| D: T →T
. *)
+(*
let rec is_dummy M ≝
match M with
[D P ⇒ true
|_ ⇒ false
- ].
+ ]. *)
let rec is_lambda M ≝
match M with
[Lambda P Q ⇒ true
|_ ⇒ false
].
-
+
+(*
theorem is_dummy_to_exists: ∀M. is_dummy M = true →
∃N. M = D N.
#M (cases M) normalize
[1,2: #n #H destruct|3,4,5: #P #Q #H destruct
|#N #_ @(ex_intro … N) //
]
-qed.
+qed.*)
theorem is_lambda_to_exists: ∀M. is_lambda M = true →
∃P,N. M = Lambda P N.
[1,2,6: #n #H destruct|3,5: #P #Q #H destruct
|#P #N #_ @(ex_intro … P) @(ex_intro … N) //
]
-qed.
+qed.
inductive pr : T →T → Prop ≝
| beta: ∀P,M,N,M1,N1. pr M M1 → pr N N1 →
pr (App (Lambda P M) N) (M1[0 ≝ N1])
- | dapp: ∀M,N,P. pr (App M N) P →
- pr (App (D M) N) (D P)
- | dlam: ∀M,N,P. pr (Lambda M N) P → pr (Lambda M (D N)) (D P)
| none: ∀M. pr M M
| appl: ∀M,M1,N,N1. pr M M1 → pr N N1 → pr (App M N) (App M1 N1)
| lam: ∀P,P1,M,M1. pr P P1 → pr M M1 →
lemma prSort: ∀M,n. pr (Sort n) M → M = Sort n.
#M #n #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
+ [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
|//
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
lemma prRel: ∀M,n. pr (Rel n) M → M = Rel n.
#M #n #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
+ [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
|//
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
lemma prD: ∀M,N. pr (D N) M → ∃P.M = D P ∧ pr N P.
#M #N #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#R #eqR <eqR #_ @(ex_intro … N) /2/
+ [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
+ |#M #eqM #_ @(ex_intro … N) /2/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
]
qed.
-lemma prApp_not_dummy_not_lambda:
-∀M,N,P. pr (App M N) P → is_dummy M = false → is_lambda M = false →
-∃M1,N1. (P = App M1 N1 ∧ pr M M1 ∧ pr N N1).
+lemma prApp_not_lambda:
+∀M,N,P. pr (App M N) P → is_lambda M = false →
+ ∃M1,N1. (P = App M1 N1 ∧ pr M M1 ∧ pr N N1).
#M #N #P #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct #_ #_ #H1 destruct
- |#M1 #N1 #P1 #_ #_ #H destruct #_ #H1 destruct
- |#M #N #P1 #_ #_ #H destruct
- |#Q #eqProd #_ #_ #_ @(ex_intro … M) @(ex_intro … N) /3/
- |#M1 #N1 #M2 #N2 #pr1 #pr2 #_ #_ #H #H1 #_ #_ destruct
+ [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct #_ #H1 destruct
+ |#M1 #eqM1 #_ #_ @(ex_intro … M) @(ex_intro … N) /3/
+ |#M1 #N1 #M2 #N2 #pr1 #pr2 #_ #_ #H #H1 #_ destruct
@(ex_intro … N1) @(ex_intro … N2) /3/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
]
qed.
-lemma prApp_D:
-∀M,N,P. pr (App (D M) N) P →
- (∃Q. (P = D Q ∧ pr (App M N) Q)) ∨
- (∃M1,N1.(P = (App (D M1) N1) ∧ pr M M1 ∧ pr N N1)).
-#M #N #P #prH (inversion prH)
- [#R #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M1 #N1 #P1 #pr1 #_ #H destruct #eqP
- @or_introl @(ex_intro … P1) /2/
- |#M #N #P1 #_ #_ #H destruct
- |#R #eqR #_ @or_intror @(ex_intro … M) @(ex_intro … N) /3/
- |#M1 #N1 #M2 #N2 #pr1 #pr2 #_ #_ #H destruct #_
- cases (prD … pr1) #S * #eqN1 >eqN1 #pr3
- @or_intror @(ex_intro … S) @(ex_intro … N2) /3/
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#M #N #_ #_ #H destruct
- ]
-qed.
-
lemma prApp_lambda:
∀Q,M,N,P. pr (App (Lambda Q M) N) P →
-∃M1,N1. (P = M1[0:=N1] ∧ pr M M1 ∧ pr N N1) ∨
+ ∃M1,N1. (P = M1[0:=N1] ∧ pr M M1 ∧ pr N N1) ∨
(P = (App M1 N1) ∧ pr (Lambda Q M) M1 ∧ pr N N1).
#Q #M #N #P #prH (inversion prH)
[#R #M #N #M1 #N1 #pr1 #pr2 #_ #_ #H destruct #_
@(ex_intro … M1) @(ex_intro … N1) /4/
- |#M1 #N1 #P1 #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#R #eqR #_ @(ex_intro … (Lambda Q M)) @(ex_intro … N) /4/
+ |#M1 #eqM1 #_ @(ex_intro … (Lambda Q M)) @(ex_intro … N) /4/
|#M1 #N1 #M2 #N2 #pr1 #pr2 #_ #_ #H destruct #_
@(ex_intro … N1) @(ex_intro … N2) /4/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
]
qed.
-lemma prLambda_not_dummy: ∀M,N,P. pr (Lambda M N) P → is_dummy N = false →
-∃M1,N1. (P = Lambda M1 N1 ∧ pr M M1 ∧ pr N N1).
-#M #N #P #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct #_ #eqH destruct
- |#Q #eqProd #_ #_ @(ex_intro … M) @(ex_intro … N) /3/
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#Q #Q1 #S #S1 #pr1 #pr2 #_ #_ #H #H1 #_ destruct
- @(ex_intro … Q1) @(ex_intro … S1) /3/
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#M #N #_ #_ #H destruct
- ]
-qed.
-
-lemma prLambda_dummy: ∀M,N,P. pr (Lambda M (D N)) P →
- (∃M1,N1. P = Lambda M1 (D N1) ∧ pr M M1 ∧ pr N N1) ∨
- (∃Q. (P = D Q ∧ pr (Lambda M N) Q)).
+lemma prLambda: ∀M,N,P. pr (Lambda M N) P →
+ ∃M1,N1. (P = Lambda M1 N1 ∧ pr M M1 ∧ pr N N1).
#M #N #P #prH (inversion prH)
[#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M1 #N1 #P1 #prM #_ #eqlam destruct #H @or_intror
- @(ex_intro … P1) /3/
- |#Q #eqLam #_ @or_introl @(ex_intro … M) @(ex_intro … N) /3/
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#Q #Q1 #S #S1 #pr1 #pr2 #_ #_ #H #H1 destruct
- cases (prD …pr2) #S2 * #eqS1 #pr3 >eqS1 @or_introl
- @(ex_intro … Q1) @(ex_intro … S2) /3/
+ |#Q #eqQ #_ @(ex_intro … M) @(ex_intro … N) /3/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#M #N #_ #_ #H destruct
- ]
-qed.
-
-lemma prLambda: ∀M,N,P. pr (Lambda M N) P →
-(∃M1,N1. (P = Lambda M1 N1 ∧ pr M M1 ∧ pr N N1)) ∨
-(∃N1,Q. (N=D N1) ∧ (P = (D Q) ∧ pr (Lambda M N1) Q)).
-#M #N #P #prH (inversion prH)
- [#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M1 #N1 #P1 #prM1 #_ #eqlam #eqP destruct @or_intror
- @(ex_intro … N1) @(ex_intro … P1) /3/
- |#Q #eqProd #_ @or_introl @(ex_intro … M) @(ex_intro … N) /3/
- |#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
- |#Q #Q1 #S #S1 #pr1 #pr2 #_ #_ #H #H1 destruct @or_introl
+ |#Q #Q1 #S #S1 #pr1 #pr2 #_ #_ #H #H1 destruct
@(ex_intro … Q1) @(ex_intro … S1) /3/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #N #_ #_ #H destruct
]
-qed.
+qed.
lemma prProd: ∀M,N,P. pr (Prod M N) P →
-∃M1,N1. P = Prod M1 N1 ∧ pr M M1 ∧ pr N N1.
+ ∃M1,N1. P = Prod M1 N1 ∧ pr M M1 ∧ pr N N1.
#M #N #P #prH (inversion prH)
[#P #M #N #M1 #N1 #_ #_ #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#M #N #P1 #_ #_ #H destruct
- |#Q #eqProd #_ @(ex_intro … M) @(ex_intro … N) /3/
+ |#Q #eqQ #_ @(ex_intro … M) @(ex_intro … N) /3/
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#M #M1 #N #N1 #_ #_ #_ #_ #H destruct
|#Q #Q1 #S #S1 #pr1 #pr2 #_ #_ #H #H1 destruct
[ Sort n ⇒ Sort n
| Rel n ⇒ Rel n
| App P Q ⇒ full_app P (full Q)
- | Lambda P Q ⇒ full_lam (full P) Q
+ | Lambda P Q ⇒ Lambda (full P) (full Q)
| Prod P Q ⇒ Prod (full P) (full Q)
| D P ⇒ D (full P)
]
| App P Q ⇒ App (full_app P (full Q)) N
| Lambda P Q ⇒ (full Q) [0 ≝ N]
| Prod P Q ⇒ App (Prod (full P) (full Q)) N
- | D P ⇒ D (full_app P N)
- ]
-and full_lam M N on N≝
- match N with
- [ Sort n ⇒ Lambda M (Sort n)
- | Rel n ⇒ Lambda M (Rel n)
- | App P Q ⇒ Lambda M (full_app P (full Q))
- | Lambda P Q ⇒ Lambda M (full_lam (full P) Q)
- | Prod P Q ⇒ Lambda M (Prod (full P) (full Q))
- | D P ⇒ D (full_lam M P)
+ | D P ⇒ App (D (full P)) N
]
.
-lemma pr_lift: ∀N,N1,n. pr N N1 → ∀k. pr (lift N k n) (lift N1 k n).
+lemma pr_lift: ∀N,N1,n. pr N N1 →
+ ∀k. pr (lift N k n) (lift N1 k n).
#N #N1 #n #pr1 (elim pr1)
[#P #M1 #N1 #M2 #N2 #pr2 #pr3 #Hind1 #Hind2 #k
normalize >lift_subst_up @beta; //
- |#M1 #N1 #P #pr2 #Hind normalize #k @dapp @Hind
- |#M1 #N1 #P #pr2 #Hind normalize #k @dlam @Hind
|//
|#M1 #N1 #M2 #N2 #pr2 #pr3 #Hind1 #Hind2 #k
normalize @appl; [@Hind1 |@Hind2]
|#M1 #M2 #pr2 #Hind #k normalize @d //
]
qed.
-
+
theorem pr_subst: ∀M,M1,N,N1,n. pr M M1 → pr N N1 →
pr M[n≝N] M1[n≝N1].
@Telim_size #P (cases P)
>(subst_rel3 … ltni) >(subst_rel3 … ltni) //
]
|#Q #M #Hind #M1 #N #N1 #n #pr1 #pr2
- (cases (true_or_false (is_dummy Q)))
- [#isdummy (cases (is_dummy_to_exists … isdummy))
- #Q1 #eqM >eqM in pr1 #pr3 (cases (prApp_D … pr3))
- [* #Q2 * #eqM1 #pr4 >eqM1 @dapp @(Hind (App Q1 M)) //
- >eqM normalize //
- |* #M2 * #N2 * * #eqM1 #pr4 #pr5 >eqM1 @appl;
- [@Hind // [<eqM normalize // | @d //]
+ (cases (true_or_false (is_lambda Q)))
+ [#islambda (cases (is_lambda_to_exists … islambda))
+ #M2 * #N2 #eqQ >eqQ in pr1 #pr3 (cases (prApp_lambda … pr3))
+ #M3 * #N3 *
+ [* * #eqM1 #pr4 #pr5 >eqM1
+ >(plus_n_O n) in ⊢ (??%) >subst_lemma @beta;
+ [<plus_n_Sm <plus_n_O @Hind // >eqQ
+ @(transitive_lt ? (size (Lambda M2 N2))) normalize //
|@Hind // normalize //
]
- ]
- |#notdummy
- (cases (true_or_false (is_lambda Q)))
- [#islambda (cases (is_lambda_to_exists … islambda))
- #M2 * #N2 #eqQ >eqQ in pr1 #pr3 (cases (prApp_lambda … pr3))
- #M3 * #N3 *
- [* * #eqM1 #pr4 #pr5 >eqM1
- >(plus_n_O n) in ⊢ (??%) >subst_lemma
- @beta;
- [<plus_n_Sm <plus_n_O @Hind // >eqQ
- @(transitive_lt ? (size (Lambda M2 N2))) normalize //
- |@Hind // normalize //
- ]
- |* * #eqM1 #pr4 #pr5 >eqM1 @appl;
- [@Hind // <eqQ normalize //
- |@Hind // normalize //
- ]
+ |* * #eqM1 #pr4 #pr5 >eqM1 @appl;
+ [@Hind // <eqQ normalize //
+ |@Hind // normalize //
]
- |#notlambda (cases (prApp_not_dummy_not_lambda … pr1 ??)) //
- #M2 * #N2 * * #eqM1 #pr3 #pr4 >eqM1 @appl;
- [@Hind // normalize // |@Hind // normalize // ]
]
+ |#notlambda (cases (prApp_not_lambda … pr1 ?)) //
+ #M2 * #N2 * * #eqM1 #pr3 #pr4 >eqM1 @appl;
+ [@Hind // normalize // |@Hind // normalize // ]
]
|#Q #M #Hind #M1 #N #N1 #n #pr1 #pr2
(cases (prLambda … pr1))
- [* #M2 * #N2 * * #eqM1 #pr3 #pr4 >eqM1 @lam;
- [@Hind // normalize // | @Hind // normalize // ]
- |* #N2 * #Q1 * #eqM * #eqM1 #pr3 >eqM >eqM1 @dlam
- @(Hind (Lambda Q N2)) // >eqM normalize //
- ]
+ #N2 * #Q1 * * #eqM1 #pr3 #pr4 >eqM1 @lam;
+ [@Hind // normalize // | @Hind // normalize // ]
|#Q #M #Hind #M1 #N #N1 #n #pr1 #pr2
(cases (prProd … pr1)) #M2 * #N2 * * #eqM1 #pr3 #pr4 >eqM1
@prod; [@Hind // normalize // | @Hind // normalize // ]
#M2 * #eqM1 #pr1 >eqM1 @d @Hind // normalize //
]
qed.
-
+
lemma pr_full_app: ∀M,N,N1. pr N N1 →
(∀S.subterm S M → pr S (full S)) →
pr (App M N) (full_app M N1).
[#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @appl // @Hind1 /3/
|#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @beta /2/
|#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @appl // @prod /2/
- |#P #Hind #N1 #N2 #prN #H @dapp @Hind /3/
- ]
-qed.
-
-lemma pr_full_lam: ∀M,N,N1. pr N N1 →
- (∀S.subterm S M → pr S (full S)) →
- pr (Lambda N M) (full_lam N1 M).
-#M (elim M) normalize /2/
- [#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @lam // @pr_full_app /3/
- |#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @lam // @Hind2 /3/
- |#P #Q #Hind1 #Hind2 #N1 #N2 #prN #H @lam // @prod /2/
- |#P #Hind #N1 #N2 #prN #H @dlam @Hind /3/
+ |#P #Hind #N1 #N2 #prN #H @appl // @d /2/
]
qed.
theorem pr_full: ∀M. pr M (full M).
-@Telim #M (cases M)
+@Telim #M (cases M) normalize
[//
|//
|#M1 #N1 #H @pr_full_app /3/
- |#M1 #N1 #H @pr_full_lam /3/
+ |#M1 #N1 #H normalize /3/
|#M1 #N1 #H @prod /2/
|#P #H @d /2/
]
qed.
-
-lemma complete_beta: ∀Q,N,N1,M,M1.(* pr N N1 → *) pr N1 (full N) →
- (∀S,P.subterm S (Lambda Q M) → pr S P → pr P (full S)) →
- pr (Lambda Q M) M1 → pr (App M1 N1) ((full M) [O ≝ (full N)]).
-#Q #N #N1 #M (elim M)
- [1,2:#n #M1 #prN1 #sub #pr1
- (cases (prLambda_not_dummy … pr1 ?)) // #M2 * #N2
- * * #eqM1 #pr3 #pr4 >eqM1 @beta /3/
- |3,4,5:#M1 #M2 #_ #_ #M3 #prN1 #sub #pr1
- (cases (prLambda_not_dummy … pr1 ?)) // #M4 * #N3
- * * #eqM3 #pr3 #pr4 >eqM3 @beta /3/
- |#M1 #Hind #M2 #prN1 #sub #pr1
- (cases (prLambda_dummy … pr1))
- [* #M3 * #N3 * * #eqM2 #pr3 #pr4 >eqM2
- @beta // normalize @d @sub /2/
- |* #P * #eqM2 #pr3 >eqM2 normalize @dapp
- @Hind // #S #P #subH #pr4 @sub //
- (cases (sublam … subH)) [* [* /2/ | /2/] | /3/
- ]
- ]
-qed.
-lemma complete_beta1: ∀Q,N,M,M1.
- (∀N1. pr N N1 → pr N1 (full N)) →
- (∀S,P.subterm S (Lambda Q M) → pr S P → pr P (full S)) →
- pr (App (Lambda Q M) N) M1 → pr M1 ((full M) [O ≝ (full N)]).
-#Q #N #M #M1 #prH #subH #prApp
-(cases (prApp_lambda … prApp)) #M2 * #N2 *
- [* * #eqM1 #pr1 #pr2 >eqM1 @pr_subst; [@subH // | @prH //]
- |* * #eqM1 #pr1 #pr2 >eqM1 @(complete_beta … pr1);
- [@prH //
- |#S #P #subS #prS @subH //
- ]
- ]
-qed.
-
lemma complete_app: ∀M,N,P.
(∀S,P.subterm S (App M N) → pr S P → pr P (full S)) →
pr (App M N) P → pr P (full_app M (full N)).
#M (elim M) normalize
- [#n #P #Q #Hind #pr1
- cases (prApp_not_dummy_not_lambda … pr1 ??) //
+ [#n #P #Q #subH #pr1 cases (prApp_not_lambda … pr1 ?) //
#M1 * #N1 * * #eqQ #pr1 #pr2 >eqQ @appl;
- [@(Hind (Sort n)) // |@Hind //]
- |#n #P #Q #Hind #pr1
- cases (prApp_not_dummy_not_lambda … pr1 ??) //
+ [@(subH (Sort n)) // |@subH //]
+ |#n #P #Q #subH #pr1 cases (prApp_not_lambda … pr1 ?) //
#M1 * #N1 * * #eqQ #pr1 #pr2 >eqQ @appl;
- [@(Hind (Rel n)) // |@Hind //]
+ [@(subH (Rel n)) // |@subH //]
|#P #Q #Hind1 #Hind2 #N1 #N2 #subH #prH
- cases (prApp_not_dummy_not_lambda … prH ??) //
+ cases (prApp_not_lambda … prH ?) //
#M2 * #N2 * * #eqQ #pr1 #pr2 >eqQ @appl;
[@Hind1 /3/ |@subH //]
|#P #Q #Hind1 #Hind2 #N1 #P2 #subH #prH
- @(complete_beta1 … prH);
- [#N2 @subH // | #S #P1 #subS @subH
- (cases (sublam … subS)) [* [* /2/ | /2/] | /2/]
- ]
- |#P #Q #Hind1 #Hind2 #N1 #N2 #subH #prH
- cases (prApp_not_dummy_not_lambda … prH ??) //
+ cases (prApp_lambda … prH) #M2 * #N2 *
+ [* * #eqP2 #pr1 #pr2 >eqP2 @pr_subst /3/
+ |* * #eqP2 #pr1 #pr2 >eqP2 (cases (prLambda … pr1))
+ #M3 * #M4 * * #eqM2 #pr3 #pr4 >eqM2 @beta @subH /2/
+ ]
+ |#P #Q #Hind1 #Hind2 #N1 #N2 #subH #prH
+ cases (prApp_not_lambda … prH ?) //
#M2 * #N2 * * #eqQ #pr1 #pr2 >eqQ @appl;
[@(subH (Prod P Q)) // |@subH //]
- |#P #Hind #N1 #N2 #subH #prH
- (cut (∀S. subterm S (App P N1) → subterm S (App (D P) N1)))
- [#S #sub (cases (subapp …sub)) [* [ * /2/ | /3/] | /2/]] #Hcut
- cases (prApp_D … prH);
- [* #N3 * #eqN3 #pr1 >eqN3 @d @Hind //
- #S #P1 #sub1 #prS @subH /2/
- |* #N3 * #N4 * * #eqN2 #prP #prN1 >eqN2 @dapp @Hind;
- [#S #P1 #sub1 #prS @subH /2/ |@appl // ]
- ]
- ]
-qed.
-
-lemma complete_lam: ∀M,Q,M1.
- (∀S,P.subterm S (Lambda Q M) → pr S P → pr P (full S)) →
- pr (Lambda Q M) M1 → pr M1 (full_lam (full Q) M).
-#M (elim M)
- [#n #Q #M1 #sub #pr1 normalize
- (cases (prLambda_not_dummy … pr1 ?)) // #M2 * #N2
- * * #eqM1 #pr3 #pr4 >eqM1 @lam;
- [@sub /2/ | @(sub (Sort n)) /2/]
- |#n #Q #M1 #sub #pr1 normalize
- (cases (prLambda_not_dummy … pr1 ?)) // #M2 * #N2
- * * #eqM1 #pr3 #pr4 >eqM1 @lam;
- [@sub /2/ | @(sub (Rel n)) /2/]
- |#M1 #M2 #_ #_ #M3 #Q #sub #pr1
- (cases (prLambda_not_dummy … pr1 ?)) // #M4 * #N3
- * * #eqM3 #pr3 #pr4 >eqM3 @lam;
- [@sub // | @complete_app // #S #P1 #subS @sub
- (cases (subapp …subS)) [* [* /2/ | /2/] | /3/ ]
- ]
- |#M1 #M2 #_ #Hind #M3 #Q #sub #pr1
- (cases (prLambda_not_dummy … pr1 ?)) // #M4 * #N3
- * * #eqM3 #pr3 #pr4 >eqM3 @lam;
- [@sub // |@Hind // #S #P1 #subS @sub
- (cases (sublam …subS)) [* [* /2/ | /2/] | /3/ ]
- ]
- |#M1 #M2 #_ #_ #M3 #Q #sub #pr1
- (cases (prLambda_not_dummy … pr1 ?)) // #M4 * #N3
- * * #eqM3 #pr3 #pr4 >eqM3 @lam;
- [@sub // | (cases (prProd … pr4)) #M5 * #N4 * * #eqN3
- #pr5 #pr6 >eqN3 @prod;
- [@sub /3/ | @sub /3/]
- ]
- |#P #Hind #Q #M2 #sub #pr1 (cases (prLambda_dummy … pr1))
- [* #M3 * #N3 * * #eqM2 #pr3 #pr4 >eqM2 normalize
- @dlam @Hind;
- [#S #P1 #subS @sub (cases (sublam …subS))
- [* [* /2/ | /2/ ] |/3/ ]
- |@lam //
- ]
- |* #P * #eqM2 #pr3 >eqM2 normalize @d
- @Hind // #S #P #subH @sub
- (cases (sublam … subH)) [* [* /2/ | /2/] | /3/]
- ]
+ |#P #Hind #N1 #N2 #subH #pr1
+ cases (prApp_not_lambda … pr1 ?) //
+ #M1 * #N1 * * #eqQ #pr2 #pr3 >eqQ @appl;
+ [@(subH (D P) M1) // |@subH //]
]
qed.
|#n #Hind #N #prH normalize >(prRel … prH) //
|#M #N #Hind #Q @complete_app
#S #P #subS @Hind //
- | #P #P1 #Hind #N #Hpr @(complete_lam … Hpr)
- #S #P #subS @Hind //
- |5: #P #P1 #Hind #N #Hpr
+ |#P #P1 #Hind #N #Hpr
+ (cases (prLambda …Hpr)) #M1 * #N1 * * #eqN >eqN normalize /3/
+ |#P #P1 #Hind #N #Hpr
(cases (prProd …Hpr)) #M1 * #N1 * * #eqN >eqN normalize /3/
- |6:#N #Hind #P #prH normalize cases (prD … prH)
+ |#N #Hind #P #prH normalize cases (prD … prH)
#Q * #eqP >eqP #prN @d @Hind //
]
qed.
#P #Q #R #pr1 #pr2 @(ex_intro … (full P)) /3/
qed.
-
-
projects / helm.git / blobdiff