discriminant3

# Library discriminant3

`Require Export discriminant2.
Section Discriminant1.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Theorem TwoMoreThanOne : (1 < radix)%Z.
Hint Resolve TwoMoreThanOne.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanThree : 3 <= precision.
Theorem RoundLeNormal: forall f:float, forall r:R,
Closest bo radix r f -> (Fnormal radix bo f) ->
(Rabs f <= Rabs r / (1 - powerRZ radix (- precision)))%R.
Variables a b b' c p q t d u v dp dq:float.
Let delta := (Rabs (d-(b*b'-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fb': (Fbounded bo b').
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fp : (Fbounded bo p).
Hypothesis Fq : (Fbounded bo q).
Hypothesis Fd : (Fbounded bo d).
Hypothesis Fu : (Fbounded bo u).
Hypothesis Fv : (Fbounded bo v).
Hypothesis Cand : (Fcanonic radix bo d).
`
There is no underflow
`Hypothesis Nq:(Fnormal radix bo q).
Hypothesis Np:(Fnormal radix bo p).
Hypothesis Nv:(Fnormal radix bo v).
Hypothesis Nu:(Fnormal radix bo u).
Hypothesis U0: (- dExp bo <= Fexp p - 2)%Z.
Hypothesis Square:(0 <=b*b')%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b')%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis dDef : d=t.
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis dpEq : (FtoRradix dp=b*b'-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis Case1 : (3*(Rabs (p-q)) < p+q )%R.
Hypothesis Case2 : (v <= u )%R.
Theorem IneqEq: (FtoRradix v=u)%R.
Theorem dexact: (FtoRradix d=p-q)%R.
Theorem discri10: (q <= p)%R -> (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant1.
Section Discriminant2.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanThree : 3 <= precision.
Variables a b b' c p q t d u v dp dq:float.
Let delta := (Rabs (d-(b*b'-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fb': (Fbounded bo b').
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fp : (Fbounded bo p).
Hypothesis Fq : (Fbounded bo q).
Hypothesis Fd : (Fbounded bo d).
Hypothesis Fu : (Fbounded bo u).
Hypothesis Fv : (Fbounded bo v).
Hypothesis Cand : (Fcanonic radix bo d).
`
There is no underflow
`Hypothesis Nq:(Fnormal radix bo q).
Hypothesis Np:(Fnormal radix bo p).
Hypothesis Nv:(Fnormal radix bo v).
Hypothesis Nu:(Fnormal radix bo u).
Hypothesis U0: (- dExp bo <= Fexp p - 2)%Z.
Hypothesis U1: (- dExp bo <= Fexp q - 2)%Z.
Hypothesis Square:(0 <=b*b')%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b')%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis dDef : d=t.
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis dpEq : (FtoRradix dp=b*b'-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis Case1 : (3*(Rabs (p-q)) < p+q )%R.
Hypothesis Case2 : (v <= u )%R.
Theorem discri11: (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant2.
Section Discriminant3.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanFour : 4 <= precision.
Variables a b b' c p q t dp dq s d u v:float.
Let delta := (Rabs (d-(b*b'-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fb': (Fbounded bo b').
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fp : (Fbounded bo p).
Hypothesis Fq : (Fbounded bo q).
Hypothesis Fd : (Fbounded bo d).
Hypothesis Ft : (Fbounded bo t).
Hypothesis Fs : (Fbounded bo s).
Hypothesis Fdp: (u<v)%R -> (Fbounded bo dp).
Hypothesis Fdq: (u<v)%R -> (Fbounded bo dq).
Hypotheses Cv: Fcanonic radix bo v.
Hypothesis Cs:(Fcanonic radix bo s).
`
There is no underflow
`Hypothesis U1: (- dExp bo <= (Fexp t)-1)%Z.
Hypothesis U2: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (b*b'))%R.
Hypothesis U3: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (a*c))%R.
Hypothesis Np:(Fnormal radix bo p).
Hypothesis Nq:(Fnormal radix bo q).
Hypothesis Nd:(Fnormal radix bo d).
Hypothesis Nt:(Fnormal radix bo t).
Hypothesis Nu:(Fnormal radix bo u).
Hypothesis Nv:(Fnormal radix bo v).
Hypothesis Square:(0 <=b*b')%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b')%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis Case1 : (p+q <= 3*(Rabs (p-q)))%R.
Hypothesis Case2 : (u < v )%R.
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis dpEq : (FtoRradix dp=b*b'-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis Rounds : (EvenClosest bo radix precision (dp-dq)%R s).
Hypothesis Roundd : (EvenClosest bo radix precision (t+s)%R d).
Theorem RoundGeNormal: forall f:float, forall r:R,
Closest bo radix r f -> (Fnormal radix bo f) ->
(Rabs r <= Rabs f * (1 + powerRZ radix (- precision)))%R.
Theorem discri12: (q <= p)%R -> (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant3.
Section Discriminant4.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanFour : 4 <= precision.
Variables a b b' c p q t dp dq s d u v:float.
Let delta := (Rabs (d-(b*b'-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fb': (Fbounded bo b').
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fp : (Fbounded bo p).
Hypothesis Fq : (Fbounded bo q).
Hypothesis Fd : (Fbounded bo d).
Hypothesis Ft : (Fbounded bo t).
Hypothesis Fs : (Fbounded bo s).
Hypothesis Fdp: (u<v)%R -> (Fbounded bo dp).
Hypothesis Fdq: (u<v)%R -> (Fbounded bo dq).
Hypotheses Cv: Fcanonic radix bo v.
Hypotheses Cs: Fcanonic radix bo s.
`
There is no underflow
`Hypothesis U1: (- dExp bo <= (Fexp t)-1)%Z.
Hypothesis U2: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (b*b'))%R.
Hypothesis U3: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (a*c))%R.
Hypothesis Np:(Fnormal radix bo p).
Hypothesis Nq:(Fnormal radix bo q).
Hypothesis Nd:(Fnormal radix bo d).
Hypothesis Nt:(Fnormal radix bo t).
Hypothesis Nu:(Fnormal radix bo u).
Hypothesis Nv:(Fnormal radix bo v).
Hypothesis Square:(0 <=b*b')%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b')%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis Case1 : (p+q <= 3*(Rabs (p-q)))%R.
Hypothesis Case2 : (u < v )%R.
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis dpEq : (FtoRradix dp=b*b'-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis Rounds : (EvenClosest bo radix precision (dp-dq)%R s).
Hypothesis Roundd : (EvenClosest bo radix precision (t+s)%R d).
Theorem discri13: (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant4.
Section Discriminant5.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanFour : 4 <= precision.
Variables a b c p q t dp dq s d u v:float.
Let delta := (Rabs (d-(b*b-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fp : (Fbounded bo p).
Hypothesis Fq : (Fbounded bo q).
Hypothesis Fd : (Fbounded bo d).
Hypothesis Ft : (Fbounded bo t).
Hypothesis Fs : (u<v)%R -> (Fbounded bo s).
Hypothesis Fdp: (u<v)%R -> (Fbounded bo dp).
Hypothesis Fdq: (u<v)%R -> (Fbounded bo dq).
Hypothesis Fu: (Fbounded bo u).
Hypothesis Fv: (Fbounded bo v).
Hypothesis Cs: (u < v)%R -> (Fcanonic radix bo s).
`
There is no underflow
`Hypothesis U0: (- dExp bo <= Fexp d - 1)%Z.
Hypothesis U1: (- dExp bo <= (Fexp t)-1)%Z.
Hypothesis U2: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (b*b))%R.
Hypothesis U3: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (a*c))%R.
Hypothesis Np:(Fnormal radix bo p).
Hypothesis Nq:(Fnormal radix bo q).
Hypothesis Nd:(Fnormal radix bo d).
Hypothesis Nu:(Fnormal radix bo u).
Hypothesis Nv:(Fnormal radix bo v).
Hypothesis Nt:(Fnormal radix bo t).
Hypothesis Roundp : (EvenClosest bo radix precision (b*b)%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis FRoundd : (v <= u)%R ->
(EvenClosest bo radix precision (p-q)%R d).
Hypothesis dpEq : (FtoRradix dp=b*b-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis SRounds : (u < v)%R -> (EvenClosest bo radix precision (dp-dq)%R s).
Hypothesis SRoundd : (u < v)%R -> (EvenClosest bo radix precision (t+s)%R d).
Theorem discri14: (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant5.
Section Discriminant6.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanFour : 4 <= precision.
Variables a b c p q t dp dq s d u v:float.
Let delta := (Rabs (d-(b*b-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fdp: (u<v)%R -> (Fbounded bo dp).
Hypothesis Fdq: (u<v)%R -> (Fbounded bo dq).
`
There is no underflow
`
Hypothesis U1: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (b*b))%R.
Hypothesis U2: (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (a*c))%R.
Hypothesis U4: (powerRZ radix (-dExp bo+precision) <= Rabs d)%R.
Hypothesis U5: (powerRZ radix (-dExp bo+precision-1) <= Rabs u)%R.
Hypothesis U6: (powerRZ radix (-dExp bo+precision-1) <= Rabs v)%R.
Hypothesis U7: (powerRZ radix (-dExp bo+precision) <= Rabs t)%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b)%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis FRoundd : (v <= u)%R ->
(EvenClosest bo radix precision (p-q)%R d).
Hypothesis dpEq : (FtoRradix dp=b*b-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis SRounds : (u < v)%R -> (EvenClosest bo radix precision (dp-dq)%R s).
Hypothesis SRoundd : (u < v)%R -> (EvenClosest bo radix precision (t+s)%R d).
Theorem discri15: (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant6.
Section Discriminant7.
Variable bo : Fbound.
Variable precision : nat.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ TwoMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum bo) = Zpower_nat radix precision.
Hypothesis precisionGreaterThanFour : 4 <= precision.
Theorem FexpGeUnderf: forall e:Z, forall f:float,
(Fbounded bo f) ->
((powerRZ radix e) <= Rabs f)%R -> (e-precision+1 <= Fexp f)%Z.
Theorem AddExpGeUnderf: forall f1:float ,forall f2:float, forall g:float, forall e:Z,
Closest bo radix (f1+f2) g -> (Fbounded bo f1) -> (Fbounded bo f2)
-> (powerRZ radix e <= Rabs f1)%R
-> (powerRZ radix e <= Rabs f2)%R
-> ((FtoRradix g=0)%R \/ (powerRZ radix (e-precision+1) <= Rabs g)%R).
Theorem AddExpGeUnderf2: forall f1:float ,forall f2:float, forall g:float, forall e:Z,
Closest bo radix (f1+f2) g -> (Fbounded bo f1) -> (Fbounded bo f2)
-> (powerRZ radix e <= Rabs f1)%R
-> (powerRZ radix e <= Rabs f2)%R
-> (FtoRradix g <>0)%R
-> (powerRZ radix (e-precision+1) <= Rabs g)%R.
Theorem AddExpGe1Underf: forall f1:float ,forall f2:float, forall g:float, forall e:Z,
Closest bo radix (f1+f2) g -> (Fcanonic radix bo f1) -> (Fcanonic radix bo f2)
-> (powerRZ radix e <= Rabs f1)%R
-> (-dExp bo <= e-1)%Z
-> ((FtoRradix g=0)%R \/ (powerRZ radix (e-precision) <= Rabs g)%R).
Theorem AddExpGe1Underf2: forall f1:float ,forall f2:float, forall g:float, forall e:Z,
Closest bo radix (f1+f2) g -> (Fbounded bo f1) -> (Fbounded bo f2)
-> (powerRZ radix e <= Rabs f1)%R
-> (-dExp bo <= e-1)%Z
-> (FtoRradix g <>0)%R
-> (powerRZ radix (e-precision) <= Rabs g)%R.
Variables a b c p q t dp dq s d u v:float.
Let delta := (Rabs (d-(b*b-a*c)))%R.
Hypothesis Fa : (Fbounded bo a).
Hypothesis Fb : (Fbounded bo b).
Hypothesis Fc : (Fbounded bo c).
Hypothesis Fdp: (u < v)%R -> (Fbounded bo dp).
Hypothesis Fdq: (u < v)%R -> (Fbounded bo dq).
`
There is no underflow
`
Hypothesis U1: (FtoRradix b=0)%R \/
(powerRZ radix (-dExp bo+3*precision-1) <= Rabs (b*b))%R.
Hypothesis U2: (a*c=0)%R \/
(powerRZ radix (-dExp bo+3*precision-1) <= Rabs (a*c))%R.
Hypothesis Roundp : (EvenClosest bo radix precision (b*b)%R p).
Hypothesis Roundq : (EvenClosest bo radix precision (a*c)%R q).
Hypothesis Roundt : (EvenClosest bo radix precision (p-q)%R t).
Hypothesis Roundu : (EvenClosest bo radix precision (3*Rabs t)%R u).
Hypothesis Roundv : (EvenClosest bo radix precision (p+q)%R v).
Hypothesis FRoundd : (v <= u)%R ->
(EvenClosest bo radix precision (p-q)%R d).
Hypothesis dpEq : (FtoRradix dp=b*b-p)%R.
Hypothesis dqEq : (FtoRradix dq=a*c-q)%R.
Hypothesis SRounds : (u < v)%R -> (EvenClosest bo radix precision (dp-dq)%R s).
Hypothesis SRoundd : (u < v)%R -> (EvenClosest bo radix precision (t+s)%R d).
Theorem pGeUnderf: (FtoRradix b <> 0)%R ->
(powerRZ radix (-dExp bo+3*precision-1) <= Rabs (p))%R.
Theorem qGeUnderf: (a*c <> 0)%R ->
(powerRZ radix (-dExp bo+3*precision-1) <= Rabs (q))%R.
Theorem cases: (FtoRradix b=0)%R \/ (a*c=0)%R
\/ (FtoRradix d=0)%R \/ (FtoRradix v=0)%R \/ (FtoRradix t=0)%R \/
((powerRZ radix (-dExp bo+2*precision-1) <= Rabs (b*b))%R
/\ (powerRZ radix (-dExp bo+2*precision-1) <= Rabs (a*c))%R
/\ (powerRZ radix (-dExp bo+precision) <= Rabs d)%R
/\ (powerRZ radix (-dExp bo+precision-1) <= Rabs u)%R
/\ (powerRZ radix (-dExp bo+precision-1) <= Rabs v)%R
/\ (powerRZ radix (-dExp bo+precision) <= Rabs t)%R).
Theorem discri16: (FtoRradix d=0)%R \/ (delta <= 2*(Fulp bo radix precision d))%R.
End Discriminant7.
`