hcju/isabelleps huggingface dataset:
fail_t_def: shows "fail_t = {n. \<exists>q q'. state_rank q n \<noteq> None \<and> q' = \<delta> q (w n) \<and> q' \<notin> F \<and> sink q'}"
bin_mismatch_neq: shows "i.c\<^sub>0 \<noteq> i.c\<^sub>1"
simps_3: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 # x22 \<noteq> \<epsilon>"
add_base_zero: fixes k :: "nat"   and ns :: "nat list" shows "foldl (+) k ns = foldl (+) 0 ns + k"
remove: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "F A = a \<^bold>* F (A - {a})"
antisym: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<preceq>\<^sub>p a"   and "a \<preceq>\<^sub>p b" shows "a = b"
injectivity: fixes x :: "int"   and y :: "int" assumes "of_int x = of_int y" shows "x = y"
f_range_on_V_axiom_on_rationalizes_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "f_range_on A f"   and "V_axiom_on A f" shows "rationalizes_on A f (rwp_on A f)"
basic_cong_5: fixes p :: "int" assumes "2 < p" shows "[0 \<noteq> - 1] (mod p)"
inj_on_fun_updI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and y :: "'b"   and x :: "'a" assumes "inj_on f A"   and "y \<notin> f ` A" shows "inj_on (f(x := y)) A"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "L.seq g f" shows "L.cod (L.comp g f) = L.cod g"
fls_delta_nth: fixes a :: "int"   and b :: "'a"   and n :: "int" shows "fls_nth (Abs_fls (\<lambda>n. if n = a then b else (0::'a))) n = (if n = a then b else (0::'a))"
equiv_trans_comm: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "equiv_trans t u = equiv_trans u t"
S_ge: fixes x :: "real" assumes "1 / \<alpha> \<le> x" shows "\<alpha> * x \<le> S x"
nonstrict_eq: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat" assumes "cfg_doms G"   and "cfg_doms.non_strict_dominate G i j" shows "\<not> cfg_doms.strict_dominate G i j"
distinct_11: fixes x21 :: "char list"   and x22 :: "aexp"   and x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "x21 ::= x22 \<noteq> IF x41 THEN x42 ELSE x43"
irreducible_content: fixes p :: "'b poly" assumes "irreducible p" shows "Polynomial.degree p = 0 \<or> Missing_Polynomial_Factorial.primitive p"
list_rel_append2: fixes l :: "'d list"   and as :: "'e list"   and bs :: "'e list"   and R :: "('d \<times> 'e) set" shows "((l, as @ bs) \<in> \<langle>R\<rangle>list_rel) = (\<exists>cs ds. l = cs @ ds \<and> (cs, as) \<in> \<langle>R\<rangle>list_rel \<and> (ds, bs) \<in> \<langle>R\<rangle>list_rel)"
assms_2: shows "\<NN>' : \<FF>' \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>' : \<AA>' \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>'"
Pi_iff: fixes f :: "'g \<Rightarrow> 'h"   and I :: "'g set"   and X :: "'g \<Rightarrow> 'h set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
fBexI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b fset" assumes "P x"   and "x |\<in>| A" shows "fBex A P"
distinct_1: fixes x2 :: "'x" shows "None \<noteq> Some x2"
axiom_valid: fixes a :: "axiom" shows "valid (get_axiom a)"
vunion_def: fixes A :: "V"   and B :: "V" shows "A \<union>\<^sub>\<circ> B = set {x. x \<in>\<^sub>\<circ> A \<or> x \<in>\<^sub>\<circ> B}"
ordLess_or_ordIso: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "r <o r' \<or> r' <o r \<or> r =o r'"
uniform_limit_interval_integral_left: fixes a :: "ereal"   and b :: "ereal"   and g :: "real \<Rightarrow> real"   and Y :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "interval_lebesgue_integrable lborel a b g"   and "\<And>y. y \<in> Y \<Longrightarrow> set_borel_measurable borel (einterval a b) (f y)"   and "\<And>y. y \<in> Y \<Longrightarrow> AE x\<in>einterval a b in lborel. norm (f y x) \<le> g x"   and "a < b" shows "uniform_limit Y (\<lambda>a' y. interval_lebesgue_integral lborel a' b (f y)) (\<lambda>y. interval_lebesgue_integral lborel a b (f y)) (at_right a)"
flush_instr_cache_def: fixes state :: "'b sparc_state" shows "flush_instr_cache state \<equiv> state\<lparr>cache := cache state\<lparr>icache := empty_cache\<rparr>\<rparr>"
elements_matI: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat nr nc"   and "i < nr"   and "j < nc"   and "a = A $$ (i, j)" shows "a \<in> elements_mat A"
is_acc_classD: fixes G :: "prog"   and P :: "pname"   and C :: "qtname" assumes "is_acc_class G P C" shows "is_class G C \<and> G \<turnstile> Class C accessible_in P"
assm_1: shows "x \<le> z\<^sup>\<star>"
execlass_les_code: fixes cs :: "('a \<times> 'a) list"   and c1 :: "'a"   and c2 :: "'a" shows "class_les (set cs) c1 c2 = execlass_les cs c1 c2"
alive_trivial_simps_3: fixes sh :: "int"   and s :: "Store" shows "alive (shortV sh) s"
hF: shows "(\<lambda>f. f \<circ> \<pi>) ` F = F"
bounded_byD: fixes b :: "real"   and P :: "'a \<Rightarrow> real"   and x :: "'a" assumes "bounded_by b P" shows "P x \<le> b"
leg0_commutes: shows "\<nu>.cod.leg0 \<cdot> \<nu>.chine = \<nu>.dom.leg0"
psubset_insert_iff: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" shows "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
token_smallest_accepting_rank: fixes i :: "nat" assumes "\<MM>.smallest_accepting_rank\<^sub>R = Some i" shows "\<forall>\<^sub>\<infinity>n. \<forall>x. \<MM>.token_succeeds x = (n < x \<or> (\<exists>j\<ge>i. \<MM>.rank x n = Some j) \<or> \<MM>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
assms: fixes x :: "'a \<times> 'b" assumes "x \<in># A \<times># {#b#}" shows "P x = Q (fst x)"
congD: fixes a :: "'c"   and b :: "'c"   and n :: "'c" assumes "[a = b] (mod n)" shows "a mod n = b mod n"
sqn_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, kno, flg, hops, sip, {})" shows "sqn (update rt dip (dsn, kno, flg, hops, sip, {})) dip = dsn"
c'd'_in_set_2: shows "d' \<in> {a..b} \<inter> g -` {c..d}"
rbt_lookup_rbt_insertwk: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c"   and x :: "'a" assumes "rbt_sorted t" shows "rbt_lookup (rbt_insert_with_key f k v t) x = (rbt_lookup t(k \<mapsto> case rbt_lookup t k of None \<Rightarrow> v | Some w \<Rightarrow> f k w v)) x"
Vagree_empty: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real" shows "Vagree \<nu> \<nu>' {}"
Gromov_hyperbolic_subset_def: fixes delta :: "real"   and A :: "'c set" shows "Gromov_hyperbolic_subset delta A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. \<forall>t\<in>A. dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * delta)"
minus_divide_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (uminus (divide b c)) a = (if less zero c then less (uminus b) (times a c) else if less c zero then less (times a c) (uminus b) else less zero a)"
in_arcs_eq: fixes w :: "'a" shows "in_arcs G w = edge_rev M ` out_arcs G w"
eq_equiv_class_iff: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "equiv A r"   and "x \<in> A"   and "y \<in> A" shows "(r `` {x} = r `` {y}) = ((x, y) \<in> r)"
UN_Un: fixes M :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and B :: "'b set" shows "\<Union> (M ` (A \<union> B)) = \<Union> (M ` A) \<union> \<Union> (M ` B)"
Un_Int_distrib: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<union> B \<inter> C = (A \<union> B) \<inter> (A \<union> C)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a"   and const :: "const" shows "P (case const of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = ((\<forall>x1. const = Const x1 \<longrightarrow> P (f1 x1)) \<and> (const = Any \<longrightarrow> P f2))"
r_one: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
Enc_keys_clean_LtKeys: fixes K :: "msg set" assumes "K \<subseteq> range LtK" shows "Enc_keys_clean K"
emeasure_buffon_set': shows "emeasure lborel (buffon_set' l d) = ennreal (integral {0..pi} (\<lambda>x. min (d / 2) (sin x * l / 2)))"
iT_Plus_neg_Max_less_empty: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "Max I < k" shows "I \<oplus>- k = {}"
rec: fixes n :: "nat"   and old :: "nat" assumes "Suc n < |es|" shows "natToVertexListRec old v f es ! Suc n = (if es ! n = es ! Suc n then None else Some f\<^bsup>es ! Suc n\<^esup> \<bullet> v)"
3: fixes u :: "'t" assumes "v \<prec>\<^sub>t u" shows "lookup p u = lookup q u"
step: fixes Sup :: "'d set \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
implies_galois_increasing: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<leadsto> x \<sqinter> y"
not_less_i: shows "\<not> (GREATEST m. \<not> is_zero_row_upt_k m k A) < i"
sums_enn2ereal: fixes f :: "nat \<Rightarrow> ennreal"   and x :: "ennreal" shows "(\<lambda>i. enn2ereal (f i)) sums enn2ereal x = f sums x"
powser_ms_aux: fixes cs :: "real msllist"   and basis :: "(real \<Rightarrow> real) list"   and G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "convergent_powser cs"   and "basis_wf basis"   and "is_expansion_aux G g basis"   and "ms_exp_gt 0 (ms_aux_hd_exp G)" shows "is_expansion_aux (powser_ms_aux cs G) (powser cs \<circ> g) basis"
isTrue_\<phi>R: assumes "consistent" shows "isTrue \<phi>R"
ide_char': fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
types_snoc_eq: fixes e :: "nat \<Rightarrow> LambdaType.type"   and ts :: "dB list"   and t :: "dB"   and Ts :: "LambdaType.type list"   and T :: "LambdaType.type" shows "(e \<tturnstile> ts @ [t] : Ts @ [T]) = (e \<tturnstile> ts : Ts \<and> e \<turnstile> t : T)"
measure_lebesgue_linear_transformation: fixes A :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "bounded A"   and "A \<in> sets lebesgue"   and "linear f" shows "Sigma_Algebra.measure lebesgue (f ` A) = \<bar>det (matrix f)\<bar> * Sigma_Algebra.measure lebesgue A"
set_vars_list: fixes lp :: "linear_poly" shows "set (vars_list lp) = vars lp"
surj_image_vimage_eq: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'b set" assumes "surj f" shows "f ` f -` A = A"
residual_galoisI1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "f x \<le> y" shows "x \<le> residual f y"
priorities_def: fixes q :: "('d, 'c) pq" shows "\<parallel>q\<parallel> = map snd (alist_of q)"
dist_triangle_add_half: fixes x :: "'a"   and x' :: "'a"   and e :: "real"   and y :: "'a"   and y' :: "'a" assumes "dist x x' < e / 2"   and "dist y y' < e / 2" shows "dist (x + y) (x' + y') < e"
mult_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<cdot> z \<le> y \<cdot> z"
landau_flip_3: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(g \<in> \<Omega>[F](f)) = (f \<in> O[F](g))"
orderE: fixes b :: "'d upred"   and a :: "'d upred" assumes "a \<sqsubseteq> b" obtains "a = (a \<or> b)"
that: fixes g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "continuous_map (Euclidean_space m) (Euclidean_space n) g"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
strict_trans2: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "b <s a"   and "c \<le>s b" shows "c <s a"
greaterThanAtMost_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "ord.greaterThanAtMost less_eq less l u \<equiv> ord.greaterThan less l \<inter> ord.atMost less_eq u"
add_assoc': fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "x \<union> y \<union> z = x \<union> (y \<union> z)"
abs_eq: fixes x :: "'b ltl" shows "\<up>Unf (Abs x) = Abs (Unf x)"
v11_vcomp_vconverse': shows "\<FF>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ> = vid_on (\<BB>\<lparr>Obj\<rparr>)"
"3_6": shows "hd (x # y # xs) < (0::'a)"
xs_def: shows "xs = [OpUbx1 n1, OpUbx1 n2]"
locals_upd_id: fixes s :: "'a state_scheme" shows "s\<lparr>locals := locals s\<rparr> = s"
crename_id: fixes M :: "trm"   and a :: "coname" shows "M[a\<turnstile>c>a] = M"
DiffE: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
vector_in_orthonormal_basis: fixes a :: "'a" assumes "norm a = 1" obtains S :: "'a set" where "a \<in> S" and "pairwise orthogonal S" and "\<And>x. x \<in> S \<Longrightarrow> norm x = 1" and "independent S" and "card S = DIM('a)" and "span S = UNIV"
comp1: fixes g :: "'f \<Rightarrow> 'a" assumes "bounded_linear g" shows "bounded_bilinear (\<lambda>x. (**) (g x))"
AND_upper1: fixes x :: "int"   and y :: "int" assumes "0 \<le> x" shows "x AND y \<le> x"
item_rhs_simp: fixes N :: "symbol"   and \<alpha> :: "symbol list"   and d :: "nat"   and i :: "nat"   and j :: "nat" shows "item_rhs (Item (N, \<alpha>) d i j) = \<alpha>"
mono: fixes f :: "'b \<Rightarrow> 'b" shows "mono (\<lambda>p x. (\<exists>xa. x = f xa \<and> p xa) \<or> (\<exists>M. x = \<Or> M \<and> Complete_Partial_Order.chain (\<sqsubseteq>) M \<and> (\<forall>x\<in>M. p x)))"
share_append_Write\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and S :: "nat \<Rightarrow> bool option" shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W])) S = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"
advantage_def: fixes \<A> :: "(unit, 'x + 'x \<times> 'hash, 'hash + unit) gpv" shows "advantage \<A> = spmf (game \<A>) True"
better_d4_intro: fixes t1 :: "lam"   and t2 :: "lam"   and s1 :: "lam"   and s2 :: "lam"   and x :: "name" assumes " t1 \<longrightarrow>\<^sub>d t2"   and " s1 \<longrightarrow>\<^sub>d s2" shows " App (Lam [x].t1) s1 \<longrightarrow>\<^sub>d t2[x::=s2]"
supteq_ctxt_conv: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "s \<unrhd> t = (\<exists>C. s = C\<langle>t\<rangle>)"
endswith_def2: fixes pat :: "[:'r:]"   and xs :: "[:'r:]" shows "endswith\<cdot>pat\<cdot>xs = eq\<cdot>pat\<cdot>(shead\<cdot>(sfilter\<cdot>(\<Lambda> x. prefix\<cdot>x\<cdot>pat)\<cdot>(stails\<cdot>xs)))"
trcl_concat: fixes c :: "'f"   and w1 :: "'g list"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set"   and w2 :: "'g list"   and c'' :: "'f" assumes "(c, w1, c') \<in> trcl t"   and "(c', w2, c'') \<in> trcl t" shows "(c, w1 @ w2, c'') \<in> trcl t"
poly_weak_anti_mono_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "poly_weak_anti_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<beta> v \<le> \<alpha> v \<longrightarrow> eval_poly \<alpha> p \<le> eval_poly \<beta> p"
IH_1: fixes \<Gamma> :: "(name, name) fmap" assumes "fmpred (=) \<Gamma>" shows "alpha_equiv \<Gamma> t1_ t1_"
document_ptr_kinds_eq_h2: shows "document_ptr_kinds h3 = document_ptr_kinds h2"
matrix_to_iarray_snd_Gauss_Jordan_column_k_PA: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and P :: "(('b, 'd) vec, 'd) vec" assumes "i \<le> nrows A"   and "k < ncols A" shows "fst (snd (Gauss_Jordan_column_k_PA (P, i, A) k)) = fst (snd (Gauss_Jordan_column_k_iarrays_PA (matrix_to_iarray P, i, matrix_to_iarray A) k))"
"1_1": shows "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"
rv: fixes i :: "'b" assumes "i \<in> I" shows "Y i \<in> M' i \<rightarrow>\<^sub>M N i"
prod_mset_replicate_mset: fixes n :: "nat"   and a :: "'a" shows "\<Prod>\<^sub># (replicate_mset n a) = a ^ n"
zero_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "zero_hom hom \<equiv> hom (0::'b) = (0::'c)"
strict_mono_natI: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n < f (Suc n)" shows "OrdinalInduct.strict_mono f"
get_min_min: fixes h :: "'b tree"   and x :: "'b" assumes "is_root h"   and "pheap h"   and "x \<in> set_tree h" shows "get_min h \<le> x"
ladder_last_n_L': shows "ladder_last_n L' = length E"
evaln_nonstrict: fixes c :: "com"   and s :: "state"   and n :: "nat"   and t :: "state"   and m :: "nat" assumes "<c,s> -n-> t"   and "n \<le> m" shows "<c,s> -m-> t"
gf_U': shows "(gf_U' y y \<longlongrightarrow> r) (at_left 1)"
All2_Eats_I_D: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and A :: "fm" assumes "atom i \<sharp> (t, u)" shows "{All2 i t A, A(i::=u)} \<turnstile> All2 i (Eats t u) A"
orthogonal_basis_exists: fixes V :: "'b set" shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> card B = dim V \<and> pairwise orthogonal B"
fresh_1: fixes a :: "W.var"   and x1 :: "W.tvar" shows "a \<sharp> TVar x1 = a \<sharp> x1"
comp_cod_arr: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f"   and "J_C.cod f = b" shows "b \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f = f"
run_scons_elim: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and a :: "nat \<times> 'd"   and r :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd" assumes "grun A w V (a ## r) p" obtains "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" and "grun A w V r a"
major: shows "\<tau>Exec_mover ci P t e h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"
inf_left_idem: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf x y) = inf x y"
sup_bot_left: fixes x :: "'b" shows "(0::'b) + x = x"
elims: fixes x :: "'a list"   and y :: "cell list" assumes "tape_of_nat_list x = y" obtains "x = []" and "y = []"   | n :: "'a" where "x = [n__]" and "y = <n__>"   | n :: "'a" and v :: "'a" and va :: "'a list" where "x = n # v # va" and "y = <n> @ Bk # tape_of_nat_list (v # va)"
assms_1: fixes i :: "nat" assumes "i \<in> set_fml fml" shows "i < length qs"
get_owner_document_def: shows "Shadow_DOM.get_owner_document \<equiv> l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node_si get_disconnected_nodes get_host"
inverse_unique: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "V.inverse_arrows f g" shows "V.inv f = g"
set_iterator_union_foldli_conv: fixes as :: "'b list"   and bs :: "'b list" shows "set_iterator_union (foldli as) (foldli bs) = foldli (as @ bs)"
singleton: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set" assumes "linear_cf r linear_cf" shows "(x \<in> linear_cf X) = (linear_cf X = {x})"
rep_eq: fixes x :: "'a"   and xa :: "'b \<Rightarrow> 'b"   and xb :: "('a, 'b) RBT.rbt" shows "impl_of (map_entry x xa xb) = rbt_map_entry x xa (impl_of xb)"
sum_list'_noneg_eq_zero_iff: fixes xs :: "'a list"   and p :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'a) \<le> x" shows "(sum_list' p xs = (0::'a)) = (\<forall>x\<in>set xs. x = (0::'a))"
sf_sf_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> sf \<phi>" shows "sf \<psi> \<subseteq> sf \<phi>"
same_level_path_Append: fixes as :: "'edge list"   and as' :: "'edge list" assumes "same_level_path as"   and "same_level_path as'" shows "same_level_path (as @ as')"
karatsuba_cost_nonneg: fixes n :: "nat" shows "0 \<le> karatsuba_cost n"
NotL_inv: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta>"
8: shows "irreflexive ((w \<sqinter> - d)\<^sup>\<star> * e * (w \<sqinter> - d)\<^sup>\<star>)"
cases: fixes a :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state" assumes "Sstd.final a" obtains F :: "'fenv_std" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op) Std.instr fundef" and pc :: "nat" and H :: "'henv" and \<Sigma> :: "'dyn list" where "a = State F H [Frame f pc \<Sigma>]" and "Fstd_get F f = Some fd" and "pc = length (body fd)"
compileExpr_Prop1: fixes l :: "int"   and code :: "(int \<times> Instr) list"   and e :: "Expr"   and OUT :: "(int \<times> Instr) list \<times> int"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int" assumes "(l, code, e, OUT) \<in> compileExpr"   and "OUT = (code1, l1)" shows "l < l1 \<and> (\<forall>ll<l. code1\<down>ll = code\<down>ll) \<and> (\<forall>ll\<ge>l. ll < l1 \<longrightarrow> (\<exists>ins. code1\<down>ll = Some ins))"
set_sel_1: fixes a :: "'b llist" assumes "\<not> lnull a" shows "lhd a \<in> lset a"
Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion> A \<le> z"
less_succ: fixes i :: "'a"   and j :: "'a" assumes "local.aboveS i \<noteq> {}" shows "((j, succ i) \<in> r) = ((j, i) \<in> r \<or> j = succ i)"
Suc: shows "(\<Sum>k = 1..<2 ^ n. f (2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ k))"
simps_4: shows "inPorts Helper = {|plain_ant anyP|}"