManiPref.stable_on ds X" shows "inj_on Xd X"
simps_1: fixes A :: "'at" shows "positive (Pos A) = True"
int_encode_inverse: fixes x :: "int" shows "ni_bij (int_encode x) = x"
free_tv_of_substitutions_extend_to_scheme_lists: fixes n :: "nat"   and A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ" assumes "n \<in> free_tv A" shows "free_tv (S n) \<subseteq> free_tv ($ S A)"
set_less_aux_Compl2_conv_set_less_aux_Compl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_aux less_eq (set xs) (- set ys) = proper_intrvl.set_less_aux_Compl less proper_interval None xs ys"
select_convs_1: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_iiface \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_iiface"
f_join_f_shrink_iT_Plus_iT_Div_mod: fixes k :: "nat"   and I :: "nat set"   and xs :: "'b message_af list" assumes "0 < k"   and "\<forall>x\<in>I. x mod k = 0" shows "xs \<longmapsto> k \<Join>\<^sub> (I \<oplus> k - 1) = xs \<div> k \<Join>\<^sub> (I \<oslash> k)"
dist_sym: fixes x :: "fbint"   and y :: "fbint" shows "local.dist x y = local.dist y x"
simps_1: fixes v :: "'c"   and w :: "'c"   and xs :: "'c list" shows "first_edge_of_walk (v # w # xs) = (v, w)"
ag_pOp_add_l: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "c \<plusminus>\<^bsub>A\<^esub> a = c \<plusminus>\<^bsub>A\<^esub> b"
homI: fixes G :: "('f, 'g) monoid_scheme"   and h :: "'f \<Rightarrow> 'h"   and H :: "('h, 'i) monoid_scheme" assumes "\<And>x. x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> hom G H"
Un_Diff_cancel: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> (B - A) = A \<union> B"
assms_2: assumes "\<And>s r x. p s = Inr (x, r) \<Longrightarrow> length r < length s" shows "P"
sorted_sparse_row_matrix_zero: fixes m :: "nat"   and n :: "nat"   and a :: "(nat \<times> 'a) list"   and arr :: "(nat \<times> (nat \<times> 'a) list) list"   and j :: "nat" assumes "m \<le> n"   and "sorted_spvec ((n, a) # arr)" shows "Rep_matrix (sparse_row_matrix arr) m j = (0::'a)"
Union_insert: fixes a :: "'a set"   and B :: "'a set set" shows "\<Union> (insert a B) = a \<union> \<Union> B"
tab\<^sub>0_simps_1: fixes r :: "'a" assumes "ide r" shows "ide (tab\<^sub>0 r)"
surjective: fixes F :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c" assumes "su_rel_fun F f" obtains B :: "'d" where "(A, B) \<in> F"
similarity_inv: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and M1 :: "complex \<times> complex \<times> complex \<times> complex"   and M2 :: "complex \<times> complex \<times> complex \<times> complex" assumes "similarity A M1 = M2"   and "mat_det A \<noteq> 0" shows "similarity (mat_inv A) M2 = M1"
dbm_le_dbm_min2: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> b" shows "a = dmin b a"
subst_mset_union: fixes \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and M :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) multiset"   and N :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) multiset" shows "subst_mset \<sigma> (M + N) = subst_mset \<sigma> M + subst_mset \<sigma> N"
prems: shows "\<forall>j<length (t # ts). let b = (t # ts) ! j in case b of (uu_, uua_, uub_, sb\<^sub>j, uuc_, uud_, uue_) \<Rightarrow> a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)"
measure_spmf_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_fun (rel_pred A) (=)) (\<lambda>p. Sigma_Algebra.measure (measure_spmf p)) (\<lambda>q. Sigma_Algebra.measure (measure_spmf q))"
exp_cf3_def: shows "exp_cf3 \<equiv> \<lambda>x. numer_cf3 x / numer_cf3 (- x)"
last_smods_gcd: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0" shows "last (smods p q) = smult (lead_coeff (last (smods p q))) (gcd p q)"
sfirst_eq_0: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(sfirst P \<omega> = 0) = P \<omega>"
Proc_CFG_Call_Intra_edge_not_same_source: fixes prog :: "cmd"   and n :: "label"   and p :: "char list"   and es :: "expr list"   and rets :: "char list list"   and n' :: "label"   and et :: "(char list, val, char list \<times> label, char list) edge_kind"   and n'' :: "label" assumes "prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'"   and "prog \<turnstile> n -IEdge et\<rightarrow>\<^sub>p n''" shows "False"
onode_comp_def: fixes i :: "nat"   and onp :: "((nat \<Rightarrow> 'e) \<times> 'f, 'g seq_action) automaton"   and R\<^sub>i :: "nat set" shows "\<langle>i : onp : R\<^sub>i\<rangle>\<^sub>o \<equiv> \<lparr>init = {(\<sigma>, NodeS i s R\<^sub>i) |\<sigma> s. (\<sigma>, s) \<in> init onp}, trans = onode_sos (automaton.trans onp)\<rparr>"
run_reachable: fixes i :: "nat" shows "A.reachable (as i)"
arity_simps_1: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and \<zeta> :: "('a, 'b) hd" shows "arity.arity arity_sym arity_var (Hd \<zeta>) = arity.arity_hd arity_sym arity_var \<zeta>"
mem_not_sym: fixes x :: "V"   and y :: "V" shows "\<not> (x \<in> elts y \<and> y \<in> elts x)"
backwards_res_stab: fixes ts :: "traffic"   and d :: "cars"   and ts' :: "traffic"   and c :: "cars"   and v :: "view" shows "(ts \<^bold>\<midarrow>r( d ) \<^bold>\<rightarrow> ts') \<and> d \<noteq> c \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1 \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
prems_2: shows "val_ifex (sat_list_to_bdt u) as"
simps_4: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and u :: "Core.term" shows "term_ok' \<Sigma> (t $ u) = (term_ok' \<Sigma> t \<and> term_ok' \<Sigma> u)"
integral_bound: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and B :: "real" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> norm (f t) \<le> B" shows "norm (integral {a..b} f) \<le> B * (b - a)"
fls_divide_subdegree: fixes f :: "'a fls"   and g :: "'a fls" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "fls_subdegree (f / g) = fls_subdegree f - fls_subdegree g"
fBex_triv_one_point1: fixes A :: "'c fset"   and a :: "'c" shows "fBex A (\<lambda>x. x = a) = (a |\<in>| A)"
wp_assign_idx_eq: fixes x :: "char list"   and i :: "aexp"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.wp (x[i] ::= a) Q s = Q (s(x := (s x)(aval i s := aval a s)))"
mat_row_first_def: fixes A :: "'b mat"   and n :: "nat" shows "mat_row_first A n \<equiv> mat n (dim_col A) (\<lambda>(i, j). A $$ (i, j))"
pp_star_contl: fixes X :: "'a ppath set"   and Y :: "'a ppath set" shows "pp_prod X (pp_star Y) = (\<Union>n. pp_prod X (ppath_dioid.power Y n))"
select_convs_3: fixes mstate_i :: "nat"   and mstate_m :: "'a mformula"   and mstate_n :: "nat"   and more :: "'b" shows "mstate_n \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_n"
StepWhileTrue: fixes b :: "expr"   and s :: "char list \<Rightarrow> val option"   and c :: "cmd" assumes "interpret b s = Some true" shows "while (b) c \<turnstile> \<langle>while (b) c,s,0\<rangle> \<leadsto> \<langle>c;; while (b) c,s,2\<rangle>"
assms_2: shows "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier (H\<lparr>carrier := generate H {b}\<rparr>)" shows "\<exists>y\<in>carrier (H\<lparr>carrier := generate H {b}\<rparr>). y \<otimes>\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub> x = \<one>\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub>"
LLL_inv_N_pos: assumes "m \<noteq> 0" shows "0 < N"
simps_2: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and c :: "color"   and r :: "('b \<times> color) tree" shows "invc \<langle>l, (a, c), r\<rangle> = ((c = Red \<longrightarrow> color l = Black \<and> color r = Black) \<and> invc l \<and> invc r)"
in_genby_imp_in_reduced_letter_set: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> \<langle>A\<rangle>" shows "a \<in> \<langle>reduced_letter_set (A \<union> uminus ` A) a\<rangle>"
map_simp: fixes f :: "'c" assumes "C.arr f" shows "C.map f = f"
dghm_ArrMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
assms: shows "P ((\<sigma> i, p), a, \<sigma>' i, p')"
to_fun_ctrm: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and b :: "'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "b \<in> carrier R" shows "UP_cring.to_fun R (monom (UP R) (f 0) 0) b = f 0"
assms: fixes P :: "'b hrel" assumes "\<lbrace>p\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> F P \<lbrace>q\<rbrace>\<^sub>u"
elims_2: fixes x :: "nat"   and xa :: "nat list" assumes "check_prime_factors_subset x xa" obtains n :: "nat" where "x = n__" and "xa = []" and "n__ = 1"   | n :: "nat" and p :: "nat" and ps :: "nat list" where "x = n" and "xa = p # ps" and "if n = 0 then False else if 1 < p \<and> p dvd n then check_prime_factors_subset (n div p) (p # ps) else check_prime_factors_subset n ps"
pr_iso_test_def: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" shows "pr_iso_test Fs\<^sub>1 Fs\<^sub>2 = pr_iso_test_rec [] Fs\<^sub>1 Fs\<^sub>2"
rtrancl_induct2: fixes ax :: "'c"   and ay :: "'d"   and bx :: "'c"   and "by" :: "'d"   and r :: "(('c \<times> 'd) \<times> 'c \<times> 'd) set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P ax ay"   and "\<And>a b aa ba. \<lbrakk>((ax, ay), a, b) \<in> r\<^sup>*; ((a, b), aa, ba) \<in> r; P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
assms: shows "Y \<in> ipurge_tr_rev_t_set X"
tail_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "tail (with_proj G) (swap_in (parcs G) a) = head (with_proj G) a"
set_less_aux_empty: fixes A :: "'a set" shows "\<not> A \<sqsubset>' {}"
nofail_simps_3: fixes x :: "'c" shows "nofail (RETURN x) = True"
main_step_aux2: fixes qs :: "rat poly list" assumes "0 < length (fst (factorize_polys qs))" shows "set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"
S: fixes x :: "'b"   and d :: "real" assumes "x \<in> S"   and "0 < d" shows "\<exists>i. i \<in> K \<and> x \<in> ball (a i) (r i) \<and> r i < d"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. T (cod \<iota>, f))"
le_add_same_cancel2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
inject_1: fixes x1 :: "('e \<times> 'f) pmf"   and y1 :: "('e \<times> 'f) pmf" shows "(PMF x1 = PMF y1) = (x1 = y1)"
vimage2p_map_rel_prod: fixes f :: "'c \<Rightarrow> 'g"   and g :: "'d \<Rightarrow> 'h"   and f' :: "'e \<Rightarrow> 'i"   and g' :: "'f \<Rightarrow> 'j"   and A :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "vimage2p (map_prod f g) (map_prod f' g') (rel_prod A B) = rel_prod (vimage2p f f' A) (vimage2p g g' B)"
empty_P0_RP: shows "P_of_state (lhd (lmap state_of_wstate Sts)) = {}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "semiring_hom hom" shows "comm_semiring_hom hom"
q: shows "a2 = q @ \<delta> \<and> \<alpha>' = a1 @ [U] @ q"
int_o_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "U.int \<xi> = M.int \<xi>"
f1: shows "f1 = (stk1, loc1, C1, M1, pc1, ics1)"
f_Exec_take_eq_last_f_Exec_Stream_take: fixes xs :: "'d list"   and n :: "nat"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<noteq> []"   and "0 < n" shows "f_Exec_Comp trans_fun (xs \<down> n) c = last (f_Exec_Comp_Stream trans_fun (xs \<down> n) c)"
scale_left_commute: fixes a :: "complex"   and b :: "complex"   and x :: "'a" shows "a *\<^sub>C b *\<^sub>C x = b *\<^sub>C a *\<^sub>C x"
memo_same: fixes f :: "'v \<Rightarrow> 'm"   and x :: "'v"   and g :: "int \<Rightarrow> int \<Rightarrow> 'm" shows "bind (memo f x) (\<lambda>a. bind (memo f x) (g a)) = bind (memo f x) (\<lambda>a. g a a)"
repl_def: fixes n :: "'a"   and x :: "'c" shows "repl n x = rec [] (\<lambda>_. (#) x) n"
linorder_finite_sup_selective: fixes f :: "'c \<Rightarrow> 'd" shows "\<exists>i. sup_monoid.sum f {k. True} = f i"
C_Let: fixes E :: "trm \<Rightarrow> trm"   and p :: "pat"   and u :: "trm" assumes "ctxtp E" shows "ctxtp (\<lambda>t. LET p = E t IN u)"
s: shows "s = atomize_poly (idx_pm_of_pm bs s')"
Conseq': fixes P :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "('d, 'f, 'g) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s. s \<in> P \<longrightarrow> (\<exists>P' Q' A'. (\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)) \<and> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A))" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
array: fixes G :: "prog"   and S :: "ref_ty"   and T :: "ref_ty" assumes "G\<turnstile>RefT S\<preceq>RefT T" shows "G\<turnstile>RefT S.[]\<preceq>RefT T.[]"
reflexive_complement_irreflexive: fixes x :: "'a" assumes "top \<le> x" shows "- x \<le> - top"
gcd_right_idem: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (gcd a b) b = gcd a b"
sgn_prod_def: fixes x :: "'a \<times> 'b" shows "sgn x = x /\<^sub>R norm x"
rep_eq: shows "(\<squnion>) = (\<lambda>f g. (f\<^sub>\<bullet> \<squnion> g\<^sub>\<bullet>)\<^sup>\<bullet>)"
simps_4: fixes uv :: "('form, 'var) out_port" shows "hyps Helper uv = None"
that_1: fixes u :: "'a"   and v :: "'a" assumes "uv = Inl u"   and "uv' = Inr v"   and "edge \<Gamma> u v" shows "thesis"
inv_all_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" shows "local.inv_all p A \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> p A i j"
span_Basis: shows "module_on.span (*\<^sub>R) basis = dest.tangent_space (f p)"
llength_def: shows "llength = enat_unfold lnull ltl"
f2: shows "rk {A, B, C, A', B', C', P} = 3"
add_strict_right_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subset># b" shows "a + c \<subset># b + c"
simps_71: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'b \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'a" shows "(case \<tau>\<^sub>s of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f8"
fps_compose_divide_distrib: fixes g :: "'b fps"   and f :: "'b fps"   and h :: "'b fps" assumes "g dvd f"   and "h $ 0 = (0::'b)"   and "g oo h \<noteq> 0" shows "f / g oo h = (f oo h) / (g oo h)"
snth_sset_smerge: fixes ss :: "'d stream stream"   and n :: "nat"   and m :: "nat" shows "ss !! n !! m \<in> sset (smerge ss)"
subsets_smash_property: fixes n :: "nat"   and subsets1 :: "nat list list"   and subsets2 :: "nat list list" shows "\<forall>elem. List.member (subsets_smash n subsets1 subsets2) elem \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<and> elem = elem1 @ map ((+) n) elem2)"
order_decomp: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "\<exists>q. p = [:- a, 1::'b:] ^ order a p * q \<and> \<not> [:- a, 1::'b:] dvd q"
ads_d_def: fixes x :: "'a" shows "r x = ar ar x"
interaction_bounded_by_contD: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and out :: "'d"   and c :: "('e, 'd, 'f) rpv"   and input :: "'f" assumes "interaction_bounded_by consider gpv n"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "consider out" shows "0 < n \<and> interaction_bounded_by consider (c input) (n - 1)"
reduced_GB_is_monomial_set_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]"   and "is_monomial_set F" shows "is_monomial_set (punit.reduced_GB F)"
assms_2: shows "g \<turnstile> Entry g-ns\<rightarrow>Entry g"
Sr_one: fixes S :: "'a set" assumes "sr R S" shows "1\<^sub>r \<in> S"
append_single_T_imp_tickFree: fixes t :: "'c event list"   and a :: "'c event"   and P :: "'c process" assumes "t @ [a] \<in> T P" shows "tickFree t"
bdd: fixes x :: "'a" assumes "x \<in> S" shows "\<bar>x\<bar> \<le> a"
expand_fls_eq: fixes f :: "'b fls"   and g :: "'b fls" shows "(f = g) = (\<forall>n. fls_nth f n = fls_nth g n)"
fract_poly_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "map_poly to_fract (p * q) = map_poly to_fract p * map_poly to_fract q"
step_sim: fixes a :: "'b"   and a' :: "'b"   and b :: "'f" assumes "(a, a') \<in> s1.b.E"   and "(a, b) \<in> R2" shows "\<exists>b'. (b, b') \<in> s2.b.E \<and> (a', b') \<in> R2"
elems_all_maps_list: fixes M :: "('b \<times> 'b) list list"   and G :: "'b list \<times> ('b \<times> 'b) list" assumes "M \<in> set (all_maps_list G)"   and "distinct (snd G)" shows "\<Union> (sset M) = set (snd G)"