%%%%%  Completeness  %%%%%

etopen-fun : etopen Ac A -> etopen Ac A' -> etp-eq A A' -> type.
%mode etopen-fun +X1 +X2 -X3.

-	: etopen-fun etopen/t etopen/t etp-eq/i.

-	: etopen-fun (etopen/pi D2 D1) (etopen/pi D2' D1') Deq
	   <- etopen-fun D1 D1' Deq1
	   <- etopen-fun D2 D2' Deq2
	   <- epi-resp Deq1 ([_] Deq2) Deq.

%worlds (evar) (etopen-fun _ _ _).
%total D (etopen-fun D _ _).



trans-topen	: etopen Ac EA
		   -> ttrans EA A
		   -> topen Ac A -> type.
%mode trans-topen +X1 -X2 -X3.

-t	: trans-topen etopen/t ttrans/t topen/t.

-pi	: trans-topen (etopen/pi Dopen2 Dopen1) (ttrans/pi ([_] [_] [_] [_] Dtrans2) Dtrans1) (topen/pi Dopen2' Dopen1')
	   <- trans-topen Dopen1 Dtrans1 Dopen1'
	   <- trans-topen Dopen2 Dtrans2 Dopen2'.

%worlds (bind | tbind) (trans-topen _ _ _).
%total D (trans-topen D _ _).




kof-comp	: ekof C EA
		   -> ttrans EA A
		   -> kof C A -> type.
%mode kof-comp +X1 +X2 -X3.

-	: kof-comp
	   (ekof/i 
	      (Deopen : etopen As EA) 
	      (Dckof : ckof K As))
	   (Dtrans : ttrans EA A) 
	   (kof/i Dopen' Dckof)
	   <- trans-topen Deopen Dtrans' Dopen
	   <- ttrans-fun Dtrans' Dtrans Deq
	   <- topen-resp ctp-eq/i Deq Dopen Dopen'.

%worlds (bind | tbind) (kof-comp _ _ _).
%total {} (kof-comp _ _ _).




vof-comp	: evof EM EA
		   -> ttrans EA A
		   -> vtrans EM X
		   -> vof X A -> type.
%mode vof-comp +X1 -X2 -X3 -X4.

of-comp		: eof EM EA
		   -> ttrans EA A
		   -> trans EM B
		   -> subtype B A _ -> type.
%mode of-comp +X1 -X2 -X3 -X4.

wf-comp		: ewf EA
		   -> ttrans EA A -> type.
%mode wf-comp +X1 -X2.

subtp-comp	: subtp EA EB
		   -> ttrans EA A
		   -> ttrans EB B
		   -> subtype A B _ -> type.
%mode subtp-comp +X1 -X2 -X3 -X4.

tequiv-comp	: tequiv EA EB
		   -> ttrans EA A
		   -> ttrans EB A -> type.
%mode tequiv-comp +X1 -X2 -X3.

equiv-comp	: equiv EM EN EA
		   -> ttrans EA A
		   -> trans EM B
		   -> trans EN C
		   -> subtype B A M
		   -> subtype C A M -> type.
%mode equiv-comp +X1 -X2 -X3 -X4 -X5 -X6.


%block cbind	: some {x:atom} {a:tp} {ea:etp} {d_ttrans:ttrans ea a} {d_vof:vof x a}
		   block {ex:eterm} {xt:vtrans ex x} {ed:evof ex ea}
		   {thm1:vof-comp ed d_ttrans xt d_vof}.

%% combine sbind & cbind
%block scbind	: some {a:tp} {ea:etp} {d_tconvert:tconvert a ea} {d_ttrans:ttrans ea a} {d_wf:wf a}
		   block {x:atom} {d:vof x a} {ex:eterm} {xt:vtrans ex x} {ed:evof ex ea} {dv:variable ex}
		   {thm1:vtrans-variable xt dv}
		   {thm2:vsound d xt d_tconvert ed}
		   {thm3:vof-comp ed d_ttrans xt d}.



-var	: of-comp
	   (eof/var
	      _
	      (Devof : evof EX EA))
	   DtransA
	   (trans/var Dself Dexpand DwfA Dvof DtransX)
	   Dsubtype
	   <- vof-comp Devof
	      (DtransA : ttrans EA A)
	      (DtransX : vtrans EX X)
	      (Dvof : vof X A)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- can-expand X A M
	      (Dexpand : expand X A M)
	   <- expand-reg (aof/var DwfA Dvof) Dexpand
	      (DofM : of M A)
	   <- can-self DofM
	      (Dself : self M A B)
	   <- self-reg Dself DofM
	      (DofM' : of M B)
	      (Dsubtype : subtype B A ([_] M)).

-const	: of-comp
	   (eof/const 
	      (Dewf : ewf EA)
	      (Dekof : ekof C EA))
	   DtransA
	   (trans/const
	      Dself
	      Dexpand
	      DwfA
	      Dkof)
	   Dsubtype
	   <- wf-comp Dewf
	      (DtransA : ttrans EA A)
	   <- kof-comp Dekof DtransA
	      (Dkof : kof C A)
	   <- can-expand (const C) A M
	      (Dexpand : expand (const C) A M)
	   <- ttrans-reg DtransA (DwfA : wf A)
	   <- expand-reg (aof/const DwfA Dkof) Dexpand
	      (DofM : of M A)
	   <- can-self DofM
	      (Dself : self M A A')
	   <- self-reg Dself DofM
	      _ 
	      (Dsubtype : subtype A' A ([_] M)).

-forall	: of-comp
	   (eof/forall
	      (Dewf : ewf EA))
	   (ttrans/pi ([_] [_] [_] [_] ttrans/t) (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA))
	   (trans/forall
	      Dself
	      Dexpand
	      DtransA)
	   Dsubtype
	   <- wf-comp Dewf
	      (DtransA : ttrans EA A)
	   <- can-expand (forall A) (qtp A) M
	      (Dexpand : expand (forall A) (qtp A) M)
	   <- ttrans-reg DtransA (DwfA : wf A)
	   <- expand-reg (aof/forall DwfA) Dexpand
	      (DofM : of M (qtp A))
	   <- can-self DofM
	      (Dself : self M (qtp A) B)
	   <- self-reg Dself DofM
	      _ 
	      (Dsubtype : subtype B (qtp A) ([_] M)).

-lam	: of-comp
	   (eof/lam
	      (Deof : {ex} evof ex EA -> eof (EM ex) (EB ex))
	      (Dewf : ewf EA))
	   (ttrans/pi DtransB DtransA)
	   (trans/lam Dtrans DtransA)
	   (subtype/pi
	      DsubtypeB'
	      Dtsub
	      DsubtypeA)
	   <- wf-comp Dewf
	      (DtransA : ttrans EA A)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A}
		 {ex} {tr:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA tr d
		 -> of-comp (Deof ex ed)
		    (DtransB x d ex tr : ttrans (EB ex) (B x))
		    (Dtrans x d ex tr : trans (EM ex) (C x))
		    (DsubtypeB' x : subtype (C x) (B x) _))
	   <- ({x} {d:vof x A}
		 {ex} {tr:vtrans ex x}
		 trans-reg (Dtrans x d ex tr)
		 (DwfC x d : wf (C x)))
	   <- subtype-refl DwfA
	      (DsubtypeA : subtype A A X)
	      (Dexpand : {x} expand x A (X x))
	   <- tsub-expand-var DwfA DwfC Dexpand
	      (Dtsub : {x} tsub ([x] C x) (X x) (C x)).

-app	: of-comp
	   (eof/app
	      (DeofN : eof EN EA)
	      (DeofM : eof EM (epi EA EB)))
	   DtransBN
	   (trans/app
	      DsubDx
	      DsubtypeFC
	      DtransN
	      DtransM)
	   DsubtypeDxBx
	   <- of-comp DeofM
	      (ttrans/pi
		 (DtransB : {x} vof x A
			     -> {ex} vtrans ex x
			     -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM : trans EM (pi C D))
	      (subtype/pi
		 (DsubtypeEB : {x} subtype (E x) (B x) _)
		 (DsubE : {x} tsub D (Lac x) (E x))
		 (DsubtypeAC : subtype A C Lac))
	   <- of-comp DeofN
	      (DtransA' : ttrans EA A')
	      (DtransN : trans EN F)
	      (DsubtypeFA' : subtype F A' N')
	   <- ttrans-fun DtransA' DtransA
	      (DeqA : tp-eq A' A)
	   <- trans-principal DtransN
	      (DprincipalF : principal F)
	   <- principal-subtype DprincipalF DsubtypeFA'
	      (DeqN' : {y} term-eq (N' y) N)
	   <- subtype-resp tp-eq/i DeqA DeqN' DsubtypeFA'
	      (DsubtypeFA : subtype F A ([_] N))
	   <- trans-reg DtransN
	      (DwfF : wf F)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- trans-reg DtransM
	      (wf/pi
		 (DwfD : {x} vof x C -> wf (D x))
		 (DwfC : wf C))
	   <- subtype-reg DsubtypeAC DwfA DwfC
	      (DofLac : {x} vof x A -> of (Lac x) C)
	   <- subtype-reg' DsubtypeFA DwfF DwfA
	      (DofN : of N A)
	   <- can-sub DofLac DofN
	      (DsubO : sub Lac N O)
	   <- subst' DsubO DofLac DofN 
	      (DofO : of O C)
	   <- subtype-trans DwfF DwfA DwfC DsubtypeFA DsubtypeAC ([_] DsubO)
	      (DsubtypeFC : subtype F C ([_] O))
	   <- can-tsub DwfD DofO
	      (DsubDx : tsub D O Dx)
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- can-tsub DwfB DofN
	      (DsubBx : tsub B N Bx)
	   <- ttrans-sub DtransB DtransN DwfA DsubtypeFA DsubBx
	      (DtransBN : ttrans (EB EN) Bx)
	   <- ({x} {d:vof x A}
		 tsubst (DsubE x) DwfD (DofLac x d)
		 (DwfE x d : wf (E x)))
	   <- can-tsub DwfE DofN
	      (DsubEx : tsub E N Ex)
	   <- subtype-sub DsubtypeEB DwfE DwfB DofN DsubEx DsubBx
	      _
	      (DsubtypeExBx : subtype Ex Bx _)
	   <- ({y}
		 tsub-absent (D y) N
		 (DsubDabs y : tsub ([x] D y) N (D y)))
	   <- tsub-permute DofN DofLac ([x] [d:vof x A] [y] [e:vof y C] DwfD y e)
	      DsubDabs DsubO DsubDx DsubE DsubEx
	      (Deq : tp-eq Dx Ex)
	   <- tp-eq-symm Deq
	      (Deq' : tp-eq Ex Dx)
	   <- subtype-resp Deq' tp-eq/i ([_] term-eq/i) DsubtypeExBx
	      (DsubtypeDxBx : subtype Dx Bx _).

-pair	: of-comp
	   (eof/pair
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (DeofN : eof EN (EB EM))
	      (DeofM : eof EM EA))
	   (ttrans/sigma DtransB DtransA)
	   (trans/pair DtransN DtransM)
	   (subtype/sigma
	      ([_] DsubtypeDB)
	      ([_] DsubBx)
	      DsubtypeCA')
	   <- of-comp DeofM
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM C)
	      (DsubtypeCA : subtype C A Lca)
	   <- of-comp DeofN
	      (DtransE : ttrans (EB EM) E)
	      (DtransN : trans EN D)
	      (DsubtypeDE : subtype D E _)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x)))
	   <- trans-principal DtransM
	      (DprincipalC : principal C)
	   <- principal-subtype DprincipalC DsubtypeCA
	      (DeqLca : {x} term-eq (Lca x) M)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA
	      (DsubtypeCA' : subtype C A ([_] M))
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- trans-reg DtransM
	      (DwfC : wf C)
	   <- subtype-reg' DsubtypeCA' DwfC DwfA
	      (DofM : of M A)
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB DtransM DwfA DsubtypeCA' DsubBx
	      (DtransBx : ttrans (EB EM) Bx)
	   <- ttrans-fun DtransE DtransBx
	      (Deq : tp-eq E Bx)
	   <- subtype-resp tp-eq/i Deq ([_] term-eq/i) DsubtypeDE
	      (DsubtypeDB : subtype D Bx _).

-pi1	: of-comp
	   (eof/pi1
	      (Deof : eof EM (esigma EA EB)))
	   DtransA
	   (trans/pi1 DtransM')
	   DsubtypeCA
	   <- of-comp Deof
	      (ttrans/sigma
		 (DtransB : {x} vof x A -> {ex} vtrans ex x -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM : trans EM CD)
	      (DsubtypeCDAB : subtype CD (sigma A B) _)
	   <- trans-principal DtransM
	      (DprincipalCD : principal CD)
	   <- principal-subtype-sigma-invert DprincipalCD DsubtypeCDAB
	      (DeqCD : tp-eq CD (sigma C ([_] D)))
	   <- trans-resp eterm-eq/i DeqCD DtransM
	      (DtransM' : trans EM (sigma C ([_] D)))
	   <- subtype-resp DeqCD tp-eq/i ([_] term-eq/i) DsubtypeCDAB
	      (DsubtypeCDAB' : subtype (sigma C ([_] D)) (sigma A B) _)
	   <- subtype-sigma-invert DsubtypeCDAB'
	      (DsubtypeCA : subtype C A _)
	      _ _ _.

-pi2	: of-comp
	   (eof/pi2
	      (Deof : eof EM (esigma EA EB)))
	   DtransBx
	   (trans/pi2 DtransM')
	   (DsubtypeDB aca)
	   <- of-comp Deof
	      (ttrans/sigma
		 (DtransB : {x} vof x A -> {ex} vtrans ex x -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM : trans EM CD)
	      (DsubtypeCDAB : subtype CD (sigma A B) _)
	   <- trans-principal DtransM
	      (DprincipalCD : principal CD)
	   <- principal-subtype-sigma-invert DprincipalCD DsubtypeCDAB
	      (DeqCD : tp-eq CD (sigma C ([_] D)))
	   <- trans-resp eterm-eq/i DeqCD DtransM
	      (DtransM' : trans EM (sigma C ([_] D)))
	   <- subtype-resp DeqCD tp-eq/i ([_] term-eq/i) DsubtypeCDAB
	      (DsubtypeCDAB' : subtype (sigma C ([_] D)) (sigma A B) _)
	   <- subtype-sigma-invert DsubtypeCDAB'
	      (DsubtypeCA : subtype C A Lca)
	      (DsubE : {x} tsub B (Lca x) (E x))
	      (DsubtypeDE : {x} subtype D (E x) _)
	      _
	   <- principal-resp DeqCD DprincipalCD
	      (principal/sigma
		 (DprincipalD : principal D)
		 (DprincipalC : principal C))
	   <- principal-subtype DprincipalC DsubtypeCA
	      (DeqLca : {x} term-eq (Lca x) M)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA
	      (DsubtypeCA' : subtype C A ([_] M))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- trans-reg DtransM'
	      (wf/sigma
		 _
		 (DwfC : wf C))
	   <- subtype-reg' DsubtypeCA' DwfC DwfA
	      (DofM : of M A)
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB (trans/pi1 DtransM') DwfA DsubtypeCA' DsubBx
	      (DtransBx : ttrans (EB (epi1 EM)) Bx)
	   <- ({x}
		 tsub-resp ([_] tp-eq/i) (DeqLca x) tp-eq/i (DsubE x)
		 (DsubE' x : tsub B M (E x)))
	   <- ({x}
		 tsub-fun (DsubE' x) DsubBx
		 (DeqE x : tp-eq (E x) Bx))
	   <- ({x}
		 subtype-resp tp-eq/i (DeqE x) ([_] term-eq/i) (DsubtypeDE x)
		 (DsubtypeDB x : subtype D Bx _)).

-sing	: of-comp
	   (eof/sing 
	      (Deof : eof EM et))
	   (ttrans/sing Dtrans')
	   Dtrans'
	   subtype/sing
	   <- of-comp Deof
	      ttrans/t
	      (Dtrans : trans EM A)
	      (Dsubtype : subtype A t _)
	   <- trans-principal Dtrans
	      (Dprincipal : principal A)
	   <- principal-subtype-t-invert Dprincipal Dsubtype
	      (Deq : tp-eq A (sing R)) _
	   <- trans-resp eterm-eq/i Deq Dtrans
	      (Dtrans' : trans EM (sing R)).

-extpi	: of-comp
	   (eof/extpi
	      (Deof : {ex} evof ex EA -> eof (eapp EM ex) (EB ex))
	      (DeofM : eof EM (epi EA EBoth)))
	   (ttrans/pi DtransB DtransA)
	   DtransM
	   (subtype/pi
	      DsubtypeDB
	      DsubDx
	      DsubtypeAC)
	   <- of-comp DeofM
	      (ttrans/pi
		 (DtransH : {x} vof x A -> {ex} vtrans ex x -> ttrans (EBoth ex) (Both x))
		 (DtransA : ttrans EA A))
	      (DtransM : trans EM (pi C D))
	      (subtype/pi
		 (DsubtypeDBoth : {x} subtype (Dx x) (Both x) _)
		 (DsubDx : {x} tsub D (Lac x) (Dx x))
		 (DsubtypeAC : subtype A C Lac)
		 : subtype (pi C D) (pi A Both) _)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> of-comp (Deof ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (trans/app
		       (DsubFy x : tsub ([y] F x y) (M x) (Fy x))
		       (DsubtypeHE x : subtype (H x) (E x) ([_] M x))
		       (DtransX x d ex xt : trans ex (H x))
		       (DtransM' x d ex xt : trans EM (pi (E x) ([y] F x y)))
		       : trans (eapp EM ex) (Fy x))
		    (DsubtypeFB x : subtype (Fy x) (B x) _))
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 trans-fun (DtransM' x d ex xt) DtransM
		 (DeqEFCD x : tp-eq (pi (E x) ([y] F x y)) (pi C D)))
	   <- ({x}
		 tp-eq-cdr-pi (DeqEFCD x)
		 (DeqEC x : tp-eq (E x) C)
		 (DeqFD x : {y} tp-eq (F x y) (D y)))
	   <- ({x}
		 can-expand x A (X x)
		 (Dexpand x : expand x A (X x)))
	   <- ({x} {d:vof x A}
		 expand-reg (aof/var DwfA d) (Dexpand x)
		 (DofX x d : of (X x) A))
	   <- ({x} {d:vof x A}
		 can-self (DofX x d)
		 (Dself x : self (X x) A (As x)))
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 trans-fun (DtransX x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) 
		 (DeqHAs x : tp-eq (H x) (As x)))
	   <- ({x}
		 subtype-resp (DeqHAs x) (DeqEC x) ([_] term-eq/i) (DsubtypeHE x)
		 (DsubtypeAsC x : subtype (As x) C ([_] M x)))
	   <- ({x} {d:vof x A}
		 self-reg (Dself x) (DofX x d)
		 (DofX' x d : of (X x) (As x))
		 (DsubtypeAsA x : subtype (As x) A ([_] X x)))
	   <- trans-reg DtransM
	      (wf/pi
		 (DwfD : {x} vof x C -> wf (D x))
		 (DwfC : wf C))
	   <- subtype-reg DsubtypeAC DwfA DwfC
	      (DofLac : {x} vof x A -> of (Lac x) C)
	   <- ({x} {d:vof x A}
		 can-sub DofLac (DofX x d)
		 (DsubX' x : sub Lac (X x) (X' x)))
	   <- ({x} {d:vof x A}
		 sub-expand (aof/var DwfA d) DofLac (Dexpand x) (DsubX' x)
		 (DeqX x : term-eq (X' x) (Lac x)))
	   <- ({x}
		 sub-resp ([_] term-eq/i) term-eq/i (DeqX x) (DsubX' x)
		 (DsubLac x : sub Lac (X x) (Lac x)))
	   <- ({x} {d:vof x A}
		 of-reg (DofX' x d)
		 (DwfAs x d : wf (As x)))
	   <- ({x} {d:vof x A}
		 subtype-trans (DwfAs x d) DwfA DwfC (DsubtypeAsA x) DsubtypeAC ([_] DsubLac x)
		 (DsubtypeAsC' x : subtype (As x) C ([_] Lac x)))
	   <- ({x}
		 subtype-fun (DsubtypeAsC x) (DsubtypeAsC' x) 
		 (DeqM x : {y} term-eq (M x) (Lac x)))
	   <- ({x}
		 tsub-resp ([y] DeqFD x y) (DeqM x aca) tp-eq/i (DsubFy x)
		 (DsubFy' x : tsub ([y] D y) (Lac x) (Fy x)))
	   <- ({x}
		 tsub-fun (DsubFy' x) (DsubDx x)
		 (DeqDx x : tp-eq (Fy x) (Dx x)))
	   <- ({x}
		 subtype-resp (DeqDx x) tp-eq/i ([_] term-eq/i) (DsubtypeFB x)
		 (DsubtypeDB x : subtype (Dx x) (B x) _)).

-extsig	: of-comp
	   (eof/extsigma
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (Deof2 : eof (epi2 EM) (EB (epi1 EM)))
	      (Deof1 : eof (epi1 EM) EA))
	   (ttrans/sigma DtransB DtransA)
	   DtransM
	   (subtype/sigma
	      ([_] DsubtypeDB)
	      ([_] DsubBx)
	      DsubtypeCA')
	   <- of-comp Deof1
	      (DtransA : ttrans EA A)
	      (trans/pi1
		 (DtransM : trans EM (sigma C ([_] D))))
	      (DsubtypeCA : subtype C A Lca)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x)))
	   <- trans-principal DtransM
	      (principal/sigma
		 (DprincipalD : principal D)
		 (DprincipalC : principal C))
	   <- principal-subtype DprincipalC DsubtypeCA
	      (DeqLca : {x} term-eq (Lca x) M)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA
	      (DsubtypeCA' : subtype C A ([_] M))
	   <- ({x} {d:vof x A}
		 {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- trans-reg DtransM
	      (wf/sigma
		 _
		 (DwfC : wf C))
	   <- subtype-reg' DsubtypeCA' DwfC DwfA
	      (DofM : of M A)
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB (trans/pi1 DtransM) DwfA DsubtypeCA' DsubBx
	      (DtransBx : ttrans (EB (epi1 EM)) Bx)
	   <- of-comp Deof2
	      (DtransH : ttrans (EB (epi1 EM)) H)
	      (trans/pi2
		 (DtransM' : trans EM (sigma E ([_] F))))
	      (DsubtypeFH : subtype F H _)
	   <- ttrans-fun DtransH DtransBx
	      (DeqHB : tp-eq H Bx)
	   <- trans-fun DtransM' DtransM
	      (DeqEFCD : tp-eq (sigma E ([_] F)) (sigma C ([_] D)))
	   <- tp-eq-cdr-sigma DeqEFCD _
	      (DeqFD : {x} tp-eq F D)
	   <- subtype-resp (DeqFD aca) DeqHB ([_] term-eq/i) DsubtypeFH
	      (DsubtypeDB : subtype D Bx _).

-sub	: of-comp
	   (eof/subsume
	      (Dsubtp : subtp EA EB)
	      (Deof : eof EM EA))
	   DtransB
	   Dtrans
	   DsubtypeCB
	   <- of-comp Deof
	      (DtransA : ttrans EA A)
	      (Dtrans : trans EM C)
	      (DsubtypeCA : subtype C A _)
	   <- subtp-comp Dsubtp
	      (DtransA' : ttrans EA A')
	      (DtransB : ttrans EB B)
	      (DsubtypeA'B : subtype A' B _)
	   <- ttrans-fun DtransA' DtransA
	      (Deq : tp-eq A' A)
	   <- subtype-resp Deq tp-eq/i ([_] term-eq/i) DsubtypeA'B
	      (DsubtypeAB : subtype A B _)
	   <- trans-reg Dtrans
	      (DwfC : wf C)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ttrans-reg DtransB
	      (DwfB : wf B)
	   <- subtype-trans' DwfC DwfA DwfB DsubtypeCA DsubtypeAB
	      _
	      (DsubtypeCB : subtype C B _).


%%%


-t	: wf-comp ewf/t ttrans/t.

-pi	: wf-comp
	   (ewf/pi
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (DewfA : ewf EA))
	   (ttrans/pi DtransB DtransA)
	   <- wf-comp DewfA (DtransA : ttrans EA A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed) 
		    (DtransB x d ex xt : ttrans (EB ex) (B x))).

-sigma	: wf-comp
	   (ewf/sigma
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (DewfA : ewf EA))
	   (ttrans/sigma DtransB DtransA)
	   <- wf-comp DewfA (DtransA : ttrans EA A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed) 
		    (DtransB x d ex xt : ttrans (EB ex) (B x))).

-sing	: wf-comp
	   (ewf/sing
	      (DofM : eof EM et))
	   (ttrans/sing Dtrans')
	   <- of-comp DofM
	      ttrans/t
	      (Dtrans : trans EM A)
	      (Dsubtype : subtype A t _)
	   <- trans-principal Dtrans
	      (Dprincipal : principal A)
	   <- principal-subtype-t-invert Dprincipal Dsubtype
	      (Deq : tp-eq A (sing R)) _
	   <- trans-resp eterm-eq/i Deq Dtrans
	      (Dtrans' : trans EM (sing R)).


%%%


-reflex	: subtp-comp
	   (subtp/reflex
	      (Dequiv : tequiv EA EB))
	   DtransA
	   DtransB
	   Dsubtype
	   <- tequiv-comp Dequiv
	      (DtransA : ttrans EA A)
	      (DtransB : ttrans EB A)
	   <- ttrans-reg DtransA
	      (Dwf : wf A)
	   <- subtype-refl Dwf
	      (Dsubtype : subtype A A _)
	      _.

-trans	: subtp-comp
	   (subtp/trans
	      (DsubtpBC : subtp EB EC)
	      (DsubtpAB : subtp EA EB))
	   DtransA
	   DtransC
	   DsubtypeAC
	   <- subtp-comp DsubtpAB
	      (DtransA : ttrans EA A)
	      (DtransB : ttrans EB B)
	      (DsubtypeAB : subtype A B _)
	   <- subtp-comp DsubtpBC
	      (DtransB' : ttrans EB B')
	      (DtransC : ttrans EC C)
	      (DsubtypeBC : subtype B' C _)
	   <- ttrans-fun DtransB' DtransB
	      (Deq : tp-eq B' B)
	   <- subtype-resp Deq tp-eq/i ([_] term-eq/i) DsubtypeBC
	      (DsubtypeBC' : subtype B C _)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ttrans-reg DtransB
	      (DwfB : wf B)
	   <- ttrans-reg DtransC
	      (DwfC : wf C)
	   <- subtype-trans' DwfA DwfB DwfC DsubtypeAB DsubtypeBC'
	      _
	      (DsubtypeAC : subtype A C _).

-sing_t	: subtp-comp
	   (subtp/sing_t
	      (Deof : eof EM et))
	   (ttrans/sing Dtrans')
	   ttrans/t
	   subtype/sing_t
	   <- of-comp Deof
	      ttrans/t
	      (Dtrans : trans EM A)
	      (Dsubtype : subtype A t _)
	   <- trans-principal Dtrans
	      (Dprincipal : principal A)
	   <- principal-subtype-t-invert Dprincipal Dsubtype
	      (Deq : tp-eq A (sing R)) _
	   <- trans-resp eterm-eq/i Deq Dtrans
	      (Dtrans' : trans EM (sing R)).

-pi	: subtp-comp
	   (subtp/pi
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (DsubtpBD : {ex} evof ex EC -> subtp (EB ex) (ED ex))
	      (DsubtpCA : subtp EC EA))
	   (ttrans/pi DtransB DtransA)
	   (ttrans/pi DtransD DtransC)
	   (subtype/pi
	      DsubtypeBD
	      Dsub
	      DsubtypeCA)
	   <- subtp-comp DsubtpCA
	      (DtransC : ttrans EC C)
	      (DtransA : ttrans EA A)
	      (DsubtypeCA : subtype C A M)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x)))
	   <- ({x} {d:vof x C} {ex} {xt:vtrans ex x} {ed:evof ex EC}
		 vof-comp ed DtransC xt d
		 -> subtp-comp (DsubtpBD ex ed)
		    (DtransB' x d ex xt : ttrans (EB ex) (B' x))
		    (DtransD x d ex xt : ttrans (ED ex) (D x))
		    (DsubtypeBD x : subtype (B' x) (D x) ([y] N x y)))
	   <- trans-var-coerce DtransC DtransA DsubtypeCA DtransB' DtransB
	      (Dsub : {x} tsub B (M x) (B' x)).

-sigma	: subtp-comp
	   (subtp/sigma
	      (DewfD : {ex} evof ex EC -> ewf (ED ex))
	      (DsubtpBD : {ex} evof ex EA -> subtp (EB ex) (ED ex))
	      (DsubtpAC : subtp EA EC))
	   (ttrans/sigma DtransB DtransA)
	   (ttrans/sigma DtransD DtransC)
	   (subtype/sigma
	      DsubtypeBD
	      Dsub
	      DsubtypeAC)
	   <- subtp-comp DsubtpAC
	      (DtransA : ttrans EA A)
	      (DtransC : ttrans EC C)
	      (DsubtypeAC : subtype A C M)
	   <- ({x} {d:vof x C} {ex} {xt:vtrans ex x} {ed:evof ex EC}
		 vof-comp ed DtransC xt d
		 -> wf-comp (DewfD ex ed)
		    (DtransD x d ex xt : ttrans (ED ex) (D x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> subtp-comp (DsubtpBD ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (DtransD' x d ex xt : ttrans (ED ex) (D' x))
		    (DsubtypeBD x : subtype (B x) (D' x) ([y] N x y)))
	   <- trans-var-coerce DtransA DtransC DsubtypeAC DtransD' DtransD
	      (Dsub : {x} tsub D (M x) (D' x)).


%%%


-refl	: tequiv-comp
	   (tequiv/reflex
	      (Dewf : ewf EA))
	   DtransA
	   DtransA
	   <- wf-comp Dewf
	      (DtransA : ttrans EA A).

-symm	: tequiv-comp
	   (tequiv/symm
	      (Dequiv : tequiv EB EA))
	   DtransA
	   DtransB
	   <- tequiv-comp Dequiv
	      (DtransB : ttrans EB A)
	      (DtransA : ttrans EA A).

-trans	: tequiv-comp
	   (tequiv/trans
	      (DequivBC : tequiv EB EC)
	      (DequivAB : tequiv EA EB))
	   DtransA
	   DtransC'
	   <- tequiv-comp DequivAB
	      (DtransA : ttrans EA A)
	      (DtransB : ttrans EB A)
	   <- tequiv-comp DequivBC
	      (DtransB' : ttrans EB B)
	      (DtransC : ttrans EC B)
	   <- ttrans-fun DtransB' DtransB
	      (Deq : tp-eq B A)
	   <- ttrans-resp etp-eq/i Deq DtransC
	      (DtransC' : ttrans EC A).

-pi	: tequiv-comp
	   (tequiv/pi
	      (DequivB : {ex} evof ex EA -> tequiv (EB ex) (EB' ex))
	      (DequivA : tequiv EA EA'))
	   (ttrans/pi DtransB DtransA)
	   (ttrans/pi DtransB' DtransA')
	   <- tequiv-comp DequivA
	      (DtransA : ttrans EA A)
	      (DtransA' : ttrans EA' A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> tequiv-comp (DequivB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (DtransB' x d ex xt : ttrans (EB' ex) (B x))).

-sigma	: tequiv-comp
	   (tequiv/sigma
	      (DequivB : {ex} evof ex EA -> tequiv (EB ex) (EB' ex))
	      (DequivA : tequiv EA EA'))
	   (ttrans/sigma DtransB DtransA)
	   (ttrans/sigma DtransB' DtransA')
	   <- tequiv-comp DequivA
	      (DtransA : ttrans EA A)
	      (DtransA' : ttrans EA' A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> tequiv-comp (DequivB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (DtransB' x d ex xt : ttrans (EB' ex) (B x))).

-sing	: tequiv-comp
	   (tequiv/sing
	      (Dequiv : equiv EM EN et))
	   (ttrans/sing DtransM')
	   (ttrans/sing DtransN')
	   <- equiv-comp Dequiv
	      ttrans/t
	      (DtransM : trans EM A)
	      (DtransN : trans EN B)
	      (DsubtypeA : subtype A t M)
	      (DsubtypeB : subtype B t M)
	   <- trans-principal DtransM
	      (DprincipalA : principal A)
	   <- trans-principal DtransN
	      (DprincipalB : principal B)
	   <- principal-subtype-t-invert DprincipalA DsubtypeA
	      (DeqA : tp-eq A (sing R))
	      (DeqM : {x} term-eq (M x) (at R))
	   <- principal-subtype-t-invert' DprincipalB DsubtypeB DeqM
	      (DeqB : tp-eq B (sing R))
	   <- trans-resp eterm-eq/i DeqA DtransM
	      (DtransM' : trans EM (sing R))
	   <- trans-resp eterm-eq/i DeqB DtransN
	      (DtransN' : trans EN (sing R)).


%%%


-refl	: equiv-comp
	   (equiv/reflex
	      (Deof : eof EM EA))
	   DtransA
	   DtransM
	   DtransM
	   Dsubtype
	   Dsubtype
	   <- of-comp Deof
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM B)
	      (Dsubtype : subtype B A M).

-symm	: equiv-comp
	   (equiv/symm
	      (Dequiv : equiv EN EM EA))
	   DtransA
	   DtransM
	   DtransN
	   DsubtypeBA
	   DsubtypeCA
	   <- equiv-comp Dequiv
	      (DtransA : ttrans EA A)
	      (DtransN : trans EN C)
	      (DtransM : trans EM B)
	      (DsubtypeCA : subtype C A N)
	      (DsubtypeBA : subtype B A N).

-trans	: equiv-comp
	   (equiv/trans
	      (DequivNO : equiv EN EO EA)
	      (DequivMN : equiv EM EN EA))
	   DtransA
	   DtransM
	   DtransO
	   DsubtypeBA
	   DsubtypeDA'
	   <- equiv-comp DequivMN
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM B)
	      (DtransN : trans EN C)
	      (DsubtypeBA : subtype B A P)
	      (DsubtypeCA : subtype C A P)
	   <- equiv-comp DequivNO
	      (DtransA' : ttrans EA A')
	      (DtransN' : trans EN C')
	      (DtransO : trans EO D)
	      (DsubtypeCA' : subtype C' A' P')
	      (DsubtypeDA : subtype D A' P')
	   <- ttrans-fun DtransA' DtransA
	      (DeqA : tp-eq A' A)
	   <- trans-fun DtransN' DtransN
	      (DeqC : tp-eq C' C)
	   <- subtype-resp DeqC DeqA ([_] term-eq/i) DsubtypeCA'
	      (DsubtypeCA'' : subtype C A P')
	   <- subtype-fun DsubtypeCA'' DsubtypeCA
	      (DeqP : {x} term-eq (P' x) (P x))
	   <- subtype-resp tp-eq/i DeqA DeqP DsubtypeDA
	      (DsubtypeDA' : subtype D A P).

-forall	: equiv-comp
	   (equiv/forall
	      (DequivA : tequiv EA EA'))
	   (ttrans/pi ([_] [_] [_] [_] ttrans/t) (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA))
	   (trans/forall
	      Dself Dexpand DtransA)
	   (trans/forall
	      Dself Dexpand DtransA')
	   Dsubtype
	   Dsubtype
	   <- tequiv-comp DequivA
	      (DtransA : ttrans EA A)
	      (DtransA' : ttrans EA' A)
	   <- can-expand (forall A) (qtp A) M
	      (Dexpand : expand (forall A) (qtp A) M)
	   <- ttrans-reg DtransA (DwfA : wf A)
	   <- expand-reg (aof/forall DwfA) Dexpand
	      (DofM : of M (qtp A))
	   <- can-self DofM
	      (Dself : self M (qtp A) B)
	   <- self-reg Dself DofM
	      _ 
	      (Dsubtype : subtype B (qtp A) ([_] M)).

-app	: equiv-comp
	   (equiv/app
	      (DequivN : equiv EN1 EN2 EA)
	      (DequivM : equiv EM1 EM2 (epi EA EB)))
	   DtransBN
	   (trans/app
	      DsubD1x
	      DsubtypeF1C1
	      DtransN1
	      DtransM1)
	   (trans/app
	      DsubD2x
	      DsubtypeF2C2
	      DtransN2
	      DtransM2)
	   DsubtypeD1xBx
	   DsubtypeD2xBx
	   <- equiv-comp DequivM
	      (ttrans/pi
		 (DtransB : {x} vof x A -> {ex} vtrans ex x -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM1 : trans EM1 (pi C1 D1))
	      (DtransM2 : trans EM2 (pi C2 D2))
	      (DsubtypeCD1 : subtype (pi C1 D1) (pi A B) M)
	      (DsubtypeCD2 : subtype (pi C2 D2) (pi A B) M)
	   <- equiv-comp DequivN
	      (DtransA' : ttrans EA A')
	      (DtransN1 : trans EN1 F1)
	      (DtransN2 : trans EN2 F2)
	      (DsubtypeF1A' : subtype F1 A' N')
	      (DsubtypeF2A' : subtype F2 A' N')
	   <- subtype-pi-invert DsubtypeCD1
	      (DsubtypeAC1 : subtype A C1 Lac1)
	      (DsubE1 : {x} tsub D1 (Lac1 x) (E1 x))
	      (DsubtypeE1B : {x} subtype (E1 x) (B x) (Leb1 x))
	      (DeqM1 : {f} term-eq (M f) (lam ([x] (Leb1 x (app f (Lac1 x))))))
	   <- subtype-pi-invert DsubtypeCD2
	      (DsubtypeAC2 : subtype A C2 Lac2)
	      (DsubE2 : {x} tsub D2 (Lac2 x) (E2 x))
	      (DsubtypeE2B : {x} subtype (E2 x) (B x) (Leb2 x))
	      (DeqM2 : {f} term-eq (M f) (lam ([x] (Leb2 x (app f (Lac2 x))))))
	   <- ttrans-fun DtransA' DtransA
	      (DeqA : tp-eq A' A)
	   <- trans-principal DtransN1
	      (DprincipalF1 : principal F1)
	   <- principal-subtype DprincipalF1 DsubtypeF1A'
	      (DeqN' : {y} term-eq (N' y) N)
	   <- subtype-resp tp-eq/i DeqA DeqN' DsubtypeF1A'
	      (DsubtypeF1A : subtype F1 A ([_] N))
	   <- subtype-resp tp-eq/i DeqA DeqN' DsubtypeF2A'
	      (DsubtypeF2A : subtype F2 A ([_] N))
	   <- trans-reg DtransN1
	      (DwfF1 : wf F1)
	   <- trans-reg DtransN2
	      (DwfF2 : wf F2)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- trans-reg DtransM1
	      (wf/pi
		 (DwfD1 : {x} vof x C1 -> wf (D1 x))
		 (DwfC1 : wf C1))
	   <- trans-reg DtransM2
	      (wf/pi
		 (DwfD2 : {x} vof x C2 -> wf (D2 x))
		 (DwfC2 : wf C2))
	   <- subtype-reg DsubtypeAC1 DwfA DwfC1
	      (DofLac1 : {x} vof x A -> of (Lac1 x) C1)
	   <- subtype-reg DsubtypeAC2 DwfA DwfC2
	      (DofLac2 : {x} vof x A -> of (Lac2 x) C2)
	   <- subtype-reg' DsubtypeF1A DwfF1 DwfA
	      (DofN : of N A)
	   <- can-sub DofLac1 DofN
	      (DsubO1 : sub Lac1 N O1)
	   <- can-sub DofLac2 DofN
	      (DsubO2 : sub Lac2 N O2)
	   <- subst' DsubO1 DofLac1 DofN
	      (DofO1 : of O1 C1)
	   <- subst' DsubO2 DofLac2 DofN
	      (DofO2 : of O2 C2)
	   <- subtype-trans DwfF1 DwfA DwfC1 DsubtypeF1A DsubtypeAC1 ([_] DsubO1)
	      (DsubtypeF1C1 : subtype F1 C1 ([_] O1))
	   <- subtype-trans DwfF2 DwfA DwfC2 DsubtypeF2A DsubtypeAC2 ([_] DsubO2)
	      (DsubtypeF2C2 : subtype F2 C2 ([_] O2))
	   <- can-tsub DwfD1 DofO1
	      (DsubD1x : tsub D1 O1 D1x)
	   <- can-tsub DwfD2 DofO2
	      (DsubD2x : tsub D2 O2 D2x)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- can-tsub DwfB DofN
	      (DsubBx : tsub B N Bx)
	   <- ttrans-sub DtransB DtransN1 DwfA DsubtypeF1A DsubBx
	      (DtransBN : ttrans (EB EN1) Bx)
	   <- ({x} {d:vof x A}
		 tsubst (DsubE1 x) DwfD1 (DofLac1 x d)
		 (DwfE1 x d : wf (E1 x)))
	   <- ({x} {d:vof x A}
		 tsubst (DsubE2 x) DwfD2 (DofLac2 x d)
		 (DwfE2 x d : wf (E2 x)))
	   <- can-tsub DwfE1 DofN
	      (DsubE1x : tsub E1 N E1x)
	   <- can-tsub DwfE2 DofN
	      (DsubE2x : tsub E2 N E2x)
	   <- trans-principal DtransM1
	      (principal/pi
		 (DprincipalD1 : {x} principal (D1 x)))
	   <- trans-principal DtransM2
	      (principal/pi
		 (DprincipalD2 : {x} principal (D2 x)))
	   <- ({x}
		 principal-sub DprincipalD1 (DsubE1 x)
		 (DprincipalE1 x : principal (E1 x)))
	   <- ({x}
		 principal-sub DprincipalD2 (DsubE2 x)
		 (DprincipalE2 x : principal (E2 x)))
	   <- ({x}
		 principal-subtype (DprincipalE1 x) (DsubtypeE1B x)
		 (DeqLeb1 x : {y} term-eq (Leb1 x y) (P1 x)))
	   <- ({x}
		 principal-subtype (DprincipalE2 x) (DsubtypeE2B x)
		 (DeqLeb2 x : {y} term-eq (Leb2 x y) (P2 x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqLeb1 x) (DsubtypeE1B x)
		 (DsubtypeE1B' x : subtype (E1 x) (B x) ([_] P1 x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqLeb2 x) (DsubtypeE2B x)
		 (DsubtypeE2B' x : subtype (E2 x) (B x) ([_] P2 x)))
	   <- ({f}
		 term-eq-symm (DeqM1 f)
		 (DeqM1' f : term-eq (lam ([x] (Leb1 x (app f (Lac1 x))))) (M f)))
	   <- ({f}
		 term-eq-trans (DeqM1' f) (DeqM2 f)
		 (DeqLam f : term-eq (lam ([x] (Leb1 x (app f (Lac1 x))))) (lam ([x] Leb2 x (app f (Lac2 x))))))
	   <- ({f}
		 term-eq-cdr-lam (DeqLam f)
		 (DeqLeby f : {x} term-eq (Leb1 x (app f (Lac1 x))) (Leb2 x (app f (Lac2 x)))))
	   <- ({x} {y}
		 term-eq-symm (DeqLeb1 x y)
		 (DeqLeb1' x y : term-eq (P1 x) (Leb1 x y)))
	   <- ({f} {x}
		 term-eq-trans (DeqLeb1' x _) (DeqLeby f x)
		 (DeqLeby' f x : term-eq (P1 x) (Leb2 x (app f (Lac2 x)))))
	   <- ({f} {x}
		 term-eq-trans (DeqLeby' f x) (DeqLeb2 x _)
		 (DeqP f x : term-eq (P1 x) (P2 x)))
	   <- ({x} {d:vof x A}
		 subtype-reg' (DsubtypeE1B' x) (DwfE1 x d) (DwfB x d)
		 (DofP1 x d : of (P1 x) (B x)))
	   <- can-sub DofP1 DofN
	      (DsubP1x : sub P1 N Px)
	   <- sub-resp ([x] DeqP aca x) term-eq/i term-eq/i DsubP1x
	      (DsubP2x : sub P2 N Px)
	   <- subtype-sub' DsubtypeE1B' DwfE1 DwfB DofN DsubE1x DsubBx
	      ([y] DsubP1x : sub ([x] P1 x) N Px)
	      (DsubtypeE1xBx : subtype E1x Bx ([_] Px))
	   <- subtype-sub' DsubtypeE2B' DwfE2 DwfB DofN DsubE2x DsubBx
	      ([y] DsubP2x : sub ([x] P2 x) N Px)
	      (DsubtypeE2xBx : subtype E2x Bx ([_] Px))
	   <- ({y}
		 tsub-absent (D1 y) N
		 (DsubD1abs y : tsub ([x] D1 y) N (D1 y)))
	   <- ({y}
		 tsub-absent (D2 y) N
		 (DsubD2abs y : tsub ([x] D2 y) N (D2 y)))
	   <- tsub-permute DofN DofLac1 ([x] [d:vof x A] [y] [e:vof y C1] DwfD1 y e)
	      DsubD1abs DsubO1 DsubD1x DsubE1 DsubE1x
	      (Deq1 : tp-eq D1x E1x)
	   <- tp-eq-symm Deq1
	      (Deq1' : tp-eq E1x D1x)
	   <- tsub-permute DofN DofLac2 ([x] [d:vof x A] [y] [e:vof y C2] DwfD2 y e)
	      DsubD2abs DsubO2 DsubD2x DsubE2 DsubE2x
	      (Deq2 : tp-eq D2x E2x)
	   <- tp-eq-symm Deq2
	      (Deq2' : tp-eq E2x D2x)
	   <- subtype-resp Deq1' tp-eq/i ([_] term-eq/i) DsubtypeE1xBx
	      (DsubtypeD1xBx : subtype D1x Bx ([_] Px))
	   <- subtype-resp Deq2' tp-eq/i ([_] term-eq/i) DsubtypeE2xBx
	      (DsubtypeD2xBx : subtype D2x Bx ([_] Px)).

-lam	: equiv-comp
	   (equiv/lam
	      (DequivM : {ex} evof ex EA1 -> equiv (EM1 ex) (EM2 ex) (EB ex))
	      (DequivA : tequiv EA1 EA2))
	   (ttrans/pi DtransB DtransA1)
	   (trans/lam DtransM1 DtransA1)
	   (trans/lam DtransM2 DtransA2)
	   (subtype/pi DsubtypeC1B DsubC1 DsubtypeA)
	   (subtype/pi DsubtypeC2B DsubC2 DsubtypeA)
	   <- tequiv-comp DequivA
	      (DtransA1 : ttrans EA1 A)
	      (DtransA2 : ttrans EA2 A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA1}
		 vof-comp ed DtransA1 xt d
		 -> equiv-comp (DequivM ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (DtransM1 x d ex xt : trans (EM1 ex) (C1 x))
		    (DtransM2 x d ex xt : trans (EM2 ex) (C2 x))
		    (DsubtypeC1B x : subtype (C1 x) (B x) (M x))
		    (DsubtypeC2B x : subtype (C2 x) (B x) (M x)))
	   <- ttrans-reg DtransA1
	      (DwfA : wf A)
	   <- subtype-refl DwfA
	      (DsubtypeA : subtype A A X)
	      (Dexpand : {x} expand x A (X x))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-reg (DtransM1 x d ex xt)
		 (DwfC1 x d : wf (C1 x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-reg (DtransM2 x d ex xt)
		 (DwfC2 x d : wf (C2 x)))
	   <- tsub-expand-var DwfA DwfC1 Dexpand
	      (DsubC1 : {x} tsub C1 (X x) (C1 x))
	   <- tsub-expand-var DwfA DwfC2 Dexpand
	      (DsubC2 : {x} tsub C2 (X x) (C2 x)).

-pair	: equiv-comp
	   (equiv/pair
	      (DewfB : {ex} evof ex EA -> ewf (EB ex))
	      (DequivN : equiv EN1 EN2 (EB EM1))
	      (DequivM : equiv EM1 EM2 EA))
	   (ttrans/sigma DtransB DtransA)
	   (trans/pair DtransN1 DtransM1)
	   (trans/pair DtransN2 DtransM2)
	   (subtype/sigma ([_] DsubtypeD1Bx) ([_] DsubBx) DsubtypeC1A')
	   (subtype/sigma ([_] DsubtypeD2Bx) ([_] DsubBx) DsubtypeC2A')
	   <- equiv-comp DequivM
	      (DtransA : ttrans EA A)
	      (DtransM1 : trans EM1 C1)
	      (DtransM2 : trans EM2 C2)
	      (DsubtypeC1A : subtype C1 A M')
	      (DsubtypeC2A : subtype C2 A M')
	   <- equiv-comp DequivN
	      (DtransE : ttrans (EB EM1) E)
	      (DtransN1 : trans EN1 D1)
	      (DtransN2 : trans EN2 D2)
	      (DsubtypeD1E : subtype D1 E N')
	      (DsubtypeD2E : subtype D2 E N')
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (DewfB ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x)))
	   <- trans-principal DtransM1
	      (DprincipalC1 : principal C1)
	   <- principal-subtype DprincipalC1 DsubtypeC1A
	      (DeqM' : {x} term-eq (M' x) M)
	   <- subtype-resp tp-eq/i tp-eq/i DeqM' DsubtypeC1A
	      (DsubtypeC1A' : subtype C1 A ([_] M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqM' DsubtypeC2A
	      (DsubtypeC2A' : subtype C2 A ([_] M))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- trans-reg DtransM1
	      (DwfC1 : wf C1)
	   <- subtype-reg' DsubtypeC1A' DwfC1 DwfA
	      (DofM : of M A)
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB DtransM1 DwfA DsubtypeC1A' DsubBx
	      (DtransBx : ttrans (EB EM1) Bx)
	   <- ttrans-fun DtransE DtransBx
	      (DeqE : tp-eq E Bx)
	   <- subtype-resp tp-eq/i DeqE ([_] term-eq/i) DsubtypeD1E
	      (DsubtypeD1Bx : subtype D1 Bx N')
	   <- subtype-resp tp-eq/i DeqE ([_] term-eq/i) DsubtypeD2E
	      (DsubtypeD2Bx : subtype D2 Bx N').

-pi1	: equiv-comp
	   (equiv/pi1
	      (Dequiv : equiv EM1 EM2 (esigma EA EB)))
	   DtransA
	   (trans/pi1 DtransM1')
	   (trans/pi1 DtransM2')
	   DsubtypeC1A'
	   DsubtypeC2A'
	   <- equiv-comp Dequiv
	      (ttrans/sigma
		 (DtransB : {x} vof x A -> {ex} vtrans ex x -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM1 : trans EM1 CD1)
	      (DtransM2 : trans EM2 CD2)
	      (DsubtypeCD1AB : subtype CD1 (sigma A B) O)
	      (DsubtypeCD2AB : subtype CD2 (sigma A B) O)
	   <- trans-principal DtransM1
	      (DprincipalCD1 : principal CD1)
	   <- trans-principal DtransM2
	      (DprincipalCD2 : principal CD2)
	   <- principal-subtype-sigma-invert DprincipalCD1 DsubtypeCD1AB
	      (DeqCD1 : tp-eq CD1 (sigma C1 ([_] D1)))
	   <- principal-subtype-sigma-invert DprincipalCD2 DsubtypeCD2AB
	      (DeqCD2 : tp-eq CD2 (sigma C2 ([_] D2)))
	   <- trans-resp eterm-eq/i DeqCD1 DtransM1
	      (DtransM1' : trans EM1 (sigma C1 ([_] D1)))
	   <- trans-resp eterm-eq/i DeqCD2 DtransM2
	      (DtransM2' : trans EM2 (sigma C2 ([_] D2)))
	   <- subtype-resp DeqCD1 tp-eq/i ([_] term-eq/i) DsubtypeCD1AB
	      (DsubtypeCD1AB' : subtype (sigma C1 ([_] D1)) (sigma A B) O)
	   <- subtype-resp DeqCD2 tp-eq/i ([_] term-eq/i) DsubtypeCD2AB
	      (DsubtypeCD2AB' : subtype (sigma C2 ([_] D2)) (sigma A B) O)
	   <- subtype-sigma-invert DsubtypeCD1AB'
	      (DsubtypeC1A : subtype C1 A Lca1)
	      (DsubE1 : {x} tsub B (Lca1 x) (E1 x))
	      (DsubtypeD1E1 : {x} subtype D1 (E1 x) (Lde1 x))
	      (DeqO1 : {p} term-eq (O p) (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))))
	   <- subtype-sigma-invert DsubtypeCD2AB'
	      (DsubtypeC2A : subtype C2 A Lca2)
	      (DsubE2 : {x} tsub B (Lca2 x) (E2 x))
	      (DsubtypeD2E2 : {x} subtype D2 (E2 x) (Lde2 x))
	      (DeqO2 : {p} term-eq (O p) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p))))
	   <- trans-principal DtransM1'
	      (principal/sigma
		 (DprincipalD1 : principal D1)
		 (DprincipalC1 : principal C1))
	   <- trans-principal DtransM2'
	      (principal/sigma
		 (DprincipalD2 : principal D2)
		 (DprincipalC2 : principal C2))
	   <- principal-subtype DprincipalC1 DsubtypeC1A 
	      (DeqLca1 : {x} term-eq (Lca1 x) M)
	   <- principal-subtype DprincipalC2 DsubtypeC2A 
	      (DeqLca2 : {x} term-eq (Lca2 x) M')
	   <- ({p}
		 term-eq-symm (DeqO1 p)
		 (DeqO1' p : term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (O p)))
	   <- ({p}
		 term-eq-trans (DeqO1' p) (DeqO2 p)
		 (DeqPair p : term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))))
	   <- ({p}
		 term-eq-cdr-pair (DeqPair p)
		 (DeqLca p : term-eq (Lca1 (pi1 p)) (Lca2 (pi1 p)))
		 (DeqLde p : term-eq (Lde1 (pi1 p) (pi2 p)) (Lde2 (pi1 p) (pi2 p))))
	   <- ({x}
		 term-eq-symm (DeqLca2 x)
		 (DeqLca2' x : term-eq M' (Lca2 x)))
	   <- ({p}
		 term-eq-symm (DeqLca p)
		 (DeqLca' p : term-eq (Lca2 (pi1 p)) (Lca1 (pi1 p))))
	   <- ({p}
		 term-eq-trans (DeqLca2' (pi1 p)) (DeqLca' p)
		 (DeqLcay p : term-eq M' (Lca1 (pi1 p))))
	   <- ({p}
		 term-eq-trans (DeqLcay p) (DeqLca1 (pi1 p))
		 (DeqM p : term-eq M' M))
	   <- ({x}
		 term-eq-trans (DeqLca2 x) (DeqM aca)
		 (DeqLca2'' x : term-eq (Lca2 x) M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca1 DsubtypeC1A
	      (DsubtypeC1A' : subtype C1 A ([_] M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca2'' DsubtypeC2A
	      (DsubtypeC2A' : subtype C2 A ([_] M)).

-pi2	: equiv-comp
	   (equiv/pi2
	      (Dequiv : equiv EM1 EM2 (esigma EA EB)))
	   DtransBx
	   (trans/pi2 DtransM1')
	   (trans/pi2 DtransM2')
	   (DsubtypeD1Bx' aca)
	   (DsubtypeD2Bx' aca)
	   <- equiv-comp Dequiv
	      (ttrans/sigma
		 (DtransB : {x} vof x A -> {ex} vtrans ex x -> ttrans (EB ex) (B x))
		 (DtransA : ttrans EA A))
	      (DtransM1 : trans EM1 CD1)
	      (DtransM2 : trans EM2 CD2)
	      (DsubtypeCD1AB : subtype CD1 (sigma A B) O)
	      (DsubtypeCD2AB : subtype CD2 (sigma A B) O)
	   <- trans-principal DtransM1
	      (DprincipalCD1 : principal CD1)
	   <- trans-principal DtransM2
	      (DprincipalCD2 : principal CD2)
	   <- principal-subtype-sigma-invert DprincipalCD1 DsubtypeCD1AB
	      (DeqCD1 : tp-eq CD1 (sigma C1 ([_] D1)))
	   <- principal-subtype-sigma-invert DprincipalCD2 DsubtypeCD2AB
	      (DeqCD2 : tp-eq CD2 (sigma C2 ([_] D2)))
	   <- trans-resp eterm-eq/i DeqCD1 DtransM1
	      (DtransM1' : trans EM1 (sigma C1 ([_] D1)))
	   <- trans-resp eterm-eq/i DeqCD2 DtransM2
	      (DtransM2' : trans EM2 (sigma C2 ([_] D2)))
	   <- subtype-resp DeqCD1 tp-eq/i ([_] term-eq/i) DsubtypeCD1AB
	      (DsubtypeCD1AB' : subtype (sigma C1 ([_] D1)) (sigma A B) O)
	   <- subtype-resp DeqCD2 tp-eq/i ([_] term-eq/i) DsubtypeCD2AB
	      (DsubtypeCD2AB' : subtype (sigma C2 ([_] D2)) (sigma A B) O)
	   <- subtype-sigma-invert DsubtypeCD1AB'
	      (DsubtypeC1A : subtype C1 A Lca1)
	      (DsubE1 : {x} tsub B (Lca1 x) (E1 x))
	      (DsubtypeD1E1 : {x} subtype D1 (E1 x) (Lde1 x))
	      (DeqO1 : {p} term-eq (O p) (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))))
	   <- subtype-sigma-invert DsubtypeCD2AB'
	      (DsubtypeC2A : subtype C2 A Lca2)
	      (DsubE2 : {x} tsub B (Lca2 x) (E2 x))
	      (DsubtypeD2E2 : {x} subtype D2 (E2 x) (Lde2 x))
	      (DeqO2 : {p} term-eq (O p) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p))))
	   <- trans-principal DtransM1'
	      (principal/sigma
		 (DprincipalD1 : principal D1)
		 (DprincipalC1 : principal C1))
	   <- trans-principal DtransM2'
	      (principal/sigma
		 (DprincipalD2 : principal D2)
		 (DprincipalC2 : principal C2))
	   <- principal-subtype DprincipalC1 DsubtypeC1A 
	      (DeqLca1 : {x} term-eq (Lca1 x) M)
	   <- principal-subtype DprincipalC2 DsubtypeC2A 
	      (DeqLca2 : {x} term-eq (Lca2 x) M')
	   <- ({p}
		 term-eq-symm (DeqO1 p)
		 (DeqO1' p : term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (O p)))
	   <- ({p}
		 term-eq-trans (DeqO1' p) (DeqO2 p)
		 (DeqPair p : term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))))
	   <- ({p}
		 term-eq-cdr-pair (DeqPair p)
		 (DeqLca p : term-eq (Lca1 (pi1 p)) (Lca2 (pi1 p)))
		 (DeqLde p : term-eq (Lde1 (pi1 p) (pi2 p)) (Lde2 (pi1 p) (pi2 p))))
	   <- ({x}
		 term-eq-symm (DeqLca2 x)
		 (DeqLca2' x : term-eq M' (Lca2 x)))
	   <- ({p}
		 term-eq-symm (DeqLca p)
		 (DeqLca' p : term-eq (Lca2 (pi1 p)) (Lca1 (pi1 p))))
	   <- ({p}
		 term-eq-trans (DeqLca2' (pi1 p)) (DeqLca' p)
		 (DeqLcay p : term-eq M' (Lca1 (pi1 p))))
	   <- ({p}
		 term-eq-trans (DeqLcay p) (DeqLca1 (pi1 p))
		 (DeqM p : term-eq M' M))
	   <- ({x}
		 term-eq-trans (DeqLca2 x) (DeqM aca)
		 (DeqLca2'' x : term-eq (Lca2 x) M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca1 DsubtypeC1A
	      (DsubtypeC1A' : subtype C1 A ([_] M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca2'' DsubtypeC2A
	      (DsubtypeC2A' : subtype C2 A ([_] M))
	   <- trans-reg DtransM1'
	      (wf/sigma
		 (DwfD1 : {x} vof x C1 -> wf D1)
		 (DwfC1 : wf C1))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- subtype-reg' DsubtypeC1A' DwfC1 DwfA
	      (DofM : of M A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB (trans/pi1 DtransM1') DwfA DsubtypeC1A' DsubBx
	      (DtransBx : ttrans (EB (epi1 EM1)) Bx)
	   <- ({x}
		 tsub-resp ([_] tp-eq/i) (DeqLca1 x) tp-eq/i (DsubE1 x)
		 (DsubE1' x : tsub B M (E1 x)))
	   <- ({x}
		 tsub-resp ([_] tp-eq/i) (DeqLca2'' x) tp-eq/i (DsubE2 x)
		 (DsubE2' x : tsub B M (E2 x)))
	   <- ({x}
		 tsub-fun (DsubE1' x) DsubBx
		 (DeqE1 x : tp-eq (E1 x) Bx))
	   <- ({x}
		 tsub-fun (DsubE2' x) DsubBx
		 (DeqE2 x : tp-eq (E2 x) Bx))
	   <- ({x}
		 subtype-resp tp-eq/i (DeqE1 x) ([_] term-eq/i) (DsubtypeD1E1 x)
		 (DsubtypeD1Bx x : subtype D1 Bx (Lde1 x)))
	   <- ({x}
		 subtype-resp tp-eq/i (DeqE2 x) ([_] term-eq/i) (DsubtypeD2E2 x)
		 (DsubtypeD2Bx x : subtype D2 Bx (Lde2 x)))
	   <- ({x}
		 subtype-fun (DsubtypeD1Bx x) (DsubtypeD1Bx aca)
		 (DeqLde1x x : {y} term-eq (Lde1 x y) (Lde1 aca y)))
	   <- ({x}
		 subtype-fun (DsubtypeD2Bx x) (DsubtypeD2Bx aca)
		 (DeqLde2x x : {y} term-eq (Lde2 x y) (Lde2 aca y)))
	   <- principal-subtype DprincipalD1 (DsubtypeD1Bx aca)
	      (DeqLde1y : {y} term-eq (Lde1 aca y) N)
	   <- principal-subtype DprincipalD2 (DsubtypeD2Bx aca)
	      (DeqLde2y : {y} term-eq (Lde2 aca y) N')
	   <- ({x} {y}
		 term-eq-trans (DeqLde1x x y) (DeqLde1y y)
		 (DeqLde1 x y : term-eq (Lde1 x y) N))
	   <- ({x} {y}
		 term-eq-trans (DeqLde2x x y) (DeqLde2y y)
		 (DeqLde2 x y : term-eq (Lde2 x y) N'))
	   <- ({x} {y}
		 term-eq-symm (DeqLde2 x y)
		 (DeqLde2' x y : term-eq N' (Lde2 x y)))
	   <- ({p}
		 term-eq-symm (DeqLde p)
		 (DeqLde' p : term-eq (Lde2 (pi1 p) (pi2 p)) (Lde1 (pi1 p) (pi2 p))))
	   <- ({p}
		 term-eq-trans (DeqLde2' (pi1 p) (pi2 p)) (DeqLde' p)
		 (DeqLdey p : term-eq N' (Lde1 (pi1 p) (pi2 p))))
	   <- ({p}
		 term-eq-trans (DeqLdey p) (DeqLde1 (pi1 p) (pi2 p))
		 (DeqN p : term-eq N' N))
	   <- ({x} {y}
		 term-eq-trans (DeqLde2 x y) (DeqN aca)
		 (DeqLde2'' x y : term-eq (Lde2 x y) N))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqLde1 x) (DsubtypeD1Bx x)
		 (DsubtypeD1Bx' x : subtype D1 Bx ([_] N)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqLde2'' x) (DsubtypeD2Bx x)
		 (DsubtypeD2Bx' x : subtype D2 Bx ([_] N))).

-sing	: equiv-comp
	   (equiv/sing
	      (Dequiv : equiv EM EN et))
	   (ttrans/sing DtransM')
	   DtransM'
	   DtransN'
	   subtype/sing
	   subtype/sing
	   <- equiv-comp Dequiv
	      ttrans/t
	      (DtransM : trans EM A)
	      (DtransN : trans EN B)
	      (DsubtypeA : subtype A t M)
	      (DsubtypeB : subtype B t M)
	   <- trans-principal DtransM
	      (DprincipalA : principal A)
	   <- trans-principal DtransN
	      (DprincipalB : principal B)
	   <- principal-subtype-t-invert DprincipalA DsubtypeA
	      (DeqA : tp-eq A (sing R))
	      (DeqM : {x} term-eq (M x) (at R))
	   <- principal-subtype-t-invert' DprincipalB DsubtypeB DeqM
	      (DeqB : tp-eq B (sing R))
	   <- trans-resp eterm-eq/i DeqA DtransM
	      (DtransM' : trans EM (sing R))
	   <- trans-resp eterm-eq/i DeqB DtransN
	      (DtransN' : trans EN (sing R)).

-singel	: equiv-comp
	   (equiv/singelim
	      (Deof : eof EM (esing EN)))
	   ttrans/t
	   DtransM
	   DtransN
	   subtype/sing_t
	   subtype/sing_t
	   <- of-comp Deof
	      (ttrans/sing
		 (DtransN : trans EN (sing R)))
	      (DtransM : trans EM (sing R))
	      subtype/sing.

-extpi	: equiv-comp
	   (equiv/extpi
	      (Dequiv : {ex} evof ex EA -> equiv (eapp EM1 ex) (eapp EM2 ex) (EB ex))
	      (Deof2 : eof EM2 (epi EA EB2))
	      (Deof1 : eof EM1 (epi EA EB1)))
	   (ttrans/pi DtransB DtransA)
	   Dtrans1
	   Dtrans2
	   (subtype/pi DsubtypeD1xB' DsubD1x DsubtypeAC1)
	   (subtype/pi DsubtypeD2xB' DsubD2x DsubtypeAC2')
	   <- of-comp Deof1
	      (ttrans/pi
		 _
		 (DtransA : ttrans EA A))
	      (Dtrans1 : trans EM1 (pi C1 D1))
	      (subtype/pi
		 _
		 (DsubD1x : {x} tsub D1 (Lac1 x) (D1x x))
		 (DsubtypeAC1 : subtype A C1 Lac1))
	   <- of-comp Deof2
	      (ttrans/pi
		 _
		 (DtransA' : ttrans EA A'))
	      (Dtrans2 : trans EM2 (pi C2 D2))
	      (subtype/pi
		 _
		 (DsubD2x : {x} tsub D2 (Lac2 x) (D2x x))
		 (DsubtypeAC2 : subtype A' C2 Lac2))
	   <- ttrans-fun DtransA' DtransA
	      (DeqA : tp-eq A' A)
	   <- subtype-resp DeqA tp-eq/i ([_] term-eq/i) DsubtypeAC2
	      (DsubtypeAC2' : subtype A C2 Lac2)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> equiv-comp (Dequiv ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (trans/app
		       (DsubF1y x : tsub ([y] F1 x y) (M1 x) (F1y x))
		       (DsubtypeH1E1 x : subtype (H1 x) (E1 x) ([_] M1 x))
		       (DtransH1 x d ex xt : trans ex (H1 x))
		       (Dtrans1' x d ex xt : trans EM1 (pi (E1 x) ([y] F1 x y))))
		    (trans/app
		       (DsubF2y x : tsub ([y] F2 x y) (M2 x) (F2y x))
		       (DsubtypeH2E2 x : subtype (H2 x) (E2 x) ([_] M2 x))
		       (DtransH2 x d ex xt : trans ex (H2 x))
		       (Dtrans2' x d ex xt : trans EM2 (pi (E2 x) ([y] F2 x y))))
		    (DsubtypeF1yB x : subtype (F1y x) (B x) (O x))
		    (DsubtypeF2yB x : subtype (F2y x) (B x) (O x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (Dtrans1' x d ex xt) Dtrans1
		 (DeqEFCD1 x : tp-eq (pi (E1 x) (F1 x)) (pi C1 D1)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (Dtrans2' x d ex xt) Dtrans2
		 (DeqEFCD2 x : tp-eq (pi (E2 x) (F2 x)) (pi C2 D2)))
	   <- ({x}
		 tp-eq-cdr-pi (DeqEFCD1 x)
		 (DeqE1C1 x : tp-eq (E1 x) C1)
		 (DeqF1D1 x : {y} tp-eq (F1 x y) (D1 y)))
	   <- ({x}
		 tp-eq-cdr-pi (DeqEFCD2 x)
		 (DeqE2C2 x : tp-eq (E2 x) C2)
		 (DeqF2D2 x : {y} tp-eq (F2 x y) (D2 y)))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x}
		 can-expand x A (X x)
		 (Dexpand x : expand x A (X x)))
	   <- ({x} {d:vof x A}
		 expand-reg (aof/var DwfA d) (Dexpand x)
		 (DofX x d : of (X x) A))
	   <- ({x} {d:vof x A}
		 can-self (DofX x d)
		 (Dself x : self (X x) A (As x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (DtransH1 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt)
		 (DeqH1 x : tp-eq (H1 x) (As x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (DtransH2 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt)
		 (DeqH2 x : tp-eq (H2 x) (As x)))
	   <- ({x} 
		 subtype-resp (DeqH1 x) (DeqE1C1 x) ([_] term-eq/i) (DsubtypeH1E1 x)
		 (DsubtypeAsC1 x : subtype (As x) C1 ([_] M1 x)))
	   <- ({x} 
		 subtype-resp (DeqH2 x) (DeqE2C2 x) ([_] term-eq/i) (DsubtypeH2E2 x)
		 (DsubtypeAsC2 x : subtype (As x) C2 ([_] M2 x)))
	   <- ({x} {d:vof x A}
		 self-reg (Dself x) (DofX x d)
		 (DofX' x d : of (X x) (As x))
		 (DsubtypeAsA x : subtype (As x) A ([_] X x)))
	   <- trans-reg Dtrans1
	      (wf/pi
		 (DwfD1 : {x} vof x C1 -> wf (D1 x))
		 (DwfC1 : wf C1))
	   <- trans-reg Dtrans2
	      (wf/pi
		 (DwfD2 : {x} vof x C2 -> wf (D2 x))
		 (DwfC2 : wf C2))
	   <- subtype-reg DsubtypeAC1 DwfA DwfC1
	      (DofLac1 : {x} vof x A -> of (Lac1 x) C1)
	   <- subtype-reg DsubtypeAC2' DwfA DwfC2
	      (DofLac2 : {x} vof x A -> of (Lac2 x) C2)
	   <- sub-expand-var DwfA DofLac1 Dexpand
	      (DsubLac1 : {x} sub Lac1 (X x) (Lac1 x))
	   <- sub-expand-var DwfA DofLac2 Dexpand
	      (DsubLac2 : {x} sub Lac2 (X x) (Lac2 x))
	   <- ({x} {d:vof x A}
		 of-reg (DofX' x d)
		 (DwfAs x d : wf (As x)))
	   <- ({x} {d:vof x A}
		 subtype-trans (DwfAs x d) DwfA DwfC1 (DsubtypeAsA x) DsubtypeAC1 ([_] DsubLac1 x)
		 (DsubtypeAsC1' x : subtype (As x) C1 ([_] Lac1 x)))
	   <- ({x} {d:vof x A}
		 subtype-trans (DwfAs x d) DwfA DwfC2 (DsubtypeAsA x) DsubtypeAC2' ([_] DsubLac2 x)
		 (DsubtypeAsC2' x : subtype (As x) C2 ([_] Lac2 x)))
	   <- ({x}
		 subtype-fun (DsubtypeAsC1 x) (DsubtypeAsC1' x)
		 (DeqM1 x : {y} term-eq (M1 x) (Lac1 x)))
	   <- ({x}
		 subtype-fun (DsubtypeAsC2 x) (DsubtypeAsC2' x)
		 (DeqM2 x : {y} term-eq (M2 x) (Lac2 x)))
	   <- ({x}
		 tsub-resp ([y] DeqF1D1 x y) (DeqM1 x aca) tp-eq/i (DsubF1y x)
		 (DsubF1y' x : tsub D1 (Lac1 x) (F1y x))) 
	   <- ({x}
		 tsub-resp ([y] DeqF2D2 x y) (DeqM2 x aca) tp-eq/i (DsubF2y x)
		 (DsubF2y' x : tsub D2 (Lac2 x) (F2y x)))
	   <- ({x}
		 tsub-fun (DsubF1y' x) (DsubD1x x)
		 (DeqD1x x : tp-eq (F1y x) (D1x x)))
	   <- ({x}
		 tsub-fun (DsubF2y' x) (DsubD2x x)
		 (DeqD2x x : tp-eq (F2y x) (D2x x)))
	   <- trans-principal Dtrans1
	      (principal/pi
		 (DprincipalD1 : {x} principal (D1 x)))
	   <- ({x}
		 principal-sub DprincipalD1 (DsubD1x x)
		 (DprincipalD1x x : principal (D1x x)))
	   <- ({x}
		 subtype-resp (DeqD1x x) tp-eq/i ([_] term-eq/i) (DsubtypeF1yB x)
		 (DsubtypeD1xB x : subtype (D1x x) (B x) (O x)))
	   <- ({x}
		 subtype-resp (DeqD2x x) tp-eq/i ([_] term-eq/i) (DsubtypeF2yB x)
		 (DsubtypeD2xB x : subtype (D2x x) (B x) (O x)))
	   <- ({x}
		 principal-subtype (DprincipalD1x x) (DsubtypeD1xB x)
		 (DeqO x : {y} term-eq (O x y) (O' x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD1xB x)
		 (DsubtypeD1xB' x : subtype (D1x x) (B x) ([_] O' x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD2xB x)
		 (DsubtypeD2xB' x : subtype (D2x x) (B x) ([_] O' x))).

-extpiw	: equiv-comp
	   (equiv/extpiw
	      (Dequiv : {ex} evof ex EA -> equiv (eapp EM1 ex) (eapp EM2 ex) (EB ex))
	      (Dequiv12 : equiv EM1 EM2 (epi EA EB1)))
	   (ttrans/pi DtransB DtransA)
	   Dtrans1
	   Dtrans2
	   (subtype/pi DsubtypeD1xB' DsubD1x DsubtypeAC1)
	   (subtype/pi DsubtypeD2xB' DsubD2x DsubtypeAC2')
	   <- equiv-comp Dequiv12
	      (ttrans/pi
		 _
		 (DtransA : ttrans EA A))
	      (Dtrans1 : trans EM1 (pi C1 D1))
	      (Dtrans2 : trans EM2 (pi C2 D2))
	      (DsubtypeCDAB1 : subtype (pi C1 D1) (pi A B') _)
	      (DsubtypeCDAB2 : subtype (pi C2 D2) (pi A B') _)
	   <- subtype-pi-invert DsubtypeCDAB1
	      (DsubtypeAC1 : subtype A C1 Lac1)
	      (DsubD1x : {x} tsub D1 (Lac1 x) (D1x x))
	      _ _
	   <- subtype-pi-invert DsubtypeCDAB2
	      (DsubtypeAC2' : subtype A C2 Lac2)
	      (DsubD2x : {x} tsub D2 (Lac2 x) (D2x x))
	      _ _
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> equiv-comp (Dequiv ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (trans/app
		       (DsubF1y x : tsub ([y] F1 x y) (M1 x) (F1y x))
		       (DsubtypeH1E1 x : subtype (H1 x) (E1 x) ([_] M1 x))
		       (DtransH1 x d ex xt : trans ex (H1 x))
		       (Dtrans1' x d ex xt : trans EM1 (pi (E1 x) ([y] F1 x y))))
		    (trans/app
		       (DsubF2y x : tsub ([y] F2 x y) (M2 x) (F2y x))
		       (DsubtypeH2E2 x : subtype (H2 x) (E2 x) ([_] M2 x))
		       (DtransH2 x d ex xt : trans ex (H2 x))
		       (Dtrans2' x d ex xt : trans EM2 (pi (E2 x) ([y] F2 x y))))
		    (DsubtypeF1yB x : subtype (F1y x) (B x) (O x))
		    (DsubtypeF2yB x : subtype (F2y x) (B x) (O x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (Dtrans1' x d ex xt) Dtrans1
		 (DeqEFCD1 x : tp-eq (pi (E1 x) (F1 x)) (pi C1 D1)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (Dtrans2' x d ex xt) Dtrans2
		 (DeqEFCD2 x : tp-eq (pi (E2 x) (F2 x)) (pi C2 D2)))
	   <- ({x}
		 tp-eq-cdr-pi (DeqEFCD1 x)
		 (DeqE1C1 x : tp-eq (E1 x) C1)
		 (DeqF1D1 x : {y} tp-eq (F1 x y) (D1 y)))
	   <- ({x}
		 tp-eq-cdr-pi (DeqEFCD2 x)
		 (DeqE2C2 x : tp-eq (E2 x) C2)
		 (DeqF2D2 x : {y} tp-eq (F2 x y) (D2 y)))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x}
		 can-expand x A (X x)
		 (Dexpand x : expand x A (X x)))
	   <- ({x} {d:vof x A}
		 expand-reg (aof/var DwfA d) (Dexpand x)
		 (DofX x d : of (X x) A))
	   <- ({x} {d:vof x A}
		 can-self (DofX x d)
		 (Dself x : self (X x) A (As x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (DtransH1 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt)
		 (DeqH1 x : tp-eq (H1 x) (As x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-fun (DtransH2 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt)
		 (DeqH2 x : tp-eq (H2 x) (As x)))
	   <- ({x} 
		 subtype-resp (DeqH1 x) (DeqE1C1 x) ([_] term-eq/i) (DsubtypeH1E1 x)
		 (DsubtypeAsC1 x : subtype (As x) C1 ([_] M1 x)))
	   <- ({x} 
		 subtype-resp (DeqH2 x) (DeqE2C2 x) ([_] term-eq/i) (DsubtypeH2E2 x)
		 (DsubtypeAsC2 x : subtype (As x) C2 ([_] M2 x)))
	   <- ({x} {d:vof x A}
		 self-reg (Dself x) (DofX x d)
		 (DofX' x d : of (X x) (As x))
		 (DsubtypeAsA x : subtype (As x) A ([_] X x)))
	   <- trans-reg Dtrans1
	      (wf/pi
		 (DwfD1 : {x} vof x C1 -> wf (D1 x))
		 (DwfC1 : wf C1))
	   <- trans-reg Dtrans2
	      (wf/pi
		 (DwfD2 : {x} vof x C2 -> wf (D2 x))
		 (DwfC2 : wf C2))
	   <- subtype-reg DsubtypeAC1 DwfA DwfC1
	      (DofLac1 : {x} vof x A -> of (Lac1 x) C1)
	   <- subtype-reg DsubtypeAC2' DwfA DwfC2
	      (DofLac2 : {x} vof x A -> of (Lac2 x) C2)
	   <- sub-expand-var DwfA DofLac1 Dexpand
	      (DsubLac1 : {x} sub Lac1 (X x) (Lac1 x))
	   <- sub-expand-var DwfA DofLac2 Dexpand
	      (DsubLac2 : {x} sub Lac2 (X x) (Lac2 x))
	   <- ({x} {d:vof x A}
		 of-reg (DofX' x d)
		 (DwfAs x d : wf (As x)))
	   <- ({x} {d:vof x A}
		 subtype-trans (DwfAs x d) DwfA DwfC1 (DsubtypeAsA x) DsubtypeAC1 ([_] DsubLac1 x)
		 (DsubtypeAsC1' x : subtype (As x) C1 ([_] Lac1 x)))
	   <- ({x} {d:vof x A}
		 subtype-trans (DwfAs x d) DwfA DwfC2 (DsubtypeAsA x) DsubtypeAC2' ([_] DsubLac2 x)
		 (DsubtypeAsC2' x : subtype (As x) C2 ([_] Lac2 x)))
	   <- ({x}
		 subtype-fun (DsubtypeAsC1 x) (DsubtypeAsC1' x)
		 (DeqM1 x : {y} term-eq (M1 x) (Lac1 x)))
	   <- ({x}
		 subtype-fun (DsubtypeAsC2 x) (DsubtypeAsC2' x)
		 (DeqM2 x : {y} term-eq (M2 x) (Lac2 x)))
	   <- ({x}
		 tsub-resp ([y] DeqF1D1 x y) (DeqM1 x aca) tp-eq/i (DsubF1y x)
		 (DsubF1y' x : tsub D1 (Lac1 x) (F1y x))) 
	   <- ({x}
		 tsub-resp ([y] DeqF2D2 x y) (DeqM2 x aca) tp-eq/i (DsubF2y x)
		 (DsubF2y' x : tsub D2 (Lac2 x) (F2y x)))
	   <- ({x}
		 tsub-fun (DsubF1y' x) (DsubD1x x)
		 (DeqD1x x : tp-eq (F1y x) (D1x x)))
	   <- ({x}
		 tsub-fun (DsubF2y' x) (DsubD2x x)
		 (DeqD2x x : tp-eq (F2y x) (D2x x)))
	   <- trans-principal Dtrans1
	      (principal/pi
		 (DprincipalD1 : {x} principal (D1 x)))
	   <- ({x}
		 principal-sub DprincipalD1 (DsubD1x x)
		 (DprincipalD1x x : principal (D1x x)))
	   <- ({x}
		 subtype-resp (DeqD1x x) tp-eq/i ([_] term-eq/i) (DsubtypeF1yB x)
		 (DsubtypeD1xB x : subtype (D1x x) (B x) (O x)))
	   <- ({x}
		 subtype-resp (DeqD2x x) tp-eq/i ([_] term-eq/i) (DsubtypeF2yB x)
		 (DsubtypeD2xB x : subtype (D2x x) (B x) (O x)))
	   <- ({x}
		 principal-subtype (DprincipalD1x x) (DsubtypeD1xB x)
		 (DeqO x : {y} term-eq (O x y) (O' x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD1xB x)
		 (DsubtypeD1xB' x : subtype (D1x x) (B x) ([_] O' x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD2xB x)
		 (DsubtypeD2xB' x : subtype (D2x x) (B x) ([_] O' x))).

-extsig	: equiv-comp
	   (equiv/extsigma
	      (Dewf : {ex} evof ex EA -> ewf (EB ex))
	      (Dequiv2 : equiv (epi2 EM1) (epi2 EM2) (EB (epi1 EM1)))
	      (Dequiv1 : equiv (epi1 EM1) (epi1 EM2) EA))
	   (ttrans/sigma DtransB DtransA)
	   Dtrans1
	   Dtrans2
	   (subtype/sigma ([_] DsubtypeD1Bx') ([_] DsubBx) DsubtypeC1A')
	   (subtype/sigma ([_] DsubtypeD2Bx') ([_] DsubBx) DsubtypeC2A')
	   <- equiv-comp Dequiv1
	      (DtransA : ttrans EA A)
	      (trans/pi1
		 (Dtrans1 : trans EM1 (sigma C1 ([_] D1))))
	      (trans/pi1
		 (Dtrans2 : trans EM2 (sigma C2 ([_] D2))))
	      (DsubtypeC1A : subtype C1 A Lca)
	      (DsubtypeC2A : subtype C2 A Lca)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> wf-comp (Dewf ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x)))
	   <- trans-principal Dtrans1
	      (principal/sigma
		 (DprincipalD1 : principal D1)
		 (DprincipalC1 : principal C1))
	   <- principal-subtype DprincipalC1 DsubtypeC1A
	      (DeqLca : {x} term-eq (Lca x) M)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeC1A
	      (DsubtypeC1A' : subtype C1 A ([_] M))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeC2A
	      (DsubtypeC2A' : subtype C2 A ([_] M))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- trans-reg Dtrans1
	      (wf/sigma
		 _
		 (DwfC1 : wf C1))
	   <- subtype-reg' DsubtypeC1A' DwfC1 DwfA
	      (DofM : of M A)
	   <- can-tsub DwfB DofM
	      (DsubBx : tsub B M Bx)
	   <- ttrans-sub DtransB (trans/pi1 Dtrans1) DwfA DsubtypeC1A' DsubBx
	      (DtransBx : ttrans (EB (epi1 EM1)) Bx)
	   <- equiv-comp Dequiv2
	      (DtransBx' : ttrans (EB (epi1 EM1)) Bx')
	      (trans/pi2
		 (Dtrans1' : trans EM1 (sigma C1' ([_] D1'))))
	      (trans/pi2
		 (Dtrans2' : trans EM2 (sigma C2' ([_] D2'))))
	      (DsubtypeD1'Bx' : subtype D1' Bx' Ldb)
	      (DsubtypeD2'Bx' : subtype D2' Bx' Ldb)
	   <- ttrans-fun DtransBx' DtransBx
	      (DeqBx : tp-eq Bx' Bx)
	   <- trans-fun Dtrans1' Dtrans1
	      (DeqCD1 : tp-eq (sigma C1' ([_] D1')) (sigma C1 ([_] D1)))
	   <- trans-fun Dtrans2' Dtrans2
	      (DeqCD2 : tp-eq (sigma C2' ([_] D2')) (sigma C2 ([_] D2)))
	   <- tp-eq-cdr-sigma DeqCD1 _
	      (DeqD1 : {x} tp-eq D1' D1)
	   <- tp-eq-cdr-sigma DeqCD2 _
	      (DeqD2 : {x} tp-eq D2' D2)
	   <- subtype-resp (DeqD1 aca) DeqBx ([_] term-eq/i) DsubtypeD1'Bx'
	      (DsubtypeD1Bx : subtype D1 Bx Ldb)
	   <- subtype-resp (DeqD2 aca) DeqBx ([_] term-eq/i) DsubtypeD2'Bx'
	      (DsubtypeD2Bx : subtype D2 Bx Ldb)
	   <- principal-subtype DprincipalD1 DsubtypeD1Bx
	      (DeqLdb : {x} term-eq (Ldb x) N)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLdb DsubtypeD1Bx
	      (DsubtypeD1Bx' : subtype D1 Bx ([_] N))
	   <- subtype-resp tp-eq/i tp-eq/i DeqLdb DsubtypeD2Bx
	      (DsubtypeD2Bx' : subtype D2 Bx ([_] N)).

-subsum	: equiv-comp
	   (equiv/subsume
	      (Dsubtp : subtp EA EB)
	      (Dequiv : equiv EM EN EA))
	   DtransB
	   DtransM
	   DtransN
	   DsubtypeCB
	   DsubtypeDB
	   <- equiv-comp Dequiv
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM C)
	      (DtransN : trans EN D)
	      (DsubtypeCA : subtype C A O)
	      (DsubtypeDA : subtype D A O)
	   <- subtp-comp Dsubtp
	      (DtransA' : ttrans EA A')
	      (DtransB : ttrans EB B)
	      (DsubtypeA'B : subtype A' B P)
	   <- ttrans-fun DtransA' DtransA
	      (DeqA : tp-eq A' A)
	   <- subtype-resp DeqA tp-eq/i ([_] term-eq/i) DsubtypeA'B
	      (DsubtypeAB : subtype A B P)
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ttrans-reg DtransB
	      (DwfB : wf B)
	   <- trans-reg DtransM
	      (DwfC : wf C)
	   <- trans-reg DtransN
	      (DwfD : wf D)
	   <- subtype-reg DsubtypeCA DwfC DwfA
	      (DofO : {x} vof x C -> of (O x) A)
	   <- subtype-reg DsubtypeAB DwfA DwfB
	      (DofP : {x} vof x A -> of (P x) B)
	   <- ({x} {d:vof x C}
		 can-sub DofP (DofO x d)
		 (DsubPx x : sub P (O x) (Px x)))
	   <- subtype-trans DwfC DwfA DwfB DsubtypeCA DsubtypeAB DsubPx
	      (DsubtypeCB : subtype C B Px)
	   <- subtype-trans DwfD DwfA DwfB DsubtypeDA DsubtypeAB DsubPx
	      (DsubtypeDB : subtype D B Px).

-beta	: equiv-comp
	   (equiv/beta
	      (DeofN : eof EN EA)
	      (DeofM : {ex} evof ex EA -> eof (EM ex) (EB ex)))
	   DtransBx
	   (trans/app DsubDx DsubtypeCA' DtransN (trans/lam DtransM DtransA))
	   DtransMN
	   DsubtypeDxBx
	   DsubtypeEBx
	   <- of-comp DeofN
	      (DtransA : ttrans EA A)
	      (DtransN : trans EN C)
	      (DsubtypeCA : subtype C A Lca)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x} {ed:evof ex EA}
		 vof-comp ed DtransA xt d
		 -> of-comp (DeofM ex ed)
		    (DtransB x d ex xt : ttrans (EB ex) (B x))
		    (DtransM x d ex xt : trans (EM ex) (D x))
		    (DsubtypeDB x : subtype (D x) (B x) (Ldb x)))
	   <- trans-principal DtransN
	      (DprincipalC : principal C)
	   <- principal-subtype DprincipalC DsubtypeCA
	      (DeqLca : {x} term-eq (Lca x) N)
	   <- subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA
	      (DsubtypeCA' : subtype C A ([_] N))
	   <- ttrans-reg DtransA
	      (DwfA : wf A)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 ttrans-reg (DtransB x d ex xt)
		 (DwfB x d : wf (B x)))
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-reg (DtransM x d ex xt)
		 (DwfD x d : wf (D x)))
	   <- trans-reg DtransN
	      (DwfC : wf C)
	   <- subtype-reg' DsubtypeCA' DwfC DwfA
	      (DofN : of N A)
	   <- can-tsub DwfB DofN
	      (DsubBx : tsub B N Bx)
	   <- can-tsub DwfD DofN
	      (DsubDx : tsub D N Dx)
	   <- ttrans-sub DtransB DtransN DwfA DsubtypeCA' DsubBx
	      (DtransBx : ttrans (EB EN) Bx)
	   <- trans-sub DtransM DtransN DwfA DsubtypeCA' DsubDx
	      (DtransMN : trans (EM EN) E)
	      (DsubtypeEDx : subtype E Dx Led)
	   <- ({x} {d:vof x A} {ex} {xt:vtrans ex x}
		 trans-principal (DtransM x d ex xt)
		 (DprincipalD x : principal (D x)))
	   <- ({x}
		 principal-subtype (DprincipalD x) (DsubtypeDB x)
		 (DeqLdb x : {y} term-eq (Ldb x y) (M x)))
	   <- ({x}
		 subtype-resp tp-eq/i tp-eq/i (DeqLdb x) (DsubtypeDB x)
		 (DsubtypeDB' x : subtype (D x) (B x) ([_] M x)))
	   <- ({x} {d:vof x A}
		 subtype-reg' (DsubtypeDB' x) (DwfD x d) (DwfB x d)
		 (DofM x d : of (M x) (B x)))
	   <- can-sub DofM DofN
	      (DsubMx : sub M N Mx)
	   <- subtype-sub' DsubtypeDB' DwfD DwfB DofN DsubDx DsubBx ([_] DsubMx)
	      (DsubtypeDxBx : subtype Dx Bx ([_] Mx))
	   <- ({x}
		 sub-absent Mx (Led x) 
		 (DsubMxabs x : sub ([_] Mx) (Led x) Mx))
	   <- trans-reg DtransMN
	      (DwfE : wf E)
	   <- tsubst DsubDx DwfD DofN
	      (DwfDx : wf Dx)
	   <- tsubst DsubBx DwfB DofN
	      (DwfBx : wf Bx)
	   <- subtype-trans DwfE DwfDx DwfBx DsubtypeEDx DsubtypeDxBx DsubMxabs
	      (DsubtypeEBx : subtype E Bx ([_] Mx)).

-beta1	: equiv-comp
	   (equiv/beta1
	      (DeofN : eof EN EB)
	      (DeofM : eof EM EA))
	   DtransA
	   (trans/pi1 (trans/pair DtransN DtransM))
	   DtransM
	   DsubtypeCA
	   DsubtypeCA
	   <- of-comp DeofM
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM C)
	      (DsubtypeCA : subtype C A _)
	   <- of-comp DeofN
	      (DtransB : ttrans EB B)
	      (DtransN : trans EN D)
	      (DsubtypeDB : subtype D B _).

-beta2	: equiv-comp
	   (equiv/beta2
	      (DeofN : eof EN EB)
	      (DeofM : eof EM EA))
	   DtransB
	   (trans/pi2 (trans/pair DtransN DtransM))
	   DtransN
	   DsubtypeDB
	   DsubtypeDB
	   <- of-comp DeofM
	      (DtransA : ttrans EA A)
	      (DtransM : trans EM C)
	      (DsubtypeCA : subtype C A _)
	   <- of-comp DeofN
	      (DtransB : ttrans EB B)
	      (DtransN : trans EN D)
	      (DsubtypeDB : subtype D B _).

%worlds (bind | cbind | scbind) (vof-comp _ _ _ _) (of-comp _ _ _ _) (wf-comp _ _) (subtp-comp _ _ _ _) (tequiv-comp _ _ _) (equiv-comp _ _ _ _ _ _).
%total (D0 D1 D2 D3 D4 D5) (vof-comp D0 _ _ _) (of-comp D1 _ _ _) (wf-comp D2 _) (subtp-comp D3 _ _ _) (tequiv-comp D4 _ _) (equiv-comp D5 _ _ _ _ _).



of-comp'	: eof EM EA
		   -> ttrans EA A
		   -> trans EM B
		   -> subtype B A ([_] M) -> type.
%mode of-comp' +X1 -X2 -X3 -X4.

-	: of-comp' Dof DtransA DtransM Dsubtype'
	   <- of-comp Dof DtransA DtransM Dsubtype
	   <- trans-subtype DtransM Dsubtype Dsubtype' _.

%worlds (bind | cbind | scbind) (of-comp' _ _ _ _).
%total {} (of-comp' _ _ _ _).



equiv-comp'	: equiv EM EN EA
		   -> ttrans EA A
		   -> trans EM B
		   -> trans EN C
		   -> subtype B A ([_] M)
		   -> subtype C A ([_] M) -> type.
%mode equiv-comp' +X1 -X2 -X3 -X4 -X5 -X6.

-	: equiv-comp' Dequiv DtransA DtransM DtransN DsubtypeBA' DsubtypeCA'
	   <- equiv-comp Dequiv DtransA DtransM DtransN DsubtypeBA DsubtypeCA
	   <- trans-subtype DtransM DsubtypeBA DsubtypeBA' Deq
	   <- subtype-resp tp-eq/i tp-eq/i Deq DsubtypeCA DsubtypeCA'.

%worlds (bind | cbind | scbind) (equiv-comp' _ _ _ _ _ _).
%total {} (equiv-comp' _ _ _ _ _ _).