url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | use y | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ ∃ ry rz, (r = 2⁻¹ ∧ x = x ∧ y = ry ∧ z = rz) ∧ x ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4 | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x ^ 2 + y ^ 2 + rz ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ ∃ ry rz, (r = 2⁻¹ ∧ x = x ∧ y = ry ∧ z = rz) ∧ x ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | use z | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x ^ 2 + y ^ 2 + rz ^ 2 = 3 / 4 | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x ^ 2 + y ^ 2 + rz ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [and_self, and_true] | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rcases h₀ with ⟨h₁, hx, hy, hz⟩ | case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | have EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → (x₀ = 0 ∨ r = 1 / 2) := by
intro x₀ h
have hFactored : x₀ * (1 - r * 2) = 0 := by linarith
simp at hFactored
apply hFactored.imp_right
intro h
field_simp
linarith | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let hx₂ := EqSplit x hx | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let hy₂ := EqSplit y hy | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let hz₂ := EqSplit z hz | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | have hr₀ : (¬ r = 1/2) → False := by
intros hrn0
simp only [one_div] at hrn0
simp only [one_div, hrn0, or_false] at hx₂
simp only [one_div, hrn0, or_false] at hy₂
simp only [one_div, hrn0, or_false] at hz₂
simp [hx₂, hy₂, hz₂] at h₁
let h₂ := congrArg (λ (x₀ : ℝ) => x₀ - r + 1) h₁
simp only [add_sub_cancel, add_left_neg] at h₂
have hSquareNn := mul_self_nonneg (r - 1 / 2)
linarith | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | have hr₁ : r = 1/2 := by_contra hr₀ | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rw [hr₁] | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ 1 / 2 = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ r = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [one_div, true_and] | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ 1 / 2 = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ 1 / 2 = 2⁻¹ ∧ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rw [hr₁] at h₁ | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let hSphere := congrArg (λ (x₀ : ℝ) => 1 / 4 - x₀) h₁ | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : (fun x₀ => 1 / 4 - x₀) (-1 + 1 / 2) = (fun x₀ => 1 / 4 - x₀) ((1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)) :=
congrArg (fun x₀ => 1 / 4 - x₀) h₁
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | ring_nf at hSphere | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : (fun x₀ => 1 / 4 - x₀) (-1 + 1 / 2) = (fun x₀ => 1 / 4 - x₀) ((1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)) :=
congrArg (fun x₀ => 1 / 4 - x₀) h₁
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : 3 / 4 = x ^ 2 + y ^ 2 + z ^ 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : (fun x₀ => 1 / 4 - x₀) (-1 + 1 / 2) = (fun x₀ => 1 / 4 - x₀) ((1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)) :=
congrArg (fun x₀ => 1 / 4 - x₀) h₁
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rw [←hSphere] | case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : 3 / 4 = x ^ 2 + y ^ 2 + z ^ 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
r x y z : ℝ
h₁ : -1 + 1 / 2 = (1 / 2) ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hr₀ : ¬r = 1 / 2 → False
hr₁ : r = 1 / 2
hSphere : 3 / 4 = x ^ 2 + y ^ 2 + z ^ 2
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | intro x₀ h | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
⊢ ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
⊢ x₀ = 0 ∨ r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
⊢ ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | have hFactored : x₀ * (1 - r * 2) = 0 := by linarith | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
⊢ x₀ = 0 ∨ r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ * (1 - r * 2) = 0
⊢ x₀ = 0 ∨ r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
⊢ x₀ = 0 ∨ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp at hFactored | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ * (1 - r * 2) = 0
⊢ x₀ = 0 ∨ r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ x₀ = 0 ∨ r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ * (1 - r * 2) = 0
⊢ x₀ = 0 ∨ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | apply hFactored.imp_right | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ x₀ = 0 ∨ r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ 1 - r * 2 = 0 → r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ x₀ = 0 ∨ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | intro h | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ 1 - r * 2 = 0 → r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
⊢ 1 - r * 2 = 0 → r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | field_simp | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r = 1 / 2 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r * 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | linarith | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r * 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h✝ : x₀ = r * x₀ * 2
hFactored : x₀ = 0 ∨ 1 - r * 2 = 0
h : 1 - r * 2 = 0
⊢ r * 2 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | linarith | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
⊢ x₀ * (1 - r * 2) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
x₀ : ℝ
h : x₀ = r * x₀ * 2
⊢ x₀ * (1 - r * 2) = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | intros hrn0 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
⊢ ¬r = 1 / 2 → False | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 1 / 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
⊢ ¬r = 1 / 2 → False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [one_div] at hrn0 | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 1 / 2
⊢ False | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 1 / 2
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [one_div, hrn0, or_false] at hx₂ | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
⊢ False | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hx₂ : x = 0 ∨ r = 1 / 2 := EqSplit x hx
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [one_div, hrn0, or_false] at hy₂ | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
⊢ False | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hy₂ : y = 0 ∨ r = 1 / 2 := EqSplit y hy
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [one_div, hrn0, or_false] at hz₂ | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
⊢ False | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hz₂ : z = 0 ∨ r = 1 / 2 := EqSplit z hz
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp [hx₂, hy₂, hz₂] at h₁ | r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
⊢ False | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
h₁ : -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2)
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let h₂ := congrArg (λ (x₀ : ℝ) => x₀ - r + 1) h₁ | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
⊢ False | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : (fun x₀ => x₀ - r + 1) (-1 + r) = (fun x₀ => x₀ - r + 1) (r ^ 2) := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [add_sub_cancel, add_left_neg] at h₂ | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : (fun x₀ => x₀ - r + 1) (-1 + r) = (fun x₀ => x₀ - r + 1) (r ^ 2) := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : (fun x₀ => x₀ - r + 1) (-1 + r) = (fun x₀ => x₀ - r + 1) (r ^ 2) := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | have hSquareNn := mul_self_nonneg (r - 1 / 2) | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
hSquareNn : 0 ≤ (r - 1 / 2) * (r - 1 / 2)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | linarith | r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
hSquareNn : 0 ≤ (r - 1 / 2) * (r - 1 / 2)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r x y z : ℝ
hx : x = r * x * 2
hy : y = r * y * 2
hz : z = r * z * 2
EqSplit : ∀ (x₀ : ℝ), x₀ = r * x₀ * 2 → x₀ = 0 ∨ r = 1 / 2
hrn0 : ¬r = 2⁻¹
hx₂ : x = 0
hy₂ : y = 0
hz₂ : z = 0
h₁ : -1 + r = r ^ 2
h₂ : -1 + r - r + 1 = r ^ 2 - r + 1 := congrArg (fun x₀ => x₀ - r + 1) h₁
hSquareNn : 0 ≤ (r - 1 / 2) * (r - 1 / 2)
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | intros h₀ | case h.mk.mpr
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4) →
-1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | case h.mk.mpr
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4) →
-1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rcases h₀ with ⟨rx, ry, rz, hx, hSphere⟩ | case h.mk.mpr
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | case h.mk.mpr.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rcases hx with ⟨hr, hx, hy, hz⟩ | case h.mk.mpr.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp_rw [hr] | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2 | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + 2⁻¹ = 2⁻¹ ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = 2⁻¹ * x * 2 ∧ y = 2⁻¹ * y * 2 ∧ z = 2⁻¹ * z * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + r = r ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = r * x * 2 ∧ y = r * y * 2 ∧ z = r * z * 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | ring_nf | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + 2⁻¹ = 2⁻¹ ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = 2⁻¹ * x * 2 ∧ y = 2⁻¹ * y * 2 ∧ z = 2⁻¹ * z * 2 | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 1 / 4 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ True ∧ True ∧ True | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 + 2⁻¹ = 2⁻¹ ^ 2 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ x = 2⁻¹ * x * 2 ∧ y = 2⁻¹ * y * 2 ∧ z = 2⁻¹ * z * 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | simp only [Int.cast_negOfNat, Nat.cast_one, Int.ofNat_eq_coe, Int.cast_one, Nat.cast_ofNat,
one_div, neg_mul, one_mul, and_self, and_true] | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 1 / 4 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ True ∧ True ∧ True | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 1 / 4 - x ^ 2 + (-y ^ 2 - z ^ 2) ∧ True ∧ True ∧ True
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rw [←hx, ←hy, ←hz] at hSphere | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx ^ 2 + ry ^ 2 + rz ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | let hXSquare := congrArg (λ (x₀ : ℝ) => x₀ - y ^ 2 - z ^ 2) hSphere | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : (fun x₀ => x₀ - y ^ 2 - z ^ 2) (x ^ 2 + y ^ 2 + z ^ 2) = (fun x₀ => x₀ - y ^ 2 - z ^ 2) (3 / 4) :=
congrArg (fun x₀ => x₀ - y ^ 2 - z ^ 2) hSphere
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | ring_nf at hXSquare | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : (fun x₀ => x₀ - y ^ 2 - z ^ 2) (x ^ 2 + y ^ 2 + z ^ 2) = (fun x₀ => x₀ - y ^ 2 - z ^ 2) (3 / 4) :=
congrArg (fun x₀ => x₀ - y ^ 2 - z ^ 2) hSphere
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : (fun x₀ => x₀ - y ^ 2 - z ^ 2) (x ^ 2 + y ^ 2 + z ^ 2) = (fun x₀ => x₀ - y ^ 2 - z ^ 2) (3 / 4) :=
congrArg (fun x₀ => x₀ - y ^ 2 - z ^ 2) hSphere
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | rw [hXSquare] | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2) | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - (3 / 4 + (-y ^ 2 - z ^ 2)) + (-y ^ 2 - z ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - x ^ 2 + (-y ^ 2 - z ^ 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₀AndSoqqtstqm1₁ | [99, 1] | [154, 9] | ring | case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - (3 / 4 + (-y ^ 2 - z ^ 2)) + (-y ^ 2 - z ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hXSquare : x ^ 2 = 3 / 4 + (-y ^ 2 - z ^ 2)
⊢ -1 / 2 = 4⁻¹ - (3 / 4 + (-y ^ 2 - z ^ 2)) + (-y ^ 2 - z ^ 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | ext ⟨r, x, y, z⟩ | ⊢ Soqqtstqm1₁ = Soqqtstqm1₂ | case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₂ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Soqqtstqm1₁ = Soqqtstqm1₂
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | dsimp [Soqqtstqm1₁, Soqqtstqm1₂] | case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₂ | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₂
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [one_div, ext_iff] | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 1 / 2 | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 1 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | constructor | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
case h.mk.mpr
r x y z : ℝ
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ →
∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | intros h₀ | case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rcases h₀ with ⟨rx, ry, rz, hx, hSphere⟩ | case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rcases hx with ⟨hr, hx, hy, hz⟩ | case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rw [←hx, ←hy, ←hz] at hSphere | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | have hrSquare := congrArg (λ (x₀ : ℝ) => x₀ ^ 2) hr | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : (fun x₀ => x₀ ^ 2) r = (fun x₀ => x₀ ^ 2) 2⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [inv_pow] at hrSquare | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : (fun x₀ => x₀ ^ 2) r = (fun x₀ => x₀ ^ 2) 2⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : (fun x₀ => x₀ ^ 2) r = (fun x₀ => x₀ ^ 2) 2⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | have hNormSq := congrArg (λ (x₀ : ℝ) => x₀ + r ^ 2) hSphere | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + r ^ 2) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | nth_rewrite 2 [hrSquare] at hNormSq | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + r ^ 2) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + (2 ^ 2)⁻¹) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + r ^ 2) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only at hNormSq | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + (2 ^ 2)⁻¹) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x * x + y * y + z * z + r ^ 2 = 3 / 4 + (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : (fun x₀ => x₀ + r ^ 2) (x * x + y * y + z * z) = (fun x₀ => x₀ + (2 ^ 2)⁻¹) (3 / 4)
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | ring_nf at hNormSq | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x * x + y * y + z * z + r ^ 2 = 3 / 4 + (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x * x + y * y + z * z + r ^ 2 = 3 / 4 + (2 ^ 2)⁻¹
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | let hSqrtNormSquare := congrArg Real.sqrt (Quaternion.normSq_eq_norm_mul_self (@QuaternionAlgebra.mk ℝ (-1) (-1) r x y z)) | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) =
√(‖{ re := r, imI := x, imJ := y, imK := z }‖ * ‖{ re := r, imI := x, imJ := y, imK := z }‖) :=
congrArg Real.sqrt (normSq_eq_norm_mul_self { re := r, imI := x, imJ := y, imK := z })
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [norm_nonneg, Real.sqrt_mul_self] at hSqrtNormSquare | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) =
√(‖{ re := r, imI := x, imJ := y, imK := z }‖ * ‖{ re := r, imI := x, imJ := y, imK := z }‖) :=
congrArg Real.sqrt (normSq_eq_norm_mul_self { re := r, imI := x, imJ := y, imK := z })
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) =
√(‖{ re := r, imI := x, imJ := y, imK := z }‖ * ‖{ re := r, imI := x, imJ := y, imK := z }‖) :=
congrArg Real.sqrt (normSq_eq_norm_mul_self { re := r, imI := x, imJ := y, imK := z })
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rw [←hSqrtNormSquare, Quaternion.normSq_def'] | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ √({ re := r, imI := x, imJ := y, imK := z }.re ^ 2 + { re := r, imI := x, imJ := y, imK := z }.imI ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imJ ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imK ^ 2) =
1 ∧
r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [Real.sqrt_eq_one] | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ √({ re := r, imI := x, imJ := y, imK := z }.re ^ 2 + { re := r, imI := x, imJ := y, imK := z }.imI ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imJ ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imK ^ 2) =
1 ∧
r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1 ∧ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ √({ re := r, imI := x, imJ := y, imK := z }.re ^ 2 + { re := r, imI := x, imJ := y, imK := z }.imI ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imJ ^ 2 +
{ re := r, imI := x, imJ := y, imK := z }.imK ^ 2) =
1 ∧
r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | constructor | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1 ∧ r = 2⁻¹ | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.right
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r = 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1 ∧ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rw [←hNormSq] | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1 | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | ring_nf | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.left
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | exact hr | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.right
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r = 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro.right
r x y z rx ry rz : ℝ
hSphere : x * x + y * y + z * z = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
hrSquare : r ^ 2 = (2 ^ 2)⁻¹
hNormSq : x ^ 2 + y ^ 2 + z ^ 2 + r ^ 2 = 1
hSqrtNormSquare : √(normSq { re := r, imI := x, imJ := y, imK := z }) = ‖{ re := r, imI := x, imJ := y, imK := z }‖
⊢ r = 2⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | intros h₀ | case h.mk.mpr
r x y z : ℝ
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ →
∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4 | case h.mk.mpr
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr
r x y z : ℝ
⊢ ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹ →
∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | use x | case h.mk.mpr
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4 | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ ry rz, (r = 2⁻¹ ∧ x = x ∧ y = ry ∧ z = rz) ∧ x * x + ry * ry + rz * rz = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mpr
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | use y | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ ry rz, (r = 2⁻¹ ∧ x = x ∧ y = ry ∧ z = rz) ∧ x * x + ry * ry + rz * rz = 3 / 4 | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x * x + y * y + rz * rz = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ ry rz, (r = 2⁻¹ ∧ x = x ∧ y = ry ∧ z = rz) ∧ x * x + ry * ry + rz * rz = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | use z | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x * x + y * y + rz * rz = 3 / 4 | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ ∃ rz, (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = rz) ∧ x * x + y * y + rz * rz = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rcases h₀ with ⟨hNorm, hr⟩ | case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z : ℝ
h₀ : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1 ∧ r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [hr, and_self, true_and] | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ (r = 2⁻¹ ∧ x = x ∧ y = y ∧ z = z) ∧ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | have hNormSqMr := congrArg (λ (x₀ : ℝ) => x₀ * x₀ - 1 / 4) hNorm | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (fun x₀ => x₀ * x₀ - 1 / 4) ‖{ re := r, imI := x, imJ := y, imK := z }‖ = (fun x₀ => x₀ * x₀ - 1 / 4) 1
⊢ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
⊢ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [one_div, mul_one, ←Quaternion.normSq_eq_norm_mul_self, Quaternion.normSq_def'] at hNormSqMr | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (fun x₀ => x₀ * x₀ - 1 / 4) ‖{ re := r, imI := x, imJ := y, imK := z }‖ = (fun x₀ => x₀ * x₀ - 1 / 4) 1
⊢ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (fun x₀ => x₀ * x₀ - 1 / 4) ‖{ re := r, imI := x, imJ := y, imK := z }‖ = (fun x₀ => x₀ * x₀ - 1 / 4) 1
⊢ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | simp only [hr, inv_pow] at hNormSqMr | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (2 ^ 2)⁻¹ + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : r ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | ring_nf at hNormSqMr | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (2 ^ 2)⁻¹ + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x * x + y * y + z * z = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : (2 ^ 2)⁻¹ + x ^ 2 + y ^ 2 + z ^ 2 - 4⁻¹ = 1 - 4⁻¹
⊢ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | ring_nf | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x * x + y * y + z * z = 3 / 4 | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x * x + y * y + z * z = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₂ | [156, 1] | [190, 19] | rw [hNormSqMr] | case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
r x y z : ℝ
hNorm : ‖{ re := r, imI := x, imJ := y, imK := z }‖ = 1
hr : r = 2⁻¹
hNormSqMr : x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
⊢ x ^ 2 + y ^ 2 + z ^ 2 = 3 / 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | ext ⟨r, x, y, z⟩ | ⊢ Soqqtstqm1₁ = Soqqtstqm1₃ | case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₃ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Soqqtstqm1₁ = Soqqtstqm1₃
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | dsimp [Soqqtstqm1₁, Soqqtstqm1₃, Soqtstn1₁] | case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₃ | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, qim = { re := 0, imI := rx, imJ := ry, imK := rz } ∧ rx * rx + ry * ry + rz * rz = 1) ∧
{ re := r, imI := x, imJ := y, imK := z } = 1 / 2 + qim * ↑√3 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₁ ↔ { re := r, imI := x, imJ := y, imK := z } ∈ Soqqtstqm1₃
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | simp only [one_div, ext_iff, add_re, add_imI, add_imJ, add_imK] | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, qim = { re := 0, imI := rx, imJ := ry, imK := rz } ∧ rx * rx + ry * ry + rz * rz = 1) ∧
{ re := r, imI := x, imJ := y, imK := z } = 1 / 2 + qim * ↑√3 / 2 | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz,
{ re := r, imI := x, imJ := y, imK := z } = { re := 1 / 2, imI := rx, imJ := ry, imK := rz } ∧
rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, qim = { re := 0, imI := rx, imJ := ry, imK := rz } ∧ rx * rx + ry * ry + rz * rz = 1) ∧
{ re := r, imI := x, imJ := y, imK := z } = 1 / 2 + qim * ↑√3 / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | constructor | case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
case h.mk.mpr
r x y z : ℝ
⊢ (∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK) →
∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) ↔
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | intros h₀ | case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp
r x y z : ℝ
⊢ (∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4) →
∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | rcases h₀ with ⟨rx, ry, rz, hx, hSphere⟩ | case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp
r x y z : ℝ
h₀ : ∃ rx ry rz, (r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz) ∧ rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | rcases hx with ⟨hr, hx, hy, hz⟩ | case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hx : r = 2⁻¹ ∧ x = rx ∧ y = ry ∧ z = rz
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | use (@QuaternionAlgebra.mk ℝ (-1) (-1) 0 (rx * 2 / Real.sqrt 3) (ry * 2 / Real.sqrt 3) (rz * 2 / Real.sqrt 3)) | case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.re = 0 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imI = rx_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imJ = ry_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imK = rz_1) ∧
rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.mp.intro.intro.intro.intro.intro.intro.intro
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ qim,
(∃ rx ry rz, (qim.re = 0 ∧ qim.imI = rx ∧ qim.imJ = ry ∧ qim.imK = rz) ∧ rx * rx + ry * ry + rz * rz = 1) ∧
r = 2⁻¹.re + (qim * ↑√3 / 2).re ∧
x = 2⁻¹.imI + (qim * ↑√3 / 2).imI ∧ y = 2⁻¹.imJ + (qim * ↑√3 / 2).imJ ∧ z = 2⁻¹.imK + (qim * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | simp only [true_and] | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.re = 0 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imI = rx_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imJ = ry_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imK = rz_1) ∧
rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.re = 0 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imI = rx_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imJ = ry_1 ∧
{ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 }.imK = rz_1) ∧
rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | constructor | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK | case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1) ∧
r = 2⁻¹.re + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).re ∧
x = 2⁻¹.imI + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imI ∧
y = 2⁻¹.imJ + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imJ ∧
z = 2⁻¹.imK + ({ re := 0, imI := rx * 2 / √3, imJ := ry * 2 / √3, imK := rz * 2 / √3 } * ↑√3 / 2).imK
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | use rx * 2 / Real.sqrt 3 | case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1 | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ ry_1 rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry_1 * ry_1 + rz_1 * rz_1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rx_1 ry_1 rz_1,
(rx * 2 / √3 = rx_1 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧ rx_1 * rx_1 + ry_1 * ry_1 + rz_1 * rz_1 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | use ry * 2 / Real.sqrt 3 | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ ry_1 rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry_1 * ry_1 + rz_1 * rz_1 = 1 | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz_1 * rz_1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ ry_1 rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry_1 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry_1 * ry_1 + rz_1 * rz_1 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | use rz * 2 / Real.sqrt 3 | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz_1 * rz_1 = 1 | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ ∃ rz_1,
(rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz_1) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz_1 * rz_1 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | constructor | case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1 | case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ (rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3) ∧
rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | simp only [and_self] | case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 = rx * 2 / √3 ∧ ry * 2 / √3 = ry * 2 / √3 ∧ rz * 2 / √3 = rz * 2 / √3
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | ring_nf | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx * 2 / √3 * (rx * 2 / √3) + ry * 2 / √3 * (ry * 2 / √3) + rz * 2 / √3 * (rz * 2 / √3) = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | have h3g0 : (0 : ℝ) ≤ 3 := by linarith | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | simp only [inv_pow, Real.sq_sqrt h3g0] | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * (√3)⁻¹ ^ 2 * 4 + (√3)⁻¹ ^ 2 * ry ^ 2 * 4 + (√3)⁻¹ ^ 2 * rz ^ 2 * 4 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | have hSphere₂ := congrArg (λ (x₀ : ℝ) => x₀ * 4 / 3) hSphere | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (fun x₀ => x₀ * 4 / 3) (rx * rx + ry * ry + rz * rz) = (fun x₀ => x₀ * 4 / 3) (3 / 4)
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | simp only [isUnit_iff_ne_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
IsUnit.div_mul_cancel, div_self] at hSphere₂ | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (fun x₀ => x₀ * 4 / 3) (rx * rx + ry * ry + rz * rz) = (fun x₀ => x₀ * 4 / 3) (3 / 4)
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (rx * rx + ry * ry + rz * rz) * 4 / 3 = 1
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (fun x₀ => x₀ * 4 / 3) (rx * rx + ry * ry + rz * rz) = (fun x₀ => x₀ * 4 / 3) (3 / 4)
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/quaternionLemma.lean | EqualSetsSoqqtstqm1₁AndSoqqtstqm1₃ | [192, 1] | [234, 12] | rw [←hSphere₂] | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (rx * rx + ry * ry + rz * rz) * 4 / 3 = 1
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1 | case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (rx * rx + ry * ry + rz * rz) * 4 / 3 = 1
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = (rx * rx + ry * ry + rz * rz) * 4 / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
r x y z rx ry rz : ℝ
hSphere : rx * rx + ry * ry + rz * rz = 3 / 4
hr : r = 2⁻¹
hx : x = rx
hy : y = ry
hz : z = rz
h3g0 : 0 ≤ 3
hSphere₂ : (rx * rx + ry * ry + rz * rz) * 4 / 3 = 1
⊢ rx ^ 2 * 3⁻¹ * 4 + 3⁻¹ * ry ^ 2 * 4 + 3⁻¹ * rz ^ 2 * 4 = 1
TACTIC:
|
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