hcju/isabelleps huggingface dataset:
lc_in_ker: fixes D :: "'c set"   and d :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "D \<subseteq> C"   and "d \<in> D \<rightarrow> carrier K"   and "T (V.module.lincomb d D) = \<zero>\<^bsub>W\<^esub>"   and "v \<in> D"   and "d v \<noteq> \<zero>\<^bsub>K\<^esub>" shows "False"
map_sum_sel_1: fixes s :: "'c + 'd"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "isl s" shows "projl (map_sum f g s) = f (projl s)"
eq: shows "P * A * Q = A * (P $$ (0, 0) \<cdot>\<^sub>m Q)"
split: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and sum :: "'h + 'i" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
Sup_apply: fixes A :: "('c \<Rightarrow> 'b) set"   and x :: "'c" shows "(\<Squnion> A) x = (\<Squnion>f\<in>A. f x)"
r2f_def: fixes R :: "('b \<times> 'c) set" shows "\<F> R = (``) R \<circ> \<eta>"
partition_onD2: fixes A :: "'c set"   and P :: "'c set set" assumes "partition_on A P" shows "disjoint P"
distinct_13: shows "ConcatOp \<noteq> string_binop.GreaterEqOp"
PAC_checker_l_def: fixes spec :: "(char list list \<times> int) list"   and A :: "char list set \<times> (nat, (char list list \<times> int) list) fmap"   and b :: "char list code_status"   and st :: "((char list list \<times> int) list, char list, nat) pac_step list" shows "PAC_checker_l spec A b st = WHILE\<^sub>T (\<lambda>((b, A), n). \<not> is_cfailed b \<and> n \<noteq> []) (\<lambda>(bA, n). ASSERT (n \<noteq> []) \<bind> (\<lambda>_. PAC_checker_l_step spec bA (lead_coeff n) \<bind> (\<lambda>S. RETURN (S, tl n)))) ((b, A), st) \<bind> (\<lambda>(S, uu_). RETURN S)"
rows_mat_of_rows: fixes vs :: "'b vec list"   and n :: "nat" assumes "set vs \<subseteq> carrier_vec n" shows "rows (mat_of_rows n vs) = vs"
stable_on_Xd: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "Singleton_for_d.stable_on ds X"   and "x \<in> X" shows "Xd x \<in> ds"
Exit_empty: shows "lDef NewExit = {} \<and> lUse NewExit = {}"
assms_4: shows "free_vars e2 \<subseteq> V"
correctCompositionDiffLevelsS14: shows "correctCompositionDiffLevels sS14"
bisim1'_eq_bisim1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and n :: "nat" shows "bisim1'_syntax P e n h = bisim1_syntax P e h"
mset_subset_eq_mono_add_left_cancel: fixes C :: "'a multiset"   and A :: "'a multiset"   and B :: "'a multiset" shows "(C + A \<subseteq># C + B) = (A \<subseteq># B)"
finestPartUnion: fixes X :: "'a set" shows "X = \<Union> (finestpart X)"
distinct_1: fixes x1 :: "'h"   and x21 :: "'i"   and x22 :: "'j" shows "Pure x1 \<noteq> IO x21 x22"
thetaIfW_Wretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaIfW tval aval indis \<subseteq> PL_Indis.Wretr tval aval indis (PL_Indis.thetaIfW tval aval indis \<union> PL_Indis.Wbis tval aval indis)"
FunctorComp: fixes f :: "'c"   and g :: "'c" assumes "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)"
BIGTHETA_CONST'_def: fixes c :: "'b" shows "BIGTHETA_CONST' c = \<Theta>(\<lambda>x. c)"
simps_14: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> M\<^sub>c \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>c \<phi> \<and> (\<forall>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>c \<psi>))"
MGT_CALL: shows "{} \<tturnstile> {mgt. \<exists>p. mgt = MGT (CALL p)}"
CondThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
countable_Un_iff: fixes A :: "'k set"   and B :: "'k set" shows "countable (A \<union> B) = (countable A \<and> countable B)"
hyps_2: fixes trn :: "('state, 'act, 'out) trans" assumes "tgtOf trn = srcOf (hd tr_)"   and "validTrans trn" shows "valid2 (trn # tr_)"
p_inf: fixes x :: "'a" shows "- x \<sqinter> x = bot"
dot_scaleR_mult_1: fixes k :: "real"   and a :: "'a"   and b :: "'a" shows "k *\<^sub>R a \<bullet> b = k * (a \<bullet> b)"
Inter_iff: fixes A :: "'k"   and C :: "'k set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
assms_1: shows "\<forall>x\<in>U\<^sub>M\<^sub>_\<^sub>1. f x \<in> U\<^sub>M\<^sub>_\<^sub>2"
map_\<I>_\<I>_uniform: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'f"   and A :: "'g set"   and B :: "'h set" shows "map_\<I> f g (\<I>_uniform A B) = \<I>_uniform (f -` A) (g ` B)"
assms: shows "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
one_comp_one_cblinfun: shows "1 o\<^sub>C\<^sub>L 1 = 1"
ipresFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'g, 'b, 'h) model_scheme" shows "ipresFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> freshAbs ys y A \<longrightarrow> igFreshAbs MOD ys y (hA A)"
Cons_prefix_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
connected_empty: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.connected open {}"
minus_cont: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> bcontfun"   and "g \<in> bcontfun" shows "(\<lambda>x. f x - g x) \<in> bcontfun"
even_and_odd_parts: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<exists>k q. n = 2 ^ k * q \<and> odd q"
pth_6: fixes c :: "real"   and x :: "'a"   and y :: "'a" shows "c *\<^sub>R (x + y) \<equiv> c *\<^sub>R x + c *\<^sub>R y"
last_tl: fixes xs :: "'c list" assumes "xs = [] \<or> tl xs \<noteq> []" shows "last (tl xs) = last xs"
res: assumes "inj_on g (S \<inter> dom g)"   and "\<And>x y x' y'. \<lbrakk>x \<in> S; g x = Some x'; y \<in> S; g y = Some y'; True\<rbrakk> \<Longrightarrow> True" shows "set_iterator (set_iterator_image_filter g it) {y. \<exists>x. x \<in> S \<and> g x = Some y}"
ereal_minus_4: fixes x :: "ereal" shows "\<infinity> - x = \<infinity>"
quotient_of_eq: fixes a :: "int"   and b :: "int"   and p :: "int"   and q :: "int" assumes "quotient_of (Fract a b) = (p, q)" shows "Fract p q = Fract a b"
a_get_tag_name_def: fixes element_ptr :: "(_) element_ptr" shows "local.a_get_tag_name element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name"
shift1_isnpoly: fixes p :: "poly" assumes "isnpoly p"   and "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p)"
select_convs_4: fixes E_ES :: "'b set"   and I_ES :: "'b set"   and O_ES :: "'b set"   and Tr_ES :: "'b list set"   and more :: "'c" shows "Tr_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = Tr_ES"
set_map: fixes s :: "'a list" assumes "set s \<subseteq> carrier R" shows "set (map h s) \<subseteq> carrier S"
Union_of_fin_supp_sets: fixes X :: "'a set" assumes "fs TYPE('a) TYPE('b)"   and "finite X" shows "finite (\<Union> (supp ` X))"
inverse_arrows_def: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "B.VV.inverse_arrows f g \<equiv> B.VV.ide (B.VV.comp g f) \<and> B.VV.ide (B.VV.comp f g)"
num_mod_den'_def: shows "num_mod_den' = pure int \<diamondop> num_mod_den"
ivl_integral_def: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" shows "ivl_integral a b f = integral {a..b} f - integral {b..a} f"
code: fixes r :: "real"   and x :: "quat" shows "r *\<^sub>R x = Quat (r * quat.Re x) (r * Im1 x) (r * Im2 x) (r * Im3 x)"
stut_action_lemma: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a"   and v :: "'a \<Rightarrow> 'b" assumes "NSTUTINV P"   and "s \<approx> t"   and "(\<box>[P]_v) t" shows "(\<box>[P]_v) s"
in_auxlist0_le_\<tau>: fixes t :: "nat"   and X :: "event_data option list set" assumes "(t, X) \<in> set auxlist0" shows "t \<le> \<tau> \<sigma> ne"
fold_coeffs_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'c poly" shows "fold_coeffs f p = foldr f (coeffs p)"
conc_fun_RETURN: fixes R :: "('b \<times> 'c) set"   and x :: "'c" shows "\<Down> R (RETURN x) = SPEC (\<lambda>y. (y, x) \<in> R)"
simps_35: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> Stmt.If x51 x52 x53"
indicator_union_max: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" shows "indicator (A \<union> B) x = max (indicator A x) (indicator B x)"
run_one_step_basic_cvtop_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and t2 :: "t"   and x312 :: "cvtop"   and t1 :: "t"   and sx :: "sx option"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Cvtop t2 x312 t1 sx) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
drinks2_def: shows "drinks2 = {|((0, 1), select), ((1, 1), vend_nothing), ((1, 2), coin), ((2, 2), coin), ((2, 2), vend_fail), ((2, 3), vend)|}"
redT_updT_finite_dom_inv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'g, 'i) new_thread_action" shows "finite (dom (redT_updT ts ta)) = finite (dom ts)"
IH: fixes k :: "'a"   and s :: "real" assumes "(k, s) = widest_spread (k\<^sub>1 # ks) ps" shows "widest_spread_axis k (set (k\<^sub>1 # ks)) (set ps)"
permute_list_nth: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'d list"   and i :: "nat" assumes "f permutes {..<length xs}"   and "i < length xs" shows "permute_list f xs ! i = xs ! f i"
fresh_star_map_of: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list"   and a :: "atom set" assumes "x \<in> domA \<Gamma>"   and "a \<sharp>* \<Gamma>" shows "a \<sharp>* the (map_of \<Gamma> x)"
eq_\<I>_generat_conversep: fixes A :: "'e \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and D :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "eq_\<I>_generat A\<inverse>\<inverse> \<I> D\<inverse>\<inverse> = (eq_\<I>_generat A \<I> D)\<inverse>\<inverse>"
ihpex: shows "hp1 ## hp2 \<and> hp = hp1 + hp2 \<and> I hp1 \<and> I' hp2"
intro: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "category C"   and "replete_set_category S"   and "set_valued_functor C S F"   and "set_valued_functor C S G"   and "natural_transformation C S F G \<tau>" shows "set_valued_transformation C S F G \<tau>"
current_plus_current: shows "current \<Gamma> (plus_current f g)"
PREFER_tag_def: fixes x :: "bool" shows "PREFER_tag x \<equiv> x"
isstop_def: fixes F :: "'b formula" shows "isstop F \<equiv> F = \<^bold>\<not> \<bottom> \<or> F = \<top>"
ext_inject: fixes vertex :: "'b Point"   and slope :: "'b"   and more :: "'c"   and vertex' :: "'b Point"   and slope' :: "'b"   and more' :: "'c" shows "(\<lparr>vertex = vertex, slope = slope, \<dots> = more\<rparr> = \<lparr>vertex = vertex', slope = slope', \<dots> = more'\<rparr>) = (vertex = vertex' \<and> slope = slope' \<and> more = more')"
get_dom_component_get_child_nodes: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set c) = (ptr' \<in> set c)"
intro: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "mojmir_axioms \<delta> F" shows "mojmir \<Sigma> \<delta> q\<^sub>0 w F"
space_in_measure_of: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "\<Omega> \<in> sets (measure_of \<Omega> M \<mu>)"
simps_22: fixes x1 :: "'a"   and x8 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[x8\<^bold>]"
sublist_nth: fixes lo :: "nat"   and hi :: "nat"   and xs :: "'c list"   and k :: "nat" assumes "lo \<le> hi"   and "hi < length xs"   and "k + lo \<le> hi" shows "WB_Sort.sublist xs lo hi ! k = xs ! (lo + k)"
assms_4: shows "NT \<eta> : YFtor C @@ X \<Longrightarrow> F"
distinct_167: fixes x6 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Arctan x6 \<noteq> x151 ^\<^sub>e x152"
rep_eq: fixes x :: "('b, 'c) trie" shows "lookup x = trie_lookup (Rep_trie x) \<circ> to_bl"
b_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.b p q = card (QR.B p q)"
hyps_1: shows "(u, Spawn q, v) \<in> edges fg"
gt_sub: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "is_App t"   and "fun t \<ge>\<^sub>t s \<or> arg t \<ge>\<^sub>t s" shows "t >\<^sub>t s"
assms_1: fixes f :: "three \<Rightarrow> three \<Rightarrow> three"   and h :: "nat \<Rightarrow> three" assumes "associative f" shows "foldl1 f (map h js) = foldl1 f (map h [0..<k + 1])"
minus: fixes x :: "'b" shows "a *s - x = - (a *s x)"
execn_to_execn_restrict: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and P :: "'d set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>|\<^bsub>P\<^esub>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t' \<and> (t = Stuck \<longrightarrow> t' = Stuck) \<and> (\<forall>f. t = Fault f \<longrightarrow> t' \<in> {Fault f, Stuck}) \<and> (t' \<noteq> Stuck \<longrightarrow> t' = t)"
simps_1: fixes p :: "char list"   and ts :: "'a list" shows "get_atom (Pos p ts) = (p, ts)"
openin_trans: fixes T :: "'c set"   and S :: "'c set"   and U :: "'c set" assumes "openin (top_of_set T) S"   and "openin (top_of_set U) T" shows "openin (top_of_set U) S"
prm_id_def: shows "\<epsilon> \<equiv> abs_prm preprm_id"
set_disconnected_nodes_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
cast_lcl: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and v' :: "val"   and 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"   and l' :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>\<lparr>C\<rparr>Val v,(h, l)\<rangle> \<rightarrow> \<langle>Val v',(h, l)\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>Val v,(h, l')\<rangle> \<rightarrow> \<langle>Val v',(h, l')\<rangle>"
dropWhile_nth: fixes j :: "nat"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "j < length (dropWhile P xs)" shows "dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
integral_dominated_convergence: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and s :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>i. s i \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"   and "\<And>i. AE x in M. norm (s i x) \<le> w x" shows "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
multiplicity_prod_prime_powers_nat': fixes S :: "nat set"   and p :: "nat" assumes "finite S"   and "\<forall>p\<in>S. prime p"   and "prime p" shows "multiplicity p (\<Prod>S) = (if p \<in> S then 1 else 0)"
l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs \<equiv> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs \<and> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs"
extensional'_arb: fixes f :: "'a \<Rightarrow> 's"   and A :: "'a set"   and x :: "'a" assumes "f \<in> S.extensional' A"   and "x \<notin> A" shows "f x = S.unity"
restrictI: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "restrict f A \<in> Pi A B"
gen_model_worldsD: fixes w' :: "'b"   and M :: "('c, 'd, 'b) KripkeStructure"   and w :: "'b" assumes "w' \<in> worlds (gen_model M w)" shows "w' \<in> worlds M"
group_left_one: fixes inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.group inverse one times" shows "times one x = x"
return_def: fixes x :: "'b" shows "Parser_Monad.return x = (\<lambda>ts. Inr (x, ts))"
per_shift: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>"   and "n \<noteq> 0" shows "periodN w n = (drop n w \<le>p w)"
alw_iff_sdrop: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" shows "alw P \<omega> = (\<forall>m. P (sdrop m \<omega>))"
\<phi>'_le: shows "\<phi>' \<le> pi / 2"
horner_eval_eq_setsum: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" shows "horner_eval f x n = (\<Sum>i<n. f i * x ^ i)"
T_subset: shows "T \<subseteq> apply_chart d1 ` (domain d1 \<inter> domain c1)"
elims: fixes x :: "('a \<times> 'b) bound"   and xa :: "'a \<times> 'b"   and y :: "nat" assumes "idx x xa = y" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" and a :: "'a" and b :: "'b" where "x = Bound (l0, r0) (l1, r1)" and "xa = (a, b)" and "y = idx0 (Bound l0 l1) a * size1 (Bound r0 r1) + idx1 (Bound r0 r1) b"
inner_commute: fixes x :: "'a"   and y :: "'a" shows "inner x y = inner y x"
In1_inject: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" assumes "In1 M = In1 N" shows "M = N"
a_RS: shows "(a1, a2) \<in> R \<times> S"
rep_eq: fixes x :: "'c Filters.filter"   and xa :: "'c Filters.filter" shows "Filters.filter.Rep_filter (x \<squnion> xa) = filter_sup (Filters.filter.Rep_filter x) (Filters.filter.Rep_filter xa)"
inject: fixes x1 :: "'a"   and x2 :: "'a hterm list"   and y1 :: "'a"   and y2 :: "'a hterm list" shows "(HApp x1 x2 = HApp y1 y2) = (x1 = y1 \<and> x2 = y2)"
minarc_below: fixes x :: "'b \<times> 'b \<Rightarrow> 'c" shows "minarc\<^sub>M x \<preceq> \<ominus> \<ominus> x"
distinct_73: fixes x4 :: "ltkey"   and x11 :: "tag" shows "cLtK x4 \<noteq> cTag x11"
ssubseqs_refl: fixes xs :: "'b list" shows "xs \<in> ssubseqs xs"
max_fun_diff: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rt.FINFUNC"   and "g \<in> rt.FINFUNC" shows "\<exists>a b. a \<noteq> b \<and> a \<in> Field r \<and> b \<in> Field r \<and> f (t.max_fun_diff f g) = a \<and> g (t.max_fun_diff f g) = b"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"   and "a \<noteq> b" shows "<\<nu>b>P \<longmapsto>\<^sub>l a<\<nu>b> \<prec>\<^sub>l P'"
seqE: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" obtains "C.arr f" and "C.arr g" and "C.dom g = C.cod f"
ivl_disj_int_16: fixes l :: "'b"   and m :: "'b"   and u :: "'b" shows "{l..m} \<inter> {m<..u} = {}"
linear: fixes x :: "'a"   and y :: "'a" shows "r x y \<or> x = y \<or> r y x"
log4_log2: fixes x :: "real" shows "log 4 x = log 2 x / 2"
CM_axioms_def: shows "CM_axioms \<equiv> ((\<forall>x y. x \<oplus> y = (THE z. \<forall>v. O v z = (O v x \<or> O v y))) \<and> (\<forall>x y. \<exists>z. \<forall>v. O v z = (O v x \<or> O v y))) \<and> (\<forall>x y. x \<otimes> y = (THE z. \<forall>v. P v z = (P v x \<and> P v y))) \<and> (\<forall>x y. O x y \<longrightarrow> (\<exists>z. \<forall>v. P v z = (P v x \<and> P v y)))"
rel_setD2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b set"   and B :: "'c set"   and y :: "'c" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
M_measure: shows "borel_measurable M = borel_measurable borel"
not_psubset_empty: fixes A :: "'b set" shows "\<not> order.greater {} A"
Ri'_wf: shows "total_preorder_on alts' Ri'"
prod_diff1: fixes A :: "'c set"   and f1 :: "'c \<Rightarrow> 'd"   and f2 :: "'c \<Rightarrow> 'd" assumes "finite A" shows "(\<Prod>x\<in>A. f1 x - f2 x) = (\<Sum>X\<in>Pow A. (- (1::'d)) ^ card X * prod f2 X * prod f1 (A - X))"
const_exists_rev_def: fixes c :: "'b" shows "const_exists_rev c \<equiv> transl_rule (A_Cmp (A_Cmp (A_Lbl (S_Const c)) \<top>) (A_Lbl (S_Const c)) \<sqsubseteq> A_Lbl (S_Const c))"
lock_actions_ok_has_locks_upd_locks_eq_has_locks: fixes l :: "('c \<times> nat) option"   and t' :: "'c"   and Ls :: "lock_action list"   and t :: "'c" assumes "lock_actions_ok l t' Ls"   and "t \<noteq> t'" shows "has_locks (upd_locks l t' Ls) t = has_locks l t"
append_Cons: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "(x # xs) @ ys = x # xs @ ys"
assms: fixes x :: "'a \<times> 'b" assumes "x \<in># {#a#} \<times># B" shows "P x = Q (snd x)"
simps: fixes p1 :: "'a list list"   and p2 :: "'a list list" shows "UP.poly_add p1 p2 = (if length p2 \<le> length p1 then UP.normalize (map2 (\<oplus>\<^bsub>K [X]\<^esub>) p1 (replicate (length p1 - length p2) \<zero>\<^bsub>K [X]\<^esub> @ p2)) else UP.poly_add p2 p1)"
absorb2: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "min a b = b"
related_match: fixes x :: "Term_Class.term"   and u :: "sterm"   and env :: "(name, sterm) fmap"   and t :: "pterm" assumes "match x u = Some env"   and "t = sterm_to_pterm u" obtains env' :: "(name, pterm) fmap" where "match x t = Some env'" and "srelated.P_env env' env"
R_steps: shows "\<forall>i. i \<le> f i \<and> (t (f i), t (Suc (f i))) \<in> R"
mult_isol: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" assumes "x \<subseteq> y" shows "z \<parallel> x \<subseteq> z \<parallel> y"