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coqmodel.v
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(* This file contain the X-part and Delta-part, in addition to all the theorems. It imports the Y-part,
located in ymodel. Verifying this file can take a few minutes on a modern machine *)
Require Import Coq.Arith.EqNat.
Require Import Coq.Bool.Bool.
Load ymodel.
Section KripkeModel.
Inductive VX:= x.
Inductive EdgeX:= sx | e.
Inductive TriangleX:=
| sx_sx_sx
| sx_e_sx
| e_sx_sx
| e_e_sx
| sx_sx_e
| sx_e_e
| e_sx_e
| e_e_e.
Function sX (v1:VX):= sx.
Function d0X (e1:EdgeX):= x.
Function d1X (e1:EdgeX):= x.
Function se0X (e1:EdgeX):=
match e1 with
| sx => sx_sx_sx
| e => e_e_sx
end.
Function se1X (e1:EdgeX):=
match e1 with
| sx => sx_sx_sx
| e => sx_e_e
end.
Function dp0X (t1:TriangleX):=
match t1 with
| sx_sx_sx => sx
| sx_e_sx => sx
| e_sx_sx => e
| e_e_sx => e
| sx_sx_e => sx
| sx_e_e => sx
| e_sx_e => e
| e_e_e => e
end.
Function dp1X (t1:TriangleX):=
match t1 with
| sx_sx_sx => sx
| sx_e_sx => e
| e_sx_sx => sx
| e_e_sx => e
| sx_sx_e => sx
| sx_e_e => e
| e_sx_e => sx
| e_e_e => e
end.
Function dp2X (t1:TriangleX):=
match t1 with
| sx_sx_sx => sx
| sx_e_sx => sx
| e_sx_sx => sx
| e_e_sx => sx
| sx_sx_e => e
| sx_e_e => e
| e_sx_e => e
| e_e_e => e
end.
Function eqVX (day: Days)(v1 v2:VX):= true.
Lemma eqEdgeXDec: forall e1 e2: EdgeX, {e1 = e2} + {e1 <> e2}.
Proof.
decide equality.
Defined.
Function sameConstructorEdgeX (e1 e2:EdgeX):= if (eqEdgeXDec e1 e2) then true else false.
Function eqEdgeX (day: Days)(e1 e2:EdgeX):=
match day with
| d1 => sameConstructorEdgeX e1 e2
| d2 => true
end.
Lemma eqtriangleXDec: forall b1 b2: TriangleX, {b1 = b2} + {b1 <> b2}.
Proof.
decide equality.
Defined.
Function sameConstructorTriangleX (t1 t2:TriangleX):= if (eqtriangleXDec t1 t2) then true else false.
Function eqTriangleX (day: Days)(t1 t2:TriangleX):=
match day with
| d1 => sameConstructorTriangleX t1 t2
| d2 => true
end.
Function edgesToTriangleX(d0 d1 d2 :EdgeX) :=
match d0,d1,d2 with
| sx, sx, sx => sx_sx_sx
| sx, e, sx => sx_e_sx
| e, sx, sx => e_sx_sx
| e, e, sx => e_e_sx
| sx, sx, e => sx_sx_e
| sx, e, e => sx_e_e
| e, sx, e => e_sx_e
| e, e, e => e_e_e
end.
Function fill20X (t1 t2 t3 :TriangleX) := (edgesToTriangleX (dp0X t1) (dp0X t2) (dp0X t3)).
Function fill21X (t1 t2 t3 :TriangleX) := (edgesToTriangleX (dp0X t1) (dp1X t2) (dp1X t3)).
Function fill22X (t1 t2 t3 :TriangleX) := (edgesToTriangleX (dp1X t1) (dp1X t2) (dp2X t3)).
Function fill23X (t1 t2 t3 :TriangleX) := (edgesToTriangleX (dp2X t1) (dp2X t2) (dp2X t3)).
Function fill12X (t1 t2 :EdgeX) :=
match t1,t2 with
| sx, e=> sx_e_sx
| e, e=> e_e_sx
| sx, sx=> sx_sx_sx
| e, sx=> e_sx_sx
end.
Function fill11X (t1 t2 :EdgeX) :=
match t1,t2 with
| sx, e=> sx_sx_e
| e, e=> e_sx_e
| sx, sx=> sx_sx_sx
| e, sx=> e_sx_sx
end.
Function fill10X (t1 t2 :EdgeX) :=
match t1,t2 with
| sx, e=> sx_sx_e
| e, e=> sx_e_e
| sx, sx=> sx_sx_sx
| e, sx=> sx_e_sx
end.
Inductive Delta10:= delta0 | delta1 .
Inductive Delta11:= delta00 | delta01 | delta11.
Inductive Delta12:=
| delta000
| delta001
| delta011
| delta111.
Function s (v:Delta10):= match v with
| delta0 => delta00
| delta1 => delta11
end.
Function s0 (e:Delta11):= match e with
| delta00 => delta000
| delta01 => delta001
| delta11 => delta111
end.
Function s1 (e:Delta11):= match e with
| delta00 => delta000
| delta01 => delta011
| delta11 => delta111
end.
Function dv0 (e:Delta11):= match e with
| delta00 => delta0
| delta01 => delta1
| delta11 => delta1
end.
Function dv1 (e:Delta11):= match e with
| delta00 => delta0
| delta01 => delta0
| delta11 => delta1
end.
Function dp0 (t:Delta12):= match t with
| delta000 => delta00
| delta001 => delta01
| delta011 => delta11
| delta111 => delta11
end.
Function dp1 (t:Delta12):= match t with
| delta000 => delta00
| delta001 => delta01
| delta011 => delta01
| delta111 => delta11
end.
Function dp2 (t:Delta12):= match t with
| delta000 => delta00
| delta001 => delta00
| delta011 => delta01
| delta111 => delta11
end.
Lemma eqDelta10Dec: forall e1 e2: Delta10, {e1 = e2} + {e1 <> e2}.
Proof.
decide equality.
Defined.
Function eqDelta10 (day: Days)(e1 e2:Delta10):= if (eqDelta10Dec e1 e2) then true else false.
Lemma eqDelta11Dec: forall e1 e2: Delta11, {e1 = e2} + {e1 <> e2}.
Proof.
decide equality.
Defined.
Function eqDelta11 (day: Days)(e1 e2:Delta11):= if (eqDelta11Dec e1 e2) then true else false.
Lemma eqDelta12Dec: forall e1 e2: Delta12, {e1 = e2} + {e1 <> e2}.
Proof.
decide equality.
Defined.
Function eqDelta12 (day: Days)(e1 e2:Delta12):= if (eqDelta12Dec e1 e2) then true else false.
Function Fv (delt:Delta10) (v:VX):=
match delt with
| delta0 => y0
| delta1 => y1
end.
Function Fe (delt:Delta11) (e:EdgeX):=
match delt with
| delta00 => y0y0
| delta01 => y0y1
| delta11 => match e with
| sx => y1y1
| e => k
end
end.
Function Ft000 (t:TriangleX):= y0y0_y0y0_y0y0.
Function Ft001 (t:TriangleX):= y0y1_y0y1_y0y0.
Function Ft011 (t:TriangleX):=
match t with
| sx_sx_sx => y1y1_y0y1_y0y1
| sx_e_sx => y1y1_y0y1_y0y1
| e_sx_sx => k_y0y1_y0y1
| e_e_sx => k_y0y1_y0y1
| sx_sx_e => y1y1_y0y1_y0y1
| sx_e_e => y1y1_y0y1_y0y1
| e_sx_e => k_y0y1_y0y1
| e_e_e => k_y0y1_y0y1
end.
Function Ft111 (t:TriangleX):=
match t with
| sx_sx_sx => y1y1_y1y1_y1y1
| sx_e_sx => y1y1_k_y1y1
| e_sx_sx => k_y1y1_y1y1
| e_e_sx => k_k_y1y1
| sx_sx_e => y1y1_y1y1_k
| sx_e_e => y1y1_k_k
| e_sx_e => k_y1y1_k
| e_e_e => k_k_k
end.
Function Ft (delt:Delta12) (t:TriangleX):=
match delt with
| delta000 => Ft000 t
| delta001 => Ft001 t
| delta011 => Ft011 t
| delta111 => Ft111 t
end.
Definition ReflexiveFun {A:Type}(eqFun: Days -> A -> A -> bool):= forall (d:Days)(el:A), (eqFun d el el) = true.
Hint Unfold ReflexiveFun.
Definition SymetricFun {A:Type}(eqFun: Days -> A -> A -> bool):=
forall (d:Days)(elem1 elem2:A), ((eqFun d elem1 elem2)=true)->(eqFun d elem2 elem1)=true.
Hint Unfold SymetricFun.
Definition TransitiveFun {A:Type}(eqFun: Days -> A -> A -> bool):=
forall (d:Days)(elem1 elem2 elem3:A),
((eqFun d elem1 elem2)=true /\ (eqFun d elem2 elem3)=true) ->
(eqFun d elem1 elem3)=true.
Hint Unfold TransitiveFun.
Definition EquivalenceFun {A:Type}(eqFun: Days -> A -> A -> bool):=
ReflexiveFun eqFun /\ SymetricFun eqFun /\ TransitiveFun eqFun.
Hint Unfold EquivalenceFun.
Definition unaryFunctionRespectsEquality {Domain CoDomain:Type}
(eqDomain: Days -> Domain -> Domain -> bool)
(eqCoDomain: Days -> CoDomain -> CoDomain -> bool)
(function: Domain->CoDomain):=
forall d:Days, forall (elem1 elem2: Domain),
(eqDomain d elem1 elem2)=true -> (eqCoDomain d (function elem1) (function elem2))=true.
Definition binaryFunctionRespectsEquality {Domain CoDomain:Type}
(eqDomain: Days -> Domain -> Domain -> bool)
(eqCoDomain: Days -> CoDomain -> CoDomain -> bool)
(function: Domain-> Domain -> CoDomain):=
forall d:Days, forall (elem1 elem1p elem2 elem2p: Domain),
(eqDomain d elem1 elem1p)=true /\ (eqDomain d elem2 elem2p)=true
-> (eqCoDomain d (function elem1 elem2) (function elem1p elem2p))=true.
Definition tertaryFunctionRespectsEquality {Domain CoDomain:Type}
(eqDomain: Days -> Domain -> Domain -> bool)
(eqCoDomain: Days -> CoDomain -> CoDomain -> bool)
(function: Domain-> Domain-> Domain -> CoDomain):=
forall d:Days, forall (elem1 elem2 elem3 elem1p elem2p elem3p: Domain),
((eqDomain d elem1 elem1p)=true /\
(eqDomain d elem2 elem2p)=true /\
(eqDomain d elem3 elem3p)=true)
-> (eqCoDomain d (function elem1 elem2 elem3)
(function elem1p elem2p elem3p))=true.
Hint Unfold unaryFunctionRespectsEquality.
Hint Unfold binaryFunctionRespectsEquality.
Hint Unfold tertaryFunctionRespectsEquality.
(* A tertary function which respects equality on each of its 3 arguments respect it on all. *)
Theorem TertaryFunctionRespectsEqualtyPointwise (Domain CoDomain :Type)
(function : Domain -> Domain -> Domain -> CoDomain)
(eqDomain : Days-> Domain -> Domain -> bool)
(eqCoDomain :Days-> CoDomain -> CoDomain -> bool)
(eqCoDomainIsEqiv : EquivalenceFun eqCoDomain)
:
(forall (d:Days)(t1 t2 t3:Domain) (t1p:Domain),
(eqDomain d t1 t1p)=true -> (eqCoDomain d (function t1 t2 t3) (function t1p t2 t3))=true)
->
(forall (d:Days)(t1 t2 t3:Domain) (t2p:Domain),
(eqDomain d t2 t2p)=true -> (eqCoDomain d (function t1 t2 t3) (function t1 t2p t3))=true)
->
(forall (d:Days)(t1 t2 t3:Domain) (t3p:Domain),
(eqDomain d t3 t3p)=true -> (eqCoDomain d (function t1 t2 t3) (function t1 t2 t3p))=true)
-> tertaryFunctionRespectsEquality eqDomain eqCoDomain function.
Proof.
unfold tertaryFunctionRespectsEquality.
intros.
destruct H2.
destruct H3.
set(eqCoDomainTrans:= (proj2 (proj2 eqCoDomainIsEqiv ))).
assert (eqCoDomain d (function elem1 elem2 elem3) (function elem1p elem2 elem3)=true);auto.
assert (eqCoDomain d (function elem1p elem2 elem3) (function elem1p elem2p elem3)=true);auto.
assert (eqCoDomain d (function elem1p elem2p elem3) (function elem1p elem2p elem3p)=true);auto.
assert (eqCoDomain d (function elem1 elem2 elem3) (function elem1p elem2p elem3)=true);auto.
assert ( eqCoDomain d (function elem1 elem2 elem3) (function elem1p elem2 elem3) = true
/\ eqCoDomain d (function elem1p elem2 elem3) (function elem1p elem2p elem3) = true);auto.
unfold EquivalenceFun in eqCoDomainIsEqiv.
apply (eqCoDomainTrans d (function elem1 elem2 elem3) (function elem1p elem2 elem3) (function elem1p elem2p elem3));auto.
apply (eqCoDomainTrans d (function elem1 elem2 elem3) (function elem1p elem2p elem3) (function elem1p elem2p elem3p));auto.
Qed.
Definition EqFunctionMonotone {Domain:Type}
(eq: Days -> Domain -> Domain -> bool):=
forall (elem1 elem2: Domain),
(eq d1 elem1 elem2)=true -> (eq d2 elem1 elem2)=true.
Hint Unfold EqFunctionMonotone.
(* Note that for every equivalence function Eq we have that a=b implies
Eq a b = true. So if we want to proove A -> Eq a b = true we can instead prove A -> a=b. We sometimes do this for effeciency reasons.
*)
Class twoDayKripke (Points Edges Triangles :Type)
:= {
sP : Points -> Edges;
d0E : Edges -> Points;
d1E : Edges -> Points;
s0E : Edges -> Triangles;
s1E : Edges -> Triangles;
d0T : Triangles -> Edges;
d1T : Triangles -> Edges;
d2T : Triangles -> Edges;
eqV : Days -> Points -> Points -> bool;
eqEdges : Days -> Edges -> Edges -> bool;
eqTriangles : Days -> Triangles -> Triangles -> bool;
eqVisEq : EquivalenceFun eqV;
eqEdgessisEq : EquivalenceFun eqEdges;
eqTrianglesisEq : EquivalenceFun eqTriangles;
_ : EqFunctionMonotone eqV;
_ : EqFunctionMonotone eqEdges;
_ : EqFunctionMonotone eqTriangles;
sPRespectsEq : unaryFunctionRespectsEquality eqV eqEdges sP;
d0ERespectsEq : unaryFunctionRespectsEquality eqEdges eqV d0E;
d1ERespectsEq : unaryFunctionRespectsEquality eqEdges eqV d1E;
s0ERespectsEq : unaryFunctionRespectsEquality eqEdges eqTriangles s0E;
s1ERespectsEq : unaryFunctionRespectsEquality eqEdges eqTriangles s1E;
d0TRespectsEq : unaryFunctionRespectsEquality eqTriangles eqEdges d0T;
d1TRespectsEq : unaryFunctionRespectsEquality eqTriangles eqEdges d1T;
d2TRespectsEq : unaryFunctionRespectsEquality eqTriangles eqEdges d2T;
(* Simplicial identity 1 *)
_: forall t: Triangles, d0E(d1T(t)) = d0E(d0T(t));
_: forall t: Triangles, d0E(d2T(t)) = d1E(d0T(t));
_: forall t: Triangles, d1E(d2T(t)) = d1E(d1T(t));
(* Simplicial identity 2 *)
_: forall e: Edges, d0T(s1E(e)) = sP(d0E(e)) ;
(* Simplicial identity 3 *)
_: forall p: Points, d0E(sP(p)) = p;
_: forall p: Points, d1E(sP(p)) = p;
_: forall e: Edges, d0T(s0E(e)) = e;
_: forall e: Edges, d1T(s0E(e)) = e;
_: forall e: Edges, d1T(s1E(e)) = e;
_: forall e: Edges, d2T(s1E(e)) = e;
(* Simplicial identity 4 *)
_ : forall e:Edges, d2T(s0E(e)) = sP(d1E(e));
(* Simplicial identity 5 *)
_: forall p: Points, s1E(sP(p)) = s0E(sP(p))
}.
Class fillableModel {Points Edges Triangles:Type} {m: twoDayKripke Points Edges Triangles}
:= {
fill10: Edges -> Edges -> Triangles;
fill11: Edges -> Edges -> Triangles;
fill12: Edges -> Edges -> Triangles;
fill20: Triangles -> Triangles -> Triangles -> Triangles;
fill21: Triangles -> Triangles -> Triangles -> Triangles;
fill22: Triangles -> Triangles -> Triangles -> Triangles;
fill23: Triangles -> Triangles -> Triangles -> Triangles;
fill10RespectEquality: binaryFunctionRespectsEquality eqEdges eqTriangles fill10;
fill11RespectEquality: binaryFunctionRespectsEquality eqEdges eqTriangles fill11;
fill12RespectEquality: binaryFunctionRespectsEquality eqEdges eqTriangles fill12;
fill20RespectsEqualty: tertaryFunctionRespectsEquality eqTriangles eqTriangles fill20;
fill21RespectsEqualty: tertaryFunctionRespectsEquality eqTriangles eqTriangles fill21;
fill22RespectsEqualty: tertaryFunctionRespectsEquality eqTriangles eqTriangles fill22;
fill23RespectsEqualty: tertaryFunctionRespectsEquality eqTriangles eqTriangles fill23;
fill10Prop: forall (d:Days)(e1 e2:Edges), ((eqV d (d1E e1) (d1E e2))=true)->
(eqEdges d (d1T (fill10 e1 e2)) e1)=true /\
(eqEdges d (d2T (fill10 e1 e2)) e2)=true;
fill11Prop: forall (d:Days)(e1 e2:Edges), ((eqV d (d1E e1) (d0E e2))=true)->
(eqEdges d (d0T (fill11 e1 e2)) e1)=true /\
(eqEdges d (d2T (fill11 e1 e2)) e2)=true;
fill12Prop: forall (d:Days)(e0 e1:Edges), ((eqV d (d0E e0) (d0E e1))=true)->
(eqEdges d (d0T (fill12 e0 e1)) e0)=true /\
(eqEdges d (d1T (fill12 e0 e1)) e1)=true;
fill20Prop: forall (d:Days)(t1 t2 t3:Triangles), (eqEdges d (d1T t1) (d1T t2))=true /\
(eqEdges d (d2T t1) (d1T t3))=true /\
(eqEdges d (d2T t2) (d2T t3))=true ->
(
(eqEdges d (d0T t1) (d0T (fill20 t1 t2 t3)))=true /\
(eqEdges d (d0T t2) (d1T (fill20 t1 t2 t3)))=true /\
(eqEdges d (d0T t3) (d2T (fill20 t1 t2 t3)))=true);
fill21Prop: forall (d:Days)(t1 t2 t3:Triangles),
(eqEdges d (d1T t1) (d0T t2))=true /\
(eqEdges d (d2T t1) (d0T t3))=true /\
(eqEdges d (d2T t2) (d2T t3))=true ->
((eqEdges d (d0T t1) (d0T (fill21 t1 t2 t3)))=true /\
(eqEdges d (d1T t2) (d1T (fill21 t1 t2 t3)))=true /\
(eqEdges d (d1T t3) (d2T (fill21 t1 t2 t3)))=true);
fill22Prop: forall (d:Days)(t1 t2 t3:Triangles),
(eqEdges d (d0T t1) (d0T t2))=true /\
(eqEdges d (d2T t1) (d0T t3))=true /\
(eqEdges d (d2T t2) (d1T t3))=true ->
((eqEdges d (d1T t1) (d0T (fill22 t1 t2 t3)))=true /\
(eqEdges d (d1T t2) (d1T (fill22 t1 t2 t3)))=true /\
(eqEdges d (d2T t3) (d2T (fill22 t1 t2 t3)))=true);
fill23Prop: forall (d:Days)(t1 t2 t3:Triangles),
(eqEdges d (d0T t1) (d0T t2))=true /\
(eqEdges d (d1T t1) (d0T t3))=true /\
(eqEdges d (d1T t2) (d1T t3))=true ->
((eqEdges d (d2T t1) (d0T (fill23 t1 t2 t3)))=true /\
(eqEdges d (d2T t2) (d1T (fill23 t1 t2 t3)))=true /\
(eqEdges d (d2T t3) (d2T (fill23 t1 t2 t3)))=true)
}.
Ltac destructGroundElement Points Edges Triangles:=
repeat (match goal with
| [elem : Points |- _] => destruct elem
| [elem : Edges |- _] => destruct elem
| [elem : Triangles |- _] => destruct elem
| [elem : Days |- _] => destruct elem
| [_:_ |- _] => auto
end).
Program Instance DeltaModel : twoDayKripke Delta10 Delta11 Delta12 := {|
sP:= s;
eqV := eqDelta10;
eqEdges := eqDelta11;
eqTriangles := eqDelta12;
d0E := dv0;
d1E := dv1;
s0E := s0;
s1E := s1;
d0T := dp0;
d1T := dp1;
d2T :=dp2
|}.
Solve All Obligations using (autounfold; repeat split;autounfold;intros;(destructGroundElement Delta10 Delta11 Delta12);trivial;destruct H;trivial).
Program Instance XModel : twoDayKripke VX EdgeX TriangleX := {|
sP:= sX;
eqV := eqVX;
eqEdges := eqEdgeX;
eqTriangles := eqTriangleX;
d0E := d0X;
d1E := d1X;
s0E := se0X;
s1E := se1X;
d0T := dp0X;
d1T := dp1X;
d2T :=dp2X
|}.
Solve All Obligations using ( autounfold; repeat split;autounfold;intros;(destructGroundElement VX EdgeX TriangleX);trivial;destruct H;trivial).
Program Instance XModelFill : fillableModel := {|
fill10:=fill10X;
fill11:=fill11X;
fill12:=fill12X;
fill20:=fill20X;
fill21:=fill21X;
fill22:=fill22X;
fill23:=fill23X
|}.
Obligation 4.
apply TertaryFunctionRespectsEqualtyPointwise; try (apply XModel_obligation_3);
intros;(destructGroundElement VX EdgeX TriangleX);trivial.
Qed.
Obligation 5.
apply TertaryFunctionRespectsEqualtyPointwise; try (apply XModel_obligation_3);
intros;(destructGroundElement VX EdgeX TriangleX);trivial.
Qed.
Obligation 6.
apply TertaryFunctionRespectsEqualtyPointwise; try (apply XModel_obligation_3);
intros;(destructGroundElement VX EdgeX TriangleX);trivial.
Qed.
Obligation 7.
apply TertaryFunctionRespectsEqualtyPointwise; try (apply XModel_obligation_3);
intros;(destructGroundElement VX EdgeX TriangleX);trivial.
Qed.
Solve All Obligations using ( autounfold; repeat split;autounfold;intros;(destructGroundElement VX EdgeX TriangleX);trivial;destruct H;destruct H0;trivial).
Program Instance YModel : twoDayKripke VY EdgeY TriangleY := {|
sP:= sY;
eqV := eqVY;
eqEdges := eqEdgeY;
eqTriangles := eqTriangleY;
d0E := d0Y;
d1E := d1Y;
s0E := se0Y;
s1E := se1Y;
d0T := dp0Y;
d1T := dp1Y;
d2T :=dp2Y
|}.
Obligation 3.
autounfold; repeat split;autounfold;intros.
destruct d;destruct el;auto.
destruct d;destruct elem1;destruct elem2;trivial.
destruct d;destruct elem1;destruct elem2;trivial;destruct H;destruct elem3;trivial.
Qed.
Solve All Obligations using ( autounfold; repeat split;autounfold;intros;(destructGroundElement VY EdgeY TriangleY);trivial;destruct H;trivial).
Theorem eqVertexYday1Minimal: forall (p1 p2:VY), (eqVY d1 p1 p2)=true -> p1=p2.
Proof.
intros.
simpl in H.
destruct p1; destruct p2;auto; simpl in H; contradict H; apply diff_false_true.
Qed.
Theorem eqEdgeYday1Minimal: forall (e1 e2:EdgeY), (eqEdgeY d1 e1 e2)=true -> e1=e2.
Proof.
intros.
simpl in H.
unfold sameConstructorEdgeY in H.
destruct (eqEdgeYDec e1 e2);trivial.
contradict H.
apply diff_false_true.
Qed.
Theorem eqTriangleYday1Minimal: forall (t1 t2:TriangleY), (eqTriangleY d1 t1 t2)=true -> t1=t2.
Proof.
intros.
simpl in H.
unfold sameConstructorTriangleY in H.
destruct (eqtriangleYDec t1 t2);trivial.
contradict H.
apply diff_false_true.
Qed.
Theorem deEqualToOnlyItself: forall (d:Days)(t1:TriangleY), (eqTriangleY d t1 de)=true -> t1=de.
intros.
destruct d;destruct t1;trivial;try contradict H;simpl;trivial;compute;intro cont;try (apply (diff_false_true cont)).
Qed.
Theorem FCommutesdp0: forall (delt:Delta12) (e:TriangleX), Fe (dp0 delt) (dp0X e) = dp0Y (Ft delt e).
Proof.
intros.
destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesdp1: forall (delt:Delta12) (e:TriangleX), Fe (dp1 delt) (dp1X e) = dp1Y (Ft delt e).
Proof.
intros.
destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesdp2: forall (delt:Delta12) (e:TriangleX), Fe (dp2 delt) (dp2X e) = dp2Y (Ft delt e).
Proof.
intros.
destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesd0: forall (delt:Delta11) (e:EdgeX), Fv (dv0 delt) (d0X e) = d0Y (Fe delt e).
Proof.
intros.
destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesd1: forall (delt:Delta11) (e:EdgeX), Fv (dv1 delt) (d1X e) = d1Y (Fe delt e).
Proof.
intros.
destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesSv: forall (deltp:Delta10) (p:VX), Fe (s deltp) (sX p) = sY (Fv deltp p).
Proof.
destruct deltp;destruct p;trivial.
Qed.
Theorem FCommutesS0: forall (d:Days) (delt:Delta11) (e:EdgeX), eqTriangleY d (Ft (s0 delt) (se0X e))( se0Y (Fe delt e)) = true.
Proof.
intros.
destruct d; destruct delt;destruct e0;trivial.
Qed.
Theorem FCommutesS1: forall (d:Days) (delt:Delta11) (e:EdgeX), eqTriangleY d (Ft (s1 delt) (se1X e))( se1Y (Fe delt e)) = true.
Proof.
intros.
destruct d; destruct delt;destruct e0;trivial.
Qed.
Definition FCommutesS (Fpoint: Delta10 -> VX -> VY)
(FEdge: Delta11 -> EdgeX -> EdgeY)
(FTriangle: Delta12 -> TriangleX -> TriangleY):=
(forall (deltp:Delta10) (p:VX), FEdge (s deltp) (sX p) = sY (Fpoint deltp p))
/\ (forall (d:Days) (delt:Delta11) (e:EdgeX),
eqTriangleY d (FTriangle (s0 delt) (se0X e))( se0Y (FEdge delt e)) = true)
/\ (forall (d:Days) (delt:Delta11) (e:EdgeX),
eqTriangleY d (FTriangle (s1 delt) (se1X e))( se1Y (FEdge delt e)) = true).
Definition FCommutesD (Fpoint: Delta10 -> VX -> VY)
(FEdge: Delta11 -> EdgeX -> EdgeY)
(FTriangle: Delta12 -> TriangleX -> TriangleY):=
(forall (delt:Delta11) (e:EdgeX), Fpoint (dv0 delt) (d0X e) = d0Y (FEdge delt e)) /\
(forall (delt:Delta11) (e:EdgeX), Fpoint (dv1 delt) (d1X e) = d1Y (FEdge delt e)) /\
(forall (delt:Delta12) (e:TriangleX), FEdge (dp2 delt) (dp2X e) = dp2Y (FTriangle delt e)) /\
(forall (delt:Delta12) (e:TriangleX), FEdge (dp1 delt) (dp1X e) = dp1Y (FTriangle delt e)) /\
forall (delt:Delta12) (e:TriangleX), FEdge (dp0 delt) (dp0X e) = dp0Y (FTriangle delt e).
Definition FCommutes (Fpoint: Delta10 -> VX -> VY)
(FEdge: Delta11 -> EdgeX -> EdgeY)
(FTriangle: Delta12 -> TriangleX -> TriangleY):=
(FCommutesS Fpoint FEdge FTriangle) /\ (FCommutesD Fpoint FEdge FTriangle).
Theorem FOrdCommutesS: FCommutesS Fv Fe Ft.
unfold FCommutesS.
auto using FCommutesSv, FCommutesS0, FCommutesS1.
Qed.
Theorem FOrdCommutesD: FCommutesD Fv Fe Ft.
unfold FCommutesD.
auto using FCommutesdp0, FCommutesdp1, FCommutesdp2, FCommutesd0,FCommutesd1.
Qed.
Definition Finverse (FIpoint: Delta10 -> VX -> VY)
(FIEdge: Delta11 -> EdgeX -> EdgeY)
(FITriangle: Delta12 -> TriangleX -> TriangleY) :=
(forall (p:VX), FIpoint delta0 p = Fv delta1 p /\
FIpoint delta1 p = Fv delta0 p) /\
(forall (e:EdgeX), FIEdge delta00 e = Fe delta11 e /\
FIEdge delta11 e = Fe delta00 e) /\
(forall (t:TriangleX), FITriangle delta000 t = Ft delta111 t /\
FITriangle delta111 t = Ft delta000 t).
(* Verification of:
In day~1 $F^{-}(001,\_)$ would have to map eee:\ntrip{x}{x}{x}{e}{e}{e} to the triangle
$(y_{1},y_{1},y_{0};k,y_{1}y_{0},y_{1}y_{0})$
*)
Theorem allFInverseInconsistentEEE: forall (FIpoint: Delta10 -> VX -> VY)
(FIEdge: Delta11 -> EdgeX -> EdgeY)
(FITriangle: Delta12 -> TriangleX -> TriangleY), Finverse FIpoint FIEdge FITriangle ->
FCommutesS FIpoint FIEdge FITriangle ->
FCommutesD FIpoint FIEdge FITriangle ->
FITriangle delta001 e_e_e = y1y0_y1y0_k.
Proof.
intros.
unfold Finverse in H.
destruct H.
destruct H2.
pose (H2 e).
destruct a.
assert ((FIEdge delta00 e )=k);auto.
unfold FCommutesD in H1.
destruct H1; destruct H7;destruct H8;destruct H9.
assert (not (y0 = y1)).
intro.
discriminate H11.
assert (dp2Y (FITriangle delta001 e_e_e)=k);auto.
pose (H8 delta001 e_e_e).
simpl in e0.
rewrite H6 in e0;auto.
assert ((FIpoint delta0 x) = y1).
apply (H x).
assert ((FIpoint delta1 x) = y0).
apply (H x).
assert (d0Y (FIEdge delta01 e) =y0).
pose (H1 delta01 e).
simpl in e0.
unfold d0X in e0.
rewrite H14 in e0; auto.
assert (d1Y (FIEdge delta01 e) =y1).
pose (H7 delta01 e).
simpl in e0.
unfold d1X in e0.
rewrite H13 in e0; auto.
assert (( (FIEdge delta01 e)=y1y0)).
destruct (FIEdge delta01 e); simpl in H15; simpl in H16; try discriminate H15; try discriminate H16;auto.
assert (dp1Y (FITriangle delta001 e_e_e)=y1y0);auto.
pose (H9 delta001 e_e_e);auto.
simpl in e0.
rewrite<- e0;auto.
assert (dp0Y (FITriangle delta001 e_e_e)=y1y0);auto.
pose (H10 delta001 e_e_e);auto.
rewrite<- e0;auto.
destruct (FITriangle delta001 e_e_e); simpl in H12;simpl in H18;simpl in H19; try discriminate H12; try discriminate H18;try discriminate H19;auto.
Qed.
(* Verification of $F^{-}(001,sss)=di$ *)
Theorem allFInverseInconsistentsss: forall (FIpoint: Delta10 -> VX -> VY)
(FIEdge: Delta11 -> EdgeX -> EdgeY)
(FITriangle: Delta12 -> TriangleX -> TriangleY), Finverse FIpoint FIEdge FITriangle ->
FCommutesS FIpoint FIEdge FITriangle ->
FCommutesD FIpoint FIEdge FITriangle ->
FITriangle delta001 sx_sx_sx = de.
Proof.
intros.
unfold Finverse in H.
destruct H.
destruct H2.
pose (H2 e).
destruct a.
assert ((FIEdge delta00 e )=k);auto.
unfold FCommutesD in H1.
destruct H1; destruct H7;destruct H8;destruct H9.
assert (not (y0 = y1)).
intro.
discriminate H11.
unfold FCommutesS in H0.
destruct H0;destruct H12.
assert ((FIpoint delta1 x) = y0).
apply H.
assert ((FIpoint delta0 x) = y1).
apply H.
assert (d0Y (FIEdge delta01 sx)=y0).
pose (H1 delta01 sx).
simpl in e0;auto.
rewrite<- e0;auto.
assert (d1Y (FIEdge delta01 sx)=y1).
pose (H7 delta01 sx).
simpl in e0;auto.
rewrite<- e0;auto.
assert ((FIEdge delta01 sx)=y1y0);auto.
destruct ((FIEdge delta01 sx));simpl in H16;simpl in H17;try discriminate H17; try discriminate H16;auto.
assert (FITriangle delta001 (se0X sx) = se0Y (FIEdge delta01 sx) ).
apply eqTriangleYday1Minimal.
rewrite H18.
simpl.
pose (H12 d1 delta01 sx).
simpl in e0.
rewrite H18 in e0.
destruct (FIEdge delta01 sx);auto.
rewrite H18 in H19.
simpl in H19.
auto.
Qed.
(* Verification of $di \neq (y_{1},y_{1},y_{0};k,y_{1}y_{0},y_{1}y_{0})$ *)
Theorem deNotEqualToThatOtherEdge: forall (d:Days), (eqTriangleY d y1y0_y1y0_k de)=false.
Proof.
intro.
destruct d;auto.
Qed.
Theorem FVRespectsEq: forall (delt:Delta10), unaryFunctionRespectsEquality eqVX eqVY (Fv delt).
Proof.
unfold unaryFunctionRespectsEquality.
intros.
destruct d; destruct elem1;destruct elem2; destruct delt;trivial.
Qed.
Theorem FERespectsEq: forall (delt:Delta11), unaryFunctionRespectsEquality eqEdgeX eqEdgeY (Fe delt).
Proof.
unfold unaryFunctionRespectsEquality.
intros.
destruct d; destruct elem1;destruct elem2; destruct delt;trivial.
Qed.
Theorem FTRespectsEq: forall (delt:Delta12), unaryFunctionRespectsEquality eqTriangleX eqTriangleY (Ft delt).
Proof.
unfold unaryFunctionRespectsEquality.
intros.
destruct d; destruct elem1;destruct elem2; destruct delt;trivial.
Qed.
Theorem fill10YProp: forall (d:Days)(e1 e2:EdgeY), ((eqVY d (d1Y e1) (d1Y e2))=true)->
(eqEdgeY d (dp1Y (fill10Y e1 e2)) e1)=true /\
(eqEdgeY d (dp2Y (fill10Y e1 e2)) e2)=true.
Proof.
intros.
split;case d;destruct e1;destruct e2;simpl;trivial.
Qed.
Theorem fill10YRespectEquality: binaryFunctionRespectsEquality eqEdgeY eqTriangleY fill10Y.
Proof.
unfold binaryFunctionRespectsEquality.
intros.
destruct H.
destruct d;destruct elem1; destruct elem1p;auto with bool;destruct elem2;destruct elem2p;trivial.
Qed.
Theorem fill11YRespectEquality: binaryFunctionRespectsEquality eqEdgeY eqTriangleY fill11Y.
Proof.
unfold binaryFunctionRespectsEquality.
intros.
destruct H.
destruct d;destruct elem1; destruct elem1p;auto with bool;destruct elem2;destruct elem2p;trivial.
Qed.
Theorem fill11YProp: forall (d:Days)(e1 e2:EdgeY), ((eqVY d (d1Y e1) (d0Y e2))=true)->
(eqEdgeY d (dp0Y (fill11Y e1 e2)) e1)=true /\
(eqEdgeY d (dp2Y (fill11Y e1 e2)) e2)=true.
Proof.
intros.
split;destruct d;destruct e1;destruct e2;trivial.
Qed.
Theorem fill12YRespectEquality: binaryFunctionRespectsEquality eqEdgeY eqTriangleY fill12Y.
Proof.
unfold binaryFunctionRespectsEquality.
intros.
destruct H.
destruct d;destruct elem1; destruct elem1p;auto with bool;destruct elem2;destruct elem2p;trivial.
Qed.
Theorem fill12YProp: forall (d:Days)(e0 e1:EdgeY), ((eqVY d (d0Y e0) (d0Y e1))=true)-> (eqEdgeY d (dp0Y (fill12Y e0 e1)) e0)=true /\
(eqEdgeY d (dp1Y (fill12Y e0 e1)) e1)=true.
Proof.
intros.
split;destruct d;destruct e0;destruct e1;trivial.
Qed.
Theorem fill20YProp: forall (d:Days)(t1 t2 t3:TriangleY), (eqEdgeY d (dp1Y t1) (dp1Y t2))=true /\
(eqEdgeY d (dp2Y t1) (dp1Y t3))=true /\
(eqEdgeY d (dp2Y t2) (dp2Y t3))=true ->
(
(eqEdgeY d (dp0Y t1) (dp0Y (fill20Y t1 t2 t3)))=true /\
(eqEdgeY d (dp0Y t2) (dp1Y (fill20Y t1 t2 t3)))=true /\
(eqEdgeY d (dp0Y t3) (dp2Y (fill20Y t1 t2 t3)))=true).
Proof.
intros.
destruct H.
destruct H0.
destruct d.
destruct t1;destruct t2;destruct t3; (split; [trivial | split;trivial]).
destruct t1;destruct t2;destruct t3; (split; [trivial | split;trivial]).
Qed.
Definition eqTriangleYreflexive := proj1 YModel_obligation_3.
Definition eqTriangleYTrans := proj2 (proj2 YModel_obligation_3).
Lemma fill20YRespectsEqualty1: forall (d:Days)(t1 t2 t3:TriangleY) (t1p:TriangleY),
(eqTriangleY d t1 t1p)=true -> (eqTriangleY d (fill20Y t1 t2 t3) (fill20Y t1p t2 t3))=true.
destruct d.
destruct t1; destruct t1p;(auto with bool || trivial using eqTriangleYreflexive);trivial.
destruct t1; destruct t1p;(auto with bool || trivial using eqTriangleYreflexive);destruct t2; (trivial using eqTriangleYreflexive) ;destruct t3;trivial.
Qed.
Hint Resolve fill20YRespectsEqualty1.
Lemma fill20YRespectsEqualty2: forall (d:Days)(t1 t2 t3:TriangleY) (t2p:TriangleY),
(eqTriangleY d t2 t2p)=true -> (eqTriangleY d (fill20Y t1 t2 t3) (fill20Y t1 t2p t3))=true.
intros.
destruct d.
destruct t2; destruct t2p;(auto with bool || trivial using eqTriangleYreflexive);trivial.
destruct t2; destruct t2p;(auto with bool || trivial using eqTriangleYreflexive);destruct t1; (trivial using eqTriangleYreflexive) ;destruct t3;trivial.
Qed.
Hint Resolve fill20YRespectsEqualty2.
Lemma fill20YRespectsEqualty3: forall (d:Days)(t1 t2 t3:TriangleY) (t3p:TriangleY),
(eqTriangleY d t3 t3p)=true -> (eqTriangleY d (fill20Y t1 t2 t3) (fill20Y t1 t2 t3p))=true.
intros.
destruct d.
destruct t3; destruct t3p;(auto with bool || trivial using eqTriangleYreflexive);trivial.
destruct t3; destruct t3p;(auto with bool || trivial using eqTriangleYreflexive);destruct t2; (trivial using eqTriangleYreflexive) ;destruct t1;trivial.
Qed.
Hint Resolve fill20YRespectsEqualty3.
Theorem fill20YRespectsEqualty: tertaryFunctionRespectsEquality eqTriangleY eqTriangleY fill20Y.
Proof.
apply TertaryFunctionRespectsEqualtyPointwise;auto.
apply YModel_obligation_3.
Qed.
Theorem fill21YProp: forall (d:Days)(t1 t2 t3:TriangleY),
(eqEdgeY d (dp1Y t1) (dp0Y t2))=true /\
(eqEdgeY d (dp2Y t1) (dp0Y t3))=true /\
(eqEdgeY d (dp2Y t2) (dp2Y t3))=true ->
((eqEdgeY d (dp0Y t1) (dp0Y (fill21Y t1 t2 t3)))=true /\
(eqEdgeY d (dp1Y t2) (dp1Y (fill21Y t1 t2 t3)))=true /\
(eqEdgeY d (dp1Y t3) (dp2Y (fill21Y t1 t2 t3)))=true).
Proof.
intros.
destruct H.
destruct H0.
destruct d.
destruct t1;destruct t2;destruct t3; (split; [trivial | split;trivial]).
destruct t1;destruct t2;destruct t3; (split; [trivial | split;trivial]).
Qed.
Lemma fill21YRespectsEqualty1: forall (d:Days)(t1 t2 t3:TriangleY) (t1p:TriangleY),
(eqTriangleY d t1 t1p)=true -> (eqTriangleY d (fill21Y t1 t2 t3) (fill21Y t1p t2 t3))=true.
intros.
destruct d.