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https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
ring
n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : -↑↑k < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 ⊒ -↑↑k = ↑↑k * -1
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : -↑↑k < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 ⊒ -↑↑k = ↑↑k * -1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact (mul_lt_mul_left h₆).mp hβ‚…
n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : ↑↑k * -1 < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 ⊒ -1 < y
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : ↑↑k * -1 < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 ⊒ -1 < y TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
linarith
n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : ↑↑k * -1 < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 hβ‚ˆ : -1 < y ⊒ y = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ x i j : β„• hir : i < ↑k hjr : j < ↑k hβ‚€ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j) h₁ : ↑↑n β‰  0 y : β„€ h₃ : ↑i - ↑j = ↑↑k * y hβ‚„ : ↑↑k * y < ↑↑k hβ‚… : ↑↑k * -1 < ↑↑k * y h₆ : ↑↑k > 0 h₇ : y < 1 hβ‚ˆ : -1 < y ⊒ y = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
intros hβ‚€
case e_f.h.e_c.mp n k : β„•+ m : β„€ z : β„‚ x : β„• ⊒ ↑↑n ∣ ↑x + m β†’ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
case e_f.h.e_c.mp n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : ↑↑n ∣ ↑x + m ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mp n k : β„•+ m : β„€ z : β„‚ x : β„• ⊒ ↑↑n ∣ ↑x + m β†’ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
obtain ⟨w, hβ‚βŸ© := hβ‚€
case e_f.h.e_c.mp n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : ↑↑n ∣ ↑x + m ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mp n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : ↑↑n ∣ ↑x + m ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + (↑↑n * ↑i) := by intros i ring_nf
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
simp_rw [hβ‚‚, h₁]
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
use ((-w) % k).toNat
case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i
case h n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mp.intro n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
constructor
case h n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
case h.left n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
Please generate a tactic in lean4 to solve the state. STATE: case h n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
intros i
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w ⊒ βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w i : β„• ⊒ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w ⊒ βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
ring_nf
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w i : β„• ⊒ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w i : β„• ⊒ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine (Int.toNat_lt' ?h.left.hn).mpr ?h.left.a
case h.left n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k
case h.left.hn n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑k β‰  0 case h.left.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ -w % ↑↑k < ↑↑k
Please generate a tactic in lean4 to solve the state. STATE: case h.left n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ (-w % ↑↑k).toNat < ↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact PNat.ne_zero k
case h.left.hn n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑k β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left.hn n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑k β‰  0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Int.emod_lt_of_pos (-w) ?h.left.a.H
case h.left.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ -w % ↑↑k < ↑↑k
case h.left.a.H n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑↑k
Please generate a tactic in lean4 to solve the state. STATE: case h.left.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ -w % ↑↑k < ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Int.ofNat_pos.mpr ?h.left.a.H.a
case h.left.a.H n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑↑k
case h.left.a.H.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑k
Please generate a tactic in lean4 to solve the state. STATE: case h.left.a.H n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact PNat.pos k
case h.left.a.H.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑k
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left.a.H.a n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 < ↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have h₃ : ↑(-w % ↑↑k).toNat = (-w % ↑↑k) := by refine Int.toNat_of_nonneg ?_ refine Int.emod_nonneg (-w) ?_ exact Ne.symm (NeZero.ne' (k : β„€))
case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
rw [h₃]
case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
clear h₁ hβ‚‚ h₃ m z x
case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
case h.right n k : β„•+ w : β„€ ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
suffices hβ‚€ : ↑↑k ∣ w + (-w % ↑↑k)
case h.right n k : β„•+ w : β„€ ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) case hβ‚€ n k : β„•+ w : β„€ ⊒ ↑↑k ∣ w + -w % ↑↑k
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ w : β„€ ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Int.toNat_of_nonneg ?_
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑(-w % ↑↑k).toNat = -w % ↑↑k
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 ≀ -w % ↑↑k
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑(-w % ↑↑k).toNat = -w % ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Int.emod_nonneg (-w) ?_
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 ≀ -w % ↑↑k
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑k β‰  0
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ 0 ≀ -w % ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact Ne.symm (NeZero.ne' (k : β„€))
n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑k β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x : β„• w : β„€ h₁ : ↑x + m = ↑↑n * w hβ‚‚ : βˆ€ (i : β„•), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i ⊒ ↑↑k β‰  0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have h₁ := mul_dvd_mul_left (n : β„€) hβ‚€
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
ring_nf at *
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact h₁
case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right n k : β„•+ w : β„€ hβ‚€ : ↑↑k ∣ w + -w % ↑↑k h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) ⊒ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Int.dvd_of_emod_eq_zero ?hβ‚€.H
case hβ‚€ n k : β„•+ w : β„€ ⊒ ↑↑k ∣ w + -w % ↑↑k
case hβ‚€.H n k : β„•+ w : β„€ ⊒ (w + -w % ↑↑k) % ↑↑k = 0
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€ n k : β„•+ w : β„€ ⊒ ↑↑k ∣ w + -w % ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have hβ‚€ : (0 : β„€) = 0 % k := by exact rfl
case hβ‚€.H n k : β„•+ w : β„€ ⊒ (w + -w % ↑↑k) % ↑↑k = 0
case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H n k : β„•+ w : β„€ ⊒ (w + -w % ↑↑k) % ↑↑k = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
rw [hβ‚€]
case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0
case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
refine Eq.symm (Int.ModEq.eq ?hβ‚€.H.a)
case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] := by exact Int.mod_modEq (-w) ↑↑k
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have hβ‚‚ : w ≑ w [ZMOD ↑↑k] := by exact rfl
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have h₃ := Int.ModEq.add hβ‚‚ h₁
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ w + -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
simp only [add_right_neg] at h₃
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ w + -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ 0 [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ w + -w [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact h₃.symm
case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ 0 [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k]
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚€.H.a n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] hβ‚‚ : w ≑ w [ZMOD ↑↑k] h₃ : w + -w % ↑↑k ≑ 0 [ZMOD ↑↑k] ⊒ 0 ≑ w + -w % ↑↑k [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact rfl
n k : β„•+ w : β„€ ⊒ 0 = 0 % ↑↑k
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ w : β„€ ⊒ 0 = 0 % ↑↑k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact Int.mod_modEq (-w) ↑↑k
n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ -w % ↑↑k ≑ -w [ZMOD ↑↑k]
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k ⊒ -w % ↑↑k ≑ -w [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact rfl
n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] ⊒ w ≑ w [ZMOD ↑↑k]
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ w : β„€ hβ‚€ : 0 = 0 % ↑↑k h₁ : -w % ↑↑k ≑ -w [ZMOD ↑↑k] ⊒ w ≑ w [ZMOD ↑↑k] TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
intros hβ‚€
case e_f.h.e_c.mpr n k : β„•+ m : β„€ z : β„‚ x : β„• ⊒ (βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)) β†’ ↑↑n ∣ ↑x + m
case e_f.h.e_c.mpr n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) ⊒ ↑↑n ∣ ↑x + m
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr n k : β„•+ m : β„€ z : β„‚ x : β„• ⊒ (βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)) β†’ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
obtain ⟨w, _, hβ‚‚βŸ© := hβ‚€
case e_f.h.e_c.mpr n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) ⊒ ↑↑n ∣ ↑x + m
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr n k : β„•+ m : β„€ z : β„‚ x : β„• hβ‚€ : βˆƒ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) ⊒ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have h₃ := dvd_of_mul_right_dvd hβ‚‚
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
have hβ‚„ : (n : β„€) ∣ ↑↑n * ↑w := by exact Int.dvd_mul_right (↑n) w
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
rw [(show ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + ↑(x + m) by ring_nf)] at h₃
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact (Int.dvd_iff_dvd_of_dvd_add h₃).mp hβ‚„
case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m
no goals
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.e_c.mpr.intro.intro n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑↑n ∣ ↑x + m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
exact Int.dvd_mul_right (↑n) w
n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑↑n * ↑w
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) ⊒ ↑↑n ∣ ↑↑n * ↑w TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffSumOfRuesDiff
[165, 1]
[256, 48]
ring_nf
n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + (↑x + m)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n k : β„•+ m : β„€ z : β„‚ x w : β„• left✝ : w < ↑k hβ‚‚ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m) h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m) hβ‚„ : ↑↑n ∣ ↑↑n * ↑w ⊒ ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + (↑x + m) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffNthIteratedDeriv
[258, 1]
[261, 10]
rw [RuesDiffIteratedDeriv, RuesDiffMPeriodic n m 1]
n : β„•+ m : β„€ ⊒ iteratedDeriv (↑n) (RuesDiff n m) = RuesDiff n m
n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n)
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ ⊒ iteratedDeriv (↑n) (RuesDiff n m) = RuesDiff n m TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffNthIteratedDeriv
[258, 1]
[261, 10]
simp only [one_mul]
n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n)
n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n)
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffNthIteratedDeriv
[258, 1]
[261, 10]
ring_nf
n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ ⊒ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hβ‚€ : βˆ€ (x : β„•), (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) := by intros x ring_nf
n : β„•+ k : β„€ ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0
n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
simp_rw [hβ‚€, Complex.exp_nat_mul]
n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0
n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
clear hβ‚€
n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
n : β„•+ k : β„€ ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚€ : βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hem := Classical.em (↑↑n ∣ k)
n : β„•+ k : β„€ ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hβ‚‚ : (n : β„‚) β‰  0 := by exact Ne.symm (NeZero.ne' (n : β„‚))
n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k hβ‚‚ : ↑↑n β‰  0 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rcases hem with hemt | hemf
n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k hβ‚‚ : ↑↑n β‰  0 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k hβ‚‚ : ↑↑n β‰  0 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
intros x
n : β„•+ k : β„€ ⊒ βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
n : β„•+ k : β„€ x : β„• ⊒ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ ⊒ βˆ€ (x : β„•), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
ring_nf
n : β„•+ k : β„€ x : β„• ⊒ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ x : β„• ⊒ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
exact Ne.symm (NeZero.ne' (n : β„‚))
n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k ⊒ ↑↑n β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k ⊒ ↑↑n β‰  0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [h₁, if_pos hemt]
case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k h₁ : βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k h₁ : βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 ⊒ βˆ‘ x ∈ range ↑n, 1 = ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k h₁ : βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
simp only [sum_const, card_range, nsmul_eq_mul, mul_one]
case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k h₁ : βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 ⊒ βˆ‘ x ∈ range ↑n, 1 = ↑↑n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k h₁ : βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 ⊒ βˆ‘ x ∈ range ↑n, 1 = ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
congr
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1
case e_f n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = βˆ‘ x ∈ range ↑n, 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
ext1 x
case e_f n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1
case e_f.h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k x : β„• ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k ⊒ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
obtain ⟨kβ‚‚, kDiv⟩ := hemt
case e_f.h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k x : β„• ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemt : ↑↑n ∣ k x : β„• ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [kDiv]
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑(↑↑n * kβ‚‚) / ↑↑n * I)) ^ x = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
field_simp
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑(↑↑n * kβ‚‚) / ↑↑n * I)) ^ x = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑(↑↑n * kβ‚‚) / ↑↑n * I)) ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
suffices h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1 case h₃ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [h₃]
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ 1 ^ x = 1
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
simp only [one_pow]
case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ 1 ^ x = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case e_f.h.intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ h₃ : cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 ⊒ 1 ^ x = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
refine Complex.exp_eq_one_iff.mpr ?h₃.a
case h₃ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1
case h₃.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ βˆƒ n, 2 * ↑π * (↑kβ‚‚ * I) = ↑n * (2 * ↑π * I)
Please generate a tactic in lean4 to solve the state. STATE: case h₃ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ cexp (2 * ↑π * (↑kβ‚‚ * I)) = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
use kβ‚‚
case h₃.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ βˆƒ n, 2 * ↑π * (↑kβ‚‚ * I) = ↑n * (2 * ↑π * I)
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ 2 * ↑π * (↑kβ‚‚ * I) = ↑kβ‚‚ * (2 * ↑π * I)
Please generate a tactic in lean4 to solve the state. STATE: case h₃.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ βˆƒ n, 2 * ↑π * (↑kβ‚‚ * I) = ↑n * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
ring_nf
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ 2 * ↑π * (↑kβ‚‚ * I) = ↑kβ‚‚ * (2 * ↑π * I)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 x : β„• kβ‚‚ : β„€ kDiv : k = ↑↑n * kβ‚‚ ⊒ 2 * ↑π * (↑kβ‚‚ * I) = ↑kβ‚‚ * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [if_neg hemf]
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 := by by_contra h rw [Complex.exp_eq_one_iff] at h obtain ⟨m, h⟩ := h rw [(show 2 * ↑π * (↑k / ↑↑n * I) = (↑k / ↑↑n) * (2 * ↑π * I) by ring)] at h have h₃ := mul_right_cancelβ‚€ Complex.two_pi_I_ne_zero h field_simp at h₃ have hβ‚„ : k = m * n := by exact mod_cast h₃ have hβ‚… : (n : β„€) ∣ k := by exact Dvd.intro_left m (id (Eq.symm hβ‚„)) apply hemf exact hβ‚…
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [geom_sum_eq hβ‚€]
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆ‘ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
suffices h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ (n : β„•) = 1
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
by_contra h
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [Complex.exp_eq_one_iff] at h
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1 ⊒ False
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : βˆƒ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1 ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
obtain ⟨m, h⟩ := h
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : βˆƒ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I) ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k h : βˆƒ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I) ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [(show 2 * ↑π * (↑k / ↑↑n * I) = (↑k / ↑↑n) * (2 * ↑π * I) by ring)] at h
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I) ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I) ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have h₃ := mul_right_cancelβ‚€ Complex.two_pi_I_ne_zero h
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k / ↑↑n = ↑m ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
field_simp at h₃
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k / ↑↑n = ↑m ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k / ↑↑n = ↑m ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hβ‚„ : k = m * n := by exact mod_cast h₃
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
have hβ‚… : (n : β„€) ∣ k := by exact Dvd.intro_left m (id (Eq.symm hβ‚„))
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
apply hemf
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ False
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ ↑↑n ∣ k
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ False TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
exact hβ‚…
case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ ↑↑n ∣ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n hβ‚… : ↑↑n ∣ k ⊒ ↑↑n ∣ k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
ring
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I) ⊒ 2 * ↑π * (↑k / ↑↑n * I) = ↑k / ↑↑n * (2 * ↑π * I)
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I) ⊒ 2 * ↑π * (↑k / ↑↑n * I) = ↑k / ↑↑n * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
exact mod_cast h₃
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n ⊒ k = m * ↑↑n
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n ⊒ k = m * ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
exact Dvd.intro_left m (id (Eq.symm hβ‚„))
n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n ⊒ ↑↑n ∣ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k m : β„€ h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I) h₃ : ↑k = ↑m * ↑↑n hβ‚„ : k = m * ↑↑n ⊒ ↑↑n ∣ k TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [h₁]
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
simp only [sub_self, zero_div]
case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 ⊒ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
rw [(Complex.exp_nat_mul _ n).symm]
case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1
Please generate a tactic in lean4 to solve the state. STATE: case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
refine Complex.exp_eq_one_iff.mpr ?h₁.a
case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1
case h₁.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆƒ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I)
Please generate a tactic in lean4 to solve the state. STATE: case h₁ n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1 TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
use k
case h₁.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆƒ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I)
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I)
Please generate a tactic in lean4 to solve the state. STATE: case h₁.a n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ βˆƒ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
field_simp
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I)
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I)
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RouGeometricSumEqIte
[263, 1]
[308, 14]
ring_nf
case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h n : β„•+ k : β„€ hβ‚‚ : ↑↑n β‰  0 hemf : ¬↑↑n ∣ k hβ‚€ : cexp (2 * ↑π * (↑k / ↑↑n * I)) β‰  1 ⊒ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I) TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffEqualsExpSum
[310, 1]
[352, 25]
simp_rw [Complex.exp_add]
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ kβ‚€ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑kβ‚€ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑kβ‚€ / ↑↑n) * I)) / ↑↑n
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ kβ‚€ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑kβ‚€ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑kβ‚€ / ↑↑n) * I)) / ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffEqualsExpSum
[310, 1]
[352, 25]
have hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑(Nat.factorial k_1) := by intros k exact ExpTsumForm (z * cexp (2 * ↑π * (↑k / ↑↑n) * I))
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffEqualsExpSum
[310, 1]
[352, 25]
simp_rw [hβ‚€]
n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffEqualsExpSum
[310, 1]
[352, 25]
clear hβ‚€
n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ z : β„‚ hβ‚€ : βˆ€ (k : β„•), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n TACTIC:
https://github.com/Nazgand/NazgandLean4.git
a6c5455a06d14c59786b1c23c2d20dada7598be6
NazgandLean4/RootOfUnityExponentialSum.lean
RuesDiffEqualsExpSum
[310, 1]
[352, 25]
simp_rw [←tsum_mul_right]
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, βˆ‘' (x_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
Please generate a tactic in lean4 to solve the state. STATE: n : β„•+ m : β„€ z : β„‚ ⊒ RuesDiff n m z = (βˆ‘ x ∈ range ↑n, (βˆ‘' (k_1 : β„•), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n TACTIC: