hcju/isabelleps huggingface dataset:
foldr_fold: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "foldr f xs = fold f xs"
map_simp_2: fixes b :: "'c"   and f :: "'c" assumes "C.ide b"   and "C.arr f" shows "Hom.map (b, f) = S.mkArr (Hom.set (b, C.dom f)) (Hom.set (b, C.cod f)) (\<phi> (b, C.cod f) \<circ> (\<cdot>) f \<circ> \<psi> (b, C.dom f))"
case_cong_weak: fixes option :: "'c option"   and option' :: "'c option"   and f1 :: "'d"   and f2 :: "'c \<Rightarrow> 'd" assumes "option = option'" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option' of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x)"
may_lock_t_may_lock_unlock_lock_t: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "may_lock l t" shows "may_lock (unlock_lock l) t"
Infinitesimal_FreeUltrafilterNat_iff2: fixes X :: "nat \<Rightarrow> 'c" shows "(star_n X \<in> Infinitesimal) = (\<forall>m. \<forall>\<^sub>F n in \<U>. norm (X n) < inverse (real (Suc m)))"
surjective: fixes r :: "('a, 'b) monoid_scheme" shows "r = \<lparr>carrier = carrier r, monoid.mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, \<dots> = monoid.more r\<rparr>"
valuate_rat_valuate: fixes lp :: "linear_poly"   and vl :: "nat \<Rightarrow> QDelta"   and \<delta> :: "rat" shows "lp \<lbrace> \<lambda>v. val (vl v) \<delta> \<rbrace> = val lp \<lbrace> vl \<rbrace> \<delta>"
refl_onI: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<succeq>[r] x" shows "refl_on A r"
orthogonal_complement_def: fixes W :: "'b vec set" shows "orthogonal_complement W = {x \<in> carrier_vec n. \<forall>y\<in>W. x \<bullet> y = (0::'b)}"
axioms_2: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and index :: "int"   and u_block_size :: "int" assumes "k_\<Lambda>_PBD point_set block_collection index u_block_size" shows "block_design point_set block_collection u_block_size"
horizontal_chop_assoc1: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view" shows "v=v1\<parallel>v2 \<and> v2=v3\<parallel>v4 \<longrightarrow> (\<exists>v'. v=v'\<parallel>v4 \<and> v'=v1\<parallel>v3)"
set_remisodups: fixes Us :: "('b, 'g) aezfun set list" shows "set (remisodups Us) \<subseteq> set Us"
r_phi: fixes i :: "nat"   and f :: "recf"   and x :: "nat" assumes "i = encode f"   and "recfn 1 f" shows "eval r_phi [i, x] = eval f [x]"
add_le_same_cancel1: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># {#})"
next_plane0_incr_degree: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and v :: "nat" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "minGraphProps g"   and "v \<in> \<V> g" shows "degree g v \<le> degree g' v"
unit_def: fixes a :: "'c" shows "unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
UNIF1_singleton_decomp: fixes ss :: "('c, 'd) Term.term list"   and ts :: "('c, 'd) Term.term list"   and f :: "'c" assumes "length ss = length ts" shows "UNIF1 Var {#(Fun f ss, Fun f ts)#} (mset (zip ss ts))"
null_def: shows "H.null = (THE n. \<forall>f. n \<star> f = n \<and> f \<star> n = n)"
weak_ranking_index_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "finite_total_preorder_on carrier le" shows "finite_total_preorder_on.weak_ranking_index le x = find_index ((\<in>) x) (weak_ranking le)"
mult_mm_cancel_r: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0"   and "A *\<^sub>m\<^sub>m M = B *\<^sub>m\<^sub>m M" shows "A = B"
sat_slice: shows "MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<sigma>) v i \<phi>"
while_option_stop: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "while_option b c s = Some t" shows "\<not> b t"
ftip_cons_trueI: fixes cld :: "cld"   and fds :: "fd list"   and P :: "cld list"   and ctx :: "ctx"   and f :: "char list"   and ty :: "ty"   and ctxcld_list :: "(ctx \<times> cld) list" assumes "class_fields cld fds"   and "ftype_in_fds P ctx fds f (ty_opt_bot_opt (Some ty))" shows "ftype_in_path P ([(ctx, cld)] @ ctxcld_list) f (Some ty)"
similar_angle_def: fixes x :: "real"   and y :: "real" shows "(x \<sim> y) = (\<bar>x - y\<bar> < 15)"
cmidi: shows "m\<inverse> O d\<inverse> \<subseteq> b\<inverse>"
adjoint_adjoint: fixes A :: "'c mat" shows "adjoint (adjoint A) = A"
B: shows "\<forall>x>a. poly p x \<noteq> 0"
runFw_sameFw_behave: fixes W :: "'a word set set"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list"   and ip1 :: "'a word"   and ip2 :: "'a word" assumes "\<forall>A\<in>W. \<forall>a1\<in>A. \<forall>a2\<in>A. same_fw_behaviour_one a1 a2 c rs"   and "\<Union> W = UNIV"   and "\<forall>B\<in>W. \<exists>b\<in>B. runFw ip1 b c rs = runFw ip2 b c rs"   and "\<forall>B\<in>W. \<exists>b\<in>B. runFw b ip1 c rs = runFw b ip2 c rs" shows "same_fw_behaviour_one ip1 ip2 c rs"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
put_det: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b"   and \<rho> :: "'c" assumes "pbij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
rev_swap_def: fixes a :: "'b"   and b :: "'b" shows "G'.rev_swap a b = \<lparr>edge_rev = perm_swap a b (edge_rev (rev_swap a b)), edge_succ = perm_swap a b (edge_succ (rev_swap a b))\<rparr>"
map_plus_intsFrom: fixes n :: "int"   and m :: "int" shows "map\<cdot>(+MkI\<cdot>n)\<cdot>(intsFrom\<cdot>(MkI\<cdot>m)) = intsFrom\<cdot>(MkI\<cdot>(m + n))"
sum_list_zeroI: fixes xs :: "'c list" assumes "set xs \<subseteq> {0::'c}" shows "sum_list xs = (0::'c)"
subelem_set_of_high: fixes x :: "ref"   and t :: "dag"   and high :: "ref \<Rightarrow> ref"   and p :: "ref"   and low :: "ref \<Rightarrow> ref" assumes "x \<in> set_of t"   and "x \<noteq> Null"   and "high x \<noteq> Null"   and "Dag p low high t" shows "high x \<in> set_of t"
UN: fixes K :: "'b set set"   and S :: "'b set set" assumes "\<And>k. k \<in> K \<Longrightarrow> generate_topology S k" shows "generate_topology S (\<Union> K)"
R16_R12: shows "pmf (sds R16) c + pmf (sds R16) a \<le> pmf (sds R12) a"
I: fixes w :: "letter list"   and zs :: "letter list list" assumes "w # xs \<noteq> []"   and "R a (w # xs) zs" shows "bar zs"
summable_zero: shows "summable (\<lambda>n. 0::'c)"
card_eq: fixes A :: "'e set"   and B :: "'e set" assumes "A = B" shows "card A = card B"
compatible_iff_seq: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(domains g \<inter> codomains f \<noteq> {}) = seq g f"
y: fixes z :: "complex" assumes "z \<in> S" shows "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y"
iterate_add_add_eq1: fixes j :: "nat"   and i :: "nat"   and u :: "'k"   and m :: "'k"   and n :: "'k" assumes "j \<le> i" shows "iterate_add i u + m - (iterate_add j u + n) = iterate_add (i - j) u + m - n"
powr_minus: fixes x :: "'a"   and a :: "'a" shows "x powr - a = inverse (x powr a)"
euclidean_representation: fixes x :: "'a" shows "(\<Sum>b\<in>Basis. inner x b *\<^sub>R b) = x"
pred_state_run_state: fixes P :: "'b \<Rightarrow> bool"   and m :: "('c, 'b) state"   and s :: "'c" assumes "pred_state P m" shows "P (fst (run_state m s))"
le_add1: fixes n :: "nat"   and m :: "nat" shows "order.greater_eq (n + m) n"
targetsI: fixes b :: "'a"   and \<mu> :: "'a" assumes "weak_unit b"   and "b \<star> \<mu> \<noteq> null" shows "b \<in> targets \<mu>"
padic_a_inv_simp: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and n :: "nat" shows "padic_a_inv p f n \<equiv> \<ominus>\<^bsub>residue_ring (p ^ n)\<^esub> f n"
ferrack2_def: fixes p :: "fm" shows "ferrack2 p \<equiv> let q = simpfm p; mp = minusinf q; pp = plusinf q in if mp = T \<or> pp = T then T else let U = remdups (uset q) in decr0 (list_disj [mp, pp, simpfm (subst0 (CP 0\<^sub>p) q), evaldjf (\<lambda>(c, t). msubst2 q (c *\<^sub>p (- 2)\<^sub>p) t) U, evaldjf (\<lambda>((b, a), d, c). msubst2 q ((- 2)\<^sub>p *\<^sub>p b *\<^sub>p d) (tm.Add (tm.Mul d a) (tm.Mul b c))) (alluopairs U)])"
cxt_seq_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt" assumes "c\<^sub>1 ;; c\<^sub>2 = cxt_to_stmt E c" shows "E = [] \<and> c = c\<^sub>1 ;; c\<^sub>2 \<or> (\<exists>c' cs. E = c' # cs \<and> c = c\<^sub>1 \<and> c\<^sub>2 = cxt_to_stmt cs c')"
distinct_Basis_list: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Basis_list :: "'b list"   and eucl_down :: "int \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_down :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_up :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "distinct Basis_list"
induced_arrow_def: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" shows "induced_arrow a' \<chi>' = (THE f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> D.cones_map f \<chi> = \<chi>')"
ccos_eq_minus1: fixes z :: "complex" shows "(cos z = - 1) = (\<exists>x. z = complex_of_real (of_real (2 * real_of_int x * pi) + pi))"
dnso6: shows "ad (ad (zero_class.zero::'b)) = (zero_class.zero::'b)"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.cod f = (if B.VVV.arr f then B.VxVxV.cod f else B.VxVxV.null)"
rel_gpv''_map_gpv_1: fixes A :: "'t \<Rightarrow> 'q \<Rightarrow> bool"   and C :: "'u \<Rightarrow> 'r \<Rightarrow> bool"   and R :: "'v \<Rightarrow> 's \<Rightarrow> bool"   and f :: "'w \<Rightarrow> 't"   and g :: "'x \<Rightarrow> 'u"   and gpv :: "('w, 'x, 'v) gpv" shows "rel_gpv'' A C R (map_gpv f g gpv) \<equiv> rel_gpv'' (\<lambda>a. A (f a)) (\<lambda>c. C (g c)) R gpv"
lunit'_simps_2: fixes f :: "'d" assumes "D.ide f" shows "src\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f] = src\<^sub>D f"
strong_confluentp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "strong_confluentp r = (r\<inverse>\<inverse> OO r \<le> r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>=\<^sup>=)"
Quotient_to_transfer: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'b"   and c' :: "'c" assumes "Quotient R Abs Rep T"   and "R c c"   and "c' \<equiv> Abs c" shows "T c c'"
eq0: shows "set ps - set (ps\<^sub>+) \<union> set (ps\<^sub>+) = set ps"
remove_rels_def: shows "remove_rels \<equiv> map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, sb, \<D>, \<O>, ()))"
normalize_monic: fixes p :: "'b poly" assumes "monic p" shows "normalize p = p"
weakenL_set: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and F :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "F + \<Gamma> \<Rightarrow> \<Delta>"
preserves_arr: fixes f :: "'c" assumes "Cop.arr f" shows "S.arr (Cones.map f)"
reduce_nested_mpoly_extract_var: fixes p :: "'a mpoly"   and v :: "nat" shows "reduce_nested_mpoly (extract_var p v) = p"
g1_2: shows "g1 ` T \<subseteq> c"
cons2: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s1 :: "'b state"   and d :: "dec"   and s2 :: "'b state"   and new_env :: "v sem_env"   and ds :: "dec list"   and s3 :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r)" shows "evaluate_decs ck mn env s1 (d # ds) (s3, combine_dec_result new_env r)"
assms: shows "unifier\<^sub>l\<^sub>s \<sigma> L"
distinct_3: fixes x1 :: "'a"   and x3 :: "'c" shows "In1 x1 \<noteq> In3 x3"
bij_induces_invertible_functor: fixes \<phi> :: "'b \<Rightarrow> nat"   and S :: "'b set"   and n :: "'b" assumes "bij_betw \<phi> S (Collect \<phi>.A.arr)"   and "n \<notin> S" shows "\<exists>C'a. Collect (partial_magma.arr C'a) = S \<and> invertible_functor C'a C' (\<lambda>i. if partial_magma.arr C'a i then \<phi> i else \<phi>.A.null)"
simps: fixes p :: "nat"   and q :: "nat"   and u :: "nat"   and ns :: "nat list"   and xs :: "'c list" shows "bn_inv p q (u, ns, xs) = Ball (set ns) (case_nat True (case_nat True (\<lambda>m. bn_valid m p q)))"
depthZero: fixes P :: "pi" assumes "depth P = 0"   and "uhnf P" shows "P = \<zero>"
fun_upd_image: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'b"   and A :: "'c set" shows "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)"
uniformly_continuous_on_def: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "uniformly_continuous_on s f = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
read_only_share_all_acquired_in': fixes \<O> :: "nat set"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and a :: "nat" assumes "\<O> \<inter> read_only S = {}"   and "weak_sharing_consistent \<O> sb"   and "a \<in> read_only (share sb S)" shows "a \<in> read_only (share sb Map.empty) \<or> a \<in> read_only S \<and> a \<notin> acquired True sb \<O> \<and> a \<notin> all_shared sb"
Sup_insert_finite: fixes S :: "'a set"   and x :: "'a" assumes "finite S" shows "\<Squnion> insert x S = (if S = {} then x else max x (\<Squnion> S))"
extended_distance_middle_1: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c Bonk_Schramm_extension_unfolded"   and b :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open"   and "a \<in> metric_space.extended_distance_set dist"   and "b \<in> metric_space.extended_distance_set dist" shows "metric_space.extended_distance dist a (middle a b) = metric_space.extended_distance dist a b / 2"
H2: shows "Bet A B C \<and> Bet A' B' C' \<and> Cong A C A' C' \<and> Cong B C B' C' \<and> Cong A D A' D' \<and> Cong C D C' D'"
lunit\<^sub>F\<^sub>M\<^sub>C_def: fixes a :: "'c local.term set" shows "\<l>[a] = mkarr \<^bold>\<l>\<^bold>[rep a\<^bold>]"
strict_assoc': fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "ide \<a>\<^sup>-\<^sup>1[f, g, h]"
natural_transformation_axioms: shows "natural_transformation A C (H \<circ> F) (K \<circ> F) (\<tau> \<circ> F)"
indets_pp_def: fixes p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'b" shows "indets_pp p = remdups (concat (map keys_pp_to_list (punit.keys_to_list p)))"
assms_1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "d z + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> d z \<le> y"
conv_complement_sub_leq: fixes x :: "'a"   and y :: "'a" shows "x[[- (x * y)]] \<le> - y"
atLeastLessThan_empty_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastLessThan less_eq less a b = {}) = (\<not> less a b)"
complex_rotations_invariants: fixes r :: "Vertex \<Rightarrow> Vertex"   and v :: "Vertex"   and w :: "Vertex" assumes "r \<in> complex_rotations" shows "((\<forall>v. r v \<noteq> v) \<longrightarrow> r v = w \<longrightarrow> r w = v) \<and> (r v = v \<longrightarrow> r w = w \<longrightarrow> v \<noteq> w \<longrightarrow> r = id)"
llist_all2_conv_all_lnth: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>n. enat n < llength ys \<longrightarrow> P (xs $ n) (ys $ n)))"
open_ennreal_def: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
y_in_upper_cont: shows "y \<in> at_least_as_good y carrier relation"
res_prime_units_eq: shows "Units (residue_ring (int p)) = {1..int (p - 1)}"
cases: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" obtains "x = \<langle>\<rangle>"   | l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and c :: "tcolor" and uu_ :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (a, c, uu_), r\<rangle>"
inf_neqE: fixes x :: "'b llist"   and A :: "'b set"   and y :: "'b llist" assumes "x \<in> A\<^sup>\<omega>"   and "y \<in> A\<^sup>\<omega>"   and "x \<noteq> y" obtains s :: "'b llist" where "s \<in> A\<^sup>\<star>" and "s \<le> x" and "\<not> s \<le> y"
heq: shows "(\<lambda>t. sterm_sem I \<theta> (fst (mk_v I ODE (sol 0, b) (sol t)))) = sterm_sem I \<theta> \<circ> (\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t)))"
z_less_i: fixes x :: "int" shows "ant x < \<infinity>"
all_distinct_left: fixes l :: "'b tree"   and x :: "'b"   and b :: "bool"   and r :: "'b tree" assumes "all_distinct (Node l x b r)" shows "all_distinct l"
scale_minus_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c" assumes "Modules.module scale" shows "scale (- a) x = - scale a x"
simps: fixes p :: "real poly"   and qs :: "real poly list"   and m :: "rat mat"   and subs :: "nat list list"   and signs :: "rat list list" shows "reduce_system p (qs, m, subs, signs) = reduction_step m signs subs (solve_for_lhs p qs subs m)"
timpl_closure_set_Var_in_iff: fixes x :: "'d"   and M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "(Var x \<in> timpl_closure_set M TI) = (Var x \<in> M)"
subst_apply_stateful_strand_def: fixes S :: "('a, 'b) stateful_strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
inorder_bal_list: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "inorder (bal_list n xs) = take n xs"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v) = (\<exists>p. H.awalk u p v)"
simps_4: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and oper :: "pdf_operator"   and e :: "expr" shows "expr_type \<Gamma> (oper $$ e) = (case expr_type \<Gamma> e of None \<Rightarrow> None | Some x \<Rightarrow> op_type oper x)"
exists_fun_notin_funs_terms: fixes M :: "('fun, 'a) Term.term set" assumes "finite M" shows "\<exists>f. f \<notin> \<Union> (funs_term ` M)"
MtransC_Trans: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and cf'' :: "('a, 'c) com \<times> 'b" assumes "PL.MtransC tval aval cf cf'"   and "PL.MtransC tval aval cf' cf''" shows "PL.MtransC tval aval cf cf''"
SDG_edge_sum_SDG_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt n'" shows "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt False n'"
Aeta_expand_0: fixes e :: "exp" shows "Aeta_expand 0 e = e"
expand: fixes state :: "('a, 'b) state"   and state' :: "('a, 'b) state" assumes "\<T> state = \<T> state' \<and> \<B>\<^sub>i\<^sub>l state = \<B>\<^sub>i\<^sub>l state' \<and> \<B>\<^sub>i\<^sub>u state = \<B>\<^sub>i\<^sub>u state' \<and> \<V> state = \<V> state' \<and> \<U> state = \<U> state' \<and> \<U>\<^sub>c state = \<U>\<^sub>c state'" shows "state = state'"
plain_RETURN: fixes a :: "'a" shows "RETURN a \<le> RETURN a"
borel_measurable_nn_transfer_operator: fixes f :: "'a \<Rightarrow> ennreal" shows "nn_transfer_operator f \<in> borel_measurable M"
ax4: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>[P \<longrightarrow> Q]_v \<longrightarrow> \<box>[P]_v \<longrightarrow> \<box>[Q]_v"
eq_refl: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" assumes "x = y" shows "less_eq_prod x y"
intros_1: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and R :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and H :: "nat set"   and K :: "nat set"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<in> S"   and "((\<sigma>, t), H\<not>K:arrive(m), \<sigma>', t') \<in> T"   and "H \<subseteq> R"   and "K \<inter> R = {}" shows "((\<sigma>, SubnetS s t), R:*cast(m), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"