hcju/isabelleps huggingface dataset:
compact_Icc: fixes a :: "real"   and b :: "real" shows "compact {a..b}"
uset_l: fixes p :: "fm" assumes "isrlfm p" shows "\<forall>(t, k)\<in>set (uset p). numbound0 t \<and> 0 < k"
listEqualsButlastAppendLast: fixes list :: "'b list" assumes "list \<noteq> []" shows "list = butlast list @ [last list]"
gram_schmidt_sub_triv: fixes n :: "nat"   and us :: "rat vec list"   and ws :: "rat vec list" shows "gram_schmidt_sub_triv n (map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) us) ws = map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) (gram_schmidt_sub n us ws)"
correct_game_def: fixes key_gen :: "('a \<times> 'b) spmf"   and commit :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and verify :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'c" shows "abstract_commitment.correct_game key_gen commit verify m = key_gen \<bind> (\<lambda>(ck, vk). commit ck m \<bind> (\<lambda>(c, d). return_spmf (verify vk m c d)))"
ext: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "g \<cdot> f \<noteq> \<F>C.null" shows "\<F>C.seq g f"
mset_set_empty_iff: fixes A :: "'k set" shows "(mset_set A = {#}) = (A = {} \<or> infinite A)"
negligible_Un: fixes S :: "'b set"   and T :: "'b set" assumes "negligible S"   and "negligible T" shows "negligible (S \<union> T)"
table_of_map_SomeI: fixes t :: "('b \<times> 'a) list"   and k :: "'b"   and x :: "'a"   and f :: "'a \<Rightarrow> 'c" assumes "table_of t k = Some x" shows "table_of (map (\<lambda>(k, x). (k, f x)) t) k = Some (f x)"
neg_smult: fixes r :: "'r"   and m :: "'m" assumes "r \<in> R"   and "m \<in> M" shows "- r \<cdot> m = - (r \<cdot> m)"
nth_default_map_eq: fixes f :: "'c \<Rightarrow> 'b"   and dflt' :: "'c"   and dflt :: "'b"   and xs :: "'c list"   and n :: "nat" assumes "f dflt' = dflt" shows "nth_default dflt (map f xs) n = f (nth_default dflt' xs n)"
lookup_sym_mpoly_aux_monom_singleton: fixes X :: "nat set"   and x :: "nat"   and k :: "nat" assumes "finite X"   and "x \<in> X"   and "0 < k" shows "lookup (powsum_mpoly_aux X k) (Poly_Mapping.single x k) = (1::'b)"
arctan_ms_aux_def: fixes xs :: "('a \<times> real) msllist" shows "arctan_ms_aux xs = times_ms_aux xs (powser_ms_aux' arctan_series_stream (times_ms_aux xs xs))"
WT_ideal: shows "\<I>_ideal \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res ideal_resource \<surd>"
monadic_FOREACH_itsl: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d set"   and c :: "'c \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" shows "it_to_sorted_list R s \<bind> (\<lambda>l. monadic_nfoldli l c f \<sigma>) \<le> monadic_FOREACH R I s c f \<sigma>"
comp_null_2: fixes f :: "'d" shows "f \<cdot>\<^sub>D D.null = D.null"
advantage_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "advantage \<A> = spmf (game \<A>) True"
coreflexive_comp_top_inf: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x" shows "x * top \<sqinter> y = x * y"
decE_decE: fixes n :: "nat"   and k :: "nat"   and n' :: "nat"   and \<Gamma> :: "binding list" shows "\<down>\<^sub>e n k (\<down>\<^sub>e n' (k + n) \<Gamma>) = \<down>\<^sub>e (n + n') k \<Gamma>"
Jordan_curve_R2: fixes c :: "real \<Rightarrow> 'a" assumes "simple_path c"   and "pathfinish c = pathstart c" obtains inside :: "'a set" and outside :: "'a set" where "inside \<noteq> {}" and "open inside" and "connected inside" and "outside \<noteq> {}" and "open outside" and "connected outside" and "bounded inside" and "\<not> bounded outside" and "inside \<inter> outside = {}" and "inside \<union> outside = - path_image c" and "frontier inside = path_image c" and "frontier outside = path_image c"
rec_all1_less_def: fixes f :: "recf" shows "rec_all1_less f = (let cond1 = CN rec_eq [recf.Id 3 0, recf.Id 3 1]; cond2 = CN f [recf.Id 3 0, recf.Id 3 2] in CN (rec_all2 (CN rec_disj [cond1, cond2])) [recf.Id 2 0, recf.Id 2 0, recf.Id 2 1])"
neg64: shows "SCAST(32 \<rightarrow> 64) w2 + 0xFFFFFFFF80000001 = SCAST(32 \<rightarrow> 64) w2 + - 0x7FFFFFFF"
is_heap_of_list: fixes i :: "'a list" shows "is_heap (as_tree (of_list i))"
mon_iff_hom_one: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "Group.group H" shows "(f \<in> mon G H) = (f \<in> hom G H \<and> (\<forall>x. x \<in> carrier G \<and> f x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>))"
is_extensional: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" assumes "\<not> A2_BxA2.arr f" shows "E.map f = B.null"
capacity_const: shows "\<forall>e. 0 \<le> max_flow e \<and> max_flow e \<le> c e"
canon_ang_sum: fixes \<alpha> :: "real"   and \<beta> :: "real" shows "\<downharpoonright>\<alpha> + \<beta>\<downharpoonleft> = \<downharpoonright>\<downharpoonright>\<alpha>\<downharpoonleft> + \<downharpoonright>\<beta>\<downharpoonleft>\<downharpoonleft>"
indep_in_supI: fixes X :: "'a set"   and \<E>' :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>'"   and "\<E>' \<subseteq> \<E>"   and "X_insert.indep_in \<E> X" shows "X_insert.indep_in \<E>' X"
scaleC_diff_left: fixes a :: "complex"   and b :: "complex"   and x :: "'a" shows "(a - b) *\<^sub>C x = a *\<^sub>C x - b *\<^sub>C x"
simps_3: fixes k :: "nat" shows "\<P> (Suc k) 0 = \<Q> k"
Iy: shows "M \<le> Gromov_product_at py x y"
flat_lub_def: fixes b :: "'a"   and A :: "'a set" shows "flat_lub b A = (if A \<subseteq> {b} then b else THE x. x \<in> A - {b})"
divide_zero_left: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "field R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "\<zero>\<^bsub>R\<^esub> \<oslash>\<^bsub>R\<^esub> a = \<zero>\<^bsub>R\<^esub>"
degree_poly_x_mult_y: fixes p :: "'a poly" shows "degree (poly_x_mult_y p) = degree p"
star_aci_2: fixes a :: "assn"   and b :: "assn" shows "a * b = b * a"
set_i_il_eq: fixes i' :: "nat" shows "(i' \<in> set (i # il)) = (i' < length xs \<and> P (xs ! i'))"
weaken_post: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}"   and "\<forall>l s. Q l s \<longrightarrow> Q' l s" shows "\<turnstile>\<^sub>1 {P} c { e \<Down> Q'}"
type_wf_preserved: fixes SW :: "((_) heap, 'p, 'q) prog set"   and setter :: "((_) heap, 'p, 'r) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>object_ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> \<forall>node_ptr. preserved (get_M\<^sub>N\<^sub>o\<^sub>d\<^sub>e node_ptr RNode.nothing) h h'" shows "NodeClass.type_wf h = NodeClass.type_wf h'"
square_free_iff_separable: fixes f :: "'c poly" shows "square_free f = separable f"
map_nth_eq: fixes xs :: "'c list"   and ys :: "'d list"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> f (xs ! i) = g (ys ! i)" shows "map f xs = map g ys"
connected_sing: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.topological_space open" shows "topological_space.connected open {x}"
f6: shows "min (List.gen_length 0 (tl i2)) (List.gen_length 0 (tl i1)) = List.gen_length 0 (tl i2)"
tape_of_nat_def: fixes n :: "nat" shows "<n> \<equiv> Oc \<up> Suc n"
direct_lim_axioms: shows "direct_lim Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec {U. is_zariski_open U \<and> \<pp> \<in> U}"
pred_prod_beta: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and xy :: "'b \<times> 'c" shows "pred_prod P Q xy = (P (fst xy) \<and> Q (snd xy))"
map_eq_on_tvs_imp_map_eq_on_term: fixes t :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "\<And>p. p \<in> tvs t \<Longrightarrow> subs p = subs' p" shows "tsubst t (convert_subs subs) = tsubst t (convert_subs subs')"
antisym_conv2: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
bind_tell: fixes w :: "'w"   and m :: "'m"   and f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (tell w m) f = tell w (bind m f)"
empty: shows "\<Union> {g - {x} |g. g \<in> G \<and> g \<noteq> {x}} = \<Union> ({g - {x} |g. g \<in> G} - {{}})"
Un_empty_left: fixes B :: "'d set" shows "{} \<union> B = B"
domInterMT: fixes a :: "'b \<Rightarrow> 'c option"   and b :: "'b \<Rightarrow> 'd option"   and x :: "'b" assumes "dom a \<inter> dom b = {}"   and "x \<in> dom a" shows "x \<notin> dom b"
types_preserved_unop_testop_cvtop: fixes v :: "v"   and e :: "b_e"   and v' :: "v"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and t :: "t"   and iop :: "unop_i"   and fop :: "unop_f"   and testop :: "testop"   and t2 :: "t"   and cvtop :: "cvtop"   and sx :: "sx option" assumes "\<lparr>[$C v, $e]\<rparr> \<leadsto> \<lparr>[$C v']\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C v, $e] : ts _> ts'"   and "e = Unop_i t iop \<or> e = Unop_f t fop \<or> e = Testop t testop \<or> e = Cvtop t2 cvtop t sx" shows "\<S>\<bullet>\<C> \<turnstile> [$C v'] : ts _> ts'"
wf_subtype_acc: fixes wf_mb :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog wf_mb P" shows "acc (subtype P)"
assms_4: fixes x :: "'a" assumes "x \<in> space M" shows "f x = v (g x)"
case_prod_curry: fixes f :: "'a \<times> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). curry f x y) = f"
conjugate_zero: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate zero = zero"
maximalI: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "x \<in> S"   and "\<And>y. y \<in> S \<Longrightarrow> f y \<le> f x" shows "x \<in> maximal f S"
elts_of_set: fixes X :: "V set" shows "elts (set X) = (if small X then X else {})"
w_def: shows "w = justifying_ws (J (Suc n)) a''"
remove_strictly_subsumed_clauses_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 Q :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N"   and "\<forall>D\<in>fst ` set P. \<not> strictly_subsume [C] D" shows "wstate_of_dstate (N, P @ P', Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P @ filter (Not \<circ> strictly_subsume [C] \<circ> fst) P', Q, n)"
unannotated: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "eval\<^sub>w_simplep c mem c' mem'" shows "eval\<^sub>wp (cxt_to_stmt E c) mds mem (cxt_to_stmt E c') mds mem'"
imp_ex: fixes P :: "'b \<Rightarrow> bool"   and Q :: "bool" shows "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)"
frontier_of_components_subset: fixes C :: "'b set"   and S :: "'b set" assumes "C \<in> components S" shows "frontier C \<subseteq> frontier S"
adjoint_funcs_skip: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" assumes "SFuncs \<sigma> f = undeft" shows "Funcs (USubst.adjoint \<sigma> I \<omega>) f = Funcs I f"
lemma2_6_6_b': fixes r :: "('b \<times> 'b) set"   and Q :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset" assumes "trans r"   and "irrefl r"   and "(Q + M, Q + N) \<in> mul_eq r" shows "(M, N) \<in> mul_eq r"
BlockRedSome: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V := None))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = \<lfloor>v\<rfloor>"   and "\<not> assigned V e" shows "P \<turnstile> \<langle>{V:T; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v;; e'},(h', l'(V := l V))\<rangle>"
Liminf_const: fixes F :: "'b filter"   and c :: "'d" assumes "F \<noteq> bot" shows "Liminf F (\<lambda>x. c) = c"
subst_Betas: fixes s :: "Lambda_Z.term"   and s' :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and t' :: "Lambda_Z.term"   and x :: "name" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "subst x s t \<rightarrow>\<^sub>\<beta>\<^sup>* subst x s' t'"
prod_in_terms_of_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "arr g" shows "HF'.prod f g = tuple (local.comp f (pr1 (local.dom f) (local.dom g))) (local.comp g (pr0 (local.dom f) (local.dom g)))"
divsor_prime_ex: fixes x :: "nat" assumes "\<not> Prime x"   and "Suc 0 < x" shows "\<exists>p. Prime p \<and> p dvd x"
AR_imp_absolute_retract: fixes S :: "'a set"   and S' :: "'b set"   and U :: "'b set" assumes "AR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" shows "S' retract_of U"
beta_subset_par_beta: shows "(\<rightarrow>\<^sub>\<beta>) \<le> (=>)"
simps: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b" shows "pairself f (x, y) = (f x, f y)"
remove_is_empty_imp_mul: fixes M :: "'a multiset"   and r :: "('a \<times> 'a) set"   and \<alpha> :: "'a" assumes "M -s r \<down>m {#\<alpha>#} = {#}" shows "(M, {#\<alpha>#}) \<in> mul r"
\<phi>_biuniversal_prop_1: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "\<guillemotleft>\<mu> : k \<star> v \<Rightarrow> h \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : v \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (h \<star> \<theta>) \<cdot> \<a>[h, p\<^sub>0, w] \<cdot> (\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w] \<cdot> (k \<star> \<nu>) = \<mu>"
fmdom_notI: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" assumes "fmlookup m x = None" shows "x |\<notin>| fmdom m"
Nil1: fixes ys :: "'a list"   and zs :: "'b list" assumes "as = []"   and "bs = ys"   and "cs = zs" shows "P"
pseudominimal_inD2: fixes P :: "'b set"   and x :: "'b"   and z :: "'b" assumes "codomain.pseudominimal_in P x"   and "z \<in> P"   and "codomain.greater x z" shows "z = codomain.bottom P"
bind_merge_merge: fixes y :: "'a cset"   and f :: "'a \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (merge y f) (\<lambda>A. merge A g) = merge y (\<lambda>x. bind (f x) (\<lambda>A. merge A g))"
naturality: fixes f :: "'c" assumes "\<tau>.A.arr f" shows "B (\<tau>' (\<tau>.A.cod f)) (G f) = B (H f) (\<tau>' (\<tau>.A.dom f))"
adjunction_unit_determines_counit: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<epsilon>' :: "'a" assumes "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"   and "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>'" shows "\<epsilon> = \<epsilon>'"
AtParallelExprs: fixes i :: "nat"   and cs :: "('a, 'b, 'c) com list"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and as :: "(('a, 'b, 'c) ann \<times> 'a set \<times> 'a set) list"   and a :: "'a set \<times> ('a, 'b, 'c) com" assumes "i < length cs"   and "atomicsR \<Gamma> \<Theta> (fst (as ! i)) (cs ! i) a" shows "atomicsR \<Gamma> \<Theta> (AnnPar as) (Parallel cs) a"
integ1: shows "set_integrable M1 A (\<lambda>x. \<integral>y\<in>B. norm (f (x, y))\<partial>M2)"
match_list_semantics: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms1 :: "'b match_expr list"   and a :: "action"   and p :: "'c"   and ms2 :: "'b match_expr list"   and s :: "state" assumes "match_list \<gamma> ms1 a p = match_list \<gamma> ms2 a p" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms1) s = approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms2) s"
steps_non_empty': fixes E :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "Graph_Defs.steps E xs" shows "xs \<noteq> []"
ide_cod: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.ide (VoV.cod f)"
abs_fun_if: fixes pi :: "('b \<times> 'b) list"   and c :: "bool"   and x :: "'a"   and y :: "'a" shows "pi \<bullet> (if c then x else y) = (if c then pi \<bullet> x else pi \<bullet> y)"
Elim_1: fixes F :: "'b"   and S :: "'b set"   and G :: "'b" assumes "F \<in> S \<and> G \<in> S" shows "{F, G} \<subseteq> S"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "msort f [] = []"
of_nat_power_eq_of_nat_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero x = power.power one times (semiring_1.of_nat one plus zero b) w) = (x = b ^ w)"
carrier_vec_conjugate: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "conjugate v \<in> carrier_vec n"
gcd_mod_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm"   and "b \<noteq> zero" shows "gcd (modulo a b) b = gcd a b"
valid_node_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and n :: "'g" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.valid_node sourcenode targetnode valid_edge n \<equiv> \<exists>a. valid_edge a \<and> (n = sourcenode a \<or> n = targetnode a)"
length_input2state_repeat: fixes i :: "value list"   and x :: "nat"   and a :: "value"   and y :: "nat" assumes "input2state i $ x = Some a" shows "y < length (i @ repeat y a)"
bit_word_eqI: fixes a :: "'b word"   and b :: "'b word" assumes "\<And>n. n < LENGTH('b) \<Longrightarrow> bit a n = bit b n" shows "a = b"
DERIV: shows "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"
numeral_power_le_of_nat_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and i :: "num"   and n :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (power.power one times (numeral.numeral one plus i) n) (semiring_1.of_nat one plus zero x) = (numeral i ^ n \<le> x)"
not_outside_3: fixes i :: "'a"   and x :: "'b"   and y :: "'b" assumes "R i x y" shows "y \<in> alts"
assms: shows "ev (alw (\<lambda>xs. \<not> P xs)) \<omega>"
fsubsetD: fixes A :: "'c fset"   and B :: "'c fset"   and c :: "'c" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q Q' A\<^sub>Q \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (P \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
perm_2: fixes pi :: "(name \<times> name) list"   and x3 :: "'d"   and x2 :: "name list"   and x1 :: "'d" shows "pi \<bullet> x3\<lparr>\<nu>*x2\<rparr>\<langle>x1\<rangle> = (pi \<bullet> x3)\<lparr>\<nu>*(pi \<bullet> x2)\<rparr>\<langle>(pi \<bullet> x1)\<rangle>"
lateEarlyBoundOutput: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'" shows "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"
path_join_set: fixes G :: "graph"   and p :: "nat list"   and q :: "nat list" assumes "joinable G p q" shows "path_join G p q \<in> path_set G (hd p) (last q)"
cmp_rule2: fixes s :: "'a"   and ni :: "'b"   and ni' :: "'b"   and i :: "'c ifex"   and i' :: "'c ifex" assumes "I s"   and "cmp ni ni'"   and "(ni, i) \<in> R s"   and "(ni', i') \<in> R s" shows "i = i'"
iteratei_map_op_list_it_lm_ops_def: shows "iteratei_map_op_list_it_lm_ops \<equiv> lm.iteratei"