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.
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)
+ ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
+ nnormalize; #H;
+ napply False_ind;
+ napply (quatern_destruct … H)
##]
nqed.