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Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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17 declarations, 0 contain sorry, 0 declared as axioms

Isomorphism of point groups induced by a change of variables

Material for Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point; see the section header below for details.

Imports (1)

Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Imported by 3 files

lemma WeierstrassCurve.Affine.variableChange_negY line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Eq, WeierstrassCurve.Affine.negY, WeierstrassCurve.toAffine, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t, HSMul.hSMul
Proof uses: Eq.mpr, HSub.hSub, Neg.neg, WeierstrassCurve.a₁, WeierstrassCurve.a₃, Inv.inv, id, congr, congrArg, congrFun', Eq.trans, neg_add_rev, Units, Units.val_inv_eq_inv_val, inv_pow, inferInstance, CommGroupWithZero, GroupWithZero, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval and 97 more
109 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.variableChange_negY_ne The image of a pair of points under the change of variables satisfies the y₁ = -y₂ degeneracy condition (negY) only if the original pair does line 63 source
Source documentation
The image of a pair of points under the change of variables satisfies the y₁ = -y₂ degeneracy condition (negY) only if the original pair does.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Not, And, Eq, WeierstrassCurve.Affine.negY, WeierstrassCurve.toAffine, HSMul.hSMul, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t
Proof uses: Ne, Units.ne_zero, False, mul_left_cancel₀, pow_ne_zero, Mathlib.Tactic.LinearCombination.eq_of_eq, Mathlib.Tactic.LinearCombination.eq_rearrange, Mathlib.Tactic.Ring.of_eq, HSub.hSub, Mathlib.Tactic.Ring.Common.sub_congr, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm and 46 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
lemma variableChange_negY_ne {x₁ x₂ y₁ y₂ : F}
    (hxy : ¬(x₁ = x₂ ∧ y₁ = (C • W).toAffine.negY x₂ y₂)) :
    ¬((C.u : F) ^ 2 * x₁ + C.r = (C.u : F) ^ 2 * x₂ + C.r ∧
      (C.u : F) ^ 3 * y₁ + (C.u : F) ^ 2 * C.s * x₁ + C.t = W.toAffine.negY
        ((C.u : F) ^ 2 * x₂ + C.r) ((C.u : F) ^ 3 * y₂ + (C.u : F) ^ 2 * C.s * x₂ + C.t)) := by
…
```
lemma WeierstrassCurve.Affine.variableChange_addX line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Eq, WeierstrassCurve.Affine.addX, WeierstrassCurve.toAffine, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, HSMul.hSMul
Proof uses: Eq.mpr, HSub.hSub, WeierstrassCurve.a₁, WeierstrassCurve.a₂, Inv.inv, id, congrArg, congrFun', congr, Units, Units.val_inv_eq_inv_val, Eq.trans, inv_pow, inferInstance, CommGroupWithZero, GroupWithZero, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval, List.nil, Prod and 95 more
98 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.variableChange_negAddY line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Eq, WeierstrassCurve.Affine.negAddY, WeierstrassCurve.toAffine, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t, HSMul.hSMul, WeierstrassCurve.Affine.addX
Proof uses: Eq.mpr, HSub.hSub, WeierstrassCurve.a₁, WeierstrassCurve.a₂, Inv.inv, id, congrArg, congrFun', congr, Units, Units.val_inv_eq_inv_val, Eq.trans, inv_pow, inferInstance, CommGroupWithZero, GroupWithZero, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval, List.nil, Prod and 96 more
99 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.variableChange_addY line 89 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Eq, WeierstrassCurve.Affine.addY, WeierstrassCurve.toAffine, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t, HSMul.hSMul, WeierstrassCurve.Affine.addX
Proof uses: of_eq_true, WeierstrassCurve.Affine.negY, WeierstrassCurve.Affine.negAddY, Eq.trans, True, congrFun', congrArg, congr, WeierstrassCurve.Affine.variableChange_addX, WeierstrassCurve.Affine.variableChange_negAddY, WeierstrassCurve.Affine.variableChange_negY, eq_self
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.variableChange_slope line 96 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, DecidableEq, WeierstrassCurve.Affine.Equation, WeierstrassCurve.toAffine, HSMul.hSMul, Not, And, Eq, WeierstrassCurve.Affine.negY, WeierstrassCurve.Affine.slope, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s and 1 more
Proof uses: Ne, Units.ne_zero, Or, eq_or_ne, And.intro, rfl, Eq.mpr, id, congrArg, WeierstrassCurve.Affine.variableChange_negY, mul_left_cancel₀, pow_ne_zero, Mathlib.Tactic.LinearCombination.eq_of_eq, Mathlib.Tactic.LinearCombination.eq_rearrange, Mathlib.Tactic.Ring.of_eq, HSub.hSub, Mathlib.Tactic.Ring.Common.sub_congr, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr and 120 more
127 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.variableChange_equation A point (x, y) lies on C • W if and only if (u²x + r, u³y + u²sx + t) lies on W: the change of variables scales the Weierstrass polynomial by u⁶ line 126 source
Source documentation
A point (x, y) lies on C • W if and only if (u²x + r, u³y + u²sx + t) lies on W: the change of variables scales the Weierstrass polynomial by u⁶.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, Iff, WeierstrassCurve.Affine.Equation, WeierstrassCurve.toAffine, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t, HSMul.hSMul
Proof uses: Ne, Units.ne_zero, Eq.mpr, Eq, HSub.hSub, inferInstance, CommGroupWithZero, GroupWithZero, WeierstrassCurve.a₁, WeierstrassCurve.a₃, WeierstrassCurve.a₂, WeierstrassCurve.a₄, WeierstrassCurve.a₆, id, Eq.trans, congr, congrArg, Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq, Mathlib.Tactic.FieldSimp.NF.eval, List.nil and 101 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
lemma variableChange_equation (x y : F) :
    W.toAffine.Equation ((C.u : F) ^ 2 * x + C.r)
        ((C.u : F) ^ 3 * y + (C.u : F) ^ 2 * C.s * x + C.t)
      ↔ (C • W).toAffine.Equation x y := by
…
```
def WeierstrassCurve.Affine.Point.mapVariableChangeFun The underlying map (C • W).Point → W.Point of the change of variables, sending 0 to 0 and (x, y) to (u²x + r, u³y + u²sx + t) line 143 source
Source documentation
The underlying map (C • W).Point → W.Point of the change of variables, sending 0 to 0 and (x, y) to (u²x + r, u³y + u²sx + t).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, HSMul.hSMul
Proof uses: WeierstrassCurve.Affine.Point.mapVariableChangeFun.match_1, Unit, WeierstrassCurve.Affine.Point.zero, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.Affine.Point.some, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t
22 instances, coercions and auxiliary constants left out.
```
def mapVariableChangeFun : (C • W).toAffine.Point → W.toAffine.Point
  | .zero => .zero
  | .some x y h => .some ((C.u : F) ^ 2 * x + C.r)
      ((C.u : F) ^ 3 * y + (C.u : F) ^ 2 * C.s * x + C.t)
      (equation_iff_nonsingular.mp
        ((variableChange_equation W C x y).mpr (equation_iff_nonsingular.mpr h)))
```
Mentions: WeierstrassCurve.Affine.variableChange_equation
lemma WeierstrassCurve.Affine.Point.mapVariableChangeFun_zero line 150 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, Eq, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, WeierstrassCurve.Affine.Point.mapVariableChangeFun, OfNat.ofNat, HSMul.hSMul
Proof uses: rfl
10 instances, coercions and auxiliary constants left out.
lemma WeierstrassCurve.Affine.Point.mapVariableChangeFun_some line 152 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.toAffine, HSMul.hSMul, Eq, WeierstrassCurve.Affine.Point, WeierstrassCurve.Affine.Point.mapVariableChangeFun, WeierstrassCurve.Affine.Point.some, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t and 5 more
Proof uses: rfl
24 instances, coercions and auxiliary constants left out.
lemma WeierstrassCurve.Affine.Point.some_eq_some line 158 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, Eq, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.toAffine, WeierstrassCurve.Affine.Point, WeierstrassCurve.Affine.Point.some
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma WeierstrassCurve.Affine.Point.mapVariableChangeFun_injective line 163 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, Function.Injective, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, HSMul.hSMul, WeierstrassCurve.Affine.Point.mapVariableChangeFun
Proof uses: Ne, WeierstrassCurve.VariableChange.u, OfNat.ofNat, Units.ne_zero, Eq, WeierstrassCurve.Affine.Point.zero, Eq.refl, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.Affine.Point.some, False.elim, Eq.mp, False, eq_false', HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.r, WeierstrassCurve.VariableChange.s, WeierstrassCurve.VariableChange.t and 72 more
65 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
def WeierstrassCurve.Affine.Point.equivOfEq Transport of the affine point group along an equality of Weierstrass curves line 179 source
Source documentation
Transport of the affine point group along an equality of Weierstrass curves.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, DecidableEq, WeierstrassCurve, Eq, AddEquiv, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine
Proof uses: AddEquiv.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def equivOfEq {V V' : WeierstrassCurve F} (h : V = V') :
    V.toAffine.Point ≃+ V'.toAffine.Point := by
…
```
lemma WeierstrassCurve.Affine.Point.equivOfEq_some line 183 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, DecidableEq, WeierstrassCurve, Eq, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.toAffine, WeierstrassCurve.Affine.Point, AddEquiv, WeierstrassCurve.Affine.Point.equivOfEq, WeierstrassCurve.Affine.Point.some, Eq.rec
Proof uses: Eq.refl
10 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def WeierstrassCurve.Affine.Point.mapVariableChange The group homomorphism (C • W).Point →+ W.Point induced by the admissible change of variables (x, y) ↦ (u²x + r, u³y + u²sx + t) line 190 source
Source documentation
The group homomorphism (C • W).Point →+ W.Point induced by the admissible change of variables (x, y) ↦ (u²x + r, u³y + u²sx + t).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, DecidableEq, AddMonoidHom, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, HSMul.hSMul
Proof uses: AddMonoidHom.mk, ZeroHom.mk, WeierstrassCurve.Affine.Point.mapVariableChangeFun
13 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
```
def mapVariableChange : (C • W).toAffine.Point →+ W.toAffine.Point where
…
```
def WeierstrassCurve.Affine.Point.equivVariableChange The group isomorphism (C • W).Point ≃+ W.Point induced by the admissible change of variables (x, y) ↦ (u²x + r, u³y + u²sx + t), with inverse coming from C⁻¹ line 209 source
Source documentation
The group isomorphism (C • W).Point ≃+ W.Point induced by the admissible change of variables (x, y) ↦ (u²x + r, u³y + u²sx + t), with inverse coming from C⁻¹.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, DecidableEq, AddEquiv, WeierstrassCurve.Affine.Point, WeierstrassCurve.toAffine, HSMul.hSMul
Proof uses: Eq, WeierstrassCurve.Affine.Point.mapVariableChangeFun, Inv.inv, WeierstrassCurve.Affine.Point.equivOfEq, AddEquiv.mk, Equiv.mk
25 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
def equivVariableChange : (C • W).toAffine.Point ≃+ W.toAffine.Point :=
…
```
lemma WeierstrassCurve.Affine.Point.equivVariableChange_some line 225 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, WeierstrassCurve, WeierstrassCurve.VariableChange, WeierstrassCurve.IsElliptic, DecidableEq, WeierstrassCurve.Affine.Nonsingular, WeierstrassCurve.toAffine, HSMul.hSMul, Eq, WeierstrassCurve.Affine.Point, AddEquiv, WeierstrassCurve.Affine.Point.equivVariableChange, WeierstrassCurve.Affine.Point.some, HAdd.hAdd, HMul.hMul, HPow.hPow, Nat, WeierstrassCurve.VariableChange.u, OfNat.ofNat, WeierstrassCurve.VariableChange.r and 7 more
Proof uses: rfl
32 instances, coercions and auxiliary constants left out.