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lean_github

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https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_4
[70, 1]
[77, 7]
rw [hx]
case h f g : ℝ β†’ ℝ x✝ : ℝ hf : SurjectiveFunction f y x : ℝ hx : f x = (y - 1) / 2 ⊒ 1 + f x * 2 = y
case h f g : ℝ β†’ ℝ x✝ : ℝ hf : SurjectiveFunction f y x : ℝ hx : f x = (y - 1) / 2 ⊒ 1 + (y - 1) / 2 * 2 = y
Please generate a tactic in lean4 to solve the state. STATE: case h f g : ℝ β†’ ℝ x✝ : ℝ hf : SurjectiveFunction f y x : ℝ hx : f x = (y - 1) / 2 ⊒ 1 + f x * 2 = y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_4
[70, 1]
[77, 7]
ring
case h f g : ℝ β†’ ℝ x✝ : ℝ hf : SurjectiveFunction f y x : ℝ hx : f x = (y - 1) / 2 ⊒ 1 + (y - 1) / 2 * 2 = y
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h f g : ℝ β†’ ℝ x✝ : ℝ hf : SurjectiveFunction f y x : ℝ hx : f x = (y - 1) / 2 ⊒ 1 + (y - 1) / 2 * 2 = y TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_5
[97, 1]
[102, 9]
intro k
s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => n * s n) s
s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * s n ≀ (fun n => n * s n) n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => n * s n) s TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_5
[97, 1]
[102, 9]
use k
s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * s n ≀ (fun n => n * s n) n
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * s n ≀ (fun n => n * s n) n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * s n ≀ (fun n => n * s n) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_5
[97, 1]
[102, 9]
intro n hnk
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * s n ≀ (fun n => n * s n) n
case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ (fun n => n * s n) n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * s n ≀ (fun n => n * s n) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_5
[97, 1]
[102, 9]
simp
case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ (fun n => n * s n) n
case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ n * s n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ (fun n => n * s n) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_5
[97, 1]
[102, 9]
gcongr
case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ n * s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hnk : n β‰₯ k ⊒ k * s n ≀ n * s n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact
[106, 1]
[106, 71]
simp
s t : β„• β†’ β„• k a b c : β„• ⊒ c β‰₯ max a b ↔ c β‰₯ a ∧ c β‰₯ b
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k a b c : β„• ⊒ c β‰₯ max a b ↔ c β‰₯ a ∧ c β‰₯ b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
intro k
s t : β„• β†’ β„• k : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ ⊒ EventuallyGrowsFaster (s₁ + sβ‚‚) (t₁ + tβ‚‚)
s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ ⊒ EventuallyGrowsFaster (s₁ + sβ‚‚) (t₁ + tβ‚‚) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
obtain ⟨N₁, hNβ‚βŸ© := h₁ k
s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
obtain ⟨Nβ‚‚, hNβ‚‚βŸ© := hβ‚‚ k
case intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case intro.intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: case intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
use max N₁ Nβ‚‚
case intro.intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆ€ n β‰₯ max N₁ Nβ‚‚, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
intro n hn
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆ€ n β‰₯ max N₁ Nβ‚‚, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ max N₁ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n ⊒ βˆ€ n β‰₯ max N₁ Nβ‚‚, k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
rw [useful_fact] at hn
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ max N₁ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ max N₁ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
simp
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ n + tβ‚‚ n) ≀ s₁ n + sβ‚‚ n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ + tβ‚‚) n ≀ (s₁ + sβ‚‚) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
rw [mul_add]
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ n + tβ‚‚ n) ≀ s₁ n + sβ‚‚ n
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n + k * tβ‚‚ n ≀ s₁ n + sβ‚‚ n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * (t₁ n + tβ‚‚ n) ≀ s₁ n + sβ‚‚ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
gcongr
case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n + k * tβ‚‚ n ≀ s₁ n + sβ‚‚ n
case h.h₁ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n ≀ s₁ n case h.hβ‚‚ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * tβ‚‚ n ≀ sβ‚‚ n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n + k * tβ‚‚ n ≀ s₁ n + sβ‚‚ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
apply hN₁
case h.h₁ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n ≀ s₁ n
case h.h₁.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ N₁
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * t₁ n ≀ s₁ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
exact hn.1
case h.h₁.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ N₁
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h₁.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ N₁ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
apply hNβ‚‚
case h.hβ‚‚ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * tβ‚‚ n ≀ sβ‚‚ n
case h.hβ‚‚.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ Nβ‚‚
Please generate a tactic in lean4 to solve the state. STATE: case h.hβ‚‚ s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ k * tβ‚‚ n ≀ sβ‚‚ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_6
[108, 1]
[123, 15]
exact hn.2
case h.hβ‚‚.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ Nβ‚‚
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.hβ‚‚.a s t : β„• β†’ β„• k✝ : β„• s₁ sβ‚‚ t₁ tβ‚‚ : β„• β†’ β„• h₁ : EventuallyGrowsFaster s₁ t₁ hβ‚‚ : EventuallyGrowsFaster sβ‚‚ tβ‚‚ k N₁ : β„• hN₁ : βˆ€ n β‰₯ N₁, k * t₁ n ≀ s₁ n Nβ‚‚ : β„• hNβ‚‚ : βˆ€ n β‰₯ Nβ‚‚, k * tβ‚‚ n ≀ sβ‚‚ n n : β„• hn : n β‰₯ N₁ ∧ n β‰₯ Nβ‚‚ ⊒ n β‰₯ Nβ‚‚ TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
use fun n ↦ n ^ n
s t : β„• β†’ β„• k : β„• ⊒ βˆƒ s, EventuallyGrowsFaster (fun n => s (n + 1)) s ∧ βˆ€ (n : β„•), s n β‰  0
case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n) ∧ βˆ€ (n : β„•), n ^ n β‰  0
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• ⊒ βˆƒ s, EventuallyGrowsFaster (fun n => s (n + 1)) s ∧ βˆ€ (n : β„•), s n β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
constructor
case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n) ∧ βˆ€ (n : β„•), n ^ n β‰  0
case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ^ n β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n) ∧ βˆ€ (n : β„•), n ^ n β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
intro k
case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n
case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1) ^ (n + 1)) fun n => n ^ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
use k
case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
intro n hn
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
simp
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ^ n ≀ (n + 1) ^ (n + 1)
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n ^ n) n ≀ (fun n => (n + 1) ^ (n + 1)) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
calc k * n ^ n ≀ n * n ^ n := by gcongr _ = n ^ (n + 1) := by ring _ ≀ (n + 1) ^ (n + 1) := by gcongr simp
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ^ n ≀ (n + 1) ^ (n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ^ n ≀ (n + 1) ^ (n + 1) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
gcongr
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ^ n ≀ n * n ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ^ n ≀ n * n ^ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
ring
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n * n ^ n = n ^ (n + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n * n ^ n = n ^ (n + 1) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
gcongr
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ^ (n + 1) ≀ (n + 1) ^ (n + 1)
case hab s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ^ (n + 1) ≀ (n + 1) ^ (n + 1) TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
simp
case hab s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hab s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
intro n
case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ^ n β‰  0
case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ^ n β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ^ n β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7
[127, 1]
[140, 27]
apply pow_self_ne_zero
case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ^ n β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ^ n β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
use fun n ↦ n!
s t : β„• β†’ β„• k : β„• ⊒ βˆƒ s, EventuallyGrowsFaster (fun n => s (n + 1)) s ∧ βˆ€ (n : β„•), s n β‰  0
case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n !) ∧ βˆ€ (n : β„•), n ! β‰  0
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• ⊒ βˆƒ s, EventuallyGrowsFaster (fun n => s (n + 1)) s ∧ βˆ€ (n : β„•), s n β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
constructor
case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n !) ∧ βˆ€ (n : β„•), n ! β‰  0
case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n ! case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ! β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k : β„• ⊒ (EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n !) ∧ βˆ€ (n : β„•), n ! β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
intro k
case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n !
case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left s t : β„• β†’ β„• k : β„• ⊒ EventuallyGrowsFaster (fun n => (n + 1)!) fun n => n ! TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
use k
case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
intro n hn
case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k : β„• ⊒ βˆ€ n β‰₯ k, k * (fun n => n !) n ≀ (fun n => (n + 1)!) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
simp
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n !) n ≀ (fun n => (n + 1)!) n
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ! ≀ (n + 1)!
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * (fun n => n !) n ≀ (fun n => (n + 1)!) n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
calc k * n! ≀ n * n! := by gcongr _ ≀ (n + 1) * n! := by gcongr exact? _ = (n + 1)! := by rfl
case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ! ≀ (n + 1)!
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ! ≀ (n + 1)! TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
gcongr
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ! ≀ n * n !
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ k * n ! ≀ n * n ! TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
gcongr
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n * n ! ≀ (n + 1) * n !
case bc s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n * n ! ≀ (n + 1) * n ! TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
exact?
case bc s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case bc s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ n ≀ n + 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
rfl
s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ (n + 1) * n ! = (n + 1)!
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ k n : β„• hn : n β‰₯ k ⊒ (n + 1) * n ! = (n + 1)! TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
intro n
case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ! β‰  0
case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ! β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k : β„• ⊒ βˆ€ (n : β„•), n ! β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_7'
[143, 1]
[156, 11]
exact?
case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ! β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k n : β„• ⊒ n ! β‰  0 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
use n - 1
s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ βˆƒ k β‰₯ m, k + 1 = n
case h s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m ∧ n - 1 + 1 = n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ βˆƒ k β‰₯ m, k + 1 = n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
constructor
case h s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m ∧ n - 1 + 1 = n
case h.left s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 + 1 = n
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m ∧ n - 1 + 1 = n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
exact?
case h.left s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 β‰₯ m TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
have : 1 ≀ n := by exact?
case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 + 1 = n
case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 this : 1 ≀ n ⊒ n - 1 + 1 = n
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ n - 1 + 1 = n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
exact?
case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 this : 1 ≀ n ⊒ n - 1 + 1 = n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 this : 1 ≀ n ⊒ n - 1 + 1 = n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
useful_fact2
[166, 1]
[171, 11]
exact?
s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ 1 ≀ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k n m : β„• hn : n β‰₯ m + 1 ⊒ 1 ≀ n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
have h3s : βˆ€ n, 1 ≀ s n := by intro n exact?
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 ⊒ βˆ€ (k : β„•), βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n ⊒ βˆ€ (k : β„•), βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 ⊒ βˆ€ (k : β„•), βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
intro k
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n ⊒ βˆ€ (k : β„•), βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n ⊒ βˆ€ (k : β„•), βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
obtain ⟨N, hN⟩ := hs k
s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ n β‰₯ N, k * s n ≀ (fun n => s (n + 1)) n ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k : β„• ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
simp at hN
case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ n β‰₯ N, k * s n ≀ (fun n => s (n + 1)) n ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ n β‰₯ N, k * s n ≀ (fun n => s (n + 1)) n ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
use N + 1
case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k
case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆ€ n β‰₯ N + 1, s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆƒ N, βˆ€ n β‰₯ N, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
intros n hn
case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆ€ n β‰₯ N + 1, s n β‰₯ k
case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 ⊒ s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) ⊒ βˆ€ n β‰₯ N + 1, s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
obtain ⟨n', hn'N, hn'n⟩ := useful_fact2 hn
case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 ⊒ s n β‰₯ k
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n ⊒ s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case h s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 ⊒ s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
specialize hN n' hn'N
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n ⊒ s n β‰₯ k
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s (n' + 1) ⊒ s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N : β„• hN : βˆ€ (n : β„•), N ≀ n β†’ k * s n ≀ s (n + 1) n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n ⊒ s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
rw [hn'n] at hN
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s (n' + 1) ⊒ s n β‰₯ k
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ s n β‰₯ k
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s (n' + 1) ⊒ s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
calc k = k * 1 := by simp _ ≀ k * s n' := by gcongr apply h3s _ ≀ s n := by assumption
case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ s n β‰₯ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.intro.intro s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ s n β‰₯ k TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
intro n
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 ⊒ βˆ€ (n : β„•), 1 ≀ s n
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 n : β„• ⊒ 1 ≀ s n
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 ⊒ βˆ€ (n : β„•), 1 ≀ s n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
exact?
s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 n : β„• ⊒ 1 ≀ s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 n : β„• ⊒ 1 ≀ s n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
simp
s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k = k * 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k = k * 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
gcongr
s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k * 1 ≀ k * s n'
case bc s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ 1 ≀ s n'
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k * 1 ≀ k * s n' TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
apply h3s
case bc s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ 1 ≀ s n'
no goals
Please generate a tactic in lean4 to solve the state. STATE: case bc s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ 1 ≀ s n' TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Solutions/Solutions2.lean
exercise2_8
[173, 1]
[190, 29]
assumption
s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k * s n' ≀ s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: s t : β„• β†’ β„• k✝ : β„• hs : EventuallyGrowsFaster (fun n => s (n + 1)) s h2s : βˆ€ (n : β„•), s n β‰  0 h3s : βˆ€ (n : β„•), 1 ≀ s n k N n : β„• hn : n β‰₯ N + 1 n' : β„• hn'N : n' β‰₯ N hn'n : n' + 1 = n hN : k * s n' ≀ s n ⊒ k * s n' ≀ s n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_1
[25, 1]
[26, 86]
sorry
n : β„• ⊒ βˆ‘ i in range (n + 1), ↑(i ^ 3) = (βˆ‘ i in range (n + 1), ↑i) ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• ⊒ βˆ‘ i in range (n + 1), ↑(i ^ 3) = (βˆ‘ i in range (n + 1), ↑i) ^ 2 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_2
[36, 1]
[39, 8]
have h := wf.wf.has_min
ΞΉ : Type u_1 Ξ± : Type u_2 inst✝ : LinearOrder ΞΉ wf : WellFoundedLT ΞΉ A C : ΞΉ β†’ Set Ξ± hC : βˆ€ (i : ΞΉ), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j ⊒ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
ΞΉ : Type u_1 Ξ± : Type u_2 inst✝ : LinearOrder ΞΉ wf : WellFoundedLT ΞΉ A C : ΞΉ β†’ Set Ξ± hC : βˆ€ (i : ΞΉ), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j h : βˆ€ (s : Set ΞΉ), Set.Nonempty s β†’ βˆƒ a ∈ s, βˆ€ x ∈ s, Β¬x < a ⊒ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 Ξ± : Type u_2 inst✝ : LinearOrder ΞΉ wf : WellFoundedLT ΞΉ A C : ΞΉ β†’ Set Ξ± hC : βˆ€ (i : ΞΉ), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j ⊒ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_2
[36, 1]
[39, 8]
sorry
ΞΉ : Type u_1 Ξ± : Type u_2 inst✝ : LinearOrder ΞΉ wf : WellFoundedLT ΞΉ A C : ΞΉ β†’ Set Ξ± hC : βˆ€ (i : ΞΉ), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j h : βˆ€ (s : Set ΞΉ), Set.Nonempty s β†’ βˆƒ a ∈ s, βˆ€ x ∈ s, Β¬x < a ⊒ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i
no goals
Please generate a tactic in lean4 to solve the state. STATE: ΞΉ : Type u_1 Ξ± : Type u_2 inst✝ : LinearOrder ΞΉ wf : WellFoundedLT ΞΉ A C : ΞΉ β†’ Set Ξ± hC : βˆ€ (i : ΞΉ), C i = A i \ ⋃ j, ⋃ (_ : j < i), A j h : βˆ€ (s : Set ΞΉ), Set.Nonempty s β†’ βˆƒ a ∈ s, βˆ€ x ∈ s, Β¬x < a ⊒ Pairwise (Disjoint on C) ∧ ⋃ i, C i = ⋃ i, A i TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_4
[62, 1]
[63, 85]
sorry
n : β„• ⊒ Β¬Nat.Prime n ↔ n = 0 ∨ n = 1 ∨ βˆƒ a b, 2 ≀ a ∧ 2 ≀ b ∧ n = a * b
no goals
Please generate a tactic in lean4 to solve the state. STATE: n : β„• ⊒ Β¬Nat.Prime n ↔ n = 0 ∨ n = 1 ∨ βˆƒ a b, 2 ≀ a ∧ 2 ≀ b ∧ n = a * b TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
by_contra h2n
n : β„• hn : Nat.Prime (2 ^ n - 1) ⊒ Nat.Prime n
n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : Β¬Nat.Prime n ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : Nat.Prime (2 ^ n - 1) ⊒ Nat.Prime n TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
rw [exercise4_4] at h2n
n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : Β¬Nat.Prime n ⊒ False
n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : n = 0 ∨ n = 1 ∨ βˆƒ a b, 2 ≀ a ∧ 2 ≀ b ∧ n = a * b ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : Β¬Nat.Prime n ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
obtain rfl|rfl|⟨a, b, ha, hb, rfl⟩ := h2n
n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : n = 0 ∨ n = 1 ∨ βˆƒ a b, 2 ≀ a ∧ 2 ≀ b ∧ n = a * b ⊒ False
case inl hn : Nat.Prime (2 ^ 0 - 1) ⊒ False case inr.inl hn : Nat.Prime (2 ^ 1 - 1) ⊒ False case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: n : β„• hn : Nat.Prime (2 ^ n - 1) h2n : n = 0 ∨ n = 1 ∨ βˆƒ a b, 2 ≀ a ∧ 2 ≀ b ∧ n = a * b ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h : (2 : β„€) ^ a - 1 ∣ (2 : β„€) ^ (a * b) - 1
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ False
case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h2 : 2 ^ 2 ≀ 2 ^ a := by sorry
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h3 : 1 ≀ 2 ^ a := by sorry
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h4 : 2 ^ a - 1 β‰  1 := by zify; simp [h3]; linarith
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 := by apply tsub_lt_tsub_right_of_le h3 sorry
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h6' : 2 ^ 0 ≀ 2 ^ (a * b) := by sorry
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h6 : 1 ≀ 2 ^ (a * b) := h6'
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
have h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) ⊒ False
case h' a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) ⊒ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
rw [Nat.prime_def_lt] at hn
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : 2 ≀ 2 ^ (a * b) - 1 ∧ βˆ€ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 β†’ m = 1 h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : 2 ≀ 2 ^ (a * b) - 1 ∧ βˆ€ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 β†’ m = 1 h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr.inr.intro.intro.intro.intro a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : 2 ≀ 2 ^ (a * b) - 1 ∧ βˆ€ m < 2 ^ (a * b) - 1, m ∣ 2 ^ (a * b) - 1 β†’ m = 1 h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 h5 : 2 ^ a - 1 < 2 ^ (a * b) - 1 h6' : 2 ^ 0 ≀ 2 ^ (a * b) h6 : 1 ≀ 2 ^ (a * b) h' : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
case inl hn : Nat.Prime (2 ^ 0 - 1) ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl hn : Nat.Prime (2 ^ 0 - 1) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
case inr.inl hn : Nat.Prime (2 ^ 1 - 1) ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr.inl hn : Nat.Prime (2 ^ 1 - 1) ⊒ False TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
rw [← Int.modEq_zero_iff_dvd]
case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1
case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ (a * b) - 1 ≑ 0 [ZMOD 2 ^ a - 1]
Please generate a tactic in lean4 to solve the state. STATE: case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
calc (2 : β„€) ^ (a * b) - 1 ≑ ((2 : β„€) ^ a) ^ b - 1 [ZMOD (2 : β„€) ^ a - 1] := by sorry _ ≑ (1 : β„€) ^ b - 1 [ZMOD (2 : β„€) ^ a - 1] := by sorry _ ≑ 0 [ZMOD (2 : β„€) ^ a - 1] := by sorry
case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ (a * b) - 1 ≑ 0 [ZMOD 2 ^ a - 1]
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ (a * b) - 1 ≑ 0 [ZMOD 2 ^ a - 1] TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ (a * b) - 1 ≑ (2 ^ a) ^ b - 1 [ZMOD 2 ^ a - 1]
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 2 ^ (a * b) - 1 ≑ (2 ^ a) ^ b - 1 [ZMOD 2 ^ a - 1] TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ (2 ^ a) ^ b - 1 ≑ 1 ^ b - 1 [ZMOD 2 ^ a - 1]
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ (2 ^ a) ^ b - 1 ≑ 1 ^ b - 1 [ZMOD 2 ^ a - 1] TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 1 ^ b - 1 ≑ 0 [ZMOD 2 ^ a - 1]
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) ⊒ 1 ^ b - 1 ≑ 0 [ZMOD 2 ^ a - 1] TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ 2 ^ 2 ≀ 2 ^ a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 ⊒ 2 ^ 2 ≀ 2 ^ a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a ⊒ 1 ≀ 2 ^ a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a ⊒ 1 ≀ 2 ^ a TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
zify
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ 2 ^ a - 1 β‰  1
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ ↑(2 ^ a - 1) β‰  1
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ 2 ^ a - 1 β‰  1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
simp [h3]
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ ↑(2 ^ a - 1) β‰  1
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ Β¬2 ^ a - 1 = 1
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ ↑(2 ^ a - 1) β‰  1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
linarith
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ Β¬2 ^ a - 1 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a ⊒ Β¬2 ^ a - 1 = 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
apply tsub_lt_tsub_right_of_le h3
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ 2 ^ a - 1 < 2 ^ (a * b) - 1
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ 2 ^ a < 2 ^ (a * b)
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ 2 ^ a - 1 < 2 ^ (a * b) - 1 TACTIC:
https://github.com/fpvandoorn/LeanCourse23.git
7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d
LeanCourse/Assignments/Assignment4.lean
exercise4_5
[66, 1]
[89, 8]
sorry
a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ 2 ^ a < 2 ^ (a * b)
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„• ha : 2 ≀ a hb : 2 ≀ b hn : Nat.Prime (2 ^ (a * b) - 1) h : 2 ^ a - 1 ∣ 2 ^ (a * b) - 1 h2 : 2 ^ 2 ≀ 2 ^ a h3 : 1 ≀ 2 ^ a h4 : 2 ^ a - 1 β‰  1 ⊒ 2 ^ a < 2 ^ (a * b) TACTIC: