hcju/isabelleps huggingface dataset:
CFG_SSA_Transformed_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and oldDefs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and oldUses :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'h set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'h set"   and phis :: "'d \<Rightarrow> 'e \<times> 'h \<Rightarrow> 'h list option"   and var :: "'d \<Rightarrow> 'h \<Rightarrow> 'g" shows "CFG_SSA_Transformed_axioms \<alpha>n invar inEdges' oldDefs oldUses defs uses phis var \<equiv> ((\<forall>g n. oldDefs g n = var g ` defs g n) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> oldUses g n = var g ` uses g n)) \<and> (\<forall>g n ns m v x v'. graph_path_base.path2 \<alpha>n invar inEdges' g n ns m \<longrightarrow> n \<notin> set (tl ns) \<longrightarrow> v \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v \<in> CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g m \<longrightarrow> x \<in> set (tl ns) \<longrightarrow> v' \<in> CFG_SSA_base.allDefs defs phis g x \<longrightarrow> var g v' \<noteq> var g v) \<and> (\<forall>g n v vs v'. phis g (n, v) = Some vs \<longrightarrow> v' \<in> set vs \<longrightarrow> var g v' = var g v) \<and> (\<forall>n g v v'. n \<in> set (\<alpha>n g) \<longrightarrow> v \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v' \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v \<noteq> v' \<longrightarrow> var g v' \<noteq> var g v)"
UP_ring_subring_deg: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "deg R g = deg (R\<lparr>carrier := S\<rparr>) g"
distinct_78: fixes pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "!pi' \<noteq> [name1\<noteq>name2]pi"
comm_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a :: "'c" assumes "comm_monoid f z" shows "f a z = a"
card_set: fixes xs :: "'e list" shows "card (set xs) = length (remdups xs)"
M_neg_imp_K2: fixes v :: "(real, 3) vec" assumes "v \<bullet> (M *v v) < 0" shows "proj2_abs v \<in> hyp2"
adv_P2_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool"   and D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "etp.OT_12.adv_P2 m1 m2 D = \<bar>spmf (etp.R2 m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). etp.S2 m2 o2 \<bind> D)) True\<bar>"
fine_exists: fixes a :: "real"   and b :: "real"   and \<delta> :: "real \<Rightarrow> real" assumes "a \<le> b"   and "gauge {a..b} \<delta>" shows "\<exists>D. fine \<delta> (a, b) D"
assms: shows "eval r_result1 [t, i, x] \<down>= Suc v"
closure_ball_lemma: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
kbo_instance_of_wpo_with_assms: shows "wpo_with_assms weight_S weight_NS (\<lambda>f g. (pr_strict f g, pr_weak f g)) (\<lambda>(f, n). n = 0 \<and> least f) full_status False (\<lambda>f. False)"
octo_of_real_eq_iff: fixes x :: "real"   and y :: "real" shows "(octo_of_real x = octo_of_real y) = (x = y)"
psubst_Var_simp2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "goodEnv rho"   and "rho xs x = Some X" shows "Var xs x #[rho] = X"
succs: fixes g' :: "'b"   and succs :: "'b \<Rightarrow> 'b list"   and g :: "'b"   and g'' :: "'b" assumes "g' \<in> set (succs g)"   and "RTranClp succs g' g''" shows "RTranClp succs g g''"
p: fixes x :: "int"   and y :: "int" assumes "(x, y) \<in> set ((0, 0) # (2, 1) # xs_ys)" shows "poly p x = y"
semiring_simprules_13: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
subst_dom_comp_inI: fixes y :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "y \<notin> subst_domain \<sigma>"   and "y \<in> subst_domain \<delta>" shows "y \<in> subst_domain (\<sigma> \<circ>\<^sub>s \<delta>)"
Var_Eq_imp_subst_Iff: fixes H :: "fm set"   and i :: "name"   and t :: "tm"   and A :: "fm" assumes "H \<turnstile> Var i EQ t" shows "H \<turnstile> A(i::=t) IFF A"
bs2_def: shows "bs2 = [ram2] @ between vs ram2 ram1 @ [ram1]"
HNext_def: fixes s :: "state"   and s' :: "state" shows "HNext s s' = (Next s s' \<and> HNextPart s s')"
gbinomial_prod_rev: fixes a :: "'a"   and k :: "nat" shows "a gchoose k = (\<Prod>i = 0..<k. a - of_nat i) div fact k"
free_hull_def: shows "free_hull \<equiv> \<lambda>G. {x. free_hullp (\<lambda>x. x \<in> G) x}"
Abs_wf_prog_inverse: fixes y :: "cmd \<times> (char list \<times> char list list \<times> char list list \<times> cmd) list" assumes "y \<in> wf_prog" shows "Rep_wf_prog (Abs_wf_prog y) = y"
ar_coset_def: fixes a :: "'e"   and A :: "('e, 'f) aGroup_scheme"   and H :: "'e set" shows "a \<uplus>\<^bsub>A\<^esub> H = H \<bullet>\<^bsub>b_ag A\<^esub> a"
abs_triangle_ineq4: fixes abs :: "'a \<Rightarrow> '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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus a b)) (plus (abs a) (abs b))"
sel_4: fixes x21 :: "'a"   and x22 :: "nat"   and x23 :: "'a interval" shows "op_idx (DEL x21 x22 x23) = x22"
SUP_upper2: fixes i :: "'b"   and A :: "'b set"   and u :: "'c \<times> 'd"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "i \<in> A"   and "less_eq_prod u (f i)" shows "less_eq_prod u (Sup_prod (f ` A))"
convex_real_interval_2: fixes b :: "real" shows "convex {..b}"
prems_2: shows "snd ` set (tx # txs) \<subseteq> var"
roots: shows "{x. x ^ n = 1} = {x. poly (monom 1 n - 1) x = 0}"
next'_def: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration" shows "local.next' c0 c1 = (next_propagate c0 c1 \<or> next_change_multiplicity c0 c1 \<or> c1 = c0)"
abs_conv_abscissa_mono: fixes g :: "'a fds"   and f :: "'a fds" assumes "\<And>s. fds_abs_converges g s \<Longrightarrow> fds_abs_converges f s" shows "abs_conv_abscissa f \<le> abs_conv_abscissa g"
indicator_sums: fixes A :: "nat \<Rightarrow> 'c set"   and x :: "'c" assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}" shows "(\<lambda>i. indicat_real (A i) x) sums indicat_real (\<Union> (range A)) x"
aux_lemma: fixes t :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list" assumes "exact_seq t" shows "group_hom (hd (tl (fst t))) (hd (fst t)) (hd (snd t))"
VarI: fixes n :: "nat"   and A :: "type_scheme list"   and t :: "typ" assumes "n < length A"   and "t <| A ! n" shows "A |- Var n :: t"
produces_imp_in_interp: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "Neg A \<in># C"   and "production D = {A}" shows "A \<in> interp C"
ri'': shows "P \<turnstile> \<langle>RI (C,e'') ; Cs \<leftarrow> e\<^sub>0,s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
lagrange: fixes H :: "'a set" assumes "subgroup H H'" shows "card (rcosets\<^bsub>H'\<^esub> H) * card H = order H'"
dist_eqs_1: fixes a' :: "'c" shows "Nothing \<noteq> Just\<cdot>a'"
Casts_Nil: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P \<turnstile> [] Casts [] to [] "
assms: shows "rel_mset (\<le>\<^sub>\<complex>) A B"
N2_def: shows "N2 \<equiv> \<lambda>B. trms\<^sub>s\<^sub>s\<^sub>t B \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t B"
simps_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a" shows "ins n d x \<langle>\<rangle> = (if bal_i (n + 1) (d + 1) then Bal \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> else Unbal \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>)"
Iff_sym: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IFF B" shows "H \<turnstile> B IFF A"
degree_mult_le: fixes p :: "'b poly"   and q :: "'b poly" shows "degree (p * q) \<le> degree p + degree q"
emeasure_spmf_map_pmf_Some: fixes p :: "'c pmf"   and A :: "'c set" shows "emeasure (measure_spmf (map_pmf Some p)) A = emeasure (measure_pmf p) A"
sort_upto_def: fixes a :: "'b array" shows "sort_upto a = (\<lambda>l. for [1..<l] (insert_elt a))"
WhileE: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (WHILE b DO c, s) \<Rightarrow> t" obtains "t = s" and "\<not> bval b s"   | s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int" where "bval b s" and "\<pi>: (c, s) \<Rightarrow> s\<^sub>2" and "\<pi>: (WHILE b DO c, s\<^sub>2) \<Rightarrow> t"
g_to_sorted_list_impl: shows "MapSpec.map_to_sorted_list \<alpha> invar g_to_sorted_list"
dim_basis: fixes A :: "'c set" assumes "finite A"   and "V.basis A" shows "V.dim = card A"
wf_eq_append''': fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and t' :: "('b, 'a) Term.term"   and a :: "poscheckvariant"   and t :: "('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
chambercomplex_opp_fold_A_im: shows "ChamberComplex (opp_fold_A \<turnstile> A)"
right_divisibility: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (\<exists>c. a = b * c)"
assms_2: fixes r :: "complex"   and x :: "'a" shows "f (r *\<^sub>C x) = r *\<^sub>C f x"
rev_iterateoi_def: fixes list_rev_it :: "'c \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and S :: "'c" shows "poly_set_rev_iterateoi_defs.rev_iterateoi list_rev_it S \<equiv> it_to_it (list_rev_it S)"
pos_pow2: fixes n :: "nat" shows "0 < 2 ^ n"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "zero_hom hom"   and "injective hom" shows "inj_zero_hom hom"
that: fixes i :: "nat" assumes "i < num_args t"   and "args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at p d t) ! i"   and "\<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args (emb_step_at p d t) ! j" shows "thesis"
knows_decomposition_1_s: fixes a :: "Secrets"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "sKS a \<notin> specKeysSecrets P"   and "sKS a \<notin> specKeysSecrets Q"   and "subcomponents PQ = {P, Q}"   and "knows PQ [sE a]"   and "correctCompositionIn PQ"   and "correctCompositionLoc PQ" shows "knows P [sE a] \<or> knows Q [sE a]"
renamings_admit_inverse: fixes V :: "'b set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "finite V"   and "renaming \<sigma> V" shows "\<exists>\<theta>. (\<forall>x\<in>V. Var x \<lhd> \<sigma> \<lhd> \<theta> = Var x) \<and> (\<forall>x. x \<notin> subst_codomain \<sigma> V \<longrightarrow> Var x \<lhd> \<theta> = Var x)"
UN: fixes K :: "'d set set"   and S :: "'d set set" assumes "\<And>k. k \<in> K \<Longrightarrow> generate_topology S k" shows "generate_topology S (\<Union> K)"
bisimSubstSymmetric: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim>\<^sub>s Q" shows "\<Psi> \<rhd> Q \<sim>\<^sub>s P"
F_def: shows "F \<equiv> \<lambda>gen_lossless_gpv gpv. \<exists>pa. gpv = GPV pa \<and> lossless_spmf pa \<and> (\<forall>out c input. IO out c \<in> set_spmf pa \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> gen_lossless_gpv (c input))"
simps_3: fixes \<sigma> :: "nat \<Rightarrow> val"   and e1 :: "expr"   and e2 :: "expr" shows "expr_sem \<sigma> (LET e1 IN e2) = expr_sem \<sigma> e1 \<bind> (\<lambda>v. expr_sem (case_nat v \<sigma>) e2)"
simps: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
div_mult_div_if_dvd: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c"   and d :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b a"   and "dvd.dvd times d c" shows "times (divide a b) (divide c d) = divide (times a c) (times b d)"
lower_AQ: shows "lower_triangular (A * Matrix.mat 2 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then s else if i = 0 \<and> j = 1 then - b1 else if i = 1 \<and> j = 0 then t else a1))"
Mp_shrink_modulus: fixes k :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.Mp (p * k) f = poly_mod.Mp (p * k) g"   and "k \<noteq> 0" shows "f =m g"
simps_3: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "decr (x # y # xs) = (y \<le> x \<and> decr (y # xs))"
mult_ac_2: fixes a :: "'d"   and b :: "'d" shows "a * b = b * a"
subgraph_awalk_imp_awalk: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk H u p v"   and "subgraph H G" shows "pre_digraph.awalk G u p v"
left_translations_of_group_axioms: shows "left_translations_of_group G (\<cdot>) \<one>"
mkarr_eqI: fixes f :: "'c local.term"   and g :: "'c local.term" assumes "Par f g"   and "\<^bold>\<lfloor>f\<^bold>\<rfloor> = \<^bold>\<lfloor>g\<^bold>\<rfloor>" shows "mkarr f = mkarr g"
rep_eq: fixes x :: "('d, 'e) fmap"   and xa :: "('d, 'e) fmap" shows "(x \<subseteq>\<^sub>f xa) = (fmlookup x \<subseteq>\<^sub>m fmlookup xa)"
le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
limitin_def: fixes X :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" shows "limitin X f l F \<equiv> l \<in> topspace X \<and> (\<forall>U. openin X U \<and> l \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> U))"
Rep_pos_inverse: fixes x :: "'a\<^sup>+" shows "Abs_pos (Rep_pos x) = x"
INF_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'd"   and D :: "'b \<Rightarrow> 'd"   and Inf :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
ce_sets_def: shows "ce_sets = {fn_to_set p |p. p \<in> PrimRec2}"
cases: fixes x :: "'b up\<^sub>i" obtains t :: "'b btree" where "x = T\<^sub>i t"   | l :: "'b btree" and a :: "'b" and r :: "'b btree" where "x = Up\<^sub>i l a r"
map_idI: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
assoc_in_hom: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> : (f \<star> g) \<star> h \<Rightarrow> f \<star> g \<star> h\<guillemotright>"
case_option_compute_def: fixes opt :: "'b option"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a" shows "case_option_compute opt a f = (case opt of None \<Rightarrow> a | Some x \<Rightarrow> f x)"
subset: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A" shows "gcd (Gcd\<^sub>f\<^sub>i\<^sub>n B) (Gcd\<^sub>f\<^sub>i\<^sub>n A) = Gcd\<^sub>f\<^sub>i\<^sub>n A"
simps: fixes g :: "(location \<times> location) set"   and a :: "location \<Rightarrow> char list set"   and c :: "actor \<Rightarrow> char list set \<times> char list set"   and l :: "location \<Rightarrow> char list \<times> ((actor \<times> actor set) \<times> nat) set" shows "lgra (Lgraph g a c l) = l"
ide_implies_incl: fixes a :: "'s" assumes "ide a" shows "incl a"
exhaust: fixes y :: "'a ltln" obtains "y = true\<^sub>n"   | "y = false\<^sub>n"   | x3 :: "'a" where "y = prop\<^sub>n(x3)"   | x4 :: "'a" where "y = nprop\<^sub>n(x4)"   | x51 :: "'a ltln" and x52 :: "'a ltln" where "y = x51 and\<^sub>n x52"   | x61 :: "'a ltln" and x62 :: "'a ltln" where "y = x61 or\<^sub>n x62"   | x7 :: "'a ltln" where "y = X\<^sub>n x7"   | x81 :: "'a ltln" and x82 :: "'a ltln" where "y = x81 U\<^sub>n x82"   | x91 :: "'a ltln" and x92 :: "'a ltln" where "y = x91 R\<^sub>n x92"   | x101 :: "'a ltln" and x102 :: "'a ltln" where "y = x101 W\<^sub>n x102"   | x111 :: "'a ltln" and x112 :: "'a ltln" where "y = x111 M\<^sub>n x112"
inside_convex: fixes S :: "'a set" assumes "convex S" shows "inside S = {}"
subject_reduction: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and T :: "LambdaType.type"   and t' :: "dB" assumes "e \<turnstile> t : T"   and "t \<rightarrow>\<^sub>\<beta> t'" shows "e \<turnstile> t' : T"
disc_1: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "p a" shows "lnull (unfold_llist p g21 g22 a)"
isolated_verts_def: fixes G :: "('b, 'c) pre_digraph" shows "pre_digraph.isolated_verts G \<equiv> {v \<in> verts G. out_arcs G v = {}}"
zero_min: fixes t :: "'x \<Rightarrow>\<^sub>0 nat" shows "0 \<preceq> t"
min_elems5: fixes x :: "'c list"   and A :: "'c list set" assumes "x \<in> A" shows "\<exists>s\<le>x. s \<in> min_elems A"
value_g: shows "value_flow \<Psi> g = \<alpha>"
left_neutral: fixes a :: "'d" shows "\<top> \<sqinter> a = a"
Form_0_cases_raw: fixes u :: "nat list set" assumes "Form 0 u" obtains xs :: "nat list" and ys :: "nat list" where "u = {xs, ys}" and "length xs = length ys" and "xs \<noteq> ys"
upper_triangular_one: fixes n :: "nat" shows "upper_triangular (1\<^sub>m n)"
IH: shows "(\<exists>x. s = Var x \<and> t = Var x) \<or> (\<exists>f g S T. s = Fun f S \<and> t = Fun g T \<and> length S = length T \<and> (\<forall>i<length T. (S ! i, T ! i) \<in> (timpl_closure'_step TI)\<^sup>*) \<and> (f \<noteq> g \<longrightarrow> is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> TI\<^sup>+))"
update_by_pair_id: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and v :: "'c" assumes "oalist_inv_raw xs"   and "lookup_pair xs k = v" shows "update_by_pair (k, v) xs = xs"
ring_finite_field_ops_integer: fixes pp :: "integer" assumes "p = int_of_integer pp" shows "ring_ops (finite_field_ops_integer pp) mod_ring_rel_integer"
strict_trans2: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" assumes "b \<subset># a"   and "c \<subseteq># b" shows "c \<subset># a"
EsE_s_f\<C>: shows "\<forall>A\<in>set (Es @ [E]). \<forall>(f, g)\<in>fundfoldpairs. s = Abs_induced_automorph f g \<longrightarrow> A \<in> f \<turnstile> \<C>"
pick_asTerm: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood qX" shows "pick (asTerm qX) #= qX"
Abs_circline_mat_inverse: fixes y :: "complex \<times> complex \<times> complex \<times> complex" assumes "y \<in> hermitean_nonzero" shows "Rep_circline_mat (Abs_circline_mat y) = y"
distrib: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a + b) * c = a * c + b * c"
comp: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow>\<^sub>L 'f"   and g :: "'g \<Rightarrow> 'e" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_bilinear (\<lambda>x y. blinfun_apply (f x) (g y))"
inj_ring_hom_axioms: shows "inj_ring_hom hom"
DirProd_commute_iso: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "G \<times>\<times> H \<cong> H \<times>\<times> G"
transpose_mat: fixes n :: "'c" shows "Finite_Cartesian_Product.transpose (mat n) = mat n"
imp_trans: fixes A :: "'a set"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "A \<turnstile> p .\<longrightarrow>. q"   and "A \<turnstile> q .\<longrightarrow>. r" shows "A \<turnstile> p .\<longrightarrow>. r"
subset_eqvt: fixes p :: "perm"   and S :: "'a set"   and T :: "'a set" shows "p \<bullet> (S \<subseteq> T) = (p \<bullet> S \<subseteq> p \<bullet> T)"
matchC_MC_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_MC theta c d \<equiv> \<forall>s t c' s'. s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> (\<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta)"
pair_pseudo_graph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)" shows "pair_pseudo_graph (subdivide (subdivide G e w) e w)"
assms_1: shows "enat x < llength (\<pi>\<^bsub>c\<^esub>t)"
c_def: shows "c = Cn 1 (r_smn 1 1) [r_const (encode psi), recf.Id 1 0]"
powers_5: fixes a :: "'b"   and b :: "'b" shows "a * b = b * a"
wqo_on_imp_transp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "transp_on P A"
xfer_locks_ticket: fixes ts :: "'a \<Rightarrow> thread"   and t :: "'a"   and tki :: "nat" assumes "locks_ticket (map_ticket (\<lambda>t. t mod N) (ts t)) tki" obtains tk :: "nat" where "tki = tk mod N" and "locks_ticket (ts t) tk"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
ran_def: fixes m :: "'b \<Rightarrow> 'a option" shows "ran m = {b. \<exists>a. m a = Some b}"
vector_scaleR_component: fixes r :: "real"   and x :: "('b, 'c) vec"   and i :: "'c" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
approx_minus_cancel: fixes a :: "'a star"   and b :: "'a star" shows "(- a \<approx> - b) = (a \<approx> b)"
Integer_arith_strict_5: fixes x :: "Integer" shows "\<bottom> - x = \<bottom>"
1: shows "finite (\<Union> (edge_as_set ` matching_i i V E M L))"
sgn_real_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "sgn x \<le> sgn y"
S5: shows "mod7 (c_fst (c_pair 4 (c_pair n1 n2))) = 4"
quant_less_iff_magn_less: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" shows "(x < y) = (\<lbrakk>x\<rbrakk>\<^sub>Q < \<lbrakk>y\<rbrakk>\<^sub>Q)"
le_term: fixes x :: "'a"   and y :: "'a" assumes "rep_nat_term x \<le> rep_nat_term y" shows "x \<le> y"
IH1_2: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "f \<in> Mapping.keys M1"
Abs_zet_inject: fixes x :: "'b set"   and y :: "'b set" assumes "x \<in> zet"   and "y \<in> zet" shows "(Abs_zet x = Abs_zet y) = (x = y)"
UniverseProd: fixes u :: "ZF"   and v :: "ZF" assumes "Elem u U"   and "Elem v U" shows "Elem (CartProd u v) U"
arctan_monotone: fixes x :: "real"   and y :: "real" assumes "x < y" shows "arctan x < arctan y"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. \<lbrakk>x \<le> y \<Longrightarrow> P xs (y # ys); \<not> x \<le> y \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
poly_smult: fixes a :: "'c"   and p :: "'c poly"   and x :: "'c" shows "poly (smult a p) x = a * poly p x"
even_pow_cong: fixes n :: "nat"   and m :: "nat" assumes "1 < n"   and "even m" shows "[(n - 1) ^ m = 1] (mod n)"
g_def: shows "g \<equiv> Cn n r_normal_form [r_constn (n - 1) i, r_list_encode (n - 1)]"
mon_group_mul_eq: shows "((\<lambda>(x, y). x \<otimes> y) \<in> mon (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B = {\<one>} \<and> (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x))"