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laoziSolution2.v
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laoziSolution2.v
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Require Import borceuxSolution_half_old.
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp Require Import ssrbool ssrfun eqtype ssrnat seq.
Require Import Setoid.
Require Omega.
Set Implicit Arguments.
Unset Strict Implicits.
Unset Printing Implicit Defensive.
Module COPARAM.
Import LOGIC.
Import LOGIC.Ex_Notations3.
Parameter obMod_gen : Set.
Parameter Mod_gen : forall {log : logic}, obV log -> obMod_gen -> obMod_gen -> Set.
Inductive obMod : Set :=
| GenObMod : forall A : obMod_gen, obMod
| DeClass0 : forall A : obMod, obMod.
Notation "#0| A" := (GenObMod A) (at level 4, right associativity).
Notation "'D0| A" := (DeClass0 A) (at level 4, right associativity).
Reserved Notation "''Mod' (0 V |- [0 A1 ~> A2 ]0 )0"
(at level 25, format "''Mod' (0 V |- [0 A1 ~> A2 ]0 )0").
Reserved Notation "''D' (0 V |- [0 A1 ~> A2 ]0 )0"
(at level 25, format "''D' (0 V |- [0 A1 ~> A2 ]0 )0").
Inductive Mod00 {log : logic} : obV log -> obMod -> obMod -> Type :=
| PolyV_Mod : forall (V V' : obV log),
V(0 V' |- V )0 -> forall (A1 A2 : obMod), 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 ->
'Mod(0 V' |- [0 A1 ~> A2 ]0 )0
| GenArrowsMod : forall (V V' : obV log), forall A1 A2 : obMod_gen
, Mod_gen V A1 A2 -> V(0 V' |- V )0 ->
'Mod(0 V' |- [0 (#0| A1) ~> (#0| A2) ]0 )0
| UnitMod : forall (V : obV log), forall {A : obMod}
, V(0 V |- log.-I )0 -> 'Mod(0 V |- [0 A ~> A ]0 )0
| PolyMod : forall (V : obV log) (A2 : obMod) (A1 : obMod)
, 'Mod(0 V |- [0 A2 ~> A1 ]0 )0 -> forall A1' : obMod, forall (W WV : obV log),
V(0 WV |- (0 W & V )0 )0 ->
'Mod(0 W |- [0 A1 ~> A1' ]0 )0 -> 'Mod(0 WV |- [0 A2 ~> A1' ]0 )0
| UnitDeClass : forall (A : obMod) (A' : obMod) (W W' : obV log)
, V(0 W' |- (0 W & log.-I)0 )0 ->
'Mod(0 W |- [0 A ~> A' ]0 )0 -> 'D(0 W' |- [0 'D0| A ~> A' ]0 )0
| PolyDeClass : forall (V : obV log) (B : obMod) (A : obMod),
'D(0 V |- [0 B ~> A ]0 )0 -> forall A' : obMod, forall (W WV : obV log),
V(0 WV |- (0 W & V )0 )0 ->
'Mod(0 W |- [0 A ~> A' ]0 )0 -> 'D(0 WV |- [0 B ~> A' ]0 )0
(* common CoUnit, errata: Unit *)
| Classifying : forall (V V' : obV log), forall (A1 A2 : obMod),
V(0 V' |- V )0 ->
'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> 'Mod(0 V' |- [0 ('D0| A1) ~> A2 ]0 )0
| DeClassifying : forall (V V' : obV log), forall (A1 A2 : obMod),
V(0 V' |- V )0 ->
'D(0 V |- [0 A1 ~> A2 ]0 )0 -> 'D(0 V' |- [0 A1 ~> ('D0| A2) ]0 )0
where
"''Mod' (0 V |- [0 A1 ~> A2 ]0 )0"
:= (@Mod00 _ V A1 A2) and "''D' (0 V |- [0 A1 ~> A2 ]0 )0"
:= (@Mod00 _ V A1 ('D0| A2)).
Parameter Mod'00 : forall {log : logic}, obMod -> obMod -> obV log.
Parameter decode : forall {log : logic}, forall {A1 A2 : obMod},
forall {V : obV log}, 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> V(0 V |- Mod'00 A1 A2 )0.
Parameter encode : forall {log : logic}, forall {A1 A2 : obMod},
forall {V : obV log}, V(0 V |- Mod'00 A1 A2 )0 -> 'Mod(0 V |- [0 A1 ~> A2 ]0 )0.
Axiom decodeK : forall {log : logic}, forall (A1 A2 : obMod) (V : obV log),
cancel (@decode _ A1 A2 V) (@encode _ A1 A2 V).
Axiom encodeK : forall {log : logic}, forall (A1 A2 : obMod) (V : obV log),
cancel (@encode _ A1 A2 V) (@decode _ A1 A2 V).
Axiom decode_metaPoly : forall {log : logic}, forall (A1 A2 : obMod),
forall (V V' : obV log) (v : V(0 V' |- V )0) (f : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( v o> (decode f) )
~~ ( decode (PolyV_Mod v f)
: log.-V(0 V' |- Mod'00 A1 A2 )0 ) .
Definition PolyV_Mod_rewrite log V V' A1 A2 v a :=
(@PolyV_Mod log V V' v A1 A2 a).
Notation "v o>' a" := (@PolyV_Mod_rewrite _ _ _ _ _ v a)
(at level 25, right associativity, format "v o>' a").
Notation "v o>| #1| a" :=
(@GenArrowsMod _ _ _ _ _ a v) (at level 25, right associativity).
Notation "v o>| 'uMod'" := (@UnitMod _ _ _ v)(at level 25).
Notation "v o>| @ 'uMod' A" :=
(@UnitMod _ _ A v) (at level 25, only parsing).
Definition PolyMod_rewrite log V A2 A1 A1' W WV wv a_ a' :=
(@PolyMod log V A2 A1 a_ A1' W WV wv a').
Notation "v o>| a_ o>Mod a'" :=
(@PolyMod_rewrite _ _ _ _ _ _ _ v a_ a')
(at level 25, right associativity, a_ at next level,
format "v o>| a_ o>Mod a'").
Notation "v o>| ''D1|' a" := (@UnitDeClass _ _ _ _ _ v a)
(at level 25, right associativity).
Definition PolyDeClass_rewrite log V B A A' W WV wv b a :=
(@PolyDeClass log V B A b A' W WV wv a).
Notation "v o>| b o>D a" :=
(@PolyDeClass_rewrite _ _ _ _ _ _ _ v b a)
(at level 25, right associativity, b at next level, format "v o>| b o>D a").
Notation "v o>| 'clfy o>Mod a'" :=
(@Classifying _ _ _ _ _ v a') (at level 25, right associativity).
Notation "v o>| a_ o>Mod 'declfy" :=
(@DeClassifying _ _ _ _ _ v a_) (at level 25, a_ at next level,
right associativity).
Parameter PolyV_unitPre :
forall {log : logic} {V V' : obV log} (v : log.-V(0 V |- V')0), log.-1 o> v ~~ v.
Parameter PolyV_unitPost :
forall {log : logic} {V V' : obV log} (v : log.-V(0 V |- V')0), v o> log.-1 ~~ v.
Definition desIdenObLK :
forall {log : logic} {V : obV log}, log.-V(0 log.-(0 log.-I & V )0 |- V )0
:= fun log V => Des (log.-uV) .
Parameter desIdenObLKV :
forall {log : logic} {V : obV log}, log.-V(0 V |- log.-(0 log.-I & V )0 )0 .
Axiom desIdenObLK_K : forall {log : logic} {V : obV log},
log.-1 ~~ (@desIdenObLK log V) o> (@desIdenObLKV log V).
Axiom desIdenObLKV_K : forall {log : logic} {V : obV log},
log.-1 ~~ (@desIdenObLKV log V) o> (@desIdenObLK log V).
Axiom desIdenObLKV_Assoc_Rev_desIdenObLK : forall {log : logic} (V W : obV log),
log.-1 ~~ ( ( log.-(1 desIdenObLKV & V )0 o> Assoc_Rev ) o> desIdenObLK
: log.-V(0 log.-(0 W & V )0 |- log.-(0 W & V )0 )0 ).
Parameter desIdenObRK :
forall {log : logic} {V : obV log}, log.-V(0 log.-(0 V & log.-I )0 |- V )0.
Parameter desIdenObRKV :
forall {log : logic} {V : obV log}, log.-V(0 V |- log.-(0 V & log.-I )0 )0.
Parameter desV01 : forall {log : logic} {V2 V2' V1 : obV log},
log.-V(0 V2 |- V2' )0 -> log.-V(0 log.-(0 V1 & V2 )0 |- log.-(0 V1 & V2' )0 )0.
Notation "dat .-(0 V1 & v )1" := (@desV01 dat _ _ V1 v)
(at level 30, format "dat .-(0 V1 & v )1").
Notation "(0 V1 & v )1" := (_ .-(0 V1 & v )1)
(at level 30, format "(0 V1 & v )1").
Axiom desV01_consV10 :
forall {log : logic} (V2 V2' V1 : obV log) (v : log.-V(0 V2' |- V2 )0) (W : obV log)
(w : log.-V(0 log.-(0 V1 & V2 )0 |- W )0),
Des( [1 v ~> W ]0 <o (Cons w) ) ~~ w <o log.-(0 V1 & v )1 .
Axiom desIdenObLKV_IdenOb_Assoc_Rev_desIdenObLK : forall {log : logic} (V : obV log),
log.-1 ~~ ( ( (log.-(1 desIdenObLKV & V )0)
o> Assoc_Rev ) o> (log.-(0 log.-I & desIdenObLK )1)
: log.-V(0 log.-(0 log.-I & V )0 |- log.-(0 log.-I & V )0 )0 ).
Reserved Notation "f2 ~~~ f1" (at level 70).
Inductive convMod {log : logic} : forall (V : obV log) (A1 A2 : obMod),
'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> 'Mod(0 V |- [0 A1 ~> A2 ]0 )0 -> Prop :=
| Mod_ReflV : forall (V : obV log) (A1 A2 : obMod) (a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
a ~~~ a
| Mod_TransV : forall (V : obV log) (A1 A2 : obMod)
(uTrans a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
uTrans ~~~ a -> forall (a0 : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
a0 ~~~ uTrans -> a0 ~~~ a
| Mod_SymV : forall (V : obV log) (A1 A2 : obMod)
(a a0 : 'Mod(0 V |- [0 A1 ~> A2]0 )0),
a ~~~ a0 -> a0 ~~~ a
| PolyV_Mod_cong : forall (A1 A2 : obMod) (V V' : obV log) (v v0 : V(0 V' |- V )0)
(a a0 : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> ( v0 o>' a0 ) ~~~ ( v o>' a )
| GenArrowsMod_cong : forall (V V' : obV log), forall (A1 A2 : obMod_gen)
(aGen : Mod_gen V A1 A2)
, forall (v v0 : V(0 V' |- V )0), v0 ~~ v -> v0 o>| #1| aGen ~~~ v o>| #1| aGen
| UnitMod_cong : forall (V : obV log), forall {A : obMod} (v v0 : V(0 V |- log.-I )0),
v0 ~~ v -> v0 o>| @uMod A ~~~ v o>| @uMod A
| Mod_cong :
forall (V : obV log) (A A' : obMod) (a_ a_0 : 'Mod(0 V |- [0 A ~> A' ]0 )0),
forall (W : obV log) (A'' : obMod) (a' a'0 : 'Mod(0 W |- [0 A' ~> A'' ]0 )0),
forall (WV : obV log) (v v0 : V(0 WV |- (0 W & V )0 )0),
v0 ~~ v -> a_0 ~~~ a_ -> a'0 ~~~ a' ->
( v0 o>| a_0 o>Mod a'0 ) ~~~ ( v o>| a_ o>Mod a' )
| UnitDeClass_cong :
forall (A : obMod) (A' : obMod) (W W' : obV log)
(v v0 : V(0 W' |- (0 W & log.-I )0 )0) (a a0 : 'Mod(0 W |- [0 A ~> A' ]0 )0),
v0 ~~ v -> a0 ~~~ a -> ( v0 o>| 'D1| a0 ) ~~~ ( v o>| 'D1| a )
| PolyDeClass_cong :
forall (V : obV log) (B : obMod) (A : obMod) (b b0 : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a a0 : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV : obV log) (v v0 : V(0 WV |- (0 W & V )0 )0),
v0 ~~ v -> b0 ~~~ b -> a0 ~~~ a -> ( v0 o>| b0 o>D a0 ) ~~~ ( v o>| b o>D a )
| Classifying_cong :
forall (V V' : obV log) (v v0 : V(0 V' |- V )0) (A1 A2 : obMod)
(a a0 : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> (v0 o>| 'clfy o>Mod a0 ) ~~~ (v o>| 'clfy o>Mod a )
| DeClassifying_cong :
forall (V V' : obV log) (v v0 : V(0 V' |- V )0) (A1 A2 : obMod)
(a a0 : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
v0 ~~ v -> a0 ~~~ a -> (v0 o>| a0 o>Mod 'declfy ) ~~~ (v o>| a o>Mod 'declfy )
| GenArrowsMod_arrowLog : forall (V V' V'' : obV log) (A1 A2 : obMod_gen)
(aGen : Mod_gen V A1 A2)
(v : V(0 V' |- V )0) (v' : V(0 V'' |- V' )0),
( ( v' o> v) o>| #1| aGen )
~~~ (v' o>' (v o>| #1| aGen)
: 'Mod(0 V'' |- [0 #0| A1 ~> #0| A2 ]0)0 )
| UnitMod_arrowLog : forall (V V' : obV log) (A : obMod) (v : V(0 V |- log.-I )0)
(v' : V(0 V' |- V )0),
( ( v' o> v ) o>| @uMod A )
~~~ (v' o>' (v o>| @uMod A)
: 'Mod(0 V' |- [0 A ~> A ]0)0 )
| Mod_arrowLog :
forall (V : obV log) (A0 : obMod) (A : obMod)
(a_ : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a' : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV : obV log) (v : V(0 WV |- (0 W & V )0 )0),
forall (WV0 : obV log) (v0 : V(0 WV0 |- WV )0),
( ( v0 o> v ) o>| a_ o>Mod a' )
~~~ ( v0 o>' ( v o>| a_ o>Mod a' )
: 'Mod(0 WV0 |- [0 A0 ~> A' ]0)0 )
| Mod_arrowPre :
forall (V V' : obV log) (v : V(0 V' |- V )0) (A0 : obMod) (A : obMod)
(a_ : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a' : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV' : obV log) (v0 : V(0 WV' |- (0 W & V' )0 )0),
( ( v0 o> log.-(0 _ & v )1 ) o>| a_ o>Mod a' )
~~~ ( v0 o>| ( v o>' a_ ) o>Mod a'
: 'Mod(0 WV' |- [0 A0 ~> A' ]0)0 )
| Mod_arrowPost :
forall (V : obV log) (A0 : obMod) (A : obMod) (a_ : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
forall (W W' : obV log) (w : V(0 W' |- W )0) (A' : obMod)
(a' : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (W'V : obV log) (w0 : V(0 W'V |- (0 W' & V )0 )0),
( ( w0 o> log.-(1 w & _ )0 ) o>| a_ o>Mod a' )
~~~ ( w0 o>| a_ o>Mod ( w o>' a' )
: 'Mod(0 W'V |- [0 A0 ~> A' ]0)0 )
| UnitDeClass_arrowLog :
forall (W W' W'' : obV log) (w : V(0 W' |- (0 W & log.-I )0 )0)
(w' : V(0 W'' |- W' )0)
(A A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
( ( w' o> w ) o>| 'D1| a )
~~~ ( w' o>' ( w o>| 'D1| a )
: 'D(0 W'' |- [0 'D0| A ~> A' ]0)0 )
| UnitDeClass_arrow :
forall (W W' W'' : obV log) (w : V(0 W' |- W )0)
(w' : V(0 W'' |- (0 W' & log.-I )0 )0)
(A A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
( ( w' o> log.-(1 w & log.-I )0 ) o>| 'D1| a )
~~~ ( w' o>| 'D1| ( w o>' a)
: 'D(0 W'' |- [0 'D0| A ~> A' ]0)0 )
| DeClass_arrowLog :
forall (V : obV log) (B : obMod) (A : obMod)
(b : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV : obV log) (v : V(0 WV |- (0 W & V )0 )0),
forall (WV0 : obV log) (v0 : V(0 WV0 |- WV )0),
( ( v0 o> v ) o>| b o>D a )
~~~ ( v0 o>' ( v o>| b o>D a )
: 'D(0 WV0 |- [0 B ~> A' ]0)0 )
| DeClass_arrowPre :
forall (V V' : obV log) (v : V(0 V' |- V )0) (B : obMod) (A : obMod)
(b : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV' : obV log) (v0 : V(0 WV' |- (0 W & V' )0 )0),
( ( v0 o> log.-(0 _ & v )1 ) o>| b o>D a )
~~~ ( v0 o>| ( v o>' b ) o>D a
: 'D(0 WV' |- [0 B ~> A' ]0)0 )
| DeClass_arrowPost :
forall (V : obV log) (B : obMod) (A : obMod) (b : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W W' : obV log) (w : V(0 W' |- W )0) (A' : obMod)
(a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (W'V : obV log) (w0 : V(0 W'V |- (0 W' & V )0 )0),
( ( w0 o> log.-(1 w & _ )0 ) o>| b o>D a )
~~~ ( w0 o>| b o>D ( w o>' a )
: 'D(0 W'V |- [0 B ~> A' ]0)0 )
| Classifying_arrowLog : forall (V V' V'' : obV log) (v : V(0 V' |- V )0)
(v0 : V(0 V'' |- V' )0) (A1 A2 : obMod)
(a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| 'clfy o>Mod a )
~~~ ( v0 o>' ( v o>| 'clfy o>Mod a )
: 'Mod(0 V'' |- [0 'D0| A1 ~> A2 ]0)0 )
| Classifying_arrow : forall (V V' V'' : obV log) (v : V(0 V' |- V )0)
(v0 : V(0 V'' |- V' )0) (A1 A2 : obMod)
(a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| 'clfy o>Mod a )
~~~ ( v0 o>| 'clfy o>Mod ( v o>' a )
: 'Mod(0 V'' |- [0 'D0| A1 ~> A2 ]0)0 )
| DeClassifying_arrowLog :
forall (V V' V'' : obV log) (v : V(0 V' |- V )0) (v0 : V(0 V'' |- V' )0)
(A1 A2 : obMod) (a : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| a o>Mod 'declfy )
~~~ ( v0 o>' ( v o>| a o>Mod 'declfy )
: 'D(0 V'' |- [0 A1 ~> 'D0| A2 ]0)0 )
| DeClassifying_arrow :
forall (V V' V'' : obV log) (v : V(0 V' |- V )0) (v0 : V(0 V'' |- V' )0)
(A1 A2 : obMod) (a : 'D(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v0 o> v ) o>| a o>Mod 'declfy )
~~~ ( v0 o>| ( v o>' a ) o>Mod 'declfy
: 'D(0 V'' |- [0 A1 ~> 'D0| A2 ]0)0 )
| PolyV_Mod_arrowLog :
forall (V'' V' : obV log) (v' : V(0 V'' |- V' )0) (V : obV log)
(v : V(0 V' |- V )0) (A1 A2 : obMod) (a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( ( v' o> v ) o>' a )
~~~ ( v' o>' ( v o>' a )
: 'Mod(0 V'' |- [0 A1 ~> A2 ]0)0 )
(* non for reduction *)
| Mod_morphism :
forall (V : obV log) (B : obMod) (A : obMod) (b : 'Mod(0 V |- [0 B ~> A ]0 )0)
(W_ : obV log) (A' : obMod) (a_ : 'Mod(0 W_ |- [0 A ~> A' ]0 )0)
(W' : obV log) (A'' : obMod) (a' : 'Mod(0 W' |- [0 A' ~> A'' ]0 )0),
forall (W_V : obV log) (v : V(0 W_V |- (0 W_ & V )0 )0),
forall (W'W_V : obV log) (v0 : V(0 W'W_V |- (0 W' & W_V )0 )0),
( ( v0 o> (0 W' & v )1 o> Assoc ) o>| b o>Mod ( log.-1 o>| a_ o>Mod a' ) )
~~~ ( v0 o>| ( v o>| b o>Mod a_ ) o>Mod a'
: 'Mod(0 W'W_V |- [0 B ~> A'' ]0)0 )
| DeClass_morphismPost :
forall (A : obMod)
(W_ W_' : obV log) (v : V(0 W_' |- (0 W_ & log.-I )0 )0) (A' : obMod)
(a_ : 'Mod(0 W_ |- [0 A ~> A' ]0 )0)
(W' : obV log) (A'' : obMod) (a' : 'Mod(0 W' |- [0 A' ~> A'' ]0 )0),
forall (W'W_' : obV log) (v0 : V(0 W'W_' |- (0 W' & W_' )0 )0),
( ( v0 o> desIdenObRKV ) o>|
'D1| ( (log.-1) o>| ( ( v o> desIdenObRK ) o>' a_ ) o>Mod a' ) )
~~~ ( v0 o>| ( v o>| 'D1| a_ ) o>D a'
: 'D(0 W'W_' |- [0 'D0| A ~> A'' ]0)0 )
| DeClass_morphismPre :
forall (A : obMod) (V' : obV log) (B' : obMod) (b' : 'D(0 V' |- [0 B' ~> A ]0 )0),
forall (W W' : obV log) (v : V(0 W' |- (0 W & log.-I )0 )0)
(A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (W'V' : obV log) (v0 : V(0 W'V' |- (0 W' & V' )0 )0),
( v0 o>| b' o>D ( ( v o> desIdenObRK ) o>' a ) )
~~~ ( v0 o>| b' o>Mod ( v o>| 'D1| a )
: 'D(0 W'V' |- [0 B' ~> A' ]0)0 )
| PolyV_Mod_unit :
forall (V : obV log) (A1 A2 : obMod) (a : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0),
( a ) ~~~ ( log.-1 o>' a
: 'Mod(0 V |- [0 A1 ~> A2 ]0)0 )
| Mod_unit :
forall (A : obMod) (V : obV log) (v : V(0 V |- log.-I )0)
(W : obV log) (A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV : obV log) (v0 : V(0 WV |- (0 W & V )0 )0),
( ( v0 o> log.-(0 W & v )1 o> desIdenObRK ) o>' a )
~~~ ( v0 o>| ( v o>| uMod ) o>Mod a
: 'Mod(0 WV |- [0 A ~> A' ]0)0 )
| Mod_inputUnitMod :
forall (V : obV log) (B : obMod) (A : obMod) (b : 'Mod(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (w : V(0 W |- log.-I )0),
forall (WV : obV log) (w0 : V(0 WV |- (0 W & V )0 )0),
( ( w0 o> log.-(1 w & V )0 o> desIdenObLK ) o>' b )
~~~ ( w0 o>| b o>Mod ( w o>| uMod )
: 'Mod(0 WV |- [0 B ~> A ]0)0 )
| DeClass_unit :
forall (V : obV log) (v : V(0 V |- log.-I )0) (A : obMod) (A' : obMod)
(W : obV log) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV : obV log) (v0 : V(0 WV |- (0 W & V )0 )0),
( ( v0 o> log.-(0 W & v )1 ) o>| 'D1| a )
~~~ ( v0 o>| ( v o>| uMod ) o>D a
: 'D(0 WV |- [0 'D0| A ~> A' ]0 )0 )
| DeClass_inputUnitMod :
forall (V : obV log) (B : obMod) (A : obMod) (b : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (w : V(0 W |- log.-I )0),
forall (WV : obV log) (w0 : V(0 WV |- (0 W & V )0 )0),
( ( w0 o> ( log.-(1 w & _ )0 ) o> desIdenObLK ) o>' b )
~~~ ( w0 o>| b o>D ( w o>| uMod )
: 'D(0 WV |- [0 B ~> A ]0)0 )
| Classifying_morphismPre :
forall (V V' : obV log) (v : V(0 V' |- V )0 ) (A1 A2 : obMod)
(a_ : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0)
(W : obV log) (A3 : obMod) (a' : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
forall (WV' : obV log) (v0 : V(0 WV' |- (0 W & V' )0 )0),
( ( log.-1 ) o>| 'clfy o>Mod ( ( v0 o> (0 _ & v )1 ) o>| a_ o>Mod a' ) )
~~~ ( v0 o>| (v o>| 'clfy o>Mod a_ ) o>Mod a'
: 'Mod(0 WV' |- [0 'D0| A1 ~> A3 ]0)0 )
(* non-necessary, deductible *)
| Classifying_morphismPre_DeClass :
forall (V V' : obV log) (v : V(0 V' |- V )0 ) (A1 A2 : obMod)
(b : 'D(0 V |- [0 A1 ~> A2 ]0 )0)
(W : obV log) (A3 : obMod) (a' : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
forall (WV' : obV log) (v0 : V(0 WV' |- (0 W & V' )0 )0),
( ( log.-1 ) o>| 'clfy o>Mod ( ( v0 o> (0 _ & v )1 ) o>| b o>D a' ) )
~~~ ( v0 o>| (v o>| 'clfy o>Mod b ) o>D a'
: 'D(0 WV' |- [0 'D0| A1 ~> A3 ]0)0 )
| Classifying_morphismPost :
forall (V V' : obV log) (v : V(0 V' |- (0 V & log.-I )0 )0) (A1 A2 : obMod)
(a_ : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0) (W W' : obV log)
(w : V(0 W' |- W )0) (A3 : obMod) (a' : 'Mod(0 W |- [0 A2 ~> A3 ]0 )0),
forall (W'V' : obV log) (v0 : V(0 W'V' |- (0 W' & V' )0 )0),
( ( log.-1 )
o>| 'clfy o>Mod ( v0 o>| ( ( v o> desIdenObRK ) o>' a_ ) o>Mod (w o>' a') ) )
~~~ ( v0 o>| ( v o>| 'D1| a_ ) o>Mod ( w o>| 'clfy o>Mod a' )
: 'Mod(0 W'V' |- [0 'D0| A1 ~> A3 ]0)0 )
| DeClassifying_morphismPost :
forall (V : obV log) (A1 A2 : obMod) (b_ : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0)
(W W' : obV log) (w : V(0 W' |- W )0) (A3 : obMod)
(b' : 'D(0 W |- [0 A2 ~> A3 ]0 )0),
forall (W'V : obV log) (w0 : V(0 W'V |- (0 W' & V )0 )0),
( log.-1 o>| ( ( w0 o> (1 w & _ )0 ) o>| b_ o>Mod b' ) o>Mod 'declfy )
~~~ ( w0 o>| b_ o>Mod ( w o>| b' o>Mod 'declfy )
: 'D(0 W'V |- [0 A1 ~> 'D0| A3 ]0)0 )
| DeClassifying_morphismPre :
forall (V V' : obV log) (v : V(0 V' |- V )0) (A1 A2 : obMod)
(b_ : 'D(0 V |- [0 A1 ~> A2 ]0 )0) (W W' : obV log)
(w : V(0 W' |- (0 W & log.-I )0 )0) (A4 : obMod)
(b' : 'Mod(0 W |- [0 A2 ~> A4 ]0 )0),
forall (W'V' : obV log) (wv : V(0 W'V' |- (0 W' & V' )0 )0),
( log.-1 o>|
( wv o>| ( v o>' b_ ) o>D ( ( w o> desIdenObRK ) o>' b') ) o>Mod 'declfy )
~~~ ( wv o>| ( v o>| b_ o>Mod 'declfy ) o>D ( w o>| 'D1| b' )
: 'D(0 W'V' |- [0 A1 ~> 'D0| A4 ]0)0 )
| CancelOuter : forall (V V' : obV log) (v : V(0 V' |- V )0) (B : obMod) (A : obMod)
(b : 'D(0 V |- [0 B ~> A ]0 )0) (A' : obMod)
(W W' : obV log) (w : V(0 W' |- W )0) (a : 'Mod(0 W |- [0 'D0| A ~> A' ]0 )0),
forall (W'V' : obV log) (wv : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| ( v o>' b ) o>Mod ( w o>' a ) )
~~~ ( wv o>| ( v o>| b o>Mod 'declfy ) o>Mod ( w o>| 'clfy o>Mod a )
: 'Mod(0 W'V' |- [0 B ~> A' ]0)0 )
| CancelInner : forall (V V' : obV log) (v : V(0 V' |- V )0) (B : obMod) (A : obMod)
(b : 'D(0 V |- [0 B ~> A ]0 )0) (A' : obMod)
(W W' : obV log) (w : V(0 W' |- W )0) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (W'V' : obV log) (wv : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| (v o>' b) o>D (w o>' a) )
~~~ ( wv o>| (v o>| b o>Mod 'declfy ) o>D (w o>| 'clfy o>Mod a )
: 'D(0 W'V' |- [0 B ~> A' ]0)0 )
| PermOuterInner : forall (V V' : obV log) (v : V(0 V' |- V )0) (B : obMod)
(A : obMod) (b : 'D(0 V |- [0 B ~> A ]0 )0) (W W' : obV log)
(w : V(0 W' |- W )0) (u : V(0 W |- log.-I )0),
forall (W'V' : obV log) (wv : V(0 W'V' |- (0 W' & V' )0 )0),
( wv o>| ( ( log.-(1 w & _ )0 o> log.-(1 u & _ )0 o> desIdenObLK ) o>|
( v o>' b ) o>Mod 'declfy ) o>Mod 'declfy )
~~~ ( wv o>| ( v o>| b o>Mod 'declfy ) o>D ( w o>| (u o>| uMod) o>Mod 'declfy )
: 'D(0 W'V' |- [0 B ~> 'D0| 'D0| A ]0)0 )
where "f2 ~~~ f1" := (@convMod _ _ _ _ f2 f1).
Hint Constructors convMod.
Hint Extern 4 (_ ~~?lo` _) => eapply (@ReflV lo _).
Ltac rewriterMod := repeat match goal with
| [ HH : @eq (Mod00 _ _ _) _ _ |- _ ] => try rewrite -> HH in *; clear HH end.
Add Parametric Relation {log : logic} (V : obV log) (A1 A2 : obMod) :
('Mod(0 V |- [0 A1 ~> A2 ]0 )0) (@convMod log V A1 A2)
reflexivity proved by (@Mod_ReflV log V A1 A2)
symmetry proved by (@Mod_SymV log V A1 A2)
transitivity proved by
(fun x y z r1 r2 => ((@Mod_TransV log V A1 A2) y z r2 x r1))
as convMod_rewrite.
Add Parametric Relation {log : logic} (V1 V2 : obV log) :
(log.-V(0 V1 |- V2 )0) (@convV log V1 V2)
reflexivity proved by (@ReflV log log V1 V2)
symmetry proved by (fun x => (@SymV log log V1 V2)^~ x)
transitivity proved by
(fun x y z r1 r2 => ((@TransV log log V1 V2) y z r2 x r1))
as convV_rewrite.
Add Parametric Morphism {log : logic} (V V' : obV log) (A1 A2 : obMod) :
(@PolyV_Mod_rewrite log V V' A1 A2) with
signature ((@convV log V' V)
==> (@convMod log V A1 A2)
==> (@convMod log V' A1 A2))
as PolyV_Mod_cong_rewrite.
by move => *; apply: PolyV_Mod_cong. Qed.
Add Parametric Morphism {log : logic} (V V' : obV log) (A1 A2 : obMod_gen)
(aGen : Mod_gen V A1 A2) :
(@GenArrowsMod log V V' A1 A2 aGen) with
signature ((@convV log V' V)
==> (@convMod log V' (#0| A1) (#0| A2)))
as GenArrowsMod_cong_rewrite.
by move => *; apply: GenArrowsMod_cong. Qed.
Add Parametric Morphism {log : logic} (V : obV log) (A : obMod)
: (@UnitMod log V A) with
signature ((@convV log V log.-I)
==> (@convMod log V A A))
as UnitMod_cong_rewrite.
by move => *; apply: UnitMod_cong. Qed.
Add Parametric Morphism {log : logic} (V : obV log) (A A' : obMod)
(W : obV log) (A'' : obMod) (WV : obV log) :
(@PolyMod_rewrite log V A A' A'' W WV ) with
signature ((@convV log WV ((0 W & V)0) )
==>(@convMod log V A A')
==> (@convMod log W A' A'')
==> (@convMod log WV A A''))
as Mod_cong_rewrite.
by move => *; apply: Mod_cong. Qed.
Add Parametric Morphism {log : logic} (A A' : obMod) (W W' : obV log) :
(@UnitDeClass log A A' W W') with
signature ( (@convV log W' ((0 W & log.-I )0) )
==> (@convMod log W A A')
==> (@convMod log W' ('D0| A) ('D0| A')))
as UnitDeClass_cong_rewrite.
by move => *; apply: UnitDeClass_cong. Qed.
Add Parametric Morphism {log : logic} (V : obV log) (B A A' : obMod)
(W WV : obV log) :
(@PolyDeClass_rewrite log V B A A' W WV) with
signature ((@convV log WV ((0 W & V )0) )
==> (@convMod log V B ('D0| A))
==> (@convMod log W A A')
==> (@convMod log WV B ('D0| A')))
as PolyDeClass_cong_rewrite.
by move => *; apply: PolyDeClass_cong. Qed.
Add Parametric Morphism {log : logic} (V V' : obV log) (A1 A2 : obMod) :
(@Classifying log V V' A1 A2) with
signature ((@convV log V' V)
==> (@convMod log V A1 A2)
==> (@convMod log V' ('D0| A1) A2))
as Classifying_cong_rewrite.
by move => *; apply: Classifying_cong. Qed.
Add Parametric Morphism {log : logic} (V V' : obV log) (A1 A2 : obMod) :
(@DeClassifying log V V' A1 A2) with
signature ((@convV log V' V)
==> (@convMod log V A1 ('D0| A2))
==> (@convMod log V' A1 ('D0| ('D0| A2))))
as DeClassifying_cong_rewrite.
by move => *; apply: DeClassifying_cong. Qed.
Module Import Destruct_hlist.
Inductive hlist (A : Type) (B : A -> Type)
: list A -> Type :=
| HNil : hlist B nil
| HCons : forall (x : A) (ls : list A), B x -> hlist B ls -> hlist B (x :: ls).
Implicit Arguments HNil [A B].
Implicit Arguments HCons [A B x ls].
Infix ":::" := HCons (right associativity, at level 60).
Section Section1.
Variable (A : Type) (B1 B2 : A -> Type).
Variable f : forall x, B1 x -> B2 x.
Fixpoint hmap (ls : list A) (hl : hlist B1 ls) : hlist B2 ls :=
match hl with
| HNil => HNil
| HCons _ _ x hl' => f x ::: hmap hl'
end.
End Section1.
Implicit Arguments hmap [A B1 B2 ls].
Section Section2.
Variables (A : Type) (B : A -> Type).
Inductive hlist_nil : hlist B ( [::]) -> Type :=
| Hlist_nil : hlist_nil (HNil : hlist B [::]).
Inductive hlist_cons : forall (a : A) (ls' : list A), hlist B (a :: ls') -> Type :=
| Hlist_cons : forall a ls' (ba : B a) (hls' : hlist B ls'),
@hlist_cons a ls' (ba ::: hls' : hlist B ([:: a & ls'])).
Definition hlist_destructP_type : forall (ls : list A) (hls : hlist B ls), Type.
Proof.
move => ls. case : ls => [ | a ls'] hls.
- refine (@hlist_nil hls).
- refine (@hlist_cons a ls' hls).
Defined.
Definition hlist_destructP : forall (ls : list A) (hls : hlist B ls),
@hlist_destructP_type ls hls.
Proof.
move => ls hls. case: ls / hls.
rewrite /hlist_destructP_type /=.
- constructor.
- constructor.
Defined.
(* type indeed computes: ;;;
Definition hlist_cons_destructP (a : A) (ls' : list A)
(hls : hlist B (a :: ls')) :
(@hlist_cons a ls' hls).(**)
Proof.(**)
apply: (hlist_destructP hls ).(* *)
Defined.(**) *)
Definition hlist_eta_type : forall (ls : list A) (hls : hlist B ( ls)), Type.
Proof.
move => ls hls. refine (hls = _ ). move: hls. case: ls.
- move => _ . exact: HNil.
- move => a ls' hls'. refine (_ ::: _).
+ case: a ls' hls' / (hlist_destructP hls') => a ls' ba _.
exact: ba. (* hhd *)
+ case: a ls' hls' / (hlist_destructP hls') => a ls' _ hls'.
exact: hls'. (* htl *)
Defined.
Lemma hlist_eta : forall (ls : list A) (hls : hlist B ( ls)),
@hlist_eta_type ls hls.
Proof.
move => ls hls. case: ls / hls.
rewrite /hlist_eta_type. reflexivity.
rewrite /hlist_eta_type. reflexivity.
Defined.
(* memo: may ;;; *)
Definition tl_hlist_type : forall (ls : list A) (hls : hlist B ( ls)), Type.
Proof.
move => ls. case: ls.
- move => _ . refine (hlist B [::]).
- move => a ls' hls'. refine (hlist B ls').
Defined.
Definition tl_hlist : forall (ls : list A) (hls : hlist B ( ls)), tl_hlist_type hls.
Proof.
move => ls. case: ls => /=.
- move => hls. exact: hls.
- move => a ls' hls. case: a ls' hls / (hlist_destructP hls) => a ls' _ hls'.
exact: hls'.
Defined.
End Section2.
End Destruct_hlist.
Fixpoint chain (T : Type) (ls : list T) {struct ls} : list (prod T T) :=
match ls with
| nil => nil
| cons t0 ls' => match ls' with
| nil => nil
| cons t1 ls'' => (t0, t1) :: chain ls'
end
end.
Arguments chain : simpl nomatch.
Eval compute in chain [:: 0; 11; 2; 3].
Inductive chain_graph (T : Type) : list T -> list (prod T T) -> Type :=
| Chain_nil : chain_graph [::] (chain [::])
| Chain_cons_nil : forall t0 : T, chain_graph [:: t0] (chain [:: t0])
| Chain_cons_cons : forall (t0 t1 : T) (ls'' : list T),
chain_graph (t1 :: ls'') (chain ([:: t1 & ls'']))
-> chain_graph (t0 :: t1 :: ls'') ((t0 , t1) :: chain ([:: t1 & ls''])) .
Lemma chain_graphP (T : Type) :
forall (ls : list T), chain_graph ls (chain ls).
Proof.
induction ls as [|t0 ls']. constructor 1.
destruct ls' as [|t1 ls'']. constructor 2.
simpl. constructor 3. exact: IHls'.
Defined.
Definition toArrowV {log : logic} {trf : obV log -> obV log}
(V1V2 : prod (obV log) (obV log))
:= V(0 trf V1V2.1 |- trf V1V2.2 )0.
Definition arrowList {log : logic} {trf : obV log -> obV log} ls
:= (hlist (@toArrowV log trf) (chain ls)).
Module Import Destruct_arrowList.
Inductive arrowList_nil {log : logic} {trf : obV log -> obV log}
: hlist (toArrowV (trf:=trf)) (chain [::]) -> Type :=
| ArrowList_nil : arrowList_nil (HNil : arrowList [::]).
Inductive arrowList_cons_nil {log : logic} {trf : obV log -> obV log}
: forall V0 : (obV log), hlist (toArrowV (trf:=trf)) (chain [:: V0]) -> Type :=
| ArrowList_cons_nil :
forall V0, @arrowList_cons_nil log trf V0 (HNil : arrowList [:: V0]).
Inductive arrowList_cons_cons {log : logic} {trf : obV log -> obV log}
: forall (V0 V1 : obV log) (Vs'' : list (obV log)),
hlist (toArrowV (trf:=trf)) ((V0 , V1) :: (chain (V1 :: Vs''))) -> Type :=
| ArrowList_cons_cons :
forall V0 V1 Vs'' (v01 : toArrowV (V0 , V1))
(vs' : hlist (toArrowV (trf:=trf)) (chain [:: V1 & Vs''])),
@arrowList_cons_cons log trf V0 V1 Vs''
(v01 ::: vs' : hlist (toArrowV (trf:=trf)) ((V0 , V1) :: (chain [:: V1 & Vs'']))).
Definition arrowList_destructP_type {log : logic}{trf : obV log -> obV log} :
forall (Vs : list (obV log)) (vs : hlist (toArrowV (trf:=trf)) (chain Vs)), Type.
Proof.
move => Vs. case: Vs (chain Vs) / (chain_graphP Vs) =>
[ | V0 | V0 V1 Vs'' V1Vs''_chain_graph ] vs.
- refine (@arrowList_nil log trf vs).
- refine (@arrowList_cons_nil log trf V0 vs).
- refine (@arrowList_cons_cons log trf V0 V1 Vs'' vs).
Defined.
Definition arrowList_destructP {log : logic}{trf : obV log -> obV log} :
forall (Vs : list (obV log)) (vs : hlist (toArrowV (trf:=trf)) (chain Vs)),
@arrowList_destructP_type log trf Vs vs.
Proof.
move => Vs. rewrite /arrowList_destructP_type.
case: Vs (chain Vs) / (chain_graphP Vs) =>
[ | V0 | V0 V1 Vs'' V1Vs''_chain_graph ] vs.
- case : vs / (hlist_destructP vs). constructor.
- case : vs / (hlist_destructP vs). constructor.
- (* /!\ *) rewrite (hlist_eta vs). constructor.
Defined.
(* indeed computes:
Definition arrowList_cons_cons_destructP {log : logic}{trf : obV log -> obV log} :
forall V0 V1 Vs'' (vs : hlist (toArrowV (trf:=trf)) (chain [:: V0, V1 & Vs''])),
@arrowList_cons_cons log trf V0 V1 Vs'' vs.(**)
Proof.(**)
move => V0 V1 Vs'' vs.(**) exact: (arrowList_destructP vs).(**)
Defined.(**) *)
End Destruct_arrowList.
Inductive arrowList_prop {log : logic} {trf : obV log -> obV log}
: forall ls : list (obV log),
hlist (toArrowV (trf:=trf)) (chain ls) -> Type :=
| ArrowList_nil : arrowList_prop (HNil : arrowList [::])
| ArrowList_cons_nil : forall V0, arrowList_prop (HNil : arrowList [:: V0])
| ArrowList_cons_cons :
forall V0 V1 (v01 : toArrowV (V0, V1)) Vs'' (vs' : arrowList (V1 :: Vs'')),
arrowList_prop vs' ->
arrowList_prop (v01 ::: vs' : arrowList (V0 :: V1 :: Vs'')).
Lemma arrowListP {log : logic}{trf : obV log -> obV log} :
forall (Vs : list (obV log)) (vs :arrowList Vs),
(@arrowList_prop log trf Vs vs).
Proof.
move => Vs. move: (chain_graphP Vs) => Vs_chainInputP.
elim : Vs {-}(chain Vs) / Vs_chainInputP.
- move => vs. case: (arrowList_destructP vs).
apply: ArrowList_nil.
- move => V0 vs. case: V0 vs / (arrowList_destructP vs) => V0.
apply: ArrowList_cons_nil.
- intros V0 V1 Vs'' (*ch_V1Vs''_P*) _ IHVs' vs''. move: IHVs'.
case: V0 V1 Vs'' vs'' / (arrowList_destructP vs'') =>
V0 V1 Vs'' v01 vs' IHVs'.
apply: (ArrowList_cons_cons v01 (IHVs' vs')).
Defined.
Definition iterDeClass0 (n : nat) : obMod -> obMod
:= iter n DeClass0 .
Definition iterDeClassifying {log : logic} {trf : obV log -> obV log}
(V_dft : obV log) (B A : obMod)
: forall (Vs : list (obV log)) (vs : (hlist (toArrowV (trf:=trf)) (chain Vs)))
(b : 'D(0 trf (last V_dft Vs) |- [0 B ~> A ]0 )0),
'D(0 trf(head V_dft Vs) |- [0 B ~> iterDeClass0 (length Vs).-1 A ]0 )0.
Proof.
move => Vs vs. move: (arrowListP (trf:=trf) vs) => vs_arrowListP.
elim : vs_arrowListP => /= .
- move => b; exact: b.
- move => V0 b; exact: b.
- move => V0 V1 v01 Vs'' vs' vs'_arrowListP vs'_IH b.
refine (v01 o>| (vs'_IH b) o>Mod 'declfy).
Defined.
Notation "vs o>|| a o>Mod ''declfy" :=
(@iterDeClassifying _ _ _ _ _ _ vs a)
(at level 25, a at next level, right associativity).
Definition iterDeClassifying_rewrite_type {log : logic}{trf : obV log -> obV log}
(V_dft : obV log) (B A : obMod) (Vs : list (obV log))
(vs : (hlist (toArrowV (trf:=trf)) (chain Vs))) : Prop.
Proof.
case: (arrowListP vs).
- refine ( forall (b : 'D(0 trf(last V_dft [::]) |- [0 B ~> A ]0 )0),
iterDeClassifying (V_dft := V_dft) (HNil : arrowList ([::])) b = b ).
- move => V0.
refine ( forall (b : 'D(0 trf(last V0 [::]) |- [0 B ~> A ]0 )0),
iterDeClassifying (V_dft := V_dft) (HNil : arrowList ([:: V0])) b = b ).
- move => V0 V1 v01 Vs'' vs' _ .
refine ( forall (b : 'D(0 trf(last V1 Vs'') |- [0 B ~> A ]0 )0),
iterDeClassifying (V_dft := V_dft) (v01 ::: vs' : arrowList (V0 :: V1 :: Vs'')) b
= v01 o>| ( iterDeClassifying (V_dft := V_dft) vs' b ) o>Mod 'declfy ).
Defined.
Lemma iterDeClassifying_rewrite {log : logic}{trf : obV log -> obV log}
{V_dft : obV log} {B A : obMod}
(Vs : list (obV log)) (vs : arrowList Vs) :
@iterDeClassifying_rewrite_type log trf V_dft B A Vs vs.
Proof.
rewrite /iterDeClassifying_rewrite_type.
case: (arrowListP vs); reflexivity.
Defined.
Notation RHSc := (X in _ ~~~ X)%pattern.
Notation LHSc := (X in X ~~~ _)%pattern.
Definition tac_arrows := (@Mod_arrowPre, @Mod_arrowPost,
@UnitDeClass_arrow, @DeClass_arrowPre,
@DeClass_arrowPost, @Classifying_arrow, @DeClassifying_arrow).
Hint Extern 4 (_ ~~?lo` _) => eapply (@SymV lo _) : logic_hints.
Hint Resolve PolyV_unitPost : logic_hints.
Hint Resolve PolyV_unitPre : logic_hints.
Lemma GenArrowsMod_arrowLog_id {log : logic} :
forall (V' V'' : obV log) (A1 A2 : obMod_gen) (aGen : Mod_gen V' A1 A2)
(v' : V(0 V'' |- V' )0),
( ( v' ) o>| #1| aGen )
~~~ (v' o>' (log.-1 o>| #1| aGen) ).
Proof. eauto with logic_hints. Qed.
Lemma UnitMod_arrowLog_id {log : logic} :
forall (V' : obV log) (A : obMod)
(v' : V(0 V' |- log.-I )0),
( ( v' ) o>| @uMod A )
~~~ (v' o>' (log.-1 o>| @uMod A) ).
Proof. eauto with logic_hints. Qed.
Lemma Mod_arrowLog_id {log : logic} :
forall (V : obV log) (A0 : obMod) (A : obMod)
(a_ : 'Mod(0 V |- [0 A0 ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a' : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV0 : obV log) (v0 : V(0 WV0 |- (0 W & V )0 )0),
( ( v0 ) o>| a_ o>Mod a' )
~~~ ( v0 o>' ( log.-1 o>| a_ o>Mod a' ) ).
Proof. eauto with logic_hints. Qed.
Lemma UnitDeClass_arrowLog_id {log : logic} :
forall (W W'' : obV log)
(w' : V(0 W'' |- (0 W & log.-I )0 )0)
(A A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
( ( w' ) o>| 'D1| a )
~~~ ( w' o>' ( log.-1 o>| 'D1| a ) ).
Proof. eauto with logic_hints. Qed.
Lemma DeClass_arrowLog_id {log : logic} :
forall (V : obV log) (B : obMod) (A : obMod)
(b : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV0 : obV log) (v0 : V(0 WV0 |- (0 W & V )0 )0),
( ( v0 ) o>| b o>D a )
~~~ ( v0 o>' ( log.-1 o>| b o>D a ) ).
Proof. eauto with logic_hints. Qed.
Lemma Classifying_arrowLog_id {log : logic} :
forall (V' V'' : obV log) (v0 : V(0 V'' |- V' )0)
(A1 A2 : obMod)
(a : 'Mod(0 V' |- [0 A1 ~> A2 ]0 )0),
( ( v0 ) o>| 'clfy o>Mod a )
~~~ ( v0 o>' ( log.-1 o>| 'clfy o>Mod a ) ).
Proof. eauto with logic_hints. Qed.
Lemma DeClassifying_arrowLog_id {log : logic} :
forall (V' V'' : obV log) (v0 : V(0 V'' |- V' )0) (A1 A2 : obMod)
(a : 'D(0 V' |- [0 A1 ~> A2 ]0 )0),
( ( v0 ) o>| a o>Mod 'declfy )
~~~ ( v0 o>' ( log.-1 o>| a o>Mod 'declfy ) ).
Proof. eauto with logic_hints. Qed.
Lemma logic_decidable0 : forall {log : logic} (V : obV log),
log.-1 ~~log` desIdenObLKV o>`log`>
log.-(1 log.-1 & V )0 o>`log`> desIdenObLK .
Admitted.
Hint Resolve logic_decidable0 : logic_hints.
Lemma logic_decidable1 : forall {log : logic}
(V W _WV WV : obV log)
(w : log.-V(0 WV |- log.-(0 W & V )0 )0)
(w0 : log.-V(0 _WV |- WV )0),
w0 o>`log`> w ~~log` log.-1 o>`log`> (w0 o>`log`> w) o>`log`> log.-(1 log.-1 & V )0.
Admitted.
Hint Resolve logic_decidable1 : logic_hints.
Lemma logic_decidable2 : forall {log : logic}
(V W WV' : obV log)
(v0 : log.-V(0 WV' |- log.-(0 W & V )0 )0),
v0 ~~log` (v0 o>`log`> log.-(1 desIdenObRKV & V )0)
o>`log`> log.-(1 log.-1 o>`log`> desIdenObRK & V )0.
Admitted.
Hint Resolve logic_decidable2 : logic_hints.
Lemma logic_decidable3 : forall {log : logic}
(W' W_ W'W_ : obV log)
(v : log.-V(0 W'W_ |- log.-(0 W' & W_ )0 )0)
(W'W_I : obV log)
(v0 : log.-V(0 W'W_I |- log.-(0 W'W_ & log.-I )0 )0),
v0 o>`log`> log.-(1 v & log.-I )0 ~~log`
((v0 o>`log`> log.-(1 v & log.-I )0 o>`log`> Assoc_Rev) o>`log`> desIdenObRKV)
o>`log`> log.-(1 log.-1 o>`log`> (0 W' & log.-1 o>`log`> desIdenObRK )1 & log.-I )0.
Admitted.
Hint Resolve logic_decidable3 : logic_hints.
Lemma logic_decidable4 : forall {log : logic}
(V W_ W' W'W_ : obV log)
(v : log.-V(0 W'W_ |- log.-(0 W' & W_ )0 )0)
(W'W_V : obV log)
(v0 : log.-V(0 W'W_V |- log.-(0 W'W_ & V )0 )0),
v0 o>`log`> log.-(1 v & V )0 ~~log` (v0 o>`log`> log.-(1 v & V )0 o>`log`> Assoc_Rev)
o>`log`> (0 W' & log.-1 )1 o>`log`> Assoc .
Admitted.
Hint Resolve logic_decidable4 : logic_hints.
Lemma logic_decidable7 : forall {log : logic} (Vb Vb' : obV log)
(vb : log.-V(0 Vb' |- Vb )0) (V0Vb' trfV0 trfV1 : obV log)
(v01 : log.-V(0 trfV0 |- trfV1 )0)
(v0b : log.-V(0 V0Vb' |- log.-(0 trfV0 & Vb' )0 )0),
(v0b o>`log`> (0 trfV0 & vb )1) o>`log`> log.-(1 v01 & Vb )0 ~~log`
(((((v0b o>`log`> log.-(1 desIdenObRKV & Vb' )0) o>`log`>
log.-(1 (log.-1 o>`log`> log.-(1 v01 o>`log`> desIdenObLKV & log.-I )0
o>`log`> Assoc_Rev) o>`log`>
log.-(1 desIdenObRKV & log.-(0 trfV1 & log.-I )0 )0 & Vb' )0
o>`log`> Assoc_Rev) o>`log`> (0 log.-(0 log.-I & log.-I )0 &
(log.-1 o>`log`> log.-(1 log.-1 o>`log`> desIdenObRK & Vb' )0)
o>`log`> log.-(1 desIdenObRKV & Vb' )0 )1 o>`log`> Assoc)
o>`log`> (0 log.-(0 log.-(0 log.-I & log.-I )0 &
log.-(0 trfV1 & log.-I )0 )0 & vb )1) o>`log`>
log.-(1 (log.-1 o>`log`> desIdenObRKV) o>`log`> desIdenObRK & Vb )0)
o>`log`> log.-(1 (log.-1 o>`log`> ((log.-1 o>`log`>
log.-(1 log.-1 o>`log`> desIdenObRK & log.-(0 trfV1 & log.-I )0 )0)
o>`log`> log.-(1 log.-1 & log.-(0 trfV1 & log.-I )0 )0)
o>`log`> log.-(1 log.-1 & log.-(0 trfV1 & log.-I )0 )0
o>`log`> desIdenObLK) o>`log`> log.-1 o>`log`> desIdenObRK & Vb )0 .
Admitted.
Hint Resolve logic_decidable7 : logic_hints.
Lemma Mod_inputUnitMod_rev {log : logic} :
forall (V : obV log) (B : obMod) (A : obMod) (b : 'Mod(0 V |- [0 B ~> A ]0 )0),
( b )
~~~ ( desIdenObLKV o>| b o>Mod ( log.-1 o>| uMod ) ).
Proof. eauto with logic_hints.
(* intros.(**) rewrite -Mod_inputUnitMod.(**) rewrite [LHSc]PolyV_Mod_unit.(**)
eapply PolyV_Mod_cong; [| reflexivity].(**)
clear.(**) exact: logic_decidable0.(**) *) Qed.
Lemma DeClassifying_morphismPost_rev {log : logic} :
forall (V : obV log) (A1 A2 : obMod) (b_ : 'Mod(0 V |- [0 A1 ~> A2 ]0 )0)
(W _WV : obV log) (A3 : obMod) (b' : 'D(0 W |- [0 A2 ~> A3 ]0 )0),
forall (WV : obV log) (w : V(0 WV |- (0 W & V )0 )0) (w0 : V(0 _WV |- WV )0),
( w0 o>| ( w o>| b_ o>Mod b' ) o>Mod 'declfy )
~~~ ( (w0 o> w) o>| b_ o>Mod ( log.-1 o>| b' o>Mod 'declfy ) ).
Proof.
intros. rewrite -[in RHSc]DeClassifying_morphismPost.
rewrite [in LHSc]Mod_arrowLog_id. rewrite -[in LHSc]DeClassifying_arrow.
rewrite [in RHSc]Mod_arrowLog_id. rewrite -[in RHSc]DeClassifying_arrow.
eauto with logic_hints.
Qed.
Lemma DeClass_morphismPre_rev {log : logic} :
forall (A : obMod) (V : obV log) (B : obMod) (b' : 'D(0 V |- [0 B ~> A ]0 )0),
forall (W : obV log) (A' : obMod) (a : 'Mod(0 W |- [0 A ~> A' ]0 )0),
forall (WV' : obV log) (v0 : V(0 WV' |- (0 W & V )0 )0),
( v0 o>| b' o>D ( a ) )
~~~ ( (v0 o> ((1 desIdenObRKV & _ )0)) o>| b' o>Mod ( log.-1 o>| 'D1| a )
: 'D(0 WV' |- [0 B ~> A' ]0)0 ).
Proof.
intros. rewrite -[in RHSc]DeClass_morphismPre.
rewrite -DeClass_arrowPost. eauto with logic_hints.
Qed.
Lemma DeClass_morphismPost_rev {log : logic}:
forall (A : obMod)
(W' W_ W'W_ : obV log) (v : V(0 W'W_ |- (0 W' & W_ )0 )0) (A' : obMod)
(a_ : 'Mod(0 W_ |- [0 A ~> A' ]0 )0)
(A'' : obMod) (a' : 'Mod(0 W' |- [0 A' ~> A'' ]0 )0),
forall (W'W_I : obV log) (v0 : V(0 W'W_I |- (0 W'W_ & log.-I )0 )0),
( v0 o>| 'D1| ( v o>| a_ o>Mod a' ) )
~~~ ( (v0 o> (1 v & _ )0 o> Assoc_Rev) o>| ( (log.-1) o>| 'D1| a_ ) o>D a'