hcju/isabelleps huggingface dataset:
af_trans_extension_imp_wf: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "\<And>x y. P x y \<Longrightarrow> Q x y"   and "almost_full_on P A"   and "transp_on Q A" shows "wfp_on (strict Q) A"
satL_snvL_transE: fixes l :: "('fsym, 'psym) lit"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtL l"   and "I.wtE \<xi>"   and "\<And>\<sigma>. infTp \<sigma> \<or> (\<forall>x\<in>nvL l. tpOfV x \<noteq> \<sigma>)"   and "Ik.satL (transE \<xi>) l" shows "I.satL \<xi> l"
succ_of_pred_def: fixes s :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "succ_of_pred s \<equiv> \<lambda>a. {b. s a b}"
rl_comp_lv_1: shows "rl_rel O lv_rel \<subseteq> lv_rel"
mono_gamma_s: fixes f1 :: "char list \<Rightarrow> 'av"   and f2 :: "char list \<Rightarrow> 'av" assumes "f1 \<le> f2" shows "\<gamma>\<^sub>s f1 \<subseteq> \<gamma>\<^sub>s f2"
prod_6: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N"   and f :: "'a N" shows "((a r\<rightarrow> b) r\<rightarrow> b) * (a * c r\<rightarrow> a * f) * (b * c r\<rightarrow> b * f) \<le> c r\<rightarrow> f"
assms_4: shows "c \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
induct: fixes P :: "'b wordinterval \<Rightarrow> bool"   and a0 :: "'b wordinterval" assumes "\<And>r1 r2. \<lbrakk>P r1; P r2\<rbrakk> \<Longrightarrow> P (RangeUnion r1 r2)"   and "\<And>s e. P (WordInterval s e)" shows "P a0"
E: shows "offs (enum xs index key n mi ma) k ! j < offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i"
accfield_accessibleD: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname"   and fn :: "vname"   and f :: "qtname \<times> Decl.field" assumes "accfield G S C fn = Some f" shows "G \<turnstile>Field fn  f of C accessible_from S"
bigo_def: fixes F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
red': shows "if \<tau>moves2 (compP2 P) h stk es pc xcp then h' = h \<and> (if xcp' = None \<longrightarrow> pc < pc' - length (compE2 e) then \<tau>reds1r else \<tau>reds1t) P t h (es', xs) (es'', xs'') else \<exists>ta' es'a xs'. \<tau>reds1r P t h (es', xs) (es'a, xs') \<and> True,P,t \<turnstile>1 \<langle>es'a,(h, xs')\<rangle> [-ta'\<rightarrow>] \<langle>es'',(h', xs'')\<rangle> \<and> ta_bisim wbisim1 (extTA2J1 P ta') ta \<and> \<not> \<tau>moves1 P h es'a \<and> (calls1 es' = None \<or> no_calls2 es pc \<or> es'a = es' \<and> xs' = xs)"
prems_2: shows "\<Gamma>\<turnstile>Catch a1 a2 \<down> s"
assms_1: shows "x = [a, b]\<^sub>\<circ>"
fmlookup_add: fixes m :: "('e, 'd) fmap"   and n :: "('e, 'd) fmap"   and x :: "'e" shows "fmlookup (m ++\<^sub>f n) x = (if x |\<in>| fmdom n then fmlookup n x else fmlookup m x)"
span_superset: fixes S :: "('b, 'c) vec set" shows "S \<subseteq> vec.span S"
eq: shows "sets (count_space A \<Otimes>\<^sub>M count_space B) = sets (count_space (A \<times> B))"
snd_comp_apfst: fixes f :: "'b \<Rightarrow> 'd" shows "(snd \<circ>\<circ> apfst) f = snd"
rat_to_int_factor_explicit: fixes p :: "int poly"   and g :: "rat poly"   and h :: "rat poly"   and r :: "rat"   and rg :: "int poly" assumes "of_int_poly p = g * h"   and "rat_to_normalized_int_poly g = (r, rg)" shows "\<exists>r. p = rg * smult (content p) r"
Mem_imp_subtract: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and k :: "name" assumes "H \<turnstile> x IN y"   and "H \<turnstile> OrdP y"   and "atom k \<sharp> (x, y)" shows "H \<turnstile> SyntaxN.Ex k (HaddP x (Var k) y AND Zero IN Var k)"
monom_eq_iff': fixes c :: "'b"   and n :: "nat"   and d :: "'b"   and m :: "nat" shows "(monom c n = monom d m) = (c = d \<and> (c = (0::'b) \<or> n = m))"
hyps_1: fixes s :: "'a regex" assumes "local.safe_regex Past Lax r1"   and "s \<in> rpd test i r1" shows "local.safe_regex Past Lax s"
case_cmp_if: fixes l :: "'c"   and e :: "'c"   and g :: "'c"   and c :: "cmp_val" shows "(case c of LT \<Rightarrow> l | EQ \<Rightarrow> e | GT \<Rightarrow> g) = (if c = LT then l else if c = GT then g else e)"
Ana_subst': fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "Ana (Fun f T) = (K, M)" shows "Ana (Fun f T \<cdot> \<delta>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>, M \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
intro: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "dpo le"   and "dlat_axioms le" shows "dlat le"
Gromov_product_commute: fixes e :: "'c"   and x :: "'c"   and y :: "'c" shows "Gromov_product_at e x y = Gromov_product_at e y x"
icheck_init_class: fixes P :: "'a prog"   and C :: "char list"   and e :: "'b exp" assumes "icheck P C e" shows "init_class P e = \<lfloor>C\<rfloor>"
msb_word_eq: fixes w :: "'a word" shows "msb w = bit w (LENGTH('a) - 1)"
closure_minimal: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "closed T" shows "closure S \<subseteq> T"
br_algo_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br_algo \<delta> \<equiv> \<lparr>wa_cond = br_cond, wa_step = br_step \<delta>, wa_initial = {br_initial \<delta>}, wa_invar = br_invar \<delta>\<rparr>"
elims: fixes x :: "checker_type"   and xa :: "checker_type"   and y :: "checker_type" assumes "produce x xa = y" obtains ts :: "ct list" and ts' :: "t list" where "x = TopType ts__" and "xa = Type ts'__" and "y = TopType (ts__ @ to_ct_list ts'__)"   | ts :: "t list" and ts' :: "t list" where "x = Type tsa__" and "xa = Type ts'a__" and "y = Type (tsa__ @ ts'a__)"   | ts' :: "t list" and ts :: "ct list" where "x = Type ts'b__" and "xa = TopType tsb__" and "y = TopType tsb__"   | ts' :: "ct list" and ts :: "ct list" where "x = TopType ts'" and "xa = TopType ts" and "y = TopType ts"   | v :: "t list" where "x = Type v__" and "xa = Bot" and "y = Bot"   | uv_ :: "checker_type" where "x = Bot" and "xa = uv_" and "y = Bot"   | v :: "ct list" where "x = Bot" and "xa = TopType v" and "y = Bot"   | uu_ :: "checker_type" where "x = uu_" and "xa = Bot" and "y = Bot"
cbi_alpha1ialpha4mi: shows "b\<inverse> O (e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>) \<subseteq> b\<inverse>"
nth_find_index: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "find_index P xs < length xs" shows "P (xs ! find_index P xs)"
normalize_gcd_left: fixes a :: "'a"   and b :: "'a" shows "gcd (normalize a) b = gcd a b"
1: fixes uua :: "nat" shows "degree (p * monom (1::'a) uua) = degree p + degree (monom (1::'a) uua)"
Ej_take_subset: fixes j :: "nat"   and k :: "nat"   and x :: "nat list" shows "Ej j (take k x) \<subseteq> Ej j x"
eeq: fixes i :: "nat" assumes "Suc i < K" shows "multiplicity p (Suc i) = multiplicity p (Suc (j + i))"
3: shows "e\<^sup>T * top \<le> - (w\<^sup>T\<^sup>\<star> * d\<^sup>T * top)"
Inl_inject: fixes x :: "'c"   and y :: "'c" assumes "Inl x = Inl y" shows "x = y"
vdomain_fflip: fixes f :: "V" shows "\<D>\<^sub>\<circ> (fflip f) = (\<D>\<^sub>\<circ> f)\<inverse>\<^sub>\<bullet>"
map_id: fixes t :: "('a, 'b, 'c) generat" shows "map_generat id id id t = t"
map_mapping_of_pps_to_list: fixes T :: "('x, nat) pp set" shows "map pp.mapping_of (punit.pps_to_list T) = pp_pm.punit.pps_to_list (pp.mapping_of ` T)"
sv_TS_defs_3: fixes Quorum :: "process set set" assumes "quorum_process Quorum" shows "quorum_process.sv_trans Quorum = (\<Union>r S v. \<Union> (range (quorum_process.sv_round Quorum r S v)))\<^sup>="
cont_applyI: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'c"   and x :: "'d" assumes "cont luba orda lubb ordb g" shows "cont (fun_lub luba) (fun_ord orda) lubb ordb (\<lambda>f. g (f x))"
intros_7: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
cupcake_list_correct_rerr: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and err :: "v error_result"   and s :: "'b state" assumes "list_all2_shortcircuit (\<lambda>e r. cupcake_evaluate_single env e r \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> (\<forall>s. \<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', r)))) es rs"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result rs = Rerr err" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s es (s', Rerr err)"
set_nths_subset: fixes xs :: "'a list"   and I :: "nat set" shows "set (nths xs I) \<subseteq> set xs"
finite_measure_mono: fixes A :: "('s \<times> 's option) stream set"   and B :: "('s \<times> 's option) stream set"   and s :: "'s \<times> 's option" assumes "A \<subseteq> B"   and "B \<in> KN.T.events s" shows "KN.T.prob s A \<le> KN.T.prob s B"
inj_inv: fixes b :: "bool" shows "inj_on (\<lambda>x. inversion (Some i, x)) {ys. ys ! 0 = b \<and> length ys = length xs}"
ord_le_eq_trans: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> b"   and "b = c" shows "a \<preceq> c"
preserves_cod: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "cod (src f) = src (cod f)"
matrix_invertible_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f" shows "invertible (matrix f) = (\<exists>g. linear (*s) (*s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
elliptic_or_parabolic_or_loxodromic: fixes f :: "'b \<Rightarrow> 'b" assumes "isometry f" shows "elliptic_isometry f \<or> parabolic_isometry f \<or> loxodromic_isometry f"
weight2: fixes "oracle" :: "'b \<Rightarrow> bool list \<Rightarrow> ((bool list \<times> bool list) \<times> 'b) spmf"   and cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a"   and s :: "'b" assumes "\<And>s x. lossless_spmf (oracle s x)" shows "weight_spmf (exec_gpv oracle (\<A>2 cipher \<sigma>) s) = 1"
set_plus_mono3: fixes a :: "'c"   and C :: "'c set"   and D :: "'c set" assumes "a \<in> C" shows "a +o D \<subseteq> C + D"
length_nths': fixes xs :: "'m list"   and I :: "nat set" shows "length (nths xs I) = card {i \<in> I. i < length xs}"
MakeFaceFinal_minGraphProps: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "minGraphProps (makeFaceFinal f g)"
r3eq: shows "r ^ 3 - 3 * r = 1"
Rep_unit2_inject: fixes x :: "unit2"   and y :: "unit2" shows "(Rep_unit2 x = Rep_unit2 y) = (x = y)"
simps_1: fixes f :: "('b \<Rightarrow> 'a) stream"   and x :: "'b stream" shows "shd (f \<diamondop> x) = shd f (shd x)"
preserves_reflects_arr: fixes f :: "'a" shows "Left_a.arr (Left_a.L f) = Left_a.arr f"
octo_of_real_sel_8: fixes x :: "real" shows "Im7 (octo_of_real x) = 0"
hmem_0_Ord: fixes k :: "hf" assumes "Ord k"   and "k \<noteq> 0" shows "0 \<^bold>\<in> k"
word_arith_nat_div: fixes a :: "'b word"   and b :: "'b word" shows "a div b = word_of_nat (unat a div unat b)"
removeAt_distinct: fixes fss :: "'a list"   and i :: "nat" assumes "distinct fss" shows "distinct (removeAt i fss)"
zor_poly_def: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" shows "zor_poly f z = (SOME h. \<exists>r>0. h holomorphic_on cball z r \<and> h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr of_int (zorder f z) \<and> h w \<noteq> 0))"
true_cls_mset_empty: fixes I :: "'a set" shows "I \<Turnstile>m {#}"
mono_lex2: fixes r :: "('c \<times> 'c) set" shows "mono ((<*lex*>) r)"
extend_not_mem: fixes f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c"   and n :: "nat"   and A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set" assumes "f n \<notin> extend A S f (Suc n)" shows "\<not> consistent A ({f n} \<union> extend A S f n)"
deterministic_def: fixes R :: "('a \<times> 'b) set" shows "deterministic R = (R\<inverse> O R \<subseteq> Id)"
det_perm_rows_If: fixes B :: "('c, 'd) sq_matrix"   and f :: "'d \<Rightarrow> 'd" shows "Square_Matrix.det (perm_rows B f) = (if f permutes UNIV then of_int (sign f) * Square_Matrix.det B else (0::'c))"
image_Union: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
factorize_rat_poly_main_wf_rel_def: shows "factorize_rat_poly_main_wf_rel = inv_image (mult1 {(x, y). x < y}) (\<lambda>(c, irr, ps). mset (map degree ps))"
extend_to_affine_basis: fixes S :: "'a set"   and V :: "'a set" assumes "\<not> affine_dependent S"   and "S \<subseteq> V" obtains T :: "'a set" where "\<not> affine_dependent T" and "S \<subseteq> T" and "T \<subseteq> V" and "affine hull T = affine hull V"
of_int_2: shows "\<guillemotleft>2\<guillemotright> = \<one> \<oplus> \<one>"
pure_cf: fixes x :: "dB" shows "Pure x \<in> CF"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'d list"   and x :: "'d" shows "sinks_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then insert (D x) (sinks_aux I D U xs) else sinks_aux I D U xs)"
alpha_preserves_qGood: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "X #= X'" shows "qGood X = qGood X'"
assms_2: shows "ineq_model \<theta> X (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
assms: shows "P \<subseteq> {s. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c s Q',A' \<and> P \<subseteq> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A}"
vimage_Union: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'e set set" shows "f -` \<Union> A = \<Union> ((-`) f ` A)"
sets_lim: shows "sets PF.lim = sets (Pi\<^sub>M UNIV (\<lambda>_. M))"
sorted_find_indices: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "sorted (find_indices P xs)"
hom: shows "(\<lambda>x. the_elem (f ` x)) \<circ> (#>) H \<in> hom G (image_group f G)"
empty_sets: fixes M :: "'c measure" shows "{} \<in> sets M"
R'_def: fixes n :: "nat" shows "R' n = Restr R {..n} \<union> {(n + 1, y) |y. \<exists>x'>n. (x', y) \<in> R \<and> y \<le> n} \<union> {(x, n + 1) |x. \<exists>y'>n. (x, y') \<in> R \<and> x \<le> n} \<union> (if \<exists>x>n. \<exists>y>n. (x, y) \<in> R then {(n + 1, n + 1)} else {})"
ih_2: assumes "n \<noteq> 0" shows "local.well_base\<^sub>h (local.encode (e + 1) (n div base))"
Quotient3_Bonk_Schramm_extension: shows "Quotient3 (\<lambda>x y. x \<in> extended_distance_set \<and> y \<in> extended_distance_set \<and> extended_distance x y = 0) abs_Bonk_Schramm_extension rep_Bonk_Schramm_extension"
C'_def: shows "C' = complex_of_real a powr (1 - s) / (1 - s)"
comp_arr_ide: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a"   and "seq f a" shows "f \<cdot> a = f"
assms_2: shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"
distinct_1: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "Leaf \<noteq> Node x21 x22 x23 x24 x25"
eq: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' s) (Abs \<tau>' t)"
wprepare_loop_goon_Bk2: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list"   and a :: "cell"   and lista :: "cell list" assumes "lm \<noteq> []"   and "wprepare_loop_start m lm (b, Bk # a # lista)" shows "wprepare_loop_goon m lm (Bk # b, a # lista)"
test_comp_test_top: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y \<sqinter> (x \<sqinter> top \<sqinter> top \<sqinter> (z \<sqinter> top)) = x \<sqinter> top \<sqinter> y \<sqinter> (z \<sqinter> top)"
amd_lookup_def: fixes dflt :: "'b"   and k :: "nat"   and a :: "'b array" shows "amd_lookup dflt k a \<equiv> array_get_dyn dflt a k"
is_map: shows "is_left_adjoint THfgh_HHfgh.chine"
UniqueSucState_Conf_Source_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto"   and T :: "'d"   and U :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "A \<in> SAs (HA ST)"   and "A \<in> the (CompFun (HA ST) S)"   and "T \<in> States A"   and "U \<in> States A"   and "T \<in> StepConf (HA ST) (Conf ST) TS"   and "T \<noteq> U"   and "U \<in> Conf ST" shows "U \<in> ChiStar (HA ST) `` Source TS"
fempty_is_fimage: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b fset" shows "({||} = f |`| A) = (A = {||})"
fresh_Cons: fixes a :: "atom"   and x :: "'a"   and xs :: "'a list" shows "a \<sharp> x # xs = (a \<sharp> x \<and> a \<sharp> xs)"
left_commute: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "inf_prod b (inf_prod a c) = inf_prod a (inf_prod b c)"
ND_sem: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "\<Gamma> \<TTurnstile> F = (\<not> sat (\<^bold>\<not> F \<triangleright> \<Gamma>))"
finite_Trn: fixes rhs :: "'b trm set"   and Y :: "'b list set" assumes "finite rhs" shows "finite {r. Trn Y r \<in> rhs}"
swap_msgs_Recv_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
right_lemma: fixes c :: "nat" shows "rec_exec rec_right [c] = rght c"
EWP: fixes x :: "'a list set" shows "l_ewp x = (\<exists>y. x = 1 + y \<and> \<not> l_ewp y)"
semiconnex_axioms: shows "semiconnex A (\<lambda>x y. y \<sqsubset> x)"
Int_subset_iff: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
it_b: fixes k :: "'b"   and v :: "'c" assumes "m k = Some v" shows "set_iterator (it_b v) (S_b v)"
THE_default1_equality: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and d :: "'b" assumes "\<exists>!x. P x"   and "P a" shows "THE_default d P = a"
primitive_extractor_correct_3: fixes m :: "'d match_expr"   and disc :: "'d \<Rightarrow> bool"   and sel :: "'d \<Rightarrow> 'e"   and C :: "'e \<Rightarrow> 'd"   and as :: "'e negation_type list"   and ms :: "'d match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "\<not> has_disc disc ms"
sorted_wrt_dist_take_drop: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat" assumes "sorted_wrt_dist p ps" shows "\<forall>p\<^sub>0\<in>set (take n ps). \<forall>p\<^sub>1\<in>set (drop n ps). dist p\<^sub>0 p \<le> dist p\<^sub>1 p"
lens_member_def: fixes x :: "'a \<Longrightarrow> 'b"   and A :: "'b scene" shows "(x \<in>\<^sub>S A) = ((\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on A \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 on A) \<and> (\<forall>b b'. get\<^bsub>x\<^esub> (b \<oplus>\<^sub>S b' on A) = get\<^bsub>x\<^esub> b'))"
select_convs_1: fixes mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a" shows "mbal \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = mbal"
assms: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> ev \<psi> xs) xs"
elims: fixes x :: "('state \<times> 'aprop set) stream list"   and y :: "'state" assumes "pointerOf x = y" obtains \<pi>l :: "('state \<times> 'aprop set) stream list" where "x = \<pi>l" and "y = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0)"
supchamberD_1: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "chamber (supchamber v)"
simps_2: fixes c :: "'b" shows "get_const (Lt c) = c"
wfPUNIVI: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>P x. \<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wfP r"
group_hom_axioms_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
mono_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<sqsubseteq> f y)"
inf_sup_distrib1: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
i_satisfies_atom_set_mono: fixes as :: "('c \<times> 'd atom) set"   and as' :: "('c \<times> 'd atom) set"   and v :: "'c set \<times> (nat \<Rightarrow> 'd)" assumes "as \<subseteq> as'"   and "v \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as'" shows "v \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as"
fconverse_fid_on: fixes A :: "V" shows "(fid_on A)\<inverse>\<^sub>\<bullet> = fid_on A"
zero_eq_of_rat_iff: fixes a :: "rat" shows "((0::'c) = of_rat a) = (0 = a)"
Pair_inject: fixes a :: "'f"   and b :: "'g"   and a' :: "'f"   and b' :: "'g" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
fin_join_ti_hint2: fixes s :: "'a list list"   and x :: "nat"   and i :: "nat" assumes "fin_join_ti s x (Suc i) = []" shows "s ! (x + Suc i) = []"
Cauchy_theorem_starlike: fixes S :: "complex set"   and k :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "open S"   and "starlike S"   and "finite k"   and "continuous_on S f"   and "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x"   and "valid_path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g" shows "(f has_contour_integral 0) g"
word_shift_nonzero: fixes x :: "'b word"   and m :: "nat"   and n :: "nat" assumes "x \<le> 2 ^ m"   and "m + n < LENGTH('b)"   and "x \<noteq> 0" shows "x << n \<noteq> 0"
UNF_onD: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "UNF_on r A"   and "a \<in> A"   and "(a, b) \<in> r\<^sup>!"   and "(a, c) \<in> r\<^sup>!" shows "b = c"
step_2: fixes gpv :: "('a, 'e, 'f) gpv"   and s :: "'b" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv \<le> expectation_gpv2 (\<lambda>a. case a of (x, s) \<Rightarrow> f x) (inline callee gpv s)"
evalc_WHILE_case: fixes b :: "state \<Rightarrow> bool"   and c :: "com"   and s :: "state"   and t :: "state" assumes "<WHILE b DO c,s> -c-> t" obtains "t = s" and "\<not> b s"   | s1 :: "state" where "b s" and "<c,s> -c-> s1" and "<WHILE b DO c,s1> -c-> t"
assms_4: shows "(\<lambda>n. u n * indicat_real A n) \<longlonglongrightarrow> 0"
pow_slide: fixes u :: "'b list"   and v :: "'b list"   and n :: "nat" shows "u \<cdot> (v \<cdot> u) \<^sup>@ n \<cdot> v = (u \<cdot> v) \<^sup>@ Suc n"
simps: fixes a1 :: "lam"   and a2 :: "name list"   and a3 :: "lam" shows "a1 \<mapsto> a2,a3 = ((\<exists>a. a1 = Var a \<and> a2 = [] \<and> a3 = Var a) \<or> (\<exists>t1 t2. a1 = App t1 t2 \<and> a2 = [] \<and> a3 = App t1 t2) \<or> (\<exists>t xs t' x. a1 = Lam [x].t \<and> a2 = x # xs \<and> a3 = t' \<and> t \<mapsto> xs,t'))"