hcju/isabelleps huggingface dataset:
sort_key_inj_key_eq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'c" assumes "class.linorder less_eq less"   and "mset xs = mset ys"   and "inj_on f (set xs)"   and "linorder.sorted less_eq (map f ys)" shows "linorder.sort_key less_eq f xs = ys"
AssertAllState_index_valid: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "index_valid (set as) (assert_all_state t as)"
echelon_foldl_condition7: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and mb :: "'f"   and mc :: "'f" assumes "is_bezout_ext bezout"   and "echelon_form_upt_k A k"   and "k < ncols A"   and "\<not> is_zero_row_upt_k mb k A"   and "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"   and "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'f) \<le> mc"   and "A $ mc $ mod_type_class.from_nat k \<noteq> (0::'d)" shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (bezout_iterate (interchange_rows A ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'f)) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d) \<and> (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'f) \<le> n)) (nrows A - Suc 0) ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'f)) (mod_type_class.from_nat k) bezout))"
add_less_cancel_right: fixes a :: "'b zmultiset"   and c :: "'b zmultiset"   and b :: "'b zmultiset" shows "(a + c \<subset>#\<^sub>z b + c) = (a \<subset>#\<^sub>z b)"
pol_expr_of_X: shows "X = polyn_expr R X (Suc 0) (ext_cf S (Suc 0) (C\<^sub>0 S))"
map_poly_plus: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly"   and q :: "'d poly" assumes "f (0::'d) = (0::'c)"   and "\<And>a b. f (a + b) = f a + f b" shows "map_poly f (p + q) = map_poly f p + map_poly f q"
is_path_split': fixes G :: "('a, 'b) graph"   and v :: "'a"   and p1 :: "('a \<times> 'b \<times> 'a) list"   and u :: "'a"   and w :: "'b"   and u' :: "'a"   and p2 :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v (p1 @ (u, w, u') # p2) v' = (valid_graph.is_path G v p1 u \<and> (u, w, u') \<in> edges G \<and> valid_graph.is_path G u' p2 v')"
eval_if1: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and n :: "nat"   and e2 :: "exp"   and v2 :: "bval"   and e3 :: "exp" assumes "\<rho> \<turnstile> e1 \<Down> BNat n"   and "n \<noteq> 0"   and "\<rho> \<turnstile> e2 \<Down> v2" shows "\<rho> \<turnstile> EIf e1 e2 e3 \<Down> v2"
prems: shows "P \<oplus> Q \<longmapsto> Rs"
intervalE: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x \<and> x \<le> b" obtains "x \<in> {a..b}"
cond_actions_ok_\<tau>mred1_inv: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ct :: "'t conditional_action" assumes "\<tau>mred1 s1 s1'"   and "r1.cond_action_ok s1 t ct" shows "r1.cond_action_ok s1' t ct"
map_ext: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
bi_uniqueDr: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and z :: "'d" assumes "bi_unique A"   and "A x y"   and "A x z" shows "y = z"
lin_dep_def: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set" assumes "module R M" shows "module.lin_dep R M S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R \<and> module.lincomb M a A = \<zero>\<^bsub>M\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>\<^bsub>R\<^esub>)"
sim_f_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h" shows "sim_f M M' f \<equiv> \<forall>a u v. (u, v) \<in> relations M a \<longrightarrow> (f u, f v) \<in> relations M' a"
g_def: fixes k :: "nat" shows "g k = f (Suc ((j_of ^^ k) 0))"
False: shows "\<not> (a = b \<or> a = c \<or> b = c)"
noDA1C: fixes a :: "('c, 'd) Combinators"   and p :: "('c, 'd) Combinators list" assumes "noDenyAll1 (a # p)" shows "noDenyAll1 p"
map_of_map_restrict: fixes f :: "'b \<Rightarrow> 'c"   and ks :: "'b list" shows "map_of (list.map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
mapl_FGf_def: fixes l1 :: "'p \<Rightarrow> 'z"   and l2 :: "'q \<Rightarrow> 'aa" shows "mapl_FGf l1 l2 = map_FGf l1 l2 id id id id"
nn_integral_bernoulli_pmf: fixes p :: "real"   and f :: "bool \<Rightarrow> ennreal" assumes "0 \<le> p"   and "p \<le> 1"   and "\<And>x. 0 \<le> f x" shows "integral\<^sup>N (measure_pmf (bernoulli_pmf p)) f = f True * ennreal p + f False * ennreal (1 - p)"
concurrent_append_store_independent: fixes i :: "'a"   and e1 :: "'b"   and r :: "'a"   and e2 :: "'b"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Append i e1) (r, Store e2 mo r)"   and "\<not> hb (r, Store e2 mo r) (i, Append i e1)"   and "xs prefix of j"   and "(i, Append i e1) \<in> set (node_deliver_messages xs)"   and "(r, Store e2 mo r) \<in> set (node_deliver_messages xs)" shows "i \<noteq> mo"
pre: fixes ya :: "real" shows "(ya ^ 3 + e * ya + f = 0) = (\<exists>z. z ^ 3 = cor u \<and> cor ya = z - cor e / (3 * z))"
integ2: shows "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
gcd_real_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
is_discovered_impl_def: fixes u :: "'e"   and s :: "('e, 'f) simple_state_scheme" shows "is_discovered_impl u s \<equiv> u \<in> visited s"
line_convex_combination1: fixes u :: "real"   and a :: "'b"   and b :: "'b"   and i :: "real" shows "(1 - u) *\<^sub>R {a -- b}\<^bsub>i\<^esub> + u *\<^sub>R b = {a -- b}\<^bsub>i + u - i * u\<^esub>"
strict_iff_order: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
distinct_1: fixes x1 :: "'f option"   and x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.Done x1 \<noteq> Resumption.resumption.Pause x21 x22"
fst_gb_schema_incr: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and upd :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> 'g \<Rightarrow> 'g"   and b0 :: "('t \<Rightarrow>\<^sub>0 'e) \<times> 'f"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list"   and data :: "'g" shows "fst ` set (gb_schema_incr sel ap ab compl upd (b0 # bs) data) = (let (gs, n, data') = add_indices (gb_schema_incr sel ap ab compl upd bs data, data) (0, data); b = (fst b0, n, snd b0); data'' = upd gs b data' in fst ` set (gb_schema_aux sel ap ab compl gs (count_rem_components (b # gs), Suc n, data'') (ab gs [] [b] (Suc n, data'')) (ap gs [] [] [b] (Suc n, data''))))"
5: shows "(f \<sqinter> q)\<^sup>T\<^sup>\<star> * (f \<sqinter> - q)\<^sup>\<star> * e = (f \<sqinter> - q)\<^sup>\<star> * e"
rtrancl_mem_Sigma: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a \<noteq> b"   and "(a, b) \<in> (A \<times> A)\<^sup>*" shows "b \<in> A"
compact_space_discrete_topology: fixes X :: "'c set" shows "compact_space (discrete_topology X) = finite X"
equiv_eq_right_lang: shows "equiv (states M) eq_right_lang"
wfrec_fixpoint: fixes R :: "('c \<times> 'c) set"   and F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "wf R"   and "adm_wf R F" shows "wfrec R F = F (wfrec R F)"
intensional_simps_6: fixes P :: "'a \<Rightarrow> bool" shows "P = \<not> P \<equiv> #False"
arr_Rel_ArrDom_in_Vset: shows "F\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
fundfold_comp_trivial_f\<C>: fixes rs :: "'a permutation list"   and C :: "'a set" assumes "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set rs)"   and "\<forall>r\<in>set rs. \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>" shows "fold fst (map Spair rs) ` C = C"
empty_correct_3: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge" shows "\<alpha>w empty = (\<lambda>_. 0)"
simps_15: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "set_tree \<langle>x21, x22, x23\<rangle> = set_tree x21 \<union> {x22} \<union> set_tree x23"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (\<Omega>.map f) = \<Omega>.map (local.dom f)"
reduce_clause_in_P: fixes C :: "'a literal list"   and N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and P' :: "('a literal list \<times> nat) list"   and j :: "nat"   and D :: "'a literal list"   and D' :: "'a literal list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N"   and "\<forall>(E, k)\<in>set (P @ P'). j < k \<longrightarrow> is_irreducible [C] E" shows "wstate_of_dstate (N, P @ (D @ D', j) # P', Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P @ (D @ reduce [C] D D', j) # P', Q, n)"
complete_ht_lookup: shows "<is_hashtable l ht> hm_lookup k ht <\<lambda>r. is_hashtable l ht * \<up> (r = ls_lookup k (l ! bounded_hashcode_nat (length l) k))>"
ordinal_exp_assoc: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "(x ** y) ** z = x ** (y * z)"
simps_2: fixes v :: "'a"   and Q :: "'a \<Rightarrow> bool" shows "ospec (Some v) Q = Q v"
pos_mult_pos: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "0 < r * s"
\<Psi>_def: shows "\<Psi> \<equiv> \<lambda>(dl, a, b, M) l. dl l \<lless> a \<and> finite a \<and> dl l \<noteq> {} \<and> a \<noteq> {} \<and> (\<forall>j\<le>l. card (dl j) = Suc j) \<and> a \<lless> \<Union> (range b) \<and> range b \<subseteq> Collect finite \<and> a \<subseteq> N \<and> \<Union> (range b) \<subseteq> N \<and> infinite M \<and> b (l, l - 1) \<lless> M \<and> M \<subseteq> N"
pa_\<alpha>_def: fixes S :: "('a \<times> 'b, unit) HashMap.hashmap \<times> ('a \<times> 'b) list \<times> ('a \<times> 'b, 'c) ta_rule dlist" shows "pa_\<alpha> S \<equiv> let (Q, W, \<delta>d) = S in (hs.\<alpha> Q, W, ls.\<alpha> \<delta>d)"
top_scene_def: shows "\<top>\<^sub>S \<equiv> Abs_scene (\<lambda>x y. y)"
lc_uminus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.lc (- p) = - punit.lc p"
simps_2: fixes R :: "('c, 'd) ring_scheme"   and v :: "'c"   and va :: "'c list" assumes "ring R" shows "ring.normalize R (v # va) = (if lead_coeff (v # va) \<noteq> \<zero>\<^bsub>R\<^esub> then v # va else ring.normalize R (tl (v # va)))"
future_bounded_mfotl_def: fixes \<phi> :: "'a MFOTL.formula" shows "future_bounded_mfotl \<phi> \<equiv> MFOTL.future_reach \<phi> \<noteq> \<infinity>"
separated\<^sub>3_def: fixes g :: "graph"   and V :: "nat set" shows "separated\<^sub>3 g V \<equiv> \<forall>v\<in>V. \<forall>f\<in>set (facesAt g v). |vertices f| \<le> 4 \<longrightarrow> \<V> f \<inter> V = {v}"
remap_rta: fixes f :: "'Q \<Rightarrow> 'a" shows "ranked_tree_automaton (ta_remap f TA) A"
mset_le_decr_left1: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset" assumes "a + c \<subseteq># b" shows "a \<subseteq># b"
f_Exec_Stream_empty_conv: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "(f_Exec_Comp_Stream trans_fun xs c = []) = (xs = [])"
diff_delta_lnexp_cf6_def: shows "diff_delta_lnexp_cf6 \<equiv> \<lambda>x. - (x ^ 12) / (numer_cf6 (- x) * numer_cf6 x)"
sel_8: fixes x3 :: "int" shows "extract_int (IntVal x3) = x3"
rel_funI: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
monom_mult_punit_def: shows "monom_mult_punit \<equiv> punit.monom_mult"
assms_1: shows "z\<^sup>2 = x\<^sup>2 + y\<^sup>2"
simps_17: fixes 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 "sip \<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> = sip"
compact_eq_Heine_Borel: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "topological_space.compact open S = (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
covector_complement_closed: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv"   and "times top x = x" shows "times top (uminus x) = uminus x"
ih: fixes y :: "('s, 'v) tm" assumes "hsize y < hsize s"   and "y >\<^sub>t y" shows "False"
that: fixes p :: "pvar" assumes "p \<notin> P" shows "thesis"
supto_simp_1: fixes j :: "int"   and i :: "int" assumes "j < i" shows "supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j) = [::]"
bound_L1_Lp_1: fixes p :: "real"   and f :: "'a \<Rightarrow> real" assumes "1 \<le> p"   and "random_variable borel f"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" shows "integrable M f"
3023: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- y \<sqinter> - x) = top"
rel_fun_def: fixes A :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
below2: shows "\<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace> (x, e1) # \<Gamma> \<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace> (x, e1) # \<Gamma> \<rbrace>\<rho>\<^esub>"
UN_empty: fixes B :: "'e \<Rightarrow> 'd set" shows "\<Union> (B ` {}) = {}"
hom_add: fixes x :: "'a"   and y :: "'a" shows "hom (x + y) = hom x + hom y"
intensional_simps_1: fixes x :: "'b \<Rightarrow> 'c" shows "x = x \<equiv> #True"
lift_Suc_mono_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f n) (f (Suc n))"   and "n \<le> n'" shows "less_eq (f n) (f n')"
enc_preserves_pred_iff_indRelR_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_preserves_pred Pred = rel_preserves_pred indRelR Pred"
splitAt_not1: fixes v :: "'c"   and vs :: "'c list"   and ram :: "'c" assumes "v \<notin> set vs" shows "v \<notin> set (fst (splitAt ram vs))"
flattening_lemma: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>ops\<in>set \<psi>. \<exists>op. ops = [op]"   and "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (concat \<psi>)"
assms_5: shows "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
to_st_to_est_inv: fixes A :: "('fun, 'var) strand_step list" shows "to_st (to_est A) = A"
union_eqvt: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set"   and Y :: "'a set" shows "pi \<bullet> (X \<union> Y) = pi \<bullet> X \<union> pi \<bullet> Y"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "Matrix.vec n f \<in> carrier_vec n"
funion_finsert_right: fixes A :: "'b fset"   and a :: "'b"   and B :: "'b fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
prec: shows "atomic_step_precondition s tid (SK_IPC dir stage partner pag)"
ev_ev: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "ev (ev \<phi>) = ev \<phi>"
borel_measurable_lebesgue_integral: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'b measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M lborel)" shows "(\<lambda>x. integral\<^sup>L lborel (f x)) \<in> borel_measurable N"
assms_2: fixes x :: "'a" assumes "xs = [x]" shows "P [x]"
mult_column_iarray_def: fixes A :: "'b iarray iarray"   and n :: "nat"   and q :: "'b" shows "mult_column_iarray A n q = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>i j. if j = n then A !! i !! j * q else A !! i !! j)"
theG_def: fixes ltl :: "'b ltl" shows "theG ltl = (case ltl of G x8 \<Rightarrow> x8)"
border_butlast: fixes xs :: "'a list"   and ys :: "'a list" assumes "border xs ys" shows "border (butlast xs) (butlast ys)"
DSQSS: shows "br_dsq \<delta> q (Q, W) \<subseteq> b_accessible \<delta>"
in_q65_code: fixes x :: "nat" shows "(x mod 65 \<in> q65) = q65_array !! (x mod 65)"
lb2: shows "\<forall>x. poly p x = 0 \<longrightarrow> lb2 < x"
exhaust: fixes y :: "'a Stream" obtains "y = \<bottom>"   | a :: "'a" and Stream :: "'a Stream" where "y = a && Stream"
len_finals_makeFaceFinal: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "\<not> final f" shows "|finals (makeFaceFinal f g)| = |finals g| + 1"
cond_bind_spmf_def: fixes p :: "'h spmf"   and f :: "'h \<Rightarrow> 'i spmf"   and A :: "'i set" shows "cond_bind_spmf p f A = (if \<exists>x\<in>set_spmf p. set_spmf (f x) \<inter> A \<noteq> {} then cond_bind_pmf p (\<lambda>x. case x of None \<Rightarrow> return_pmf None | Some x \<Rightarrow> f x) (Some ` A) else return_pmf None)"
sorted_Cons_Min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (x # xs)" shows "linorder.Min less_eq (insert x (set xs)) = x"
Rg: shows "g \<in> {..m} \<rightarrow> carrier R"
mult_isol_var: fixes u :: "'c \<times> 'd"   and x :: "'c \<times> 'd"   and v :: "'c \<times> 'd"   and y :: "'c \<times> 'd" assumes "less_eq_prod u x"   and "less_eq_prod v y" shows "less_eq_prod (times_prod' u v) (times_prod' x y)"
gen_dfs_pre_def: fixes E :: "('b \<times> 'b) set"   and U :: "'b set"   and S :: "'b set"   and V :: "'b set"   and u0 :: "'b" shows "gen_dfs_pre E U S V u0 \<equiv> E `` U \<subseteq> U \<and> finite U \<and> V \<subseteq> U \<and> u0 \<in> U \<and> E `` (V - S) \<subseteq> V \<and> u0 \<notin> V \<and> S \<subseteq> V \<and> (\<forall>v\<in>S. (v, u0) \<in> (E \<inter> S \<times> UNIV)\<^sup>*)"
edomIff: fixes a :: "'b"   and m :: "'b \<Rightarrow> 'c" shows "(a \<in> edom m) = (m a \<noteq> \<bottom>)"
surjective: fixes r :: "('b, 'c) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = graph.more r\<rparr>"
open_greaterThanLessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThanLessThan less a b)"
nxtAct_eq: fixes n :: "nat"   and n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'"   and "\<parallel>c\<parallel>\<^bsub>t n'\<^esub>"   and "\<forall>n''\<ge>n. n'' < n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>" shows "n' = \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
ent_pure_pre_iff: fixes P :: "assn"   and b :: "bool"   and Q :: "assn" shows "(P * \<up> b \<Longrightarrow>\<^sub>A Q) = (b \<longrightarrow> (P \<Longrightarrow>\<^sub>A Q))"
remove_key_sum: fixes k :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" shows "remove_key k f + monomial (lookup f k) k = f"
decide_universal_R_def: fixes fml :: "real poly fml" shows "decide_universal_R fml = (let (fml_struct, polys) = convert fml in Let (find_consistent_signs_R polys) (list_all (lookup_sem fml_struct)))"
power_less_zero_eq: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (power.power one times a n) zero = (odd n \<and> less a zero)"
lower_uminus: fixes A :: "'b interval" shows "lower (- A) = - upper A"
has_max_lideal: shows "\<exists>\<ww>. max_lideal \<ww> B addB multB zeroB oneB"
mem_FL_Red_F_because_G_Red_F: fixes Cl :: "'a literal multiset \<times> label"   and N :: "('a literal multiset \<times> label) set" assumes "\<forall>D\<in>\<G>_F (fst Cl). D \<in> G.Red_F (\<Union> (\<G>_F ` fst ` N))" shows "Cl \<in> FL.Red_F_\<G> N"