Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M	input stringlengths 73 2.09M
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply Nat.mul_le_mul	case h.h₂ n : ℕ hn : 2 < n ⊢ (∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) * ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂ n : ℕ hn : 2 < n ⊢ (∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x) * ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) * ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. apply le_trans FactorizationCentralBinom.prodIocSqrtPowLeProdPrimes apply le_trans _ (@prodPrimesLePowFour (2*n/3)) rw [prodPrimes, Finset.range_eq_Ico, Nat.Ico_succ_right] apply Finset.prod_le_prod_of_subset_of_one_le' <;> rw [primes] . apply Finset.monotone_filter_left exact Finset.Ioc_subset_Iic_self . intro p hp _ simp at hp replace hp := Nat.Prime.one_lt hp.right linarith	case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. conv => lhs; rw [← Finset.prod_Ioc_consecutive _ (mulDivLeSelf (by norm_num)) (by linarith : n ≤ 2*n)] apply mulRightLeOfLeOne . apply le_of_eq apply Finset.prod_eq_one intro x hx simp at hx rw [Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul hn hx.right] . simp . apply @Nat.lt_of_div_lt_div _ _ 3 simp exact hx.left . apply FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes exact sqrtTwoMulLeSelf (le_of_lt hn)	case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	repeat rw [← Nat.Ico_succ_succ]	case hs n : ℕ hn : 2 < n ⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n)	case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ico (Nat.succ (2 * n / 3)) (Nat.succ (2 * n))	Please generate a tactic in lean4 to solve the state. STATE: case hs n : ℕ hn : 2 < n ⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n) ⊆ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ∪ Finset.Ioc (2 * n / 3) (2 * n) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply Finset.Ico_subset_Ico_union_Ico	case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ico (Nat.succ (2 * n / 3)) (Nat.succ (2 * n))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ico (Nat.succ (2 * n / 3)) (Nat.succ (2 * n)) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	rw [← Nat.Ico_succ_succ]	case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ioc (2 * n / 3) (2 * n)	case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ico (Nat.succ (2 * n / 3)) (Nat.succ (2 * n))	Please generate a tactic in lean4 to solve the state. STATE: case hs n : ℕ hn : 2 < n ⊢ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n)) ⊆ Finset.Ico (Nat.succ (Nat.sqrt (2 * n))) (Nat.succ (2 * n / 3)) ∪ Finset.Ioc (2 * n / 3) (2 * n) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply le_trans FactorizationCentralBinom.prodIocSqrtPowLeProdPrimes	case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3)	case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ 4 ^ Nat.pred (2 * n / 3)	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ 4 ^ Nat.pred (2 * n / 3) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply le_trans _ (@prodPrimesLePowFour (2*n/3))	case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ 4 ^ Nat.pred (2 * n / 3)	n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ prodPrimes (2 * n / 3)	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₁ n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ 4 ^ Nat.pred (2 * n / 3) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	rw [prodPrimes, Finset.range_eq_Ico, Nat.Ico_succ_right]	n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ prodPrimes (2 * n / 3)	n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ ∏ p in primes (Finset.Icc 0 (2 * n / 3)), p	Please generate a tactic in lean4 to solve the state. STATE: n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ prodPrimes (2 * n / 3) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply Finset.prod_le_prod_of_subset_of_one_le' <;> rw [primes]	n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ ∏ p in primes (Finset.Icc 0 (2 * n / 3)), p	case h n : ℕ hn : 2 < n ⊢ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊆ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i	Please generate a tactic in lean4 to solve the state. STATE: n : ℕ hn : 2 < n ⊢ ∏ p in primes (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)), p ≤ ∏ p in primes (Finset.Icc 0 (2 * n / 3)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. apply Finset.monotone_filter_left exact Finset.Ioc_subset_Iic_self	case h n : ℕ hn : 2 < n ⊢ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊆ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i	case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i	Please generate a tactic in lean4 to solve the state. STATE: case h n : ℕ hn : 2 < n ⊢ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊆ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. intro p hp _ simp at hp replace hp := Nat.Prime.one_lt hp.right linarith	case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply Finset.monotone_filter_left	case h n : ℕ hn : 2 < n ⊢ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊆ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3))	case h.a n : ℕ hn : 2 < n ⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ≤ Finset.Icc 0 (2 * n / 3)	Please generate a tactic in lean4 to solve the state. STATE: case h n : ℕ hn : 2 < n ⊢ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊆ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	exact Finset.Ioc_subset_Iic_self	case h.a n : ℕ hn : 2 < n ⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ≤ Finset.Icc 0 (2 * n / 3)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.a n : ℕ hn : 2 < n ⊢ Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3) ≤ Finset.Icc 0 (2 * n / 3) TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	intro p hp _	case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i	case hf n : ℕ hn : 2 < n p : ℕ hp : p ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊢ 1 ≤ p	Please generate a tactic in lean4 to solve the state. STATE: case hf n : ℕ hn : 2 < n ⊢ ∀ (i : ℕ), i ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) → ¬i ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) → 1 ≤ i TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	simp at hp	case hf n : ℕ hn : 2 < n p : ℕ hp : p ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊢ 1 ≤ p	case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : p ≤ 2 * n / 3 ∧ Nat.Prime p ⊢ 1 ≤ p	Please generate a tactic in lean4 to solve the state. STATE: case hf n : ℕ hn : 2 < n p : ℕ hp : p ∈ Finset.filter Nat.Prime (Finset.Icc 0 (2 * n / 3)) a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) ⊢ 1 ≤ p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	replace hp := Nat.Prime.one_lt hp.right	case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : p ≤ 2 * n / 3 ∧ Nat.Prime p ⊢ 1 ≤ p	case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : 1 < p ⊢ 1 ≤ p	Please generate a tactic in lean4 to solve the state. STATE: case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : p ≤ 2 * n / 3 ∧ Nat.Prime p ⊢ 1 ≤ p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	linarith	case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : 1 < p ⊢ 1 ≤ p	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf n : ℕ hn : 2 < n p : ℕ a✝ : ¬p ∈ Finset.filter Nat.Prime (Finset.Ioc (Nat.sqrt (2 * n)) (2 * n / 3)) hp : 1 < p ⊢ 1 ≤ p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	conv => lhs; rw [← Finset.prod_Ioc_consecutive _ (mulDivLeSelf (by norm_num)) (by linarith : n ≤ 2*n)]	case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ (∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i) * ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ ∏ x in Finset.Ioc (2 * n / 3) (2 * n), x ^ ↑(Nat.factorization (Nat.centralBinom n)) x ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply mulRightLeOfLeOne	case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ (∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i) * ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₂.h1 n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ 1 case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂ n : ℕ hn : 2 < n ⊢ (∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i) * ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. apply le_of_eq apply Finset.prod_eq_one intro x hx simp at hx rw [Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul hn hx.right] . simp . apply @Nat.lt_of_div_lt_div _ _ 3 simp exact hx.left	case h.h₂.h₂.h1 n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ 1 case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1 n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ 1 case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. apply FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes exact sqrtTwoMulLeSelf (le_of_lt hn)	case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	norm_num	n : ℕ hn : 2 < n ⊢ 2 ≤ 3	no goals	Please generate a tactic in lean4 to solve the state. STATE: n : ℕ hn : 2 < n ⊢ 2 ≤ 3 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	linarith	n : ℕ hn : 2 < n ⊢ n ≤ 2 * n	no goals	Please generate a tactic in lean4 to solve the state. STATE: n : ℕ hn : 2 < n ⊢ n ≤ 2 * n TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply le_of_eq	case h.h₂.h₂.h1 n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ 1	case h.h₂.h₂.h1.a n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i = 1	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1 n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply Finset.prod_eq_one	case h.h₂.h₂.h1.a n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i = 1	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n ⊢ ∀ (x : ℕ), x ∈ Finset.Ioc (2 * n / 3) n → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc (2 * n / 3) n, i ^ ↑(Nat.factorization (Nat.centralBinom n)) i = 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	intro x hx	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n ⊢ ∀ (x : ℕ), x ∈ Finset.Ioc (2 * n / 3) n → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : x ∈ Finset.Ioc (2 * n / 3) n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n ⊢ ∀ (x : ℕ), x ∈ Finset.Ioc (2 * n / 3) n → x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	simp at hx	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : x ∈ Finset.Ioc (2 * n / 3) n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : x ∈ Finset.Ioc (2 * n / 3) n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	rw [Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul hn hx.right]	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ 0 = 1 case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ ↑(Nat.factorization (Nat.centralBinom n)) x = 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. simp	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ 0 = 1 case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ 0 = 1 case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	. apply @Nat.lt_of_div_lt_div _ _ 3 simp exact hx.left	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	simp	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ 0 = 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ x ^ 0 = 1 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply @Nat.lt_of_div_lt_div _ _ 3	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < 3 * x / 3	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n < 3 * x TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	simp	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < 3 * x / 3	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < x	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < 3 * x / 3 TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	exact hx.left	case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h1.a.h n : ℕ hn : 2 < n x : ℕ hx : 2 * n / 3 < x ∧ x ≤ n ⊢ 2 * n / 3 < x TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	apply FactorizationCentralBinom.prodIocSqrtLePowLeProdPrimes	case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p	case h.h₂.h₂.h.ha n : ℕ hn : 2 < n ⊢ Nat.sqrt (2 * n) ≤ n	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h n : ℕ hn : 2 < n ⊢ ∏ i in Finset.Ioc n (2 * n), i ^ ↑(Nat.factorization (Nat.centralBinom n)) i ≤ ∏ p in primes (Finset.Ioc n (2 * n)), p TACTIC:
https://github.com/jvlmdr/from_the_book.git	9fb6080539a2f32bb24719600a9e7531abf2328d	FromTheBook/Ch02/Bertrand/CentralBinom.lean	fourPowLeMulProdPrimes	[162, 1]	[216, 45]	exact sqrtTwoMulLeSelf (le_of_lt hn)	case h.h₂.h₂.h.ha n : ℕ hn : 2 < n ⊢ Nat.sqrt (2 * n) ≤ n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.h₂.h₂.h.ha n : ℕ hn : 2 < n ⊢ Nat.sqrt (2 * n) ≤ n TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	let A := {x : N0 \| motive x}	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	have hzmem : z ∈ A := by simp only [Set.mem_setOf_eq] exact hz	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	have hind : (∀ n : N0, n ∈ A → (S n) ∈ A) := by intros n hel simp only [Set.mem_setOf_eq] simp only [Set.mem_setOf_eq] at hel specialize hs n exact hs hel	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	have heq := p3 A ⟨hzmem, hind⟩	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : Subtype.val '' A = N0 ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	simp [A, Set.ext_iff] at heq	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : Subtype.val '' A = N0 ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : ∀ (x : α), (∃ (x_1 : x ∈ N0), motive { val := x, property := (_ : x ∈ N0) }) ↔ x ∈ N0 ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : Subtype.val '' A = N0 ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	specialize heq x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : ∀ (x : α), (∃ (x_1 : x ∈ N0), motive { val := x, property := (_ : x ∈ N0) }) ↔ x ∈ N0 ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : (∃ (x_1 : ↑x ∈ N0), motive { val := ↑x, property := (_ : ↑x ∈ N0) }) ↔ ↑x ∈ N0 ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : ∀ (x : α), (∃ (x_1 : x ∈ N0), motive { val := x, property := (_ : x ∈ N0) }) ↔ x ∈ N0 ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : (∃ (x_1 : ↑x ∈ N0), motive { val := ↑x, property := (_ : ↑x ∈ N0) }) ↔ ↑x ∈ N0 ⊢ motive x	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : motive x ⊢ motive x	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : (∃ (x_1 : ↑x ∈ N0), motive { val := ↑x, property := (_ : ↑x ∈ N0) }) ↔ ↑x ∈ N0 ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	exact heq	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : motive x ⊢ motive x	no goals	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A hind : ∀ n ∈ A, S n ∈ A heq : motive x ⊢ motive x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	simp only [Set.mem_setOf_eq]	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ z ∈ A	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive z	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ z ∈ A TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	exact hz	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive z	no goals	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} ⊢ motive z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	intros n hel	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A ⊢ ∀ n ∈ A, S n ∈ A	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ S n ∈ A	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A ⊢ ∀ n ∈ A, S n ∈ A TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	simp only [Set.mem_setOf_eq]	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ S n ∈ A	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ motive (S n)	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ S n ∈ A TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	simp only [Set.mem_setOf_eq] at hel	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ motive (S n)	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n ⊢ motive (S n)	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : n ∈ A ⊢ motive (S n) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	specialize hs n	motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n ⊢ motive (S n)	motive : ↑N0 → Prop hz : motive z x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n hs : motive n → motive (S n) ⊢ motive (S n)	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z hs : ∀ (n : ↑N0), motive n → motive (S n) x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n ⊢ motive (S n) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	generic_recursor	[31, 1]	[52, 12]	exact hs hel	motive : ↑N0 → Prop hz : motive z x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n hs : motive n → motive (S n) ⊢ motive (S n)	no goals	Please generate a tactic in lean4 to solve the state. STATE: motive : ↑N0 → Prop hz : motive z x : ↑N0 A : Set ↑N0 := {x \| motive x} hzmem : z ∈ A n : ↑N0 hel : motive n hs : motive n → motive (S n) ⊢ motive (S n) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_plus_x_eq_eq	[57, 1]	[63, 19]	apply generic_recursor	⊢ ∀ (x : ↑N0), plus (z, x) = x	case hz ⊢ plus (z, z) = z case hs ⊢ ∀ (n : ↑N0), plus (z, n) = n → plus (z, S n) = S n	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (x : ↑N0), plus (z, x) = x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_plus_x_eq_eq	[57, 1]	[63, 19]	exact zplus z	case hz ⊢ plus (z, z) = z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ plus (z, z) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_plus_x_eq_eq	[57, 1]	[63, 19]	intros n hi	case hs ⊢ ∀ (n : ↑N0), plus (z, n) = n → plus (z, S n) = S n	case hs n : ↑N0 hi : plus (z, n) = n ⊢ plus (z, S n) = S n	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), plus (z, n) = n → plus (z, S n) = S n TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_plus_x_eq_eq	[57, 1]	[63, 19]	rw [splus, hi]	case hs n : ↑N0 hi : plus (z, n) = n ⊢ plus (z, S n) = S n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs n : ↑N0 hi : plus (z, n) = n ⊢ plus (z, S n) = S n TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	apply generic_recursor	⊢ ∀ (c a b : ↑N0), plus (a, plus (b, c)) = plus (plus (a, b), c)	case hz ⊢ ∀ (a b : ↑N0), plus (a, plus (b, z)) = plus (plus (a, b), z) case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), plus (a, plus (b, n)) = plus (plus (a, b), n)) → ∀ (a b : ↑N0), plus (a, plus (b, S n)) = plus (plus (a, b), S n)	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (c a b : ↑N0), plus (a, plus (b, c)) = plus (plus (a, b), c) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	intro a b	case hz ⊢ ∀ (a b : ↑N0), plus (a, plus (b, z)) = plus (plus (a, b), z)	case hz a b : ↑N0 ⊢ plus (a, plus (b, z)) = plus (plus (a, b), z)	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (a b : ↑N0), plus (a, plus (b, z)) = plus (plus (a, b), z) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	rw [zplus, zplus]	case hz a b : ↑N0 ⊢ plus (a, plus (b, z)) = plus (plus (a, b), z)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz a b : ↑N0 ⊢ plus (a, plus (b, z)) = plus (plus (a, b), z) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	intro c hi a b	case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), plus (a, plus (b, n)) = plus (plus (a, b), n)) → ∀ (a b : ↑N0), plus (a, plus (b, S n)) = plus (plus (a, b), S n)	case hs c : ↑N0 hi : ∀ (a b : ↑N0), plus (a, plus (b, c)) = plus (plus (a, b), c) a b : ↑N0 ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c)	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), plus (a, plus (b, n)) = plus (plus (a, b), n)) → ∀ (a b : ↑N0), plus (a, plus (b, S n)) = plus (plus (a, b), S n) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	specialize hi a b	case hs c : ↑N0 hi : ∀ (a b : ↑N0), plus (a, plus (b, c)) = plus (plus (a, b), c) a b : ↑N0 ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c)	case hs c a b : ↑N0 hi : plus (a, plus (b, c)) = plus (plus (a, b), c) ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c)	Please generate a tactic in lean4 to solve the state. STATE: case hs c : ↑N0 hi : ∀ (a b : ↑N0), plus (a, plus (b, c)) = plus (plus (a, b), c) a b : ↑N0 ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	plus_assoc	[66, 1]	[74, 33]	rw [splus, splus, hi, splus]	case hs c a b : ↑N0 hi : plus (a, plus (b, c)) = plus (plus (a, b), c) ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs c a b : ↑N0 hi : plus (a, plus (b, c)) = plus (plus (a, b), c) ⊢ plus (a, plus (b, S c)) = plus (plus (a, b), S c) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	apply generic_recursor	⊢ ∀ (c a b : ↑N0), mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c))	case hz ⊢ ∀ (a b : ↑N0), mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z)) case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (a, plus (b, n)) = plus (mul (a, b), mul (a, n))) → ∀ (a b : ↑N0), mul (a, plus (b, S n)) = plus (mul (a, b), mul (a, S n))	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (c a b : ↑N0), mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	intro a b	case hz ⊢ ∀ (a b : ↑N0), mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z))	case hz a b : ↑N0 ⊢ mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z))	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (a b : ↑N0), mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	rw [zplus, zmul, zplus]	case hz a b : ↑N0 ⊢ mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz a b : ↑N0 ⊢ mul (a, plus (b, z)) = plus (mul (a, b), mul (a, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	intro c hi a b	case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (a, plus (b, n)) = plus (mul (a, b), mul (a, n))) → ∀ (a b : ↑N0), mul (a, plus (b, S n)) = plus (mul (a, b), mul (a, S n))	case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) a b : ↑N0 ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c))	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (a, plus (b, n)) = plus (mul (a, b), mul (a, n))) → ∀ (a b : ↑N0), mul (a, plus (b, S n)) = plus (mul (a, b), mul (a, S n)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	specialize hi a b	case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) a b : ↑N0 ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c))	case hs c a b : ↑N0 hi : mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c))	Please generate a tactic in lean4 to solve the state. STATE: case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) a b : ↑N0 ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_distrib_add	[76, 1]	[84, 43]	rw [splus, smul, smul, hi, plus_assoc]	case hs c a b : ↑N0 hi : mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs c a b : ↑N0 hi : mul (a, plus (b, c)) = plus (mul (a, b), mul (a, c)) ⊢ mul (a, plus (b, S c)) = plus (mul (a, b), mul (a, S c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	apply generic_recursor	⊢ ∀ (c a b : ↑N0), mul (mul (a, b), c) = mul (a, mul (b, c))	case hz ⊢ ∀ (a b : ↑N0), mul (mul (a, b), z) = mul (a, mul (b, z)) case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (mul (a, b), n) = mul (a, mul (b, n))) → ∀ (a b : ↑N0), mul (mul (a, b), S n) = mul (a, mul (b, S n))	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (c a b : ↑N0), mul (mul (a, b), c) = mul (a, mul (b, c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	intro a b	case hz ⊢ ∀ (a b : ↑N0), mul (mul (a, b), z) = mul (a, mul (b, z))	case hz a b : ↑N0 ⊢ mul (mul (a, b), z) = mul (a, mul (b, z))	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (a b : ↑N0), mul (mul (a, b), z) = mul (a, mul (b, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	rw [zmul, zmul, zmul]	case hz a b : ↑N0 ⊢ mul (mul (a, b), z) = mul (a, mul (b, z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz a b : ↑N0 ⊢ mul (mul (a, b), z) = mul (a, mul (b, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	intro c hi a b	case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (mul (a, b), n) = mul (a, mul (b, n))) → ∀ (a b : ↑N0), mul (mul (a, b), S n) = mul (a, mul (b, S n))	case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (mul (a, b), c) = mul (a, mul (b, c)) a b : ↑N0 ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c))	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (a b : ↑N0), mul (mul (a, b), n) = mul (a, mul (b, n))) → ∀ (a b : ↑N0), mul (mul (a, b), S n) = mul (a, mul (b, S n)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	specialize hi a b	case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (mul (a, b), c) = mul (a, mul (b, c)) a b : ↑N0 ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c))	case hs c a b : ↑N0 hi : mul (mul (a, b), c) = mul (a, mul (b, c)) ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c))	Please generate a tactic in lean4 to solve the state. STATE: case hs c : ↑N0 hi : ∀ (a b : ↑N0), mul (mul (a, b), c) = mul (a, mul (b, c)) a b : ↑N0 ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	mul_assoc'	[86, 1]	[94, 41]	rw [smul, smul, hi, mul_distrib_add]	case hs c a b : ↑N0 hi : mul (mul (a, b), c) = mul (a, mul (b, c)) ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs c a b : ↑N0 hi : mul (mul (a, b), c) = mul (a, mul (b, c)) ⊢ mul (mul (a, b), S c) = mul (a, mul (b, S c)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	apply generic_recursor	⊢ ∀ (r m n : ↑N0), exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r))	case hz ⊢ ∀ (m n : ↑N0), exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z)) case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (m, plus (n_1, n)) = mul (exp (m, n_1), exp (m, n))) → ∀ (m n_1 : ↑N0), exp (m, plus (n_1, S n)) = mul (exp (m, n_1), exp (m, S n))	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (r m n : ↑N0), exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	intros m n	case hz ⊢ ∀ (m n : ↑N0), exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z))	case hz m n : ↑N0 ⊢ exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z))	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (m n : ↑N0), exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	rw [zplus, zexp, smul, zmul, zero_plus_x_eq_eq]	case hz m n : ↑N0 ⊢ exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz m n : ↑N0 ⊢ exp (m, plus (n, z)) = mul (exp (m, n), exp (m, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	intros r hi m n	case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (m, plus (n_1, n)) = mul (exp (m, n_1), exp (m, n))) → ∀ (m n_1 : ↑N0), exp (m, plus (n_1, S n)) = mul (exp (m, n_1), exp (m, S n))	case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) m n : ↑N0 ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r))	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (m, plus (n_1, n)) = mul (exp (m, n_1), exp (m, n))) → ∀ (m n_1 : ↑N0), exp (m, plus (n_1, S n)) = mul (exp (m, n_1), exp (m, S n)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	specialize hi m n	case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) m n : ↑N0 ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r))	case hs r m n : ↑N0 hi : exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r))	Please generate a tactic in lean4 to solve the state. STATE: case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) m n : ↑N0 ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_add_eq_mul_exp_exp	[96, 1]	[104, 43]	rw [splus, sexp, sexp, hi, mul_assoc']	case hs r m n : ↑N0 hi : exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs r m n : ↑N0 hi : exp (m, plus (n, r)) = mul (exp (m, n), exp (m, r)) ⊢ exp (m, plus (n, S r)) = mul (exp (m, n), exp (m, S r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	apply generic_recursor	⊢ ∀ (r m n : ↑N0), exp (exp (m, n), r) = exp (m, mul (n, r))	case hz ⊢ ∀ (m n : ↑N0), exp (exp (m, n), z) = exp (m, mul (n, z)) case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (exp (m, n_1), n) = exp (m, mul (n_1, n))) → ∀ (m n_1 : ↑N0), exp (exp (m, n_1), S n) = exp (m, mul (n_1, S n))	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (r m n : ↑N0), exp (exp (m, n), r) = exp (m, mul (n, r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	intros m n	case hz ⊢ ∀ (m n : ↑N0), exp (exp (m, n), z) = exp (m, mul (n, z))	case hz m n : ↑N0 ⊢ exp (exp (m, n), z) = exp (m, mul (n, z))	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (m n : ↑N0), exp (exp (m, n), z) = exp (m, mul (n, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	rw [zexp, zmul, zexp]	case hz m n : ↑N0 ⊢ exp (exp (m, n), z) = exp (m, mul (n, z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz m n : ↑N0 ⊢ exp (exp (m, n), z) = exp (m, mul (n, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	intros r hi m n	case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (exp (m, n_1), n) = exp (m, mul (n_1, n))) → ∀ (m n_1 : ↑N0), exp (exp (m, n_1), S n) = exp (m, mul (n_1, S n))	case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (exp (m, n), r) = exp (m, mul (n, r)) m n : ↑N0 ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r))	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (m n_1 : ↑N0), exp (exp (m, n_1), n) = exp (m, mul (n_1, n))) → ∀ (m n_1 : ↑N0), exp (exp (m, n_1), S n) = exp (m, mul (n_1, S n)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	specialize hi m n	case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (exp (m, n), r) = exp (m, mul (n, r)) m n : ↑N0 ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r))	case hs r m n : ↑N0 hi : exp (exp (m, n), r) = exp (m, mul (n, r)) ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r))	Please generate a tactic in lean4 to solve the state. STATE: case hs r : ↑N0 hi : ∀ (m n : ↑N0), exp (exp (m, n), r) = exp (m, mul (n, r)) m n : ↑N0 ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	exp_assoc	[107, 1]	[115, 48]	rw [smul, sexp, exp_add_eq_mul_exp_exp, hi]	case hs r m n : ↑N0 hi : exp (exp (m, n), r) = exp (m, mul (n, r)) ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs r m n : ↑N0 hi : exp (exp (m, n), r) = exp (m, mul (n, r)) ⊢ exp (exp (m, n), S r) = exp (m, mul (n, S r)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_zero_mul_eq_self'	[120, 1]	[126, 32]	apply generic_recursor	⊢ ∀ (x : ↑N0), mul (S z, x) = x	case hz ⊢ mul (S z, z) = z case hs ⊢ ∀ (n : ↑N0), mul (S z, n) = n → mul (S z, S n) = S n	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (x : ↑N0), mul (S z, x) = x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_zero_mul_eq_self'	[120, 1]	[126, 32]	rw [zmul]	case hz ⊢ mul (S z, z) = z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ mul (S z, z) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_zero_mul_eq_self'	[120, 1]	[126, 32]	intros x hi	case hs ⊢ ∀ (n : ↑N0), mul (S z, n) = n → mul (S z, S n) = S n	case hs x : ↑N0 hi : mul (S z, x) = x ⊢ mul (S z, S x) = S x	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), mul (S z, n) = n → mul (S z, S n) = S n TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_zero_mul_eq_self'	[120, 1]	[126, 32]	rw [smul, hi, splus, zplus]	case hs x : ↑N0 hi : mul (S z, x) = x ⊢ mul (S z, S x) = S x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs x : ↑N0 hi : mul (S z, x) = x ⊢ mul (S z, S x) = S x TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_mul_eq_zero	[128, 1]	[134, 37]	apply generic_recursor	⊢ ∀ (x : ↑N0), mul (z, x) = z	case hz ⊢ mul (z, z) = z case hs ⊢ ∀ (n : ↑N0), mul (z, n) = z → mul (z, S n) = z	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (x : ↑N0), mul (z, x) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_mul_eq_zero	[128, 1]	[134, 37]	rw [zmul]	case hz ⊢ mul (z, z) = z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ mul (z, z) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_mul_eq_zero	[128, 1]	[134, 37]	intro x hi	case hs ⊢ ∀ (n : ↑N0), mul (z, n) = z → mul (z, S n) = z	case hs x : ↑N0 hi : mul (z, x) = z ⊢ mul (z, S x) = z	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), mul (z, n) = z → mul (z, S n) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	zero_mul_eq_zero	[128, 1]	[134, 37]	rw [smul, hi, zero_plus_x_eq_eq]	case hs x : ↑N0 hi : mul (z, x) = z ⊢ mul (z, S x) = z	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs x : ↑N0 hi : mul (z, x) = z ⊢ mul (z, S x) = z TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	apply generic_recursor	⊢ ∀ (b a : ↑N0), plus (S a, b) = S (plus (a, b))	case hz ⊢ ∀ (a : ↑N0), plus (S a, z) = S (plus (a, z)) case hs ⊢ ∀ (n : ↑N0), (∀ (a : ↑N0), plus (S a, n) = S (plus (a, n))) → ∀ (a : ↑N0), plus (S a, S n) = S (plus (a, S n))	Please generate a tactic in lean4 to solve the state. STATE: ⊢ ∀ (b a : ↑N0), plus (S a, b) = S (plus (a, b)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	intro a	case hz ⊢ ∀ (a : ↑N0), plus (S a, z) = S (plus (a, z))	case hz a : ↑N0 ⊢ plus (S a, z) = S (plus (a, z))	Please generate a tactic in lean4 to solve the state. STATE: case hz ⊢ ∀ (a : ↑N0), plus (S a, z) = S (plus (a, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	rw [zplus, zplus]	case hz a : ↑N0 ⊢ plus (S a, z) = S (plus (a, z))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hz a : ↑N0 ⊢ plus (S a, z) = S (plus (a, z)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	intro b hi a	case hs ⊢ ∀ (n : ↑N0), (∀ (a : ↑N0), plus (S a, n) = S (plus (a, n))) → ∀ (a : ↑N0), plus (S a, S n) = S (plus (a, S n))	case hs b : ↑N0 hi : ∀ (a : ↑N0), plus (S a, b) = S (plus (a, b)) a : ↑N0 ⊢ plus (S a, S b) = S (plus (a, S b))	Please generate a tactic in lean4 to solve the state. STATE: case hs ⊢ ∀ (n : ↑N0), (∀ (a : ↑N0), plus (S a, n) = S (plus (a, n))) → ∀ (a : ↑N0), plus (S a, S n) = S (plus (a, S n)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	specialize hi a	case hs b : ↑N0 hi : ∀ (a : ↑N0), plus (S a, b) = S (plus (a, b)) a : ↑N0 ⊢ plus (S a, S b) = S (plus (a, S b))	case hs b a : ↑N0 hi : plus (S a, b) = S (plus (a, b)) ⊢ plus (S a, S b) = S (plus (a, S b))	Please generate a tactic in lean4 to solve the state. STATE: case hs b : ↑N0 hi : ∀ (a : ↑N0), plus (S a, b) = S (plus (a, b)) a : ↑N0 ⊢ plus (S a, S b) = S (plus (a, S b)) TACTIC:
https://github.com/aronerben/lean4-playground.git	5efced915ecee24cd723d28d00aa63f9c7ea0a9c	meetings/ex6.lean	succ_plus_eq_succ_plus	[136, 1]	[144, 27]	rw [splus, hi, ←splus]	case hs b a : ↑N0 hi : plus (S a, b) = S (plus (a, b)) ⊢ plus (S a, S b) = S (plus (a, S b))	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hs b a : ↑N0 hi : plus (S a, b) = S (plus (a, b)) ⊢ plus (S a, S b) = S (plus (a, S b)) TACTIC:

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