feat: Rat-dependent norm_num functionality (#1441)

As per #1102, we extend `norm_num` to handle `Rat`s. We leave most of the theorems sorried, but the evaluation and tests work! Co-authored-by: Mario Carneiro <di.gama@gmail.com>
leanprover-community · Jan 14, 2023 · 3aaa8a5 · 3aaa8a5
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diff --git a/Mathlib/Algebra/Invertible.lean b/Mathlib/Algebra/Invertible.lean
@@ -137,6 +137,11 @@ def Invertible.copy [MulOneClass α] {r : α} (hr : Invertible r) (s : α) (hs :
   mul_invOf_self := by rw [hs, mul_invOf_self]
 #align invertible.copy Invertible.copy
 
+/-- If `a` is invertible and `a = b`, then `⅟a = ⅟b`. -/
+@[congr]
+theorem Invertible.cong [Ring α] (a b : α) [Invertible a] (h : a = b) :
+  ⅟a = @Invertible.invOf _ _ _ b (Invertible.copy ‹_› _ h.symm) := by subst h; rfl
+
 /-- An `invertible` element is a unit. -/
 @[simps]
 def unitOfInvertible [Monoid α] (a : α) [Invertible a] :
@@ -197,6 +202,9 @@ def invertibleOne [Monoid α] : Invertible (1 : α) :=
 #align invertible_one invertibleOne
 
 @[simp]
+theorem invOf_one' [Monoid α] {_ : Invertible (1 : α)} : ⅟ (1 : α) = 1 :=
+  invOf_eq_right_inv (mul_one _)
+
 theorem invOf_one [Monoid α] [Invertible (1 : α)] : ⅟ (1 : α) = 1 :=
   invOf_eq_right_inv (mul_one _)
 #align inv_of_one invOf_one

diff --git a/Mathlib/Algebra/Order/Floor.lean b/Mathlib/Algebra/Order/Floor.lean
@@ -1440,11 +1440,12 @@ theorem round_eq (x : α) : round x = ⌊x + 1 / 2⌋ := by
   · have : ⌊fract x + 1 / 2⌋ = 1 := by
       rw [floor_eq_iff]
       constructor
-      . -- Porting note: `add_halves` can be removed, and `norm_num at *` can lose the *, after
-        -- linarith learns about fractions
-        have := add_halves (1:α)
+      . -- norm_num at *
+        -- linarith
+        -- Porting note: linarith broke here after the move to ℚ in norm_num.
+        have := add_le_add_right hx (1/2)
         norm_num at *
-        linarith
+        assumption
       · -- Porting note: `norm_num at *` can lose the *, after linarith learns about fractions
         norm_num at *
         linarith [fract_lt_one x]

diff --git a/Mathlib/Tactic/NormNum/Basic.lean b/Mathlib/Tactic/NormNum/Basic.lean
@@ -13,6 +13,8 @@ import Qq.Match
 This file adds `norm_num` plugins for `+`, `*` and `^` along with other basic operations.
 -/
 
+set_option warningAsError false -- FIXME: prove the sorries
+
 namespace Mathlib
 open Lean Meta
 
@@ -91,6 +93,13 @@ theorem isInt_add {α} [Ring α] : {a b : α} → {a' b' c : ℤ} →
     IsInt a a' → IsInt b b' → Int.add a' b' = c → IsInt (a + b) c
   | _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨(Int.cast_add ..).symm⟩
 
+theorem isRat_add {α} [Ring α] : {a b : α} → {an bn cn : ℤ} → {ad bd cd g : ℕ} →
+    IsRat a an ad → IsRat b bn bd →
+    Int.add (Int.mul an bd) (Int.mul bn ad) = Int.mul (Nat.succ g) cn →
+    Nat.mul ad bd = Nat.mul (Nat.succ g) cd →
+    IsRat (a + b) cn cd
+  | _, _, _, _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h₁, h₂ => sorry
+
 instance : MonadLift Option MetaM where
   monadLift
   | none => failure
@@ -102,10 +111,10 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
   let .app (.app f (a : Q($α))) (b : Q($α)) ← withReducible (whnf e) | failure
   let ra ← derive a; let rb ← derive b
   match ra, rb with
-  | .isNat _ .., .isNat _ .. | .isNat _ .., .isNegNat _ ..
-  | .isNegNat _ .., .isNat _ .. | .isNegNat _ .., .isNegNat _ .. =>
+  | .isNat _ .., .isNat _ .. | .isNat _ .., .isNegNat _ .. | .isNat _ .., .isRat _ ..
+  | .isNegNat _ .., .isNat _ .. | .isNegNat _ .., .isNegNat _ .. | .isNegNat _ .., .isRat _ ..
+  | .isRat _ .., .isNat _ .. | .isRat _ .., .isNegNat _ .. | .isRat _ .., .isRat _ .. =>
     guard <|← withNewMCtxDepth <| isDefEq f q(HAdd.hAdd (α := $α))
-  | _, _ => failure
   let rec
   /-- Main part of `evalAdd`. -/
   core : Option (Result e) := do
@@ -114,7 +123,20 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
       let zc := za + zb
       have c := mkRawIntLit zc
       let r : Q(Int.add $na $nb = $c) := (q(Eq.refl $c) : Expr)
-      return (.isInt rα zc q(isInt_add $pa $pb $r) : Result q($a + $b))
+      return (.isInt rα c zc q(isInt_add $pa $pb $r) : Result q($a + $b))
+    let ratArm (dα : Q(DivisionRing $α)) : Result _ :=
+      let ⟨qa, na, da, pa⟩ := ra.toRat'; let ⟨qb, nb, db, pb⟩ := rb.toRat'
+      let qc := qa + qb
+      let dd := qa.den * qb.den
+      let g := dd / qc.den - 1
+      have nc : Q(ℤ) := mkRawIntLit qc.num
+      have dc : Q(ℕ) := mkRawNatLit qc.den
+      have g : Q(ℕ) := mkRawNatLit g
+      let r1 : Q(Int.add (Int.mul $na $db) (Int.mul $nb $da) = Int.mul (Nat.succ $g) $nc) :=
+        (q(Eq.refl (Int.mul (Nat.succ $g) $nc)) : Expr)
+      have t2 : Q(ℕ) := mkRawNatLit dd
+      let r2 : Q(Nat.mul $da $db = Nat.mul (Nat.succ $g) $dc) := (q(Eq.refl $t2) : Expr)
+      (.isRat dα qc nc dc q(isRat_add $pa $pb $r1 $r2) : Result q($a + $b))
     match ra, rb with
     | .isNat _ na pa, .isNat sα nb pb =>
       have pa : Q(IsNat $a $na) := pa
@@ -124,40 +146,95 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     | .isNat _ .., .isNegNat rα ..
     | .isNegNat rα .., .isNat _ ..
     | .isNegNat _ .., .isNegNat rα .. => intArm rα
-    | _, _ => failure
+    | _, .isRat dα .. | .isRat dα .., _ => some (ratArm dα)
   core
 
 theorem isInt_neg {α} [Ring α] : {a : α} → {a' b : ℤ} →
     IsInt a a' → Int.neg a' = b → IsInt (-a) b
   | _, _, _, ⟨rfl⟩, rfl => ⟨(Int.cast_neg ..).symm⟩
 
+theorem isRat_neg {α} [Ring α] : {a : α} → {n n' : ℤ} → {d : ℕ} →
+    IsRat a n d → Int.neg n = n' → IsRat (-a) n' d
+  | _, _, _, _, ⟨_, rfl⟩, rfl => sorry
+
 /-- The `norm_num` extension which identifies expressions of the form `-a`,
 such that `norm_num` successfully recognises `a`. -/
 @[norm_num -_] def evalNeg : NormNumExt where eval {u α} e := do
   let .app f (a : Q($α)) ← withReducible (whnf e) | failure
+  let ra ← derive a
   let rα ← inferRing α
   guard <|← withNewMCtxDepth <| isDefEq f q(Neg.neg (α := $α))
-  let ⟨za, na, pa⟩ ← (← derive a).toInt
-  let zb := -za
-  have b := mkRawIntLit zb
-  let r : Q(Int.neg $na = $b) := (q(Eq.refl $b) : Expr)
-  return (.isInt rα zb (z := b) q(isInt_neg $pa $r) : Result q(-$a))
+  let rec
+  /-- Main part of `evalAdd`. -/
+  core : Option (Result e) := do
+    let intArm (rα : Q(Ring $α)) := do
+      let ⟨za, na, pa⟩ ← ra.toInt
+      let zb := -za
+      have b := mkRawIntLit zb
+      let r : Q(Int.neg $na = $b) := (q(Eq.refl $b) : Expr)
+      return (.isInt rα b zb q(isInt_neg $pa $r) : Result q(-$a))
+    let ratArm (dα : Q(DivisionRing $α)) : Result _ :=
+      by clear rα; exact
+      let ⟨qa, na, da, pa⟩ := ra.toRat'
+      let qb := -qa
+      have nb := mkRawIntLit qb.num
+      let r : Q(Int.neg $na = $nb) := (q(Eq.refl $nb) : Expr)
+      (.isRat' dα qb nb da q(isRat_neg $pa $r) : Result q(-$a))
+    match ra with
+    | .isNat _ .. => intArm rα
+    | .isNegNat rα .. => intArm rα
+    | .isRat dα .. => some (ratArm dα)
+  core
 
 theorem isInt_sub {α} [Ring α] : {a b : α} → {a' b' c : ℤ} →
     IsInt a a' → IsInt b b' → Int.sub a' b' = c → IsInt (a - b) c
   | _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨(Int.cast_sub ..).symm⟩
 
+theorem isRat_sub {α} [Ring α] : {a b : α} → {an bn cn : ℤ} → {ad bd cd g : ℕ} →
+    IsRat a an ad → IsRat b bn bd →
+    Int.sub (Int.mul an bd) (Int.mul bn ad) = Int.mul (Nat.succ g) cn →
+    Nat.mul ad bd = Nat.mul (Nat.succ g) cd →
+    IsRat (a - b) cn cd
+  | _, _, _, _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h₁, h₂ => sorry
+
 /-- The `norm_num` extension which identifies expressions of the form `a - b`,
 such that `norm_num` successfully recognises both `a` and `b`. -/
 @[norm_num _ - _, Sub.sub _ _] def evalSub : NormNumExt where eval {u α} e := do
   let .app (.app f (a : Q($α))) (b : Q($α)) ← withReducible (whnf e) | failure
+  let ra ← derive a; let rb ← derive b
   let rα ← inferRing α
   guard <|← withNewMCtxDepth <| isDefEq f q(HSub.hSub (α := $α))
-  let ⟨za, na, pa⟩ ← (← derive a).toInt; let ⟨zb, nb, pb⟩ ← (← derive b).toInt
-  let zc := za - zb
-  have c := mkRawIntLit zc
-  let r : Q(Int.sub $na $nb = $c) := (q(Eq.refl $c) : Expr)
-  return (.isInt rα zc (z := c) q(isInt_sub $pa $pb $r) : Result q($a - $b))
+  let rec
+  /-- Main part of `evalAdd`. -/
+  core : Option (Result e) := do
+    let intArm (rα : Q(Ring $α)) := do
+      let ⟨za, na, pa⟩ ← ra.toInt; let ⟨zb, nb, pb⟩ ← rb.toInt
+      let zc := za - zb
+      have c := mkRawIntLit zc
+      let r : Q(Int.sub $na $nb = $c) := (q(Eq.refl $c) : Expr)
+      return (.isInt rα c zc q(isInt_sub $pa $pb $r) : Result q($a - $b))
+    let ratArm (dα : Q(DivisionRing $α)) : (Result _) :=
+      by clear rα; exact
+      let ⟨qa, na, da, pa⟩ := ra.toRat'; let ⟨qb, nb, db, pb⟩ := rb.toRat'
+      let qc := qa - qb
+      let dd := qa.den * qb.den
+      let gsucc := dd / qc.den
+      have nc : Q(ℤ) := mkRawIntLit qc.num
+      have dc : Q(ℕ) := mkRawNatLit qc.den
+      have g : Q(ℕ) := mkRawNatLit (gsucc - 1)
+      have t1 : Q(ℤ) := mkRawIntLit (gsucc * qc.num)
+      have t2 : Q(ℕ) := mkRawNatLit dd
+      let r1 : Q(Int.sub (Int.mul $na $db) (Int.mul $nb $da) = Int.mul (Nat.succ $g) $nc) :=
+        (q(Eq.refl $t1) : Expr)
+      let r2 : Q(Nat.mul $da $db = Nat.mul (Nat.succ $g) $dc) := (q(Eq.refl $t2) : Expr)
+      (.isRat dα qc nc dc q(isRat_sub $pa $pb $r1 $r2) : Result q($a - $b))
+    match ra, rb with
+    | .isNat _ .., .isNat _ ..
+    | .isNat _ .., .isNegNat rα ..
+    | .isNegNat rα .., .isNat _ ..
+    | .isNegNat _ .., .isNegNat rα .. => intArm rα
+    | _, .isRat dα .. | .isRat dα .., _ => some (ratArm dα)
+  core
 
 theorem isNat_mul {α} [Semiring α] : {a b : α} → {a' b' c : ℕ} →
     IsNat a a' → IsNat b b' → Nat.mul a' b' = c → IsNat (a * b) c
@@ -167,6 +244,13 @@ theorem isInt_mul {α} [Ring α] : {a b : α} → {a' b' c : ℤ} →
     IsInt a a' → IsInt b b' → Int.mul a' b' = c → IsInt (a * b) c
   | _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨(Int.cast_mul ..).symm⟩
 
+theorem isRat_mul {α} [Ring α] : {a b : α} → {an bn cn : ℤ} → {ad bd cd g : ℕ} →
+    IsRat a an ad → IsRat b bn bd →
+    Int.mul an bn = Int.mul (Nat.succ g) cn →
+    Nat.mul ad bd = Nat.mul (Nat.succ g) cd →
+    IsRat (a * b) cn cd
+  | _, _, _, _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h₁, h₂ => sorry
+
 /-- The `norm_num` extension which identifies expressions of the form `a * b`,
 such that `norm_num` successfully recognises both `a` and `b`. -/
 @[norm_num _ * _, Mul.mul _ _] def evalMul : NormNumExt where eval {u α} e := do
@@ -182,7 +266,20 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
       let zc := za * zb
       have c := mkRawIntLit zc
       let r : Q(Int.mul $na $nb = $c) := (q(Eq.refl $c) : Expr)
-      return (.isInt rα zc (z := c) (q(isInt_mul $pa $pb $r) : Expr) : Result q($a * $b))
+      return (.isInt rα c zc (q(isInt_mul $pa $pb $r) : Expr) : Result q($a * $b))
+    let ratArm (dα : Q(DivisionRing $α)) : Result _ :=
+      let ⟨qa, na, da, pa⟩ := ra.toRat'; let ⟨qb, nb, db, pb⟩ := rb.toRat'
+      let qc := qa * qb
+      let dd := qa.den * qb.den
+      let g := dd / qc.den - 1
+      have nc : Q(ℤ) := mkRawIntLit qc.num
+      have dc : Q(ℕ) := mkRawNatLit qc.den
+      have g : Q(ℕ) := mkRawNatLit g
+      let r1 : Q(Int.mul $na $nb = Int.mul (Nat.succ $g) $nc) :=
+        (q(Eq.refl (Int.mul $na $nb)) : Expr)
+      have t2 : Q(ℕ) := mkRawNatLit dd
+      let r2 : Q(Nat.mul $da $db = Nat.mul (Nat.succ $g) $dc) := (q(Eq.refl $t2) : Expr)
+      (.isRat dα qc nc dc q(isRat_mul $pa $pb $r1 $r2) : Result q($a * $b))
     match ra, rb with
     | .isNat _ na pa, .isNat mα nb pb =>
       let pa : Q(@IsNat _ AddCommMonoidWithOne.toAddMonoidWithOne $a $na) := pa
@@ -193,7 +290,7 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     | .isNat _ .., .isNegNat rα ..
     | .isNegNat rα .., .isNat _ ..
     | .isNegNat _ .., .isNegNat rα .. => intArm rα
-    | _, _ => failure
+    | _, .isRat dα .. | .isRat dα .., _ => some (ratArm dα)
   core
 
 theorem isNat_pow {α} [Semiring α] : {a : α} → {b a' b' c : ℕ} →
@@ -204,6 +301,12 @@ theorem isInt_pow {α} [Ring α] : {a : α} → {b : ℕ} → {a' : ℤ} → {b'
     IsInt a a' → IsNat b b' → Int.pow a' b' = c → IsInt (a ^ b) c
   | _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨by simp⟩
 
+theorem isRat_pow {α} [Ring α] : {a : α} → {an cn : ℤ} → {ad b b' cd : ℕ} →
+    IsRat a an ad → IsNat b b' →
+    Int.pow an b' = cn → Nat.pow ad b' = cd →
+    IsRat (a ^ b) cn cd
+  | _, _, _, _, _, _, _, ⟨_, rfl⟩, ⟨rfl⟩, rfl, rfl => sorry
+
 /-- The `norm_num` extension which identifies expressions of the form `a ^ b`,
 such that `norm_num` successfully recognises both `a` and `b`, with `b : ℕ`. -/
 @[norm_num (_ : α) ^ (_ : ℕ), Pow.pow _ (_ : ℕ)]
@@ -228,6 +331,65 @@ def evalPow : NormNumExt where eval {u α} e := do
       let zc := za ^ nb.natLit!
       let c := mkRawIntLit zc
       let r : Q(Int.pow $na $nb = $c) := (q(Eq.refl $c) : Expr)
-      return (.isInt rα zc (z := c) (q(isInt_pow $pa $pb $r) : Expr) : Result q($a ^ $b))
-    | _ => failure
+      return (.isInt rα c zc (q(isInt_pow $pa $pb $r) : Expr) : Result q($a ^ $b))
+    | .isRat dα qa na da pa =>
+      let qc := qa ^ nb.natLit!
+      have nc : Q(ℤ) := mkRawIntLit qc.num
+      have dc : Q(ℕ) := mkRawNatLit qc.den
+      have r1 : Q(Int.pow $na $nb = $nc) := (q(Eq.refl $nc) : Expr)
+      have r2 : Q(Nat.pow $da $nb = $dc) := (q(Eq.refl $dc) : Expr)
+      return (.isRat dα qc nc dc (q(isRat_pow $pa $pb $r1 $r2) : Expr) : Result q($a ^ $b))
   core
+
+theorem isRat_inv_pos {α} [DivisionRing α] : {a : α} → {n d : ℕ} →
+    Nat.ble (nat_lit 1) n = true → IsRat a (.ofNat n) d → IsRat a⁻¹ (.ofNat d) n
+  | _, _, _, _, ⟨_, rfl⟩ => sorry
+
+theorem isRat_inv_zero {α} [DivisionRing α] : {a : α} →
+    IsNat a (nat_lit 0) → IsNat a⁻¹ (nat_lit 0)
+  | _, ⟨rfl⟩ => ⟨by simp⟩
+
+theorem isRat_inv_neg {α} [DivisionRing α] : {a : α} → {n d : ℕ} →
+    Nat.ble 1 n = true → IsRat a (.negOfNat n) d → IsRat a⁻¹ (.negOfNat d) n
+  | _, _, _, _, ⟨_, rfl⟩ => sorry
+
+/-- The `norm_num` extension which identifies expressions of the form `a⁻¹`,
+such that `norm_num` successfully recognises `a`. -/
+@[norm_num _⁻¹] def evalInv : NormNumExt where eval {u α} e := do
+  let .app f (a : Q($α)) ← withReducible (whnf e) | failure
+  let ra ← derive a
+  let dα ← inferDivisionRing α
+  guard <|← withNewMCtxDepth <| isDefEq f q(Inv.inv (α := $α))
+  let ⟨qa, na, da, pa⟩ := ra.toRat'
+  let qb := qa⁻¹
+  if qa > 0 then
+    have lit : Q(ℕ) := na.appArg!
+    let pa : Q(IsRat «$a» (Int.ofNat $lit) $da) := pa
+    let r : Q(Nat.ble (nat_lit 1) $lit = true) := (q(Eq.refl true) : Expr)
+    return (.isRat' dα qb q(.ofNat $da) lit
+      (q(isRat_inv_pos (α := $α) $r $pa) : Expr) : Result q($a⁻¹))
+  else if qa < 0 then
+    have lit : Q(ℕ) := na.appArg!
+    let pa : Q(IsRat «$a» (Int.negOfNat $lit) $da) := pa
+    let r : Q(Nat.ble (nat_lit 1) $lit = true) := (q(Eq.refl true) : Expr)
+    return (.isRat' dα qb q(.negOfNat $da) lit
+      (q(isRat_inv_neg (α := $α) $r $pa) : Expr) : Result q($a⁻¹))
+  else
+    let .isNat inst _z (pa : Q(@IsNat _ AddGroupWithOne.toAddMonoidWithOne $a (nat_lit 0))) := ra
+      | failure
+    return (.isNat inst _z (q(isRat_inv_zero $pa) : Expr) : Result q($a⁻¹))
+
+theorem isRat_div {α} [DivisionRing α] : {a b : α} → {cn : ℤ} → {cd : ℕ} → IsRat (a * b⁻¹) cn cd →
+    IsRat (a / b) cn cd
+  | _, _, _, _, h => by simp [div_eq_mul_inv]; exact h
+
+/-- The `norm_num` extension which identifies expressions of the form `a / b`,
+such that `norm_num` successfully recognises both `a` and `b`. -/
+@[norm_num _ / _, Div.div _ _] def evalDiv : NormNumExt where eval {u α} e := do
+  let .app (.app f (a : Q($α))) (b : Q($α)) ← withReducible (whnf e) | failure
+  let dα ← inferDivisionRing α
+  guard <|← withNewMCtxDepth <| isDefEq f q(HDiv.hDiv (α := $α))
+  let rab ← derive (q($a * $b⁻¹) : Q($α))
+  let ⟨qa, na, da, pa⟩ := rab.toRat'
+  let pa : Q(IsRat ($a * $b⁻¹) $na $da) := pa
+  return (.isRat' dα qa na da q(isRat_div $pa) : Result q($a / $b))