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[ refactor ] introduce NonZero

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This has a much better behaviour with respect to proof search and
the coverage checker realising we don't need to consider the Z case
than the `Not (x = Z)` we used earlier.
This commit is contained in:
Guillaume ALLAIS 2021-03-30 15:20:43 +01:00 committed by G. Allais
parent 09d8e25441
commit 5af1efb56e
3 changed files with 36 additions and 33 deletions
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35
libs/base/Data/Nat.idr
View File

@ -241,9 +241,18 @@ export
succInjective : (0 left, right : Nat) -> S left = S right -> left = right
succInjective _ _ Refl = Refl
export total
SIsNotZ : (S x = Z) -> Void
SIsNotZ Refl impossible
||| A definition of non-zero with a better behaviour than `Not (x = Z)`
||| This is amenable to proof search and `NonZero Z` is more readily
||| detected as impossible by Idris
public export
data NonZero : Nat -> Type where
SIsNonZero : NonZero (S x)
export Uninhabited (NonZero Z) where uninhabited SIsNonZero impossible
export
SIsNotZ : Not (S x = Z)
SIsNotZ = absurd
||| Auxiliary function:
||| mod' fuel a b = a `mod` (S b)
@ -258,13 +267,13 @@ mod' (S fuel) centre right =
mod' fuel (minus centre (S right)) right
public export
modNatNZ : Nat -> (y: Nat) -> Not (y = Z) -> Nat
modNatNZ left Z p = void (p Refl)
modNatNZ : Nat -> (y: Nat) -> (0 _ : NonZero y) -> Nat
modNatNZ left Z p = void (absurd p)
modNatNZ left (S right) _ = mod' left left right
export partial
modNat : Nat -> Nat -> Nat
modNat left (S right) = modNatNZ left (S right) SIsNotZ
modNat left (S right) = modNatNZ left (S right) SIsNonZero
||| Auxiliary function:
||| div' fuel a b = a `div` (S b)
@ -280,23 +289,22 @@ div' (S fuel) centre right =
-- 'public' to allow type-level division
public export
divNatNZ : Nat -> (y: Nat) -> Not (y = Z) -> Nat
divNatNZ left Z p = void (p Refl)
divNatNZ : Nat -> (y: Nat) -> (0 _ : NonZero y) -> Nat
divNatNZ left (S right) _ = div' left left right
export partial
divNat : Nat -> Nat -> Nat
divNat left (S right) = divNatNZ left (S right) SIsNotZ
divNat left (S right) = divNatNZ left (S right) SIsNonZero
export partial
divCeilNZ : Nat -> (y: Nat) -> Not (y = Z) -> Nat
divCeilNZ : Nat -> (y: Nat) -> (0 _ : NonZero y) -> Nat
divCeilNZ x y p = case (modNatNZ x y p) of
Z => divNatNZ x y p
S _ => S (divNatNZ x y p)
export partial
divCeil : Nat -> Nat -> Nat
divCeil x (S y) = divCeilNZ x (S y) SIsNotZ
divCeil x (S y) = divCeilNZ x (S y) SIsNonZero
public export
@ -310,8 +318,7 @@ divmod' (S fuel) centre right =
in (S (fst qr), snd qr)
public export
divmodNatNZ : Nat -> (y: Nat) -> Not (y = Z) -> (Nat, Nat)
divmodNatNZ left Z p = void (p Refl)
divmodNatNZ : Nat -> (y: Nat) -> (0 _ : NonZero y) -> (Nat, Nat)
divmodNatNZ left (S right) _ = divmod' left left right
@ -324,7 +331,7 @@ export partial
gcd : (a: Nat) -> (b: Nat) -> {auto ok: NotBothZero a b} -> Nat
gcd a Z = a
gcd Z b = b
gcd a (S b) = gcd (S b) (modNatNZ a (S b) SIsNotZ)
gcd a (S b) = gcd (S b) (modNatNZ a (S b) SIsNonZero)
export partial
lcm : Nat -> Nat -> Nat

30
libs/contrib/Data/Nat/Division.idr
View File

@ -74,20 +74,19 @@ mod''_eq_mod' fuel numer denom = cong snd $
divmod'_eq_div'_mod' fuel numer denom
export
divmodNatNZeqDivMod : (numer, denom : Nat) -> (prf1, prf2, prf3 : Not (denom = 0))
divmodNatNZeqDivMod : (numer, denom : Nat) -> (0 prf1, prf2, prf3 : NonZero denom)
-> (divmodNatNZ numer denom prf1) = (divNatNZ numer denom prf2, modNatNZ numer denom prf3)
divmodNatNZeqDivMod numer 0 prf1 prf2 prf3 = void $ prf1 Refl
divmodNatNZeqDivMod numer (S denom) prf1 prf2 prf3 = divmod'_eq_div'_mod' numer numer denom
export
fstDivmodNatNZeqDiv : (numer, denom : Nat) -> (prf1, prf2 : Not (denom = 0))
fstDivmodNatNZeqDiv : (numer, denom : Nat) -> (0 prf1, prf2 : NonZero denom)
-> (fst $ divmodNatNZ numer denom prf1) = divNatNZ numer denom prf2
fstDivmodNatNZeqDiv numer denom prf1 prf2 =
rewrite divmodNatNZeqDivMod numer denom prf1 prf2 prf2 in
Refl
export
sndDivmodNatNZeqMod : (numer, denom : Nat) -> (prf1, prf2 : Not (denom = 0))
sndDivmodNatNZeqMod : (numer, denom : Nat) -> (0 prf1, prf2 : NonZero denom)
-> (snd $ divmodNatNZ numer denom prf1) = modNatNZ numer denom prf2
sndDivmodNatNZeqMod numer denom prf1 prf2 =
rewrite divmodNatNZeqDivMod numer denom prf1 prf2 prf2 in
@ -106,9 +105,8 @@ bound_mod'' (S fuel) numer predDenom enough = case @@(Data.Nat.lte numer predDe
(fuelLemma numer predDenom fuel enough numer_gte_n)
export
boundModNatNZ : (numer, denom : Nat) -> (denom_nz : Not (denom = 0))
boundModNatNZ : (numer, denom : Nat) -> (0 denom_nz : NonZero denom)
-> (modNatNZ numer denom denom_nz) `LT` denom
boundModNatNZ numer 0 denom_nz = void $ denom_nz Refl
boundModNatNZ numer (S predDenom) denom_nz = LTESucc $
rewrite sym $ mod''_eq_mod' numer numer predDenom in
bound_mod'' numer numer predDenom (reflexive numer)
@ -152,15 +150,14 @@ divisionTheorem' numer predDenom (S fuel) enough with (@@(Data.Nat.lte numer pr
export
DivisionTheoremDivMod : (numer, denom : Nat) -> (prf : Not (denom = Z))
DivisionTheoremDivMod : (numer, denom : Nat) -> (0 prf : NonZero denom)
-> numer = snd ( divmodNatNZ numer denom prf)
+ (fst $ divmodNatNZ numer denom prf)*denom
DivisionTheoremDivMod numer 0 prf = void (prf Refl)
DivisionTheoremDivMod numer (S predDenom) prf
= divisionTheorem' numer predDenom numer (reflexive numer)
export
DivisionTheorem : (numer, denom : Nat) -> (prf1, prf2 : Not (denom = Z))
DivisionTheorem : (numer, denom : Nat) -> (0 prf1, prf2 : NonZero denom)
-> numer = (modNatNZ numer denom prf1) + (divNatNZ numer denom prf2)*denom
DivisionTheorem numer denom prf1 prf2
= rewrite sym $ fstDivmodNatNZeqDiv numer denom prf1 prf2 in
@ -299,13 +296,12 @@ addMultipleMod' (S fuel1) fuel2 predn a (S k) enough1 enough2 =
fuelLemma ((1+k)*n + a) predn fuel1 enough1 prf1)
enough2
addMultipleMod : (a, b, n : Nat) -> (n_neq_z1, n_neq_z2 : Not (n = 0))
addMultipleMod : (a, b, n : Nat) -> (0 n_neq_z1, n_neq_z2 : NonZero n)
-> snd (divmodNatNZ (a*n + b) n n_neq_z1) = snd (divmodNatNZ b n n_neq_z2)
addMultipleMod a b 0 n_neq_z1 n_neq_z2 = void (n_neq_z1 Refl)
addMultipleMod a b n@(S predn) n_neq_z1 n_neq_z2 =
addMultipleMod' (a*n + b) b predn b a (reflexive {po = LTE} _) (reflexive {po = LTE} _)
modBelowDenom : (r, n : Nat) -> (n_neq_z : Not (n = 0))
modBelowDenom : (r, n : Nat) -> (0 n_neq_z : NonZero n)
-> (r `LT` n)
-> snd (divmodNatNZ r n n_neq_z) = r
modBelowDenom 0 (S predn) n_neq_0 (LTESucc r_lte_predn) = Refl
@ -313,7 +309,7 @@ modBelowDenom r@(S _) (S predn) n_neq_0 (LTESucc r_lte_predn) =
rewrite LteIslte r predn r_lte_predn in
Refl
modInjective : (r1, r2, n : Nat) -> (n_neq_z1, n_neq_z2 : Not (n = 0))
modInjective : (r1, r2, n : Nat) -> (0 n_neq_z1, n_neq_z2 : NonZero n)
-> (r1 `LT` n)
-> (r2 `LT` n)
-> snd (divmodNatNZ r1 n n_neq_z1) = snd (divmodNatNZ r2 n n_neq_z2)
@ -325,7 +321,7 @@ modInjective r1 r2 n n_neq_z1 n_neq_z2 r1_lt_n r2_lt_n ri_mod_eq = Calc $
~~ r2 ...( modBelowDenom r2 n n_neq_z2 r2_lt_n)
step1 : (numer : Nat) -> (denom : Nat) -> (denom_nz : Not (denom = 0))
step1 : (numer : Nat) -> (denom : Nat) -> (0 denom_nz : NonZero denom)
-> (q, r : Nat) -> (r `LT` denom) -> (numer = q * denom + r)
-> snd (divmodNatNZ numer denom denom_nz) = r
step1 x n n_nz q r r_lt_n x_eq_qnpr = Calc $
@ -334,7 +330,7 @@ step1 x n n_nz q r r_lt_n x_eq_qnpr = Calc $
~~ snd(divmodNatNZ r n n_nz) ...(addMultipleMod q r n n_nz n_nz)
~~ r ...(modBelowDenom r n n_nz r_lt_n)
step2 : (numer : Nat) -> (denom : Nat) -> (denom_nz : Not (denom = 0))
step2 : (numer : Nat) -> (denom : Nat) -> (0 denom_nz : NonZero denom)
-> (q, r : Nat) -> (r `LT` denom) -> (numer = q * denom + r)
-> fst (divmodNatNZ numer denom denom_nz) = q
step2 x n n_nz q r r_lt_n x_eq_qnr =
@ -358,7 +354,7 @@ step2 x n n_nz q r r_lt_n x_eq_qnr =
$ two_decompositions
export
DivisionTheoremUniquenessDivMod : (numer : Nat) -> (denom : Nat) -> (denom_nz : Not (denom = 0))
DivisionTheoremUniquenessDivMod : (numer : Nat) -> (denom : Nat) -> (0 denom_nz : NonZero denom)
-> (q, r : Nat) -> (r `LT` denom) -> (numer = q * denom + r)
-> divmodNatNZ numer denom denom_nz = (q, r)
DivisionTheoremUniquenessDivMod numer denom denom_nz q r x prf =
@ -372,7 +368,7 @@ DivisionTheoremUniquenessDivMod numer denom denom_nz q r x prf =
pair_eta (x,y) = Refl
export
DivisionTheoremUniqueness : (numer : Nat) -> (denom : Nat) -> (denom_nz : Not (denom = 0))
DivisionTheoremUniqueness : (numer : Nat) -> (denom : Nat) -> (0 denom_nz : NonZero denom)
-> (q, r : Nat) -> (r `LT` denom) -> (numer = q * denom + r)
-> (divNatNZ numer denom denom_nz = q, modNatNZ numer denom denom_nz = r)
DivisionTheoremUniqueness numer denom denom_nz q r x prf =

4
libs/contrib/Data/Nat/Properties.idr
View File

@ -19,8 +19,8 @@ unfoldDoubleS = irrelevantEq $ Calc $
~~ 2 + 2 * n ...( cong (2 +) (sym unfoldDouble) )
export
multRightCancel : (a,b,r : Nat) -> Not (r = 0) -> a*r = b*r -> a = b
multRightCancel a b 0 r_nz ar_eq_br = void $ r_nz Refl
multRightCancel : (a,b,r : Nat) -> (0 _ : NonZero r) -> a*r = b*r -> a = b
multRightCancel a b 0 r_nz ar_eq_br = void (absurd r_nz)
multRightCancel 0 0 r@(S predr) r_nz ar_eq_br = Refl
multRightCancel 0 (S b) r@(S predr) r_nz ar_eq_br impossible
multRightCancel (S a) 0 r@(S predr) r_nz ar_eq_br impossible