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Translation.v
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From Coq Require Import Bool String List BinPos Compare_dec Lia Arith.
Require Import Equations.Prop.DepElim.
From Equations Require Import Equations.
From Translation
Require Import util SAst SLiftSubst Equality SCommon XTyping ITyping
ITypingInversions ITypingLemmata ITypingAdmissible Optim
PackLifts FundamentalLemma.
Import ListNotations.
Section Translation.
Context `{Sort_notion : Sorts.notion}.
Open Scope type_scope.
Open Scope x_scope.
Open Scope i_scope.
(*! Translation *)
Fact length_increl : forall {Γ Γ'}, Γ ⊂ Γ' -> #|Γ| = #|Γ'|.
Proof.
intros Γ Γ' h.
dependent induction h.
- reflexivity.
- cbn. now f_equal.
Defined.
Fact nth_error_increl :
forall Γ Γ' n A A',
Γ ⊂ Γ' ->
nth_error Γ n = Some A ->
nth_error Γ' n = Some A' ->
A ⊏ A'.
Proof.
intros Γ Γ' n A A' h e e'.
induction h in n, A, A', e, e' |- *.
1:{ destruct n. all: discriminate. }
destruct n.
- cbn in *. congruence.
- cbn in *. eapply IHh. all: eauto.
Defined.
Definition trans_snoc {Σ Γ A s Γ' A' s'} :
Σ |--i Γ' ∈ ⟦ Γ ⟧ ->
Σ ;;;; Γ' ⊢ [A'] : sSort s' ∈ ⟦ Γ ⊢ [A] : sSort s ⟧ ->
Σ |--i Γ' ,, A' ∈ ⟦ Γ ,, A ⟧.
Proof.
intros hΓ hA.
split.
- constructor ; now destruct hA as [[[? ?] ?] ?].
- econstructor.
+ now destruct hΓ.
+ now destruct hA as [[[? ?] ?] ?].
Defined.
Definition trans_Prod {Σ Γ n A B s1 s2 Γ' A' B'} :
Σ |--i Γ' ∈ ⟦ Γ ⟧ ->
Σ ;;;; Γ' ⊢ [A'] : sSort s1 ∈ ⟦ Γ ⊢ [A] : sSort s1 ⟧ ->
Σ ;;;; Γ' ,, A' ⊢ [B'] : sSort s2
∈ ⟦ Γ ,, A ⊢ [B]: sSort s2 ⟧ ->
Σ ;;;; Γ' ⊢ [sProd n A' B']: sSort (Sorts.prod_sort s1 s2)
∈ ⟦ Γ ⊢ [ sProd n A B]: sSort (Sorts.prod_sort s1 s2) ⟧.
Proof.
intros hΓ hA hB.
destruct hΓ. destruct hA as [[? ?] ?]. destruct hB as [[? ?] ?].
repeat split.
- assumption.
- constructor.
- now constructor.
- now eapply type_Prod.
Defined.
Definition trans_Sum {Σ Γ n A B s1 s2 Γ' A' B'} :
Σ |--i Γ' ∈ ⟦ Γ ⟧ ->
Σ ;;;; Γ' ⊢ [A'] : sSort s1 ∈ ⟦ Γ ⊢ [A] : sSort s1 ⟧ ->
Σ ;;;; Γ' ,, A' ⊢ [B'] : sSort s2
∈ ⟦ Γ ,, A ⊢ [B]: sSort s2 ⟧ ->
Σ ;;;; Γ' ⊢ [sSum n A' B']: sSort (Sorts.sum_sort s1 s2)
∈ ⟦ Γ ⊢ [ sSum n A B]: sSort (Sorts.sum_sort s1 s2) ⟧.
Proof.
intros hΓ hA hB.
destruct hΓ. destruct hA as [[? ?] ?]. destruct hB as [[? ?] ?].
repeat split.
- assumption.
- constructor.
- now constructor.
- now eapply type_Sum.
Defined.
Definition trans_Eq {Σ Γ A u v s Γ' A' u' v'} :
Σ |--i Γ' ∈ ⟦ Γ ⟧ ->
Σ ;;;; Γ' ⊢ [A'] : sSort s ∈ ⟦ Γ ⊢ [A] : sSort s ⟧ ->
Σ ;;;; Γ' ⊢ [u'] : A' ∈ ⟦ Γ ⊢ [u] : A ⟧ ->
Σ ;;;; Γ' ⊢ [v'] : A' ∈ ⟦ Γ ⊢ [v] : A ⟧ ->
Σ ;;;; Γ' ⊢ [sEq A' u' v'] : sSort (Sorts.eq_sort s)
∈ ⟦ Γ ⊢ [sEq A u v] : sSort (Sorts.eq_sort s) ⟧.
Proof.
intros hΓ hA hu hv.
destruct hA as [[[? ?] ?] ?].
destruct hu as [[[? ?] ?] ?].
destruct hv as [[[? ?] ?] ?].
repeat split.
- assumption.
- constructor.
- constructor ; assumption.
- apply type_Eq ; assumption.
Defined.
Definition trans_subst {Σ Γ s A B u Γ' A' B' u'} :
type_glob Σ ->
Σ |--i Γ' ∈ ⟦ Γ ⟧ ->
Σ ;;;; Γ',, A' ⊢ [B']: sSort s ∈ ⟦ Γ,, A ⊢ [B]: sSort s ⟧ ->
Σ ;;;; Γ' ⊢ [u']: A' ∈ ⟦ Γ ⊢ [u]: A ⟧ ->
Σ ;;;; Γ' ⊢ [B'{ 0 := u' }]: sSort s ∈ ⟦ Γ ⊢ [B{ 0 := u }]: sSort s ⟧.
Proof.
intros hg hΓ hB hu.
destruct hΓ.
destruct hB as [[[? ?] ?] ?]. destruct hu as [[[? ?] ?] ?].
repeat split.
- assumption.
- constructor.
- apply inrel_subst ; assumption.
- lift_sort. eapply typing_subst ; eassumption.
Defined.
(* Maybe put this together with the other translation definitions *)
Definition eqtrans Σ Γ A u v Γ' A' A'' u' v' p' :=
Γ ⊂ Γ' *
A ⊏ A' *
A ⊏ A'' *
u ⊏ u' *
v ⊏ v' *
(Σ ;;; Γ' |-i p' : sHeq A' u' A'' v').
Lemma eqtrans_trans :
forall {Σ Γ A u v Γ' A' A'' u' v' p'},
type_glob Σ ->
eqtrans Σ Γ A u v Γ' A' A'' u' v' p' ->
(Σ ;;;; Γ' ⊢ [u'] : A' ∈ ⟦ Γ ⊢ [u] : A ⟧) *
(Σ ;;;; Γ' ⊢ [v'] : A'' ∈ ⟦ Γ ⊢ [v] : A ⟧).
Proof.
intros Σ Γ A u v Γ' A' A'' u' v' p' hg h.
destruct h as [[[[[eΓ eS'] eS''] eA] eB] hp'].
repeat split ; try assumption.
all: destruct (istype_type hg hp') as [? hheq].
all: ttinv hheq.
all: assumption.
Defined.
Scheme typing_ind := Induction for XTyping.typing Sort Type
with eq_term_ind := Induction for XTyping.eq_term Sort Type.
(* Set Printing Depth 100. *)
(* Combined Scheme typing_all from typing_ind , wf_ind , eq_term_ind. *)
Definition typing_all :=
fun Σ P P0 X X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15
X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 =>
(typing_ind Sort_notion Σ P P0 X X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12
X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27,
eq_term_ind Sort_notion Σ P P0 X X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12
X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27).
Definition complete_translation {Σ} :
type_glob Σ ->
(forall {Γ t A} (h : Σ ;;; Γ |-x t : A)
{Γ'} (hΓ : Σ |--i Γ' ∈ ⟦ Γ ⟧),
∑ A' t', Σ ;;;; Γ' ⊢ [t'] : A' ∈ ⟦ Γ ⊢ [t] : A ⟧) *
(forall {Γ u v A} (h : Σ ;;; Γ |-x u ≡ v : A)
{Γ'} (hΓ : Σ |--i Γ' ∈ ⟦ Γ ⟧),
∑ A' A'' u' v' p',
eqtrans Σ Γ A u v Γ' A' A'' u' v' p').
Proof.
intro hg.
unshelve refine (
typing_all
Σ
(fun Γ t A (h : Σ ;;; Γ |-x t : A) => forall
Γ' (hΓ : Σ |--i Γ' ∈ ⟦ Γ ⟧),
∑ A' t', Σ ;;;; Γ' ⊢ [t'] : A' ∈ ⟦ Γ ⊢ [t] : A ⟧)
(fun Γ u v A (h : Σ ;;; Γ |-x u ≡ v : A) => forall
Γ' (hΓ : Σ |--i Γ' ∈ ⟦ Γ ⟧),
∑ A' A'' u' v' p',
eqtrans Σ Γ A u v Γ' A' A'' u' v' p')
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
) ; intros.
(** type_translation **)
(* type_Rel *)
+ case_eq (nth_error Γ' n).
2:{
intro h.
apply nth_error_None in h.
apply nth_error_Some_length in e.
destruct hΓ as [iΓ _]. apply length_increl in iΓ.
mylia.
}
intros B e'.
exists (lift0 (S n) B), (sRel n).
repeat split.
* now destruct hΓ.
* apply inrel_lift. eapply nth_error_increl. all: eauto.
now destruct hΓ.
* constructor.
* apply type_Rel.
-- now destruct hΓ.
-- assumption.
(* type_Sort *)
+ exists (sSort (Sorts.succ s)), (sSort s).
repeat split.
* now destruct hΓ.
* constructor.
* constructor.
* apply type_Sort. now destruct hΓ.
(* type_Prod *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S' [t' ht']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th ht') as [T' [[t'' ht''] hh]].
clear ht' t' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ ht''))
as [S' [b' hb']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hb') as [T' [[b'' hb''] hh]].
clear hb' b' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Now we conclude *)
exists (sSort (Sorts.prod_sort s1 s2)), (sProd n t'' b'').
now apply trans_Prod.
(* type_Lambda *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S' [t' ht']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th ht') as [T' [[t'' ht''] hh]].
clear ht' t' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ ht''))
as [S' [bty' hbty']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hbty') as [T' [[bty'' hbty''] hh]].
clear hbty' bty' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the term *)
destruct (X1 _ (trans_snoc hΓ ht''))
as [S' [b' hb']].
destruct (change_type hg hb' hbty'') as [b'' hb''].
clear hb' S' b'.
exists (sProd n' t'' bty''), (sLambda n t'' bty'' b'').
destruct ht'' as [[[? ?] ?] ?].
destruct hbty'' as [[[? ?] ?] ?].
destruct hb'' as [[[? ?] ?] ?].
repeat split.
* now destruct hΓ.
* constructor ; eassumption.
* constructor ; eassumption.
* eapply type_Lambda ; eassumption.
(* type_App *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S' [A'' hA'']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA'') as [T' [[A' hA'] hh]].
clear hA'' A'' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ hA'))
as [S' [B'' hB'']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB'') as [T' [[B' hB'] hh]].
clear hB'' B'' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the function *)
destruct (X1 _ hΓ) as [T'' [t'' ht'']].
assert (th : type_head (head (sProd n A B))) by constructor.
destruct (choose_type hg th ht'') as [T' [[t' ht'] hh]].
clear ht'' t'' T''.
destruct T' ; inversion hh. subst. clear hh th.
rename T'1 into A'', T'2 into B''.
destruct (change_type hg ht' (trans_Prod hΓ hA' hB')) as [t'' ht''].
clear ht' A'' B'' t'.
(* Translation of the argument *)
destruct (X2 _ hΓ) as [A'' [u'' hu'']].
destruct (change_type hg hu'' hA') as [u' hu'].
clear hu'' A'' u''.
(* We now conclude *)
exists (B'{ 0 := u' }), (sApp t'' A' B' u').
destruct hΓ.
destruct hA' as [[[? ?] ?] ?].
destruct hB' as [[[? ?] ?] ?].
destruct ht'' as [[[? ?] ?] ?].
destruct hu' as [[[? ?] ?] ?].
repeat split.
* assumption.
* now apply inrel_subst.
* now constructor.
* eapply type_App ; eassumption.
(* type_Sum *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S' [t' ht']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th ht') as [T' [[t'' ht''] hh]].
clear ht' t' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ ht''))
as [S' [b' hb']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hb') as [T' [[b'' hb''] hh]].
clear hb' b' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Now we conclude *)
exists (sSort (Sorts.sum_sort s1 s2)), (sSum n t'' b'').
now apply trans_Sum.
(* type_Pair *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S' [A'' hA'']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA'') as [T' [[A' hA'] hh]].
clear hA'' A'' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ hA'))
as [S' [B'' hB'']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB'') as [T' [[B' hB'] hh]].
clear hB'' B'' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the first component *)
destruct (X1 _ hΓ) as [A'' [u'' hu'']].
destruct (change_type hg hu'' hA') as [u' hu'].
clear hu'' A'' u''.
(* Translation of the second component *)
destruct (X2 _ hΓ) as [Bv' [v'' hv'']].
destruct (change_type hg hv'' (trans_subst hg hΓ hB' hu')) as [v' hv'].
clear hv'' Bv' v''.
(* Now we conclude *)
exists (sSum n A' B'), (sPair A' B' u' v').
destruct hΓ.
destruct hA' as [[[? ?] ?] ?].
destruct hB' as [[[? ?] ?] ?].
destruct hu' as [[[? ?] ?] ?].
destruct hv' as [[[? ?] ?] ?].
repeat split.
* assumption.
* constructor ; assumption.
* constructor ; assumption.
* eapply type_Pair' ; eassumption.
(* type_Pi1 *)
+ (* Translation of the domain *)
destruct (X0 _ hΓ) as [S' [A'' hA'']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA'') as [T' [[A' hA'] hh]].
clear hA'' A'' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X1 _ (trans_snoc hΓ hA'))
as [S' [B'' hB'']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB'') as [T' [[B' hB'] hh]].
clear hB'' B'' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the pair *)
destruct (X _ hΓ) as [T'' [p'' hp'']].
assert (th : type_head (head (sSum n A B))) by constructor.
destruct (choose_type hg th hp'') as [T' [[p' hp'] hh]].
clear hp'' p'' T''.
destruct T' ; inversion hh. subst. clear hh th.
rename T'1 into A'', T'2 into B''.
destruct (change_type hg hp' (trans_Sum hΓ hA' hB')) as [p'' hp''].
clear hp' A'' B'' p'.
(* Now we conclude *)
exists A', (sPi1 A' B' p'').
destruct hp'' as [[[? ?] ?] hp'].
destruct hA' as [[[? ?] ?] hA'].
destruct hB' as [[[? ?] ?] hB'].
repeat split.
* assumption.
* assumption.
* constructor ; assumption.
* eapply type_Pi1' ; eassumption.
(* type_Pi2 *)
+ (* Translation of the domain *)
destruct (X0 _ hΓ) as [S' [A'' hA'']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA'') as [T' [[A' hA'] hh]].
clear hA'' A'' S'.
destruct T' ; inversion hh.
subst. clear hh th.
(* Translation of the codomain *)
destruct (X1 _ (trans_snoc hΓ hA'))
as [S' [B'' hB'']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB'') as [T' [[B' hB'] hh]].
clear hB'' B'' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the pair *)
destruct (X _ hΓ) as [T'' [p'' hp'']].
assert (th : type_head (head (sSum n A B))) by constructor.
destruct (choose_type hg th hp'') as [T' [[p' hp'] hh]].
clear hp'' p'' T''.
destruct T' ; inversion hh. subst. clear hh th.
rename T'1 into A'', T'2 into B''.
destruct (change_type hg hp' (trans_Sum hΓ hA' hB')) as [p'' hp''].
clear hp' A'' B'' p'.
(* Now we conclude *)
exists (B'{ 0 := sPi1 A' B' p'' }), (sPi2 A' B' p'').
destruct hp'' as [[[? ?] ?] hp'].
destruct hA' as [[[? ?] ?] hA'].
destruct hB' as [[[? ?] ?] hB'].
repeat split.
* assumption.
* apply inrel_subst ; try assumption.
constructor ; assumption.
* constructor ; assumption.
* eapply type_Pi2' ; eassumption.
(* type_Eq *)
+ (* The type *)
destruct (X _ hΓ) as [S [A'' hA'']].
assert (th : type_head (head (sSort s))) by constructor.
destruct (choose_type hg th hA'') as [T [[A' hA'] hh]].
clear hA'' A'' S.
destruct T ; inversion hh. subst. clear hh th.
(* The first term *)
destruct (X0 _ hΓ) as [A'' [u'' hu'']].
destruct (change_type hg hu'' hA') as [u' hu'].
clear hu'' u'' A''.
(* The other term *)
destruct (X1 _ hΓ) as [A'' [v'' hv'']].
destruct (change_type hg hv'' hA') as [v' hv'].
(* Now we conclude *)
exists (sSort (Sorts.eq_sort s)), (sEq A' u' v').
apply trans_Eq ; assumption.
(* type_Refl *)
+ destruct (X0 _ hΓ) as [A' [u' hu']].
exists (sEq A' u' u'), (sRefl A' u').
destruct hu' as [[[? ?] ?] hu'].
destruct hΓ.
repeat split.
* assumption.
* constructor ; assumption.
* constructor ; assumption.
* destruct (istype_type hg hu').
eapply type_Refl ; eassumption.
(* type_Ax *)
+ exists ty, (sAx id).
repeat split.
* now destruct hΓ.
* apply inrel_refl.
eapply xcomp_ax_type ; eassumption.
* constructor.
* eapply type_Ax ; try eassumption.
now destruct hΓ.
(* type_conv *)
+ (* Translating the conversion *)
destruct (X1 _ hΓ)
as [S' [S'' [A'' [B'' [p' h']]]]].
destruct (eqtrans_trans hg h') as [hA'' hB''].
destruct h' as [[[[[eΓ eS'] eS''] eA] eB] hp'].
assert (th : type_head (head (sSort s))) by constructor.
destruct (choose_type hg th hA'') as [T [[A' hA'] hh]].
(* clear hA'' eS' eA A'' S'. *)
destruct T ; inversion hh. subst. clear hh.
destruct (choose_type hg th hB'') as [T [[B' hB'] hh]].
(* clear hB'' eS'' eB B'' S''. *)
destruct T ; inversion hh. subst. clear hh th.
(* Translating the term *)
destruct (X _ hΓ) as [A''' [t'' ht'']].
destruct (change_type hg ht'' hA') as [t' ht'].
assert (hpA : ∑ pA, Σ ;;; Γ' |-i pA : sHeq (sSort s) A' S' A'').
{ destruct hA' as [[_ eA'] hA'].
destruct hA'' as [_ hA''].
assert (hr : A' ∼ A'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [pA hpA].
exists pA. apply hpA ; assumption.
}
destruct hpA as [pA hpA].
assert (hpB : ∑ pB, Σ ;;; Γ' |-i pB : sHeq S'' B'' (sSort s) B').
{ destruct hB' as [[_ eB'] hB'].
destruct hB'' as [_ hB''].
assert (hr : B'' ∼ B').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [pB hpB].
exists pB. apply hpB ; assumption.
}
destruct hpB as [pB hpB].
assert (hq : ∑ q, Σ ;;; Γ' |-i q : sHeq (sSort s) A' (sSort s) B').
{ exists (optHeqTrans pA (optHeqTrans p' pB)).
eapply opt_HeqTrans ; try assumption.
- eassumption.
- eapply opt_HeqTrans ; try eassumption.
}
destruct hq as [q hq].
destruct (opt_sort_heq_ex hg hq) as [e' he'].
(* Now we conclude *)
exists B', (optTransport A' B' e' t').
destruct hA' as [[[? ?] ?] ?].
destruct hB' as [[[? ?] ?] ?].
destruct ht' as [[[? ?] ?] ?].
repeat split ; try assumption.
* apply inrel_optTransport. assumption.
* eapply opt_Transport ; eassumption.
(** eq_translation **)
(* eq_reflexivity *)
+ destruct (X _ hΓ) as [A' [u' hu']].
destruct hu' as [[[? ?] ?] hu'].
exists A', A', u', u', (sHeqRefl A' u').
repeat split ; try assumption.
destruct (istype_type hg hu') as [s' hA'].
eapply type_HeqRefl ; eassumption.
(* eq_symmetry *)
+ destruct (X _ hΓ)
as [A' [A'' [u' [v' [p' h']]]]].
destruct h' as [[[[[? ?] ?] ?] ?] hp'].
exists A'', A', v', u', (optHeqSym p').
repeat split ; try assumption.
eapply opt_HeqSym ; eassumption.
(* eq_transitivity *)
+ destruct (X _ hΓ)
as [A1 [A2 [u1 [v1 [p1 h1']]]]].
destruct (X0 _ hΓ)
as [A3 [A4 [v2 [w1 [p2 h2']]]]].
destruct (eqtrans_trans hg h1') as [hu1 hv1].
destruct (eqtrans_trans hg h2') as [hv2 hw1].
destruct h1' as [[[[[? ?] ?] ?] ?] hp1].
destruct h2' as [[[[[? ?] ?] ?] ?] hp2].
(* We have a missing link between (v1 : A2) and (v2 : A3) *)
assert (sim : v1 ∼ v2).
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- apply inrel_trel. assumption.
}
destruct hv1 as [_ hv1].
destruct hv2 as [_ hv2].
destruct (trel_to_heq Γ' hg sim) as [p3 hp3].
(* We can conclude *)
exists A1, A4, u1, w1.
exists (optHeqTrans p1 (optHeqTrans p3 p2)).
repeat split ; try assumption.
specialize (hp3 _ _ hv1 hv2).
eapply opt_HeqTrans ; try assumption.
* eassumption.
* eapply opt_HeqTrans ; eassumption.
(* eq_beta *)
+ (* Translation of the domain *)
destruct (X _ hΓ) as [S [A'' hA'']].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA'') as [T' [[A' hA'] hh]].
clear hA'' A'' S.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the codomain *)
destruct (X0 _ (trans_snoc hΓ hA'))
as [S' [B'' hB'']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB'') as [T' [[B' hB'] hh]].
clear hB'' B'' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Translation of the in-term *)
destruct (X1 _ (trans_snoc hΓ hA'))
as [T' [t'' ht'']].
destruct (change_type hg ht'' hB') as [t' ht'].
clear ht'' T' t''.
(* Translation of the argument *)
destruct (X2 _ hΓ) as [A'' [u'' hu'']].
destruct (change_type hg hu'' hA') as [u' hu'].
clear hu'' A'' u''.
(* Now we conclude using beta *)
exists (B'{0 := u'}), (B'{0 := u'}).
exists (sApp (sLambda n A' B' t') A' B' u'), (t'{0 := u'}).
exists (sEqToHeq (sBeta t' u')).
destruct hA' as [[[? ?] ?] ?].
destruct hB' as [[[? ?] ?] ?].
destruct ht' as [[[? ?] ?] ?].
destruct hu' as [[[? ?] ?] ?].
repeat split.
* assumption.
* eapply inrel_subst ; assumption.
* eapply inrel_subst ; assumption.
* constructor ; try assumption.
constructor ; assumption.
* eapply inrel_subst ; assumption.
* eapply type_EqToHeq' ; try eassumption.
eapply type_Beta ; eassumption.
(* eq_conv *)
+ (* Translating the conversion *)
destruct (X0 _ hΓ)
as [S' [S'' [T1'' [T2'' [p' h']]]]].
destruct (eqtrans_trans hg h') as [hT1'' hT2''].
destruct h' as [[[[[eΓ eS'] eS''] eT1] eT2] hp'].
assert (th : type_head (head (sSort s))) by constructor.
destruct (choose_type hg th hT1'') as [T [[T1' hT1'] hh]].
destruct T ; inversion hh. subst. clear hh.
destruct (choose_type hg th hT2'') as [T [[T2' hT2'] hh]].
destruct T ; inversion hh. subst. clear hh th.
(* Translation the term conversion *)
destruct (X _ hΓ)
as [T1''' [T2''' [t1'' [t2'' [q' hq']]]]].
destruct (eqtrans_trans hg hq') as [ht1'' ht2''].
destruct (change_type hg ht1'' hT1') as [t1' ht1'].
destruct (change_type hg ht2'' hT1') as [t2' ht2'].
(* clear ht1'' ht2'' hq' T1''' T2''' t1'' t2'' q'. *)
destruct hq' as [[[[[_ eT1'''] eT2'''] et1''] et2''] hq'].
(* Building the intermediary paths *)
assert (hpT1 : ∑ p1, Σ ;;; Γ' |-i p1 : sHeq (sSort s) T1' S' T1'').
{ destruct hT1' as [[_ eT1'] hT1'].
destruct hT1'' as [_ hT1''].
assert (hr : T1' ∼ T1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [p1 hp1].
exists p1. apply hp1 ; assumption.
}
destruct hpT1 as [p1 hp1].
assert (hp2 : ∑ p2, Σ ;;; Γ' |-i p2 : sHeq S'' T2'' (sSort s) T2').
{ destruct hT2' as [[_ eT2'] hT2'].
destruct hT2'' as [_ hT2''].
assert (hr : T2'' ∼ T2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [p2 hp2].
exists p2. apply hp2 ; assumption.
}
destruct hp2 as [p2 hp2].
assert (he : ∑ e, Σ ;;; Γ' |-i e : sHeq (sSort s) T1' (sSort s) T2').
{ exists (optHeqTrans p1 (optHeqTrans p' p2)).
eapply opt_HeqTrans ; try assumption.
- eassumption.
- eapply opt_HeqTrans ; try eassumption.
}
destruct he as [e' he'].
rename e into eqt.
destruct (opt_sort_heq_ex hg he') as [e he].
(* Likewise, we build paths for the terms *)
assert (hq1 : ∑ q1, Σ ;;; Γ' |-i q1 : sHeq T1' t1' T1''' t1'').
{ destruct ht1' as [[_ et1'] ht1'].
destruct ht1'' as [_ ht1''].
assert (hr : t1' ∼ t1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [q1 hq1].
exists q1. apply hq1 ; assumption.
}
destruct hq1 as [q1 hq1].
assert (hq2 : ∑ q2, Σ ;;; Γ' |-i q2 : sHeq T2''' t2'' T1' t2').
{ destruct ht2' as [[_ et2'] ht2'].
destruct ht2'' as [_ ht2''].
assert (hr : t2'' ∼ t2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [q2 hq2].
exists q2. apply hq2 ; assumption.
}
destruct hq2 as [q2 hq2].
assert (hqq : ∑ qq, Σ ;;; Γ' |-i qq : sHeq T1' t1' T1' t2').
{ exists (optHeqTrans q1 (optHeqTrans q' q2)).
eapply opt_HeqTrans ; try assumption.
- eassumption.
- eapply opt_HeqTrans ; try eassumption.
}
destruct hqq as [qq hqq].
assert (hql : ∑ ql, Σ ;;; Γ' |-i ql : sHeq T2' (sTransport T1' T2' e t1') T1' t1').
{ exists (optHeqSym (optHeqTransport e t1')).
destruct ht1' as [_ ht1'].
eapply opt_HeqSym ; try assumption.
eapply opt_HeqTransport ; eassumption.
}
destruct hql as [ql hql].
assert (hqr : ∑ qr, Σ ;;; Γ' |-i qr : sHeq T1' t2' T2' (sTransport T1' T2' e t2')).
{ exists (optHeqTransport e t2').
destruct ht2' as [_ ht2'].
eapply opt_HeqTransport ; eassumption.
}
destruct hqr as [qr hqr].
assert (hqf : ∑ qf, Σ ;;; Γ' |-i qf
: sHeq T2' (sTransport T1' T2' e t1')
T2' (sTransport T1' T2' e t2')).
{ exists (optHeqTrans (optHeqTrans ql qq) qr).
eapply opt_HeqTrans ; try assumption.
- eapply opt_HeqTrans ; eassumption.
- assumption.
}
destruct hqf as [qf hqf].
(* Now we conclude *)
exists T2', T2', (sTransport T1' T2' e t1'), (sTransport T1' T2' e t2').
exists qf.
destruct hT1' as [[[? ?] ?] ?].
destruct hT2' as [[[? ?] ?] ?].
destruct ht1' as [[[? ?] ?] ?].
destruct ht2' as [[[? ?] ?] ?].
repeat split ; try eassumption.
* econstructor. assumption.
* econstructor. assumption.
(* cong_Prod *)
+ (* The domains *)
destruct (X _ hΓ)
as [T1 [T2 [A1'' [A2'' [pA h1']]]]].
destruct (eqtrans_trans hg h1') as [hA1'' hA2''].
destruct h1' as [[[[[? ?] ?] ?] ?] hpA''].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA1'') as [T' [[A1' hA1'] hh]].
destruct T' ; inversion hh. subst.
clear hh.
destruct (choose_type hg th hA2'') as [T' [[A2' hA2'] hh]].
destruct T' ; inversion hh. subst.
clear hh th.
(* Now the codomains *)
destruct (X0 _ (trans_snoc hΓ hA1'))
as [S1 [S2 [B1'' [B2'' [pB h2']]]]].
destruct (eqtrans_trans hg h2') as [hB1'' hB2''].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB1'') as [T' [[B1' hB1'] hh]].
destruct T' ; inversion hh. subst.
clear hh.
destruct (choose_type hg th hB2'') as [T' [[B2' hB2'] hh]].
destruct T' ; inversion hh. subst.
clear hh th.
destruct h2' as [[[[[? ?] ?] ?] ?] hpB''].
(* Now we connect the paths for the domains *)
assert (hp1 : ∑ p1, Σ ;;; Γ' |-i p1 : sHeq (sSort s1) A1' (sSort s1) A2').
{ destruct hA1' as [[_ eA1'] hA1'].
destruct hA1'' as [_ hA1''].
destruct hA2' as [[_ eA2'] hA2'].
destruct hA2'' as [_ hA2''].
assert (hr : A1' ∼ A1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [pl hpl].
assert (hr' : A2'' ∼ A2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr') as [pr hpr].
exists (optHeqTrans (optHeqTrans pl pA) pr).
specialize (hpl _ _ ltac:(eassumption) ltac:(eassumption)).
specialize (hpr _ _ ltac:(eassumption) ltac:(eassumption)).
eapply opt_HeqTrans ; try assumption.
- eapply opt_HeqTrans ; eassumption.
- eassumption.
}
destruct hp1 as [p1 hp1].
(* And then the paths for the codomains *)
pose (Γ1 := nil ,, A1').
pose (Γ2 := nil ,, A2').
pose (Γm := [ (sPack A1' A2') ]).
assert (hm : ismix Σ Γ' Γ1 Γ2 Γm).
{ revert Γm.
replace A1' with (llift0 #|@nil sterm| A1')
by (cbn ; now rewrite llift00).
replace A2' with (rlift0 #|@nil sterm| A2')
by (cbn ; now rewrite rlift00).
intros.
destruct hA1' as [[? ?] ?].
destruct hA2' as [[? ?] ?].
econstructor.
- constructor.
- eassumption.
- assumption.
}
pose (Δ := Γ' ,,, Γm).
assert (hp2 : ∑ p2, Σ ;;; Γ' ,,, Γ1 |-i p2 : sHeq (sSort s2) B1'
(sSort s2) B2').
{ destruct hB1' as [[_ eB1'] hB1'].
destruct hB1'' as [_ hB1''].
destruct hB2' as [[_ eB2'] hB2'].
destruct hB2'' as [_ hB2''].
assert (hr : B1' ∼ B1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq (Γ',, A1') hg hr) as [pl hpl].
specialize (hpl _ _ ltac:(eassumption) ltac:(eassumption)).
assert (hr' : B2'' ∼ B2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq (Γ',, A1') hg hr') as [pr hpr].
specialize (hpr _ _ ltac:(eassumption) ltac:(eassumption)).
exists (optHeqTrans (optHeqTrans pl pB) pr).
eapply opt_HeqTrans ; try assumption.
- eapply opt_HeqTrans ; eassumption.
- eassumption.
}
destruct hp2 as [p2 hp2].
assert (hp3 : ∑ p3, Σ ;;; Δ |-i p3 : sHeq (sSort s2)
(llift0 #|Γm| B1')
(sSort s2)
(llift0 #|Γm| B2')
).
{ exists (llift0 #|Γm| p2).
match goal with
| |- _ ;;; _ |-i _ : ?T =>
change T with (llift0 #|Γm| (sHeq (sSort s2) B1' (sSort s2) B2'))
end.
eapply type_llift0 ; eassumption.
}
destruct hp3 as [p3 hp3].
(* Also translating the typing hypothesis for B2 *)
destruct (X2 _ (trans_snoc hΓ hA2'))
as [S' [B2''' hB2''']].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB2''') as [T' [[tB2 htB2] hh]].
clear hB2''' B2''' S'.
destruct T' ; inversion hh. subst. clear hh th.
(* Now we can use the strong version of the lemma to build a path between
B2' and tB2 !
*)
assert (hp4 : ∑ p4, Σ ;;; Δ |-i p4 : sHeq (sSort s2) (llift0 #|Γm| B2')
(sSort s2) (rlift0 #|Γm| tB2)
).
{ change (sSort s2) with (llift0 #|Γm| (sSort s2)) at 1.
change (sSort s2) with (rlift0 #|Γm| (sSort s2)) at 2.
assert (hr : B2' ∼ tB2).
{ destruct htB2 as [[? ?] ?].
destruct hB2' as [[? ?] ?].
eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- apply inrel_trel. assumption.
}
edestruct (trel_to_heq' hg hr) as [p4 hp4].
exists p4. apply hp4.
- eassumption.
- destruct hB2' as [[? ?] ?]. assumption.
- destruct htB2 as [[? ?] ?]. assumption.
}
destruct hp4 as [p4 hp4].
(* This gives us a better path *)
assert (hp5 : ∑ p5, Σ ;;; Δ |-i p5 : sHeq (sSort s2) (llift0 #|Γm| B1')
(sSort s2) (rlift0 #|Γm| tB2)
).
{ exists (optHeqTrans p3 p4).
eapply opt_HeqTrans ; eassumption.
}
destruct hp5 as [p5 hp5].
(* We can finally conclude! *)
exists (sSort (Sorts.prod_sort s1 s2)), (sSort (Sorts.prod_sort s1 s2)).
exists (sProd n1 A1' B1'), (sProd n2 A2' tB2).
exists (optCongProd B1' tB2 p1 p5).
destruct hA1' as [[[? ?] ?] ?].
destruct hB1' as [[[? ?] ?] ?].
destruct hA2' as [[[? ?] ?] ?].
destruct htB2 as [[[? ?] ?] ?].
repeat split ; [ try constructor .. |].
all: try assumption.
eapply opt_CongProd ; try assumption.
cbn in hp5. rewrite <- llift_substProj, <- rlift_substProj in hp5.
rewrite !llift00, !rlift00 in hp5.
apply hp5.
(* cong_Lambda *)
+ (* The domains *)
destruct (X _ hΓ)
as [T1 [T2 [A1'' [A2'' [pA h1']]]]].
destruct (eqtrans_trans hg h1') as [hA1'' hA2''].
destruct h1' as [[[[[? ?] ?] ?] ?] hpA''].
assert (th : type_head (head (sSort s1))) by constructor.
destruct (choose_type hg th hA1'') as [T' [[A1' hA1'] hh]].
destruct T' ; inversion hh. subst.
clear hh.
destruct (choose_type hg th hA2'') as [T' [[A2' hA2'] hh]].
destruct T' ; inversion hh. subst.
clear hh th.
(* Now the codomains *)
destruct (X0 _ (trans_snoc hΓ hA1'))
as [S1 [S2 [B1'' [B2'' [pB h2']]]]].
destruct (eqtrans_trans hg h2') as [hB1'' hB2''].
assert (th : type_head (head (sSort s2))) by constructor.
destruct (choose_type hg th hB1'') as [T' [[B1' hB1'] hh]].
destruct T' ; inversion hh. subst.
clear hh.
destruct (choose_type hg th hB2'') as [T' [[B2' hB2'] hh]].
destruct T' ; inversion hh. subst.
clear hh th.
destruct h2' as [[[[[? ?] ?] ?] ?] hpB''].
(* Now we connect the paths for the domains *)
assert (hp1 : ∑ p1, Σ ;;; Γ' |-i p1 : sHeq (sSort s1) A1' (sSort s1) A2').
{ destruct hA1' as [[_ eA1'] hA1'].
destruct hA1'' as [_ hA1''].
destruct hA2' as [[_ eA2'] hA2'].
destruct hA2'' as [_ hA2''].
assert (hr : A1' ∼ A1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr) as [pl hpl].
specialize (hpl _ _ ltac:(eassumption) ltac:(eassumption)).
assert (hr' : A2'' ∼ A2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq Γ' hg hr') as [pr hpr].
specialize (hpr _ _ ltac:(eassumption) ltac:(eassumption)).
exists (optHeqTrans (optHeqTrans pl pA) pr).
eapply opt_HeqTrans ; try assumption.
- eapply opt_HeqTrans ; eassumption.
- eassumption.
}
destruct hp1 as [p1 hp1].
(* And then the paths for the codomains *)
pose (Γ1 := nil ,, A1').
pose (Γ2 := nil ,, A2').
pose (Γm := [ sPack A1' A2' ]).
assert (hm : ismix Σ Γ' Γ1 Γ2 Γm).
{ revert Γm.
replace A1' with (llift0 #|@nil sterm| A1')
by (cbn ; now rewrite llift00).
replace A2' with (rlift0 #|@nil sterm| A2')
by (cbn ; now rewrite rlift00).
intros.
destruct hA1' as [[? ?] ?].
destruct hA2' as [[? ?] ?].
econstructor.
- constructor.
- eassumption.
- assumption.
}
pose (Δ := Γ' ,,, Γm).
assert (hp2 : ∑ p2, Σ ;;; Γ' ,,, Γ1 |-i p2 : sHeq (sSort s2) B1'
(sSort s2) B2').
{ destruct hB1' as [[_ eB1'] hB1'].
destruct hB1'' as [_ hB1''].
destruct hB2' as [[_ eB2'] hB2'].
destruct hB2'' as [_ hB2''].
assert (hr : B1' ∼ B1'').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq (Γ',, A1') hg hr) as [pl hpl].
specialize (hpl _ _ ltac:(eassumption) ltac:(eassumption)).
assert (hr' : B2'' ∼ B2').
{ eapply trel_trans.
- eapply trel_sym. eapply inrel_trel. eassumption.
- eapply inrel_trel. eassumption.
}
destruct (trel_to_heq (Γ',, A1') hg hr') as [pr hpr].
specialize (hpr _ _ ltac:(eassumption) ltac:(eassumption)).
exists (optHeqTrans (optHeqTrans pl pB) pr).
eapply opt_HeqTrans ; try assumption.
- eapply opt_HeqTrans ; eassumption.
- eassumption.
}
destruct hp2 as [p2 hp2].
assert (hp3 : ∑ p3, Σ ;;; Δ |-i p3 : sHeq (sSort s2)
(llift0 #|Γm| B1')
(sSort s2)
(llift0 #|Γm| B2')
).
{ exists (llift0 #|Γm| p2).
match goal with