ndefinition quatern_destruct_aux ≝
Πn1,n2:quatern.ΠP:Prop.n1 = n2 →
- match n1 with
- [ q0 ⇒ match n2 with [ q0 ⇒ P → P | _ ⇒ P ]
- | q1 ⇒ match n2 with [ q1 ⇒ P → P | _ ⇒ P ]
- | q2 ⇒ match n2 with [ q2 ⇒ P → P | _ ⇒ P ]
- | q3 ⇒ match n2 with [ q3 ⇒ P → P | _ ⇒ P ]
- ].
+ match eq_qu n1 n2 with [ true ⇒ P → P | false ⇒ P ].
ndefinition quatern_destruct : quatern_destruct_aux.
- #n1; #n2; #P;
+ #n1; #n2; #P; #H;
+ nrewrite < H;
nelim n1;
- ##[ ##1: nelim n2; nnormalize; #H;
- ##[ ##1: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match q0 with [ q0 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##2: nelim n2; nnormalize; #H;
- ##[ ##2: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match q1 with [ q1 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##3: nelim n2; nnormalize; #H;
- ##[ ##3: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match q2 with [ q2 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##4: nelim n2; nnormalize; #H;
- ##[ ##4: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match q3 with [ q3 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##]
+ nnormalize;
+ napply (λx.x).
nqed.
-nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
- #n1; #n2;
- nelim n1;
+nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
nelim n2;
nnormalize;
napply refl_eq.
nqed.
-nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_qu n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqqu n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
##]
nqed.
-nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
#n1; #n2;
ncases n1;
ncases n2;
nnormalize;
##[ ##1,6,11,16: #H; napply refl_eq
- ##| ##*: #H; napply (quatern_destruct … H)
+ ##| ##*: #H; napply (bool_destruct … H)
##]
nqed.
-nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
- #x; #y;
- nnormalize;
- nelim x;
- nelim y;
- ##[ ##1,6,11,16: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (quatern_destruct … H)
- ##]
+nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_qu n1 n2));
+ napply (not_to_not (eq_qu n1 n2 = true) (n1 = n2) ? H);
+ napply (eqqu_to_eq n1 n2).
nqed.
-nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; napply (bool_destruct … H)
- ##| ##*: #H; #H1; napply (quatern_destruct … H1)
+nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_qu x y = true) (eq_qu x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqqu_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqqu_to_neq … H))
##]
nqed.
-nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
#n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; nelim (H (refl_eq …))
- ##| ##*: #H; napply refl_eq
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_qu n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqqu n1 n2 H);
+ napply (symmetric_eq ? (eq_qu n2 n1) false);
+ napply (neq_to_neqqu n2 n1 (symmetric_neq ? n1 n2 H))
##]
nqed.