From 9490a4ae2615e23c9ab4b6dc13ef6c699ce33120 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Wed, 7 Dec 2011 08:13:17 +0000 Subject: [PATCH] \vee notation for boolean or --- matita/matita/lib/re/re.ma | 204 ++++++++++++++++--------------------- 1 file changed, 88 insertions(+), 116 deletions(-) diff --git a/matita/matita/lib/re/re.ma b/matita/matita/lib/re/re.ma index 388c27e70..c37b47348 100644 --- a/matita/matita/lib/re/re.ma +++ b/matita/matita/lib/re/re.ma @@ -13,21 +13,12 @@ (**************************************************************************) include "arithmetics/nat.ma". -include "basics/list.ma". +include "basics/lists/list.ma". +include "basics/sets.ma". -interpretation "iff" 'iff a b = (iff a b). +definition word ≝ λS:DeqSet.list S. -record Alpha : Type[1] ≝ { carr :> Type[0]; - eqb: carr → carr → bool; - eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y) -}. - -notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }. -interpretation "eqb" 'eqb a b = (eqb ? a b). - -definition word ≝ λS:Alpha.list S. - -inductive re (S: Alpha) : Type[0] ≝ +inductive re (S: DeqSet) : Type[0] ≝ z: re S | e: re S | s: S → re S @@ -44,7 +35,7 @@ notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}. interpretation "cat" 'pc a b = (c ? a b). (* to get rid of \middot -ncoercion c : ∀S:Alpha.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?. +ncoercion c : ∀S:DeqSet.∀p:re S. re S → re S ≝ c on _p : re ? to ∀_:?.?. *) notation < "a" non associative with precedence 90 for @{ 'ps $a}. @@ -54,16 +45,17 @@ interpretation "atom" 'ps a = (s ? a). notation "ϵ" non associative with precedence 90 for @{ 'epsilon }. interpretation "epsilon" 'epsilon = (e ?). -notation "∅" non associative with precedence 90 for @{ 'empty }. +notation "`∅" non associative with precedence 90 for @{ 'empty }. interpretation "empty" 'empty = (z ?). -let rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝ +let rec flatten (S : DeqSet) (l : list (word S)) on l : word S ≝ match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ]. -let rec conjunct (S : Alpha) (l : list (word S)) (r : word S → Prop) on l: Prop ≝ +let rec conjunct (S : DeqSet) (l : list (word S)) (r : word S → Prop) on l: Prop ≝ match l with [ nil ⇒ ? | cons w tl ⇒ r w ∧ conjunct ? tl r ]. // qed. +(* definition empty_lang ≝ λS.λw:word S.False. notation "{}" non associative with precedence 90 for @{'empty_lang}. interpretation "empty lang" 'empty_lang = (empty_lang ?). @@ -73,7 +65,7 @@ definition sing_lang ≝ λS.λx,w:word S.x=w. interpretation "sing lang" 'singl x = (sing_lang ? x). definition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.l1 w ∨ l2 w. -interpretation "union lang" 'union a b = (union ? a b). +interpretation "union lang" 'union a b = (union ? a b). *) definition cat : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2. @@ -82,9 +74,9 @@ interpretation "cat lang" 'pc a b = (cat ? a b). definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l. interpretation "star lang" 'pk l = (star ? l). -let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ +let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝ match r with -[ z ⇒ {} +[ z ⇒ ∅ | e ⇒ { [ ] } | s x ⇒ { [x] } | c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2) @@ -98,15 +90,7 @@ interpretation "in_l mem" 'mem w l = (in_l ? l w). lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*. // qed. -notation "a || b" left associative with precedence 30 for @{'orb $a $b}. -interpretation "orb" 'orb a b = (orb a b). - -definition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f]. -notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. -notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. -interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f). - -inductive pitem (S: Alpha) : Type[0] ≝ +inductive pitem (S: DeqSet) : Type[0] ≝ pz: pitem S | pe: pitem S | ps: S → pitem S @@ -132,9 +116,9 @@ interpretation "patom" 'ps a = (ps ? a). interpretation "pepsilon" 'epsilon = (pe ?). interpretation "pempty" 'empty = (pz ?). -let rec forget (S: Alpha) (l : pitem S) on l: re S ≝ +let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝ match l with - [ pz ⇒ ∅ + [ pz ⇒ `∅ | pe ⇒ ϵ | ps x ⇒ `x | pp x ⇒ `x @@ -145,11 +129,11 @@ let rec forget (S: Alpha) (l : pitem S) on l: re S ≝ (* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*) interpretation "forget" 'norm a = (forget ? a). -let rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ +let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝ match r with -[ pz ⇒ {} -| pe ⇒ {} -| ps _ ⇒ {} +[ pz ⇒ ∅ +| pe ⇒ ∅ +| ps _ ⇒ ∅ | pp x ⇒ { [x] } | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2) | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2) @@ -158,13 +142,15 @@ match r with interpretation "in_pl" 'in_l E = (in_pl ? E). interpretation "in_pl mem" 'mem w l = (in_pl ? l w). -definition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}. +(* +definition epsilon : ∀S:DeqSet.bool → word S → Prop +≝ λS,b.if b then { ([ ] : word S) } else ∅. interpretation "epsilon" 'epsilon = (epsilon ?). notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}. -interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b). +interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b). *) -definition in_prl ≝ λS : Alpha.λp:pre S. +definition in_prl ≝ λS : DeqSet.λp:pre S. if (\snd p) then \sem{\fst p} ∪ { ([ ] : word S) } else \sem{\fst p}. interpretation "in_prl mem" 'mem w l = (in_prl ? l w). @@ -205,7 +191,7 @@ lemma sem_star_w : ∀S.∀i:pitem S.∀w. lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ]. #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed. -lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e). +lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ ([ ] ∈ e). #S #e elim e normalize /2/ [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/ @@ -223,14 +209,14 @@ lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → [ ] ∈ e. #S * #i #b #btrue normalize in btrue; >btrue %2 // qed. -definition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉. +definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉. notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}. interpretation "oplus" 'oplus a b = (lo ? a b). -lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1||b2〉. +lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉. // qed. -definition pre_concat_r ≝ λS:Alpha.λi:pitem S.λe:pre S. +definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S. match e with [ pair i1 b ⇒ 〈i · i1, b〉]. notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}. @@ -244,15 +230,15 @@ lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. A = B → A =1 B. #S #A #B #H >H /2/ qed. -lemma ext_eq_trans: ∀S.∀A,B,C:word S → Prop. +(* lemma eqP_trans: ∀S.∀A,B,C:word S → Prop. A =1 B → B =1 C → A =1 C. #S #A #B #C #eqAB #eqBC #w cases (eqAB w) cases (eqBC w) /4/ -qed. +qed. lemma union_assoc: ∀S.∀A,B,C:word S → Prop. A ∪ B ∪ C =1 A ∪ (B ∪ C). #S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/] -qed. +qed. *) lemma sem_pre_concat_r : ∀S,i.∀e:pre S. \sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}. @@ -260,7 +246,7 @@ lemma sem_pre_concat_r : ∀S,i.∀e:pre S. >sem_pre_true >sem_cat >sem_pre_true /2/ qed. -definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pitem S → pre S.λe1:pre S.λi2:pitem S. +definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S. match e1 with [ pair i1 b1 ⇒ match b1 with [ true ⇒ (i1 ◂ (bcast ? i2)) @@ -270,12 +256,12 @@ definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pitem S → pre S.λe1:pre S.λi2 definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. match e with - [ pair i b ⇒ 〈\fst (f i), \snd (f i) || b〉]. + [ pair i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉]. notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}. interpretation "lc" 'lc op a b = (lc ? op a b). -definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S. +definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S. match e with [ pair i1 b1 ⇒ match b1 with @@ -285,19 +271,19 @@ definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S. ]. (* -lemma oplus_tt : ∀S: Alpha.∀i1,i2:pitem S. +lemma oplus_tt : ∀S: DeqSet.∀i1,i2:pitem S. 〈i1,true〉 ⊕ 〈i2,true〉 = 〈i1 + i2,true〉. // qed. -lemma oplus_tf : ∀S: Alpha.∀i1,i2:pitem S. +lemma oplus_tf : ∀S: DeqSet.∀i1,i2:pitem S. 〈i1,true〉 ⊕ 〈i2,false〉 = 〈i1 + i2,true〉. // qed. -lemma oplus_ft : ∀S: Alpha.∀i1,i2:pitem S. +lemma oplus_ft : ∀S: DeqSet.∀i1,i2:pitem S. 〈i1,false〉 ⊕ 〈i2,true〉 = 〈i1 + i2,true〉. // qed. -lemma oplus_ff : ∀S: Alpha.∀i1,i2:pitem S. +lemma oplus_ff : ∀S: DeqSet.∀i1,i2:pitem S. 〈i1,false〉 ⊕ 〈i2,false〉 = 〈i1 + i2,false〉. // qed. *) @@ -307,9 +293,9 @@ notation "a^⊛" non associative with precedence 90 for @{'lk eclose $a}. notation "•" non associative with precedence 60 for @{eclose ?}. -let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝ +let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝ match i with - [ pz ⇒ 〈 ∅, false 〉 + [ pz ⇒ 〈 `∅, false 〉 | pe ⇒ 〈 ϵ, true 〉 | ps x ⇒ 〈 `.x, false〉 | pp x ⇒ 〈 `.x, false 〉 @@ -320,24 +306,25 @@ let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝ notation "• x" non associative with precedence 60 for @{'eclose $x}. interpretation "eclose" 'eclose x = (eclose ? x). -lemma eclose_plus: ∀S:Alpha.∀i1,i2:pitem S. +lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S. •(i1 + i2) = •i1 ⊕ •i2. // qed. -lemma eclose_dot: ∀S:Alpha.∀i1,i2:pitem S. +lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S. •(i1 · i2) = •i1 ▸ i2. // qed. -lemma eclose_star: ∀S:Alpha.∀i:pitem S. +lemma eclose_star: ∀S:DeqSet.∀i:pitem S. •i^* = 〈(\fst(•i))^*,true〉. // qed. definition reclose ≝ λS. lift S (eclose S). interpretation "reclose" 'eclose x = (reclose ? x). -lemma epsilon_or : ∀S:Alpha.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2. +(* +lemma epsilon_or : ∀S:DeqSet.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2. #S #b1 #b2 #w % cases b1 cases b2 normalize /2/ * /2/ * ; -qed. +qed. *) (* lemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c). @@ -347,7 +334,7 @@ nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a. #S a b; napply extP; #w; @; *; nnormalize; /2/; nqed.*) (* theorem 16: 2 *) -lemma sem_oplus: ∀S:Alpha.∀e1,e2:pre S. +lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S. \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. #S * #i1 #b1 * #i2 #b2 #w % [cases b1 cases b2 normalize /2/ * /3/ * /3/ @@ -419,21 +406,6 @@ lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|. ] qed. -axiom eq_ext_sym: ∀S.∀A,B:word S →Prop. - A =1 B → B =1 A. - -axiom union_ext_l: ∀S.∀A,B,C:word S →Prop. - A =1 C → A ∪ B =1 C ∪ B. - -axiom union_ext_r: ∀S.∀A,B,C:word S →Prop. - B =1 C → A ∪ B =1 A ∪ C. - -axiom union_comm : ∀S.∀A,B:word S →Prop. - A ∪ B =1 B ∪ A. - -axiom union_idemp: ∀S.∀A:word S →Prop. - A ∪ A =1 A. - axiom cat_ext_l: ∀S.∀A,B,C:word S →Prop. A =1 C → A · B =1 C · B. @@ -449,15 +421,15 @@ qed. axiom fix_star: ∀S.∀A:word S → Prop. A^* =1 A · A^* ∪ { [ ] }. -axiom star_epsilon: ∀S:Alpha.∀A:word S → Prop. +axiom star_epsilon: ∀S:DeqSet.∀A:word S → Prop. A^* ∪ { [ ] } =1 A^*. -lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S. +lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S. \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ { [ ] }. /2/ qed. (* -lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S. +lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S. \sem{〈i^*,true〉} =1 \sem{〈i,true〉}·\sem{|i|}^* ∪ { [ ] }. /2/ qed. @@ -487,7 +459,7 @@ qed-. lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop. (A ∪ { [ ] }) · C =1 A · C ∪ C. -#S #A #C @ext_eq_trans [|@distr_cat_r |@union_ext_r @epsilon_cat_l] +#S #A #C @eqP_trans [|@distr_cat_r |@eqP_union_l @epsilon_cat_l] qed. (* axiom eplison_cut_l: ∀S.∀A:word S →Prop. @@ -511,10 +483,10 @@ lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S. \sem{e1 ▸ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}. #S * #i1 #b1 #i2 cases b1 [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/ - |#H >odot_true >sem_pre_true @(ext_eq_trans … (sem_pre_concat_r …)) - >erase_bull @ext_eq_trans [|@(union_ext_r … H)] - @ext_eq_trans [|@union_ext_r [|@union_comm ]] - @ext_eq_trans [|@eq_ext_sym @union_assoc ] /3/ + |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …)) + >erase_bull @eqP_trans [|@(eqP_union_l … H)] + @eqP_trans [|@eqP_union_l[|@union_comm ]] + @eqP_trans [|@eqP_sym @union_assoc ] /3/ ] qed. @@ -527,57 +499,57 @@ axiom sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}). (* theorem 16: 1 *) -theorem sem_bull: ∀S:Alpha. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}. +theorem sem_bull: ∀S:DeqSet. ∀e:pitem S. \sem{•e} =1 \sem{e} ∪ \sem{|e|}. #S #e elim e [#w normalize % [/2/ | * //] |/2/ |#x normalize #w % [ /2/ | * [@False_ind | //]] |#x normalize #w % [ /2/ | * // ] |#i1 #i2 #IH1 #IH2 >eclose_dot - @ext_eq_trans [|@odot_dot_aux //] >sem_cat - @ext_eq_trans - [|@union_ext_l - [|@ext_eq_trans [|@(cat_ext_l … IH1)] @distr_cat_r]] - @ext_eq_trans [|@union_assoc] - @ext_eq_trans [||@eq_ext_sym @union_assoc] - @union_ext_r // + @eqP_trans [|@odot_dot_aux //] >sem_cat + @eqP_trans + [|@eqP_union_r + [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]] + @eqP_trans [|@union_assoc] + @eqP_trans [||@eqP_sym @union_assoc] + @eqP_union_l // |#i1 #i2 #IH1 #IH2 >eclose_plus - @ext_eq_trans [|@sem_oplus] >sem_plus >erase_plus - @ext_eq_trans [|@(union_ext_r … IH2)] - @ext_eq_trans [|@eq_ext_sym @union_assoc] - @ext_eq_trans [||@union_assoc] @union_ext_l - @ext_eq_trans [||@eq_ext_sym @union_assoc] - @ext_eq_trans [||@union_ext_r [|@union_comm]] - @ext_eq_trans [||@union_assoc] /3/ + @eqP_trans [|@sem_oplus] >sem_plus >erase_plus + @eqP_trans [|@(eqP_union_l … IH2)] + @eqP_trans [|@eqP_sym @union_assoc] + @eqP_trans [||@union_assoc] @eqP_union_r + @eqP_trans [||@eqP_sym @union_assoc] + @eqP_trans [||@eqP_union_l [|@union_comm]] + @eqP_trans [||@union_assoc] /3/ |#i #H >sem_pre_true >sem_star >erase_bull >sem_star - @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]] - @ext_eq_trans [|@union_ext_l [|@distr_cat_r]] - @ext_eq_trans [|@union_assoc] @union_ext_r >erase_star @star_fix + @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@sem_fst_aux //]]] + @eqP_trans [|@eqP_union_r [|@distr_cat_r]] + @eqP_trans [|@union_assoc] @eqP_union_l >erase_star @star_fix ] qed. -definition lifted_cat ≝ λS:Alpha.λe:pre S. +definition lifted_cat ≝ λS:DeqSet.λe:pre S. lift S (lc S eclose e). notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}. interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2). -lemma sem_odot_true: ∀S:Alpha.∀e1:pre S.∀i. +lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }. #S #e1 #i -cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || true〉) [//] +cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//] #H >H cases (e1 ▸ i) #i1 #b1 cases b1 - [>sem_pre_true @ext_eq_trans [||@eq_ext_sym @union_assoc] - @union_ext_r /2/ + [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc] + @eqP_union_l /2/ |/2/ ] qed. -lemma eq_odot_false: ∀S:Alpha.∀e1:pre S.∀i. +lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. e1 ⊙ 〈i,false〉 = e1 ▸ i. #S #e1 #i -cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || false〉) [//] +cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ false〉) [//] cases (e1 ▸ i) #i1 #b1 cases b1 #H @H qed. @@ -585,8 +557,8 @@ lemma sem_odot: ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}. #S #e1 * #i2 * [>sem_pre_true - @ext_eq_trans [|@sem_odot_true] - @ext_eq_trans [||@union_assoc] @union_ext_l @odot_dot_aux // + @eqP_trans [|@sem_odot_true] + @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux // |>sem_pre_false >eq_odot_false @odot_dot_aux // ] qed. @@ -623,17 +595,17 @@ theorem sem_ostar: ∀S.∀e:pre S. \sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*. #S * #i #b cases b [>sem_pre_true >sem_pre_true >sem_star >erase_bull - @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]] - @ext_eq_trans [|@union_ext_l [|@distr_cat_r]] - @ext_eq_trans [||@eq_ext_sym @distr_cat_r] - @ext_eq_trans [|@union_assoc] @union_ext_r - @ext_eq_trans [||@eq_ext_sym @epsilon_cat_l] @star_fix + @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]] + @eqP_trans [|@eqP_union_r [|@distr_cat_r]] + @eqP_trans [||@eqP_sym @distr_cat_r] + @eqP_trans [|@union_assoc] @eqP_union_l + @eqP_trans [||@eqP_sym @epsilon_cat_l] @star_fix |>sem_pre_false >sem_pre_false >sem_star /2/ ] qed. (* -nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝ +nlet rec pre_of_re (S : DeqSet) (e : re S) on e : pitem S ≝ match e with [ z ⇒ pz ? | e ⇒ pe ? -- 2.39.2