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https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y).norm < y.norm | case H2
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ (x % y).norm < y.norm |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | rwa [norm_pos] | x y : GaussInt
hy : y ≠ 0
⊢ 0 < y.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | ext <;> simp [Int.mod'_eq, mod_def, div_def, norm] <;> ring | case H1
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
⊢ x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm } | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | calc
norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj]
_ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2
+ |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs]
_ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos
_ = norm y / 2 * (norm y / 2 * 2) := by ring
_ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num | case H2
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y).norm * y.norm ≤ y.norm / 2 * y.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | simp only [norm_mul, norm_conj] | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y).norm * y.norm = (x % y * y.conj).norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | simp [H1, norm, sq_abs] | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (x % y * y.conj).norm =
|(x.re * y.re + x.im * y.im).mod' y.norm| ^ 2 + |(-(x.re * y.im) + x.im * y.re).mod' y.norm| ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ |(x.re * y.re + x.im * y.im).mod' y.norm| ^ 2 + |(-(x.re * y.im) + x.im * y.re).mod' y.norm| ^ 2 ≤
(y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | ring | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 = y.norm / 2 * (y.norm / 2 * 2) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | gcongr | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ y.norm / 2 * (y.norm / 2 * 2) ≤ y.norm / 2 * y.norm | case h
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ y.norm / 2 * 2 ≤ y.norm |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | apply Int.ediv_mul_le | case h
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ y.norm / 2 * 2 ≤ y.norm | case h.H
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ 2 ≠ 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | norm_num | case h.H
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
⊢ 2 ≠ 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | apply Int.ediv_lt_of_lt_mul | x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ y.norm / 2 < y.norm | case H
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ 0 < 2
case H'
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ y.norm < y.norm * 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | norm_num | case H
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ 0 < 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.norm_mod_lt | [570, 1] | [587, 19] | linarith | case H'
x y : GaussInt
hy : y ≠ 0
norm_y_pos : 0 < y.norm
H1 : x % y * y.conj = { re := (x * y.conj).re.mod' y.norm, im := (x * y.conj).im.mod' y.norm }
H2 : (x % y).norm * y.norm ≤ y.norm / 2 * y.norm
⊢ y.norm < y.norm * 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.natAbs_norm_mod_lt | [603, 1] | [607, 25] | apply Int.ofNat_lt.1 | x y : GaussInt
hy : y ≠ 0
⊢ (x % y).norm.natAbs < y.norm.natAbs | x y : GaussInt
hy : y ≠ 0
⊢ ↑(x % y).norm.natAbs < ↑y.norm.natAbs |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.natAbs_norm_mod_lt | [603, 1] | [607, 25] | simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] | x y : GaussInt
hy : y ≠ 0
⊢ ↑(x % y).norm.natAbs < ↑y.norm.natAbs | x y : GaussInt
hy : y ≠ 0
⊢ (x % y).norm < y.norm |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.natAbs_norm_mod_lt | [603, 1] | [607, 25] | apply norm_mod_lt x hy | x y : GaussInt
hy : y ≠ 0
⊢ (x % y).norm < y.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | apply not_lt_of_ge | x y : GaussInt
hy : y ≠ 0
⊢ ¬(x * y).norm.natAbs < x.norm.natAbs | case h
x y : GaussInt
hy : y ≠ 0
⊢ (x * y).norm.natAbs ≥ x.norm.natAbs |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | rw [norm_mul, Int.natAbs_mul] | case h
x y : GaussInt
hy : y ≠ 0
⊢ (x * y).norm.natAbs ≥ x.norm.natAbs | case h
x y : GaussInt
hy : y ≠ 0
⊢ x.norm.natAbs * y.norm.natAbs ≥ x.norm.natAbs |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | apply le_mul_of_one_le_right (Nat.zero_le _) | case h
x y : GaussInt
hy : y ≠ 0
⊢ x.norm.natAbs * y.norm.natAbs ≥ x.norm.natAbs | case h
x y : GaussInt
hy : y ≠ 0
⊢ 1 ≤ y.norm.natAbs |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | apply Int.ofNat_le.1 | case h
x y : GaussInt
hy : y ≠ 0
⊢ 1 ≤ y.norm.natAbs | case h
x y : GaussInt
hy : y ≠ 0
⊢ ↑1 ≤ ↑y.norm.natAbs |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | rw [coe_natAbs_norm] | case h
x y : GaussInt
hy : y ≠ 0
⊢ ↑1 ≤ ↑y.norm.natAbs | case h
x y : GaussInt
hy : y ≠ 0
⊢ ↑1 ≤ y.norm |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean | GaussInt.not_norm_mul_left_lt_norm | [615, 1] | [622, 51] | exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) | case h
x y : GaussInt
hy : y ≠ 0
⊢ ↑1 ≤ y.norm | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [118, 1] | [123, 30] | induction' n with n ih | n : ℕ
⊢ 0 < fac n | case zero
⊢ 0 < fac 0
case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < fac (n + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [118, 1] | [123, 30] | rw [fac] | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < fac (n + 1) | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < (n + 1) * fac n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [118, 1] | [123, 30] | exact mul_pos n.succ_pos ih | case succ
n : ℕ
ih : 0 < fac n
⊢ 0 < (n + 1) * fac n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [118, 1] | [123, 30] | rw [fac] | case zero
⊢ 0 < fac 0 | case zero
⊢ 0 < 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | fac_pos | [118, 1] | [123, 30] | exact zero_lt_one | case zero
⊢ 0 < 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | induction' n with n ih | i n : ℕ
ipos : 0 < i
ile : i ≤ n
⊢ i ∣ fac n | case zero
i : ℕ
ipos : 0 < i
ile : i ≤ 0
⊢ i ∣ fac 0
case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ fac (n + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | rw [fac] | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ fac (n + 1) | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ (n + 1) * fac n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | rcases Nat.of_le_succ ile with h | h | case succ
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
⊢ i ∣ (n + 1) * fac n | case succ.inl
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i ≤ n
⊢ i ∣ (n + 1) * fac n
case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = n.succ
⊢ i ∣ (n + 1) * fac n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | rw [h] | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = n.succ
⊢ i ∣ (n + 1) * fac n | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = n.succ
⊢ n.succ ∣ (n + 1) * fac n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | apply dvd_mul_right | case succ.inr
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i = n.succ
⊢ n.succ ∣ (n + 1) * fac n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | exact absurd ipos (not_lt_of_ge ile) | case zero
i : ℕ
ipos : 0 < i
ile : i ≤ 0
⊢ i ∣ fac 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | dvd_fac | [133, 1] | [140, 22] | apply dvd_mul_of_dvd_right (ih h) | case succ.inl
i : ℕ
ipos : 0 < i
n : ℕ
ih : i ≤ n → i ∣ fac n
ile : i ≤ n + 1
h : i ≤ n
⊢ i ∣ (n + 1) * fac n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | rcases n with _ | n | n : ℕ
⊢ 2 ^ (n - 1) ≤ fac n | case zero
⊢ 2 ^ (0 - 1) ≤ fac 0
case succ
n : ℕ
⊢ 2 ^ (n + 1 - 1) ≤ fac (n + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | induction' n with n ih | case succ
n : ℕ
⊢ 2 ^ (n + 1 - 1) ≤ fac (n + 1) | case succ.zero
⊢ 2 ^ (0 + 1 - 1) ≤ fac (0 + 1)
case succ.succ
n : ℕ
ih : 2 ^ (n + 1 - 1) ≤ fac (n + 1)
⊢ 2 ^ (n + 1 + 1 - 1) ≤ fac (n + 1 + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | simp at * | case succ.succ
n : ℕ
ih : 2 ^ (n + 1 - 1) ≤ fac (n + 1)
⊢ 2 ^ (n + 1 + 1 - 1) ≤ fac (n + 1 + 1) | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 ^ (n + 1) ≤ fac (n + 1 + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | rw [pow_succ', fac] | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 ^ (n + 1) ≤ fac (n + 1 + 1) | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 * 2 ^ n ≤ (n + 1 + 1) * fac (n + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | apply Nat.mul_le_mul _ ih | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 * 2 ^ n ≤ (n + 1 + 1) * fac (n + 1) | n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 ≤ n + 1 + 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | repeat' apply Nat.succ_le_succ | n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 2 ≤ n + 1 + 1 | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 0 ≤ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | apply zero_le | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 0 ≤ n | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | simp [fac] | case zero
⊢ 2 ^ (0 - 1) ≤ fac 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | simp [fac] | case succ.zero
⊢ 2 ^ (0 + 1 - 1) ≤ fac (0 + 1) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | pow_two_le_fac | [153, 1] | [165, 16] | apply Nat.succ_le_succ | case a
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 1 ≤ n + 1 | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (n + 1)
⊢ 0 ≤ n |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | symm | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i ∈ range (n + 1), i = n * (n + 1) / 2 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) / 2 = ∑ i ∈ range (n + 1), i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) / 2 = ∑ i ∈ range (n + 1), i | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | induction' n with n ih | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) = 2 * ∑ i ∈ range (0 + 1), i
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = 2 * ∑ i ∈ range (n + 1 + 1), i |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | rw [Finset.sum_range_succ, mul_add 2, ← ih] | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = 2 * ∑ i ∈ range (n + 1 + 1), i | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = n * (n + 1) + 2 * (n + 1) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | ring | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) = 2 * ∑ i ∈ range (n + 1), i
⊢ (n + 1) * (n + 1 + 1) = n * (n + 1) + 2 * (n + 1) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | norm_num | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_id | [280, 1] | [285, 7] | simp | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) = 2 * ∑ i ∈ range (0 + 1), i | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | symm | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i ∈ range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) / 6 = ∑ i ∈ range (n + 1), i ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) / 6 = ∑ i ∈ range (n + 1), i ^ 2 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | induction' n with n ih | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2 | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) * (2 * 0 + 1) = 6 * ∑ i ∈ range (0 + 1), i ^ 2
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = 6 * ∑ i ∈ range (n + 1 + 1), i ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | rw [Finset.sum_range_succ, mul_add 6, ← ih] | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = 6 * ∑ i ∈ range (n + 1 + 1), i ^ 2 | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | ring | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i ∈ range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | norm_num | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 6 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | sum_sqr | [293, 1] | [302, 7] | simp | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) * (2 * 0 + 1) = 6 * ∑ i ∈ range (0 + 1), i ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [364, 1] | [367, 15] | induction' n with n ih | n : MyNat
⊢ zero.add n = n | case zero
⊢ zero.add zero = zero
case succ
n : MyNat
ih : zero.add n = n
⊢ zero.add n.succ = n.succ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [364, 1] | [367, 15] | rw [add, ih] | case succ
n : MyNat
ih : zero.add n = n
⊢ zero.add n.succ = n.succ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_add | [364, 1] | [367, 15] | rfl | case zero
⊢ zero.add zero = zero | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [369, 1] | [373, 6] | induction' n with n ih | m n : MyNat
⊢ m.succ.add n = (m.add n).succ | case zero
m : MyNat
⊢ m.succ.add zero = (m.add zero).succ
case succ
m n : MyNat
ih : m.succ.add n = (m.add n).succ
⊢ m.succ.add n.succ = (m.add n.succ).succ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [369, 1] | [373, 6] | rw [add, ih] | case succ
m n : MyNat
ih : m.succ.add n = (m.add n).succ
⊢ m.succ.add n.succ = (m.add n.succ).succ | case succ
m n : MyNat
ih : m.succ.add n = (m.add n).succ
⊢ (m.add n).succ.succ = (m.add n.succ).succ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [369, 1] | [373, 6] | rfl | case succ
m n : MyNat
ih : m.succ.add n = (m.add n).succ
⊢ (m.add n).succ.succ = (m.add n.succ).succ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_add | [369, 1] | [373, 6] | rfl | case zero
m : MyNat
⊢ m.succ.add zero = (m.add zero).succ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [375, 1] | [379, 25] | induction' n with n ih | m n : MyNat
⊢ m.add n = n.add m | case zero
m : MyNat
⊢ m.add zero = zero.add m
case succ
m n : MyNat
ih : m.add n = n.add m
⊢ m.add n.succ = n.succ.add m |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [375, 1] | [379, 25] | rw [add, succ_add, ih] | case succ
m n : MyNat
ih : m.add n = n.add m
⊢ m.add n.succ = n.succ.add m | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [375, 1] | [379, 25] | rw [zero_add] | case zero
m : MyNat
⊢ m.add zero = zero.add m | case zero
m : MyNat
⊢ m.add zero = m |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_comm | [375, 1] | [379, 25] | rfl | case zero
m : MyNat
⊢ m.add zero = m | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_assoc | [381, 1] | [388, 6] | induction' k with k ih | m n k : MyNat
⊢ (m.add n).add k = m.add (n.add k) | case zero
m n : MyNat
⊢ (m.add n).add zero = m.add (n.add zero)
case succ
m n k : MyNat
ih : (m.add n).add k = m.add (n.add k)
⊢ (m.add n).add k.succ = m.add (n.add k.succ) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_assoc | [381, 1] | [388, 6] | rw [add, ih] | case succ
m n k : MyNat
ih : (m.add n).add k = m.add (n.add k)
⊢ (m.add n).add k.succ = m.add (n.add k.succ) | case succ
m n k : MyNat
ih : (m.add n).add k = m.add (n.add k)
⊢ (m.add (n.add k)).succ = m.add (n.add k.succ) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_assoc | [381, 1] | [388, 6] | rfl | case succ
m n k : MyNat
ih : (m.add n).add k = m.add (n.add k)
⊢ (m.add (n.add k)).succ = m.add (n.add k.succ) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.add_assoc | [381, 1] | [388, 6] | rfl | case zero
m n : MyNat
⊢ (m.add n).add zero = m.add (n.add zero) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_add | [391, 1] | [397, 36] | induction' k with k ih | m n k : MyNat
⊢ m.mul (n.add k) = (m.mul n).add (m.mul k) | case zero
m n : MyNat
⊢ m.mul (n.add zero) = (m.mul n).add (m.mul zero)
case succ
m n k : MyNat
ih : m.mul (n.add k) = (m.mul n).add (m.mul k)
⊢ m.mul (n.add k.succ) = (m.mul n).add (m.mul k.succ) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_add | [391, 1] | [397, 36] | rw [add, mul, mul, ih, add_assoc] | case succ
m n k : MyNat
ih : m.mul (n.add k) = (m.mul n).add (m.mul k)
⊢ m.mul (n.add k.succ) = (m.mul n).add (m.mul k.succ) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_add | [391, 1] | [397, 36] | rfl | case zero
m n : MyNat
⊢ m.mul (n.add zero) = (m.mul n).add (m.mul zero) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_mul | [400, 1] | [407, 6] | induction' n with n ih | n : MyNat
⊢ zero.mul n = zero | case zero
⊢ zero.mul zero = zero
case succ
n : MyNat
ih : zero.mul n = zero
⊢ zero.mul n.succ = zero |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_mul | [400, 1] | [407, 6] | rw [mul, ih] | case succ
n : MyNat
ih : zero.mul n = zero
⊢ zero.mul n.succ = zero | case succ
n : MyNat
ih : zero.mul n = zero
⊢ zero.add zero = zero |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_mul | [400, 1] | [407, 6] | rfl | case succ
n : MyNat
ih : zero.mul n = zero
⊢ zero.add zero = zero | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.zero_mul | [400, 1] | [407, 6] | rfl | case zero
⊢ zero.mul zero = zero | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_mul | [410, 1] | [417, 6] | induction' n with n ih | m n : MyNat
⊢ m.succ.mul n = (m.mul n).add n | case zero
m : MyNat
⊢ m.succ.mul zero = (m.mul zero).add zero
case succ
m n : MyNat
ih : m.succ.mul n = (m.mul n).add n
⊢ m.succ.mul n.succ = (m.mul n.succ).add n.succ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_mul | [410, 1] | [417, 6] | rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] | case succ
m n : MyNat
ih : m.succ.mul n = (m.mul n).add n
⊢ m.succ.mul n.succ = (m.mul n.succ).add n.succ | case succ
m n : MyNat
ih : m.succ.mul n = (m.mul n).add n
⊢ (m.mul n).add (m.add n).succ = (m.mul n).add (m.add n.succ) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_mul | [410, 1] | [417, 6] | rfl | case succ
m n : MyNat
ih : m.succ.mul n = (m.mul n).add n
⊢ (m.mul n).add (m.add n).succ = (m.mul n).add (m.add n.succ) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.succ_mul | [410, 1] | [417, 6] | rfl | case zero
m : MyNat
⊢ m.succ.mul zero = (m.mul zero).add zero | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_comm | [420, 1] | [427, 25] | induction' n with n ih | m n : MyNat
⊢ m.mul n = n.mul m | case zero
m : MyNat
⊢ m.mul zero = zero.mul m
case succ
m n : MyNat
ih : m.mul n = n.mul m
⊢ m.mul n.succ = n.succ.mul m |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_comm | [420, 1] | [427, 25] | rw [mul, ih, succ_mul] | case succ
m n : MyNat
ih : m.mul n = n.mul m
⊢ m.mul n.succ = n.succ.mul m | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_comm | [420, 1] | [427, 25] | rw [zero_mul] | case zero
m : MyNat
⊢ m.mul zero = zero.mul m | case zero
m : MyNat
⊢ m.mul zero = zero |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C05_Elementary_Number_Theory/S02_Induction_and_Recursion.lean | MyNat.mul_comm | [420, 1] | [427, 25] | rfl | case zero
m : MyNat
⊢ m.mul zero = zero | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact1 | [387, 1] | [392, 11] | have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact1 | [387, 1] | [392, 11] | calc
a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact1 | [387, 1] | [392, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2
⊢ a * b * 2 ≤ a ^ 2 + b ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact1 | [387, 1] | [392, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact1 | [387, 1] | [392, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a - b) ^ 2 ≥ 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact2 | [394, 1] | [399, 11] | have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 | a b c d e : ℝ
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact2 | [394, 1] | [399, 11] | calc
a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring
_ ≥ 0 := by apply pow_two_nonneg | case h
a b c d e : ℝ
⊢ 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact2 | [394, 1] | [399, 11] | linarith | a b c d e : ℝ
h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2
⊢ -(a * b) * 2 ≤ a ^ 2 + b ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact2 | [394, 1] | [399, 11] | ring | a b c d e : ℝ
⊢ a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean | fact2 | [394, 1] | [399, 11] | apply pow_two_nonneg | a b c d e : ℝ
⊢ (a + b) ^ 2 ≥ 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C07_Hierarchies/S02_Morphisms.lean | map_inv_of_inv | [202, 1] | [204, 61] | rw [← MonoidHomClass₂.map_mul, h, MonoidHomClass₂.map_one] | M N F : Type
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : MonoidHomClass₂ F M N
f : F
m m' : M
h : m * m' = 1
⊢ ↑f m * ↑f m' = 1 | no goals |
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