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https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | apply generic_recursor | ⊢ ∀ (a b : ↑N0), plus (a, b) = plus (b, a) | case hz
⊢ ∀ (b : ↑N0), plus (z, b) = plus (b, z)
case hs
⊢ ∀ (n : ↑N0), (∀ (b : ↑N0), plus (n, b) = plus (b, n)) → ∀ (b : ↑N0), plus (S n, b) = plus (b, S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | intro a | case hz
⊢ ∀ (b : ↑N0), plus (z, b) = plus (b, z) | case hz
a : ↑N0
⊢ plus (z, a) = plus (a, z) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | rw [zplus, zero_plus_x_eq_eq] | case hz
a : ↑N0
⊢ plus (z, a) = plus (a, z) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | intro a hi b | case hs
⊢ ∀ (n : ↑N0), (∀ (b : ↑N0), plus (n, b) = plus (b, n)) → ∀ (b : ↑N0), plus (S n, b) = plus (b, S n) | case hs
a : ↑N0
hi : ∀ (b : ↑N0), plus (a, b) = plus (b, a)
b : ↑N0
⊢ plus (S a, b) = plus (b, S a) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | specialize hi b | case hs
a : ↑N0
hi : ∀ (b : ↑N0), plus (a, b) = plus (b, a)
b : ↑N0
⊢ plus (S a, b) = plus (b, S a) | case hs
a b : ↑N0
hi : plus (a, b) = plus (b, a)
⊢ plus (S a, b) = plus (b, S a) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | plus_comm | [146, 1] | [154, 43] | rw [splus, succ_plus_eq_succ_plus, hi] | case hs
a b : ↑N0
hi : plus (a, b) = plus (b, a)
⊢ plus (S a, b) = plus (b, S a) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | apply generic_recursor | ⊢ ∀ (b a : ↑N0), mul (S a, b) = plus (mul (a, b), b) | case hz
⊢ ∀ (a : ↑N0), mul (S a, z) = plus (mul (a, z), z)
case hs
⊢ ∀ (n : ↑N0),
(∀ (a : ↑N0), mul (S a, n) = plus (mul (a, n), n)) → ∀ (a : ↑N0), mul (S a, S n) = plus (mul (a, S n), S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | intro a | case hz
⊢ ∀ (a : ↑N0), mul (S a, z) = plus (mul (a, z), z) | case hz
a : ↑N0
⊢ mul (S a, z) = plus (mul (a, z), z) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | rw [zmul, zmul, zplus] | case hz
a : ↑N0
⊢ mul (S a, z) = plus (mul (a, z), z) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | intro b hi a | case hs
⊢ ∀ (n : ↑N0),
(∀ (a : ↑N0), mul (S a, n) = plus (mul (a, n), n)) → ∀ (a : ↑N0), mul (S a, S n) = plus (mul (a, S n), S n) | case hs
b : ↑N0
hi : ∀ (a : ↑N0), mul (S a, b) = plus (mul (a, b), b)
a : ↑N0
⊢ mul (S a, S b) = plus (mul (a, S b), S b) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | specialize hi a | case hs
b : ↑N0
hi : ∀ (a : ↑N0), mul (S a, b) = plus (mul (a, b), b)
a : ↑N0
⊢ mul (S a, S b) = plus (mul (a, S b), S b) | case hs
b a : ↑N0
hi : mul (S a, b) = plus (mul (a, b), b)
⊢ mul (S a, S b) = plus (mul (a, S b), S b) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | muls | [156, 1] | [164, 76] | rw [smul, smul, hi, splus, splus, ←plus_assoc, plus_comm b, plus_assoc] | case hs
b a : ↑N0
hi : mul (S a, b) = plus (mul (a, b), b)
⊢ mul (S a, S b) = plus (mul (a, S b), S b) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | apply generic_recursor | ⊢ ∀ (a b : ↑N0), mul (a, b) = mul (b, a) | case hz
⊢ ∀ (b : ↑N0), mul (z, b) = mul (b, z)
case hs
⊢ ∀ (n : ↑N0), (∀ (b : ↑N0), mul (n, b) = mul (b, n)) → ∀ (b : ↑N0), mul (S n, b) = mul (b, S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | intro a | case hz
⊢ ∀ (b : ↑N0), mul (z, b) = mul (b, z) | case hz
a : ↑N0
⊢ mul (z, a) = mul (a, z) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | rw [zmul, zero_mul_eq_zero] | case hz
a : ↑N0
⊢ mul (z, a) = mul (a, z) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | intro a hi b | case hs
⊢ ∀ (n : ↑N0), (∀ (b : ↑N0), mul (n, b) = mul (b, n)) → ∀ (b : ↑N0), mul (S n, b) = mul (b, S n) | case hs
a : ↑N0
hi : ∀ (b : ↑N0), mul (a, b) = mul (b, a)
b : ↑N0
⊢ mul (S a, b) = mul (b, S a) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | specialize hi b | case hs
a : ↑N0
hi : ∀ (b : ↑N0), mul (a, b) = mul (b, a)
b : ↑N0
⊢ mul (S a, b) = mul (b, S a) | case hs
a b : ↑N0
hi : mul (a, b) = mul (b, a)
⊢ mul (S a, b) = mul (b, S a) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | mul_comm' | [166, 1] | [174, 24] | rw [smul, muls, hi] | case hs
a b : ↑N0
hi : mul (a, b) = mul (b, a)
⊢ mul (S a, b) = mul (b, S a) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | apply generic_recursor | ⊢ ∀ (r m n : ↑N0), exp (mul (m, n), r) = mul (exp (m, r), exp (n, r)) | case hz
⊢ ∀ (m n : ↑N0), exp (mul (m, n), z) = mul (exp (m, z), exp (n, z))
case hs
⊢ ∀ (n : ↑N0),
(∀ (m n_1 : ↑N0), exp (mul (m, n_1), n) = mul (exp (m, n), exp (n_1, n))) →
∀ (m n_1 : ↑N0), exp (mul (m, n_1), S n) = mul (exp (m, S n), exp (n_1, S n)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | intro m n | case hz
⊢ ∀ (m n : ↑N0), exp (mul (m, n), z) = mul (exp (m, z), exp (n, z)) | case hz
m n : ↑N0
⊢ exp (mul (m, n), z) = mul (exp (m, z), exp (n, z)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | rw [zexp, zexp, zexp, succ_zero_mul_eq_self'] | case hz
m n : ↑N0
⊢ exp (mul (m, n), z) = mul (exp (m, z), exp (n, z)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | intro r hi m n | case hs
⊢ ∀ (n : ↑N0),
(∀ (m n_1 : ↑N0), exp (mul (m, n_1), n) = mul (exp (m, n), exp (n_1, n))) →
∀ (m n_1 : ↑N0), exp (mul (m, n_1), S n) = mul (exp (m, S n), exp (n_1, S n)) | case hs
r : ↑N0
hi : ∀ (m n : ↑N0), exp (mul (m, n), r) = mul (exp (m, r), exp (n, r))
m n : ↑N0
⊢ exp (mul (m, n), S r) = mul (exp (m, S r), exp (n, S r)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | specialize hi m n | case hs
r : ↑N0
hi : ∀ (m n : ↑N0), exp (mul (m, n), r) = mul (exp (m, r), exp (n, r))
m n : ↑N0
⊢ exp (mul (m, n), S r) = mul (exp (m, S r), exp (n, S r)) | case hs
r m n : ↑N0
hi : exp (mul (m, n), r) = mul (exp (m, r), exp (n, r))
⊢ exp (mul (m, n), S r) = mul (exp (m, S r), exp (n, S r)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | exp_distrib_mul | [176, 1] | [184, 98] | rw [sexp, sexp, sexp, hi, ←mul_assoc', mul_assoc' m, mul_comm' _ m, ←mul_assoc', ←mul_assoc'] | case hs
r m n : ↑N0
hi : exp (mul (m, n), r) = mul (exp (m, r), exp (n, r))
⊢ exp (mul (m, n), S r) = mul (exp (m, S r), exp (n, S r)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | false_proof | [204, 1] | [211, 16] | let baldie : Person := ⟨[]⟩ | ⊢ False | baldie : Person := { hair := [] }
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | false_proof | [204, 1] | [211, 16] | have bald : Bald baldie := rfl | baldie : Person := { hair := [] }
⊢ False | baldie : Person := { hair := [] }
bald : Bald baldie
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | false_proof | [204, 1] | [211, 16] | have still_bald := add_one_hair_still_bald bald | baldie : Person := { hair := [] }
bald : Bald baldie
⊢ False | baldie : Person := { hair := [] }
bald : Bald baldie
still_bald : Bald (add_one_hair baldie)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | false_proof | [204, 1] | [211, 16] | rw [Bald, add_one_hair] at still_bald | baldie : Person := { hair := [] }
bald : Bald baldie
still_bald : Bald (add_one_hair baldie)
⊢ False | baldie : Person := { hair := [] }
bald : Bald baldie
still_bald : { hair := Strand.strand :: baldie.hair }.hair = ∅
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | false_proof | [204, 1] | [211, 16] | contradiction | baldie : Person := { hair := [] }
bald : Bald baldie
still_bald : { hair := Strand.strand :: baldie.hair }.hair = ∅
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | intro person | ⊢ ∀ (person : Person), Bald person | person : Person
⊢ Bald person |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | rw [Bald] | person : Person
⊢ Bald person | person : Person
⊢ person.hair = ∅ |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | induction' person.hair with _ h ih | person : Person
⊢ person.hair = ∅ | case nil
person : Person
⊢ [] = ∅
case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
⊢ head✝ :: h = ∅ |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | rfl | case nil
person : Person
⊢ [] = ∅ | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | have baldie := add_one_hair_still_bald ih | case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
⊢ head✝ :: h = ∅ | case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
baldie : Bald (add_one_hair { hair := h })
⊢ head✝ :: h = ∅ |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | rw [add_one_hair, Bald] at baldie | case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
baldie : Bald (add_one_hair { hair := h })
⊢ head✝ :: h = ∅ | case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
baldie : { hair := Strand.strand :: { hair := h }.hair }.hair = ∅
⊢ head✝ :: h = ∅ |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | all_people_bald | [213, 1] | [222, 17] | exact baldie | case cons
person : Person
head✝ : Strand
h : List Strand
ih : h = ∅
baldie : { hair := Strand.strand :: { hair := h }.hair }.hair = ∅
⊢ head✝ :: h = ∅ | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | ℕplus.mul_def | [254, 1] | [261, 8] | simp only [ne_eq, mul_eq_zero, not_or] | a b : ℕplus
⊢ ↑a * ↑b ≠ 0 | a b : ℕplus
⊢ ¬↑a = 0 ∧ ¬↑b = 0 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | ℕplus.mul_def | [254, 1] | [261, 8] | exact ⟨a.property, b.property⟩ | a b : ℕplus
⊢ ¬↑a = 0 ∧ ¬↑b = 0 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | constructor | a b : ℕplus
⊢ R3 a b ↔ ↑a ∣ ↑b | case mp
a b : ℕplus
⊢ R3 a b → ↑a ∣ ↑b
case mpr
a b : ℕplus
⊢ ↑a ∣ ↑b → R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | intro hdvd | case mp
a b : ℕplus
⊢ R3 a b → ↑a ∣ ↑b | case mp
a b : ℕplus
hdvd : R3 a b
⊢ ↑a ∣ ↑b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | rcases hdvd with ⟨c, hc⟩ | case mp
a b : ℕplus
hdvd : R3 a b
⊢ ↑a ∣ ↑b | case mp.intro
a b c : ℕplus
hc : b = a * c
⊢ ↑a ∣ ↑b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | use c.val | case mp.intro
a b c : ℕplus
hc : b = a * c
⊢ ↑a ∣ ↑b | case h
a b c : ℕplus
hc : b = a * c
⊢ ↑b = ↑a * ↑c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | rw [ℕplus.mul_def] at hc | case h
a b c : ℕplus
hc : b = a * c
⊢ ↑b = ↑a * ↑c | case h
a b c : ℕplus
hc : b = { val := ↑a * ↑c, property := (_ : ↑a * ↑c ≠ 0) }
⊢ ↑b = ↑a * ↑c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | apply_fun Subtype.val at hc | case h
a b c : ℕplus
hc : b = { val := ↑a * ↑c, property := (_ : ↑a * ↑c ≠ 0) }
⊢ ↑b = ↑a * ↑c | case h
a b c : ℕplus
hc : ↑b = ↑{ val := ↑a * ↑c, property := (_ : ↑a * ↑c ≠ 0) }
⊢ ↑b = ↑a * ↑c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | exact hc | case h
a b c : ℕplus
hc : ↑b = ↑{ val := ↑a * ↑c, property := (_ : ↑a * ↑c ≠ 0) }
⊢ ↑b = ↑a * ↑c | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | intro hdvd | case mpr
a b : ℕplus
⊢ ↑a ∣ ↑b → R3 a b | case mpr
a b : ℕplus
hdvd : ↑a ∣ ↑b
⊢ R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | rcases hdvd with ⟨c, hc⟩ | case mpr
a b : ℕplus
hdvd : ↑a ∣ ↑b
⊢ R3 a b | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
⊢ R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | have hb := b.property | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
⊢ R3 a b | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑b ≠ 0
⊢ R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | rw [hc, mul_ne_zero_iff] at hb | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑b ≠ 0
⊢ R3 a b | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
⊢ R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | let lc : ℕplus := ⟨c, hb.2⟩ | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
⊢ R3 a b | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ R3 a b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | use lc | case mpr.intro
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ R3 a b | case h
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ b = a * lc |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | rw [Subtype.ext_iff] | case h
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ b = a * lc | case h
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ ↑b = ↑(a * lc) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | dvd_iff_nat_dvd | [274, 1] | [292, 13] | exact hc | case h
a b : ℕplus
c : ℕ
hc : ↑b = ↑a * c
hb : ↑a ≠ 0 ∧ c ≠ 0
lc : ℕplus := { val := c, property := (_ : c ≠ 0) }
⊢ ↑b = ↑(a * lc) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | intro hwo | ⊢ ¬WeakOrder R3 | hwo : WeakOrder R3
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | simp_rw [WeakOrder, PartialOrder', R3] at hwo | hwo : WeakOrder R3
⊢ False | hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | have hf := hwo.2 2 3 | hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
⊢ False | hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hf : 2 ∣ 3 ∨ 3 ∣ 2
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rcases hf with (hdvd3 | hdvd2) | hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hf : 2 ∣ 3 ∨ 3 ∣ 2
⊢ False | case inl
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd3 : 2 ∣ 3
⊢ False
case inr
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd2 : 3 ∣ 2
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [dvd_def] at hdvd3 | case inl
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd3 : 2 ∣ 3
⊢ False | case inl
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd3 : ∃ c, 3 = 2 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rcases hdvd3 with ⟨c, hc⟩ | case inl
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd3 : ∃ c, 3 = 2 * c
⊢ False | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : 3 = 2 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [Subtype.ext_iff] at hc | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : 3 = 2 * c
⊢ False | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | have hdvd : ∃ c : ℕ, (3 : ℕ) = 2 * c := ⟨c.val, hc⟩ | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
⊢ False | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
hdvd : ∃ c, 3 = 2 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [←dvd_def] at hdvd | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
hdvd : ∃ c, 3 = 2 * c
⊢ False | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
hdvd : 2 ∣ 3
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | contradiction | case inl.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑3 = ↑(2 * c)
hdvd : 2 ∣ 3
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [dvd_def] at hdvd2 | case inr
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd2 : 3 ∣ 2
⊢ False | case inr
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd2 : ∃ c, 2 = 3 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rcases hdvd2 with ⟨c, hc⟩ | case inr
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
hdvd2 : ∃ c, 2 = 3 * c
⊢ False | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : 2 = 3 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [Subtype.ext_iff] at hc | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : 2 = 3 * c
⊢ False | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | have hdvd : ∃ c : ℕ, (2 : ℕ) = 3 * c := ⟨c.val, hc⟩ | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
⊢ False | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
hdvd : ∃ c, 2 = 3 * c
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | rw [←dvd_def] at hdvd | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
hdvd : ∃ c, 2 = 3 * c
⊢ False | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
hdvd : 3 ∣ 2
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_weak_order | [330, 1] | [348, 18] | contradiction | case inr.intro
hwo : (∀ (a b c : ℕplus), (a ∣ b → b ∣ c → a ∣ c) ∧ (a ∣ b → b ∣ a → a = b)) ∧ ∀ (a b : ℕplus), a ∣ b ∨ b ∣ a
c : ℕplus
hc : ↑2 = ↑(3 * c)
hdvd : 3 ∣ 2
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | intro hso | ⊢ ¬StrictOrder R3 | hso : StrictOrder R3
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | rw [StrictOrder] at hso | hso : StrictOrder R3
⊢ False | hso : ∀ (a b c : ℕplus), (R3 a b → R3 b c → R3 a c) ∧ Trichotomy (R3 a b) (R3 b a) (a = b)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | specialize hso 2 3 1 | hso : ∀ (a b c : ℕplus), (R3 a b → R3 b c → R3 a c) ∧ Trichotomy (R3 a b) (R3 b a) (a = b)
⊢ False | hso : (R3 2 3 → R3 3 1 → R3 2 1) ∧ Trichotomy (R3 2 3) (R3 3 2) (2 = 3)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | rcases hso with ⟨_, (⟨_, _, heq⟩ | ⟨_, hdvd3, _⟩ | ⟨hdvd2, _, _⟩)⟩ | hso : (R3 2 3 → R3 3 1 → R3 2 1) ∧ Trichotomy (R3 2 3) (R3 3 2) (2 = 3)
⊢ False | case intro.inl.intro.intro
left✝² : R3 2 3 → R3 3 1 → R3 2 1
left✝¹ : ¬R3 2 3
left✝ : ¬R3 3 2
heq : 2 = 3
⊢ False
case intro.inr.inl.intro.intro
left✝¹ : R3 2 3 → R3 3 1 → R3 2 1
left✝ : ¬R3 2 3
hdvd3 : R3 3 2
right✝ : ¬2 = 3
⊢ False
case intro.inr.inr.intro.intro
left✝¹ : R3 2 3 → R3 3 1 → R3 2 1
hdvd2 : R3 2 3
left✝ : ¬R3 3 2
right✝ : ¬2 = 3
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | rw [←Subtype.coe_inj] at heq | case intro.inl.intro.intro
left✝² : R3 2 3 → R3 3 1 → R3 2 1
left✝¹ : ¬R3 2 3
left✝ : ¬R3 3 2
heq : 2 = 3
⊢ False | case intro.inl.intro.intro
left✝² : R3 2 3 → R3 3 1 → R3 2 1
left✝¹ : ¬R3 2 3
left✝ : ¬R3 3 2
heq : ↑2 = ↑3
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | contradiction | case intro.inl.intro.intro
left✝² : R3 2 3 → R3 3 1 → R3 2 1
left✝¹ : ¬R3 2 3
left✝ : ¬R3 3 2
heq : ↑2 = ↑3
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | contradiction | case intro.inr.inl.intro.intro
left✝¹ : R3 2 3 → R3 3 1 → R3 2 1
left✝ : ¬R3 2 3
hdvd3 : R3 3 2
right✝ : ¬2 = 3
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_not_strict_order | [350, 1] | [360, 18] | contradiction | case intro.inr.inr.intro.intro
left✝¹ : R3 2 3 → R3 3 1 → R3 2 1
hdvd2 : R3 2 3
left✝ : ¬R3 3 2
right✝ : ¬2 = 3
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | intros hpo b c | ⊢ PartialOrder' R3 | hpo b c : ℕplus
⊢ (R3 hpo b → R3 b c → R3 hpo c) ∧ (R3 hpo b → R3 b hpo → hpo = b) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | simp [R3] | hpo b c : ℕplus
⊢ (R3 hpo b → R3 b c → R3 hpo c) ∧ (R3 hpo b → R3 b hpo → hpo = b) | hpo b c : ℕplus
⊢ (hpo ∣ b → b ∣ c → hpo ∣ c) ∧ (hpo ∣ b → b ∣ hpo → hpo = b) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | constructor | hpo b c : ℕplus
⊢ (hpo ∣ b → b ∣ c → hpo ∣ c) ∧ (hpo ∣ b → b ∣ hpo → hpo = b) | case left
hpo b c : ℕplus
⊢ hpo ∣ b → b ∣ c → hpo ∣ c
case right
hpo b c : ℕplus
⊢ hpo ∣ b → b ∣ hpo → hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | intro hdvdb hdvdc | case left
hpo b c : ℕplus
⊢ hpo ∣ b → b ∣ c → hpo ∣ c | case left
hpo b c : ℕplus
hdvdb : hpo ∣ b
hdvdc : b ∣ c
⊢ hpo ∣ c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [dvd_def] at hdvdb | case left
hpo b c : ℕplus
hdvdb : hpo ∣ b
hdvdc : b ∣ c
⊢ hpo ∣ c | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : b ∣ c
⊢ hpo ∣ c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [dvd_def] at hdvdc | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : b ∣ c
⊢ hpo ∣ c | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : ∃ c_1, c = b * c_1
⊢ hpo ∣ c |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [dvd_def] | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : ∃ c_1, c = b * c_1
⊢ hpo ∣ c | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : ∃ c_1, c = b * c_1
⊢ ∃ c_1, c = hpo * c_1 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rcases hdvdb with ⟨k, hk⟩ | case left
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdc : ∃ c_1, c = b * c_1
⊢ ∃ c_1, c = hpo * c_1 | case left.intro
hpo b c : ℕplus
hdvdc : ∃ c_1, c = b * c_1
k : ℕplus
hk : b = hpo * k
⊢ ∃ c_1, c = hpo * c_1 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rcases hdvdc with ⟨l, hl⟩ | case left.intro
hpo b c : ℕplus
hdvdc : ∃ c_1, c = b * c_1
k : ℕplus
hk : b = hpo * k
⊢ ∃ c_1, c = hpo * c_1 | case left.intro.intro
hpo b c k : ℕplus
hk : b = hpo * k
l : ℕplus
hl : c = b * l
⊢ ∃ c_1, c = hpo * c_1 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [Subtype.ext_iff] at hk | case left.intro.intro
hpo b c k : ℕplus
hk : b = hpo * k
l : ℕplus
hl : c = b * l
⊢ ∃ c_1, c = hpo * c_1 | case left.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : c = b * l
⊢ ∃ c_1, c = hpo * c_1 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [Subtype.ext_iff] at hl | case left.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : c = b * l
⊢ ∃ c_1, c = hpo * c_1 | case left.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ∃ c_1, c = hpo * c_1 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | use (k * l) | case left.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ∃ c_1, c = hpo * c_1 | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ c = hpo * (k * l) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [Subtype.ext_iff] | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ c = hpo * (k * l) | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑c = ↑(hpo * (k * l)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | simp_rw [hl, ℕplus.mul_def] | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑c = ↑(hpo * (k * l)) | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑b * ↑l = ↑hpo * (↑k * ↑l) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | simp_rw [hk, ℕplus.mul_def] | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑b * ↑l = ↑hpo * (↑k * ↑l) | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑hpo * ↑k * ↑l = ↑hpo * (↑k * ↑l) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | exact Nat.mul_assoc hpo.val k.val l.val | case h
hpo b c k : ℕplus
hk : ↑b = ↑(hpo * k)
l : ℕplus
hl : ↑c = ↑(b * l)
⊢ ↑hpo * ↑k * ↑l = ↑hpo * (↑k * ↑l) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | intro hdvdb hdvdhpo | case right
hpo b c : ℕplus
⊢ hpo ∣ b → b ∣ hpo → hpo = b | case right
hpo b c : ℕplus
hdvdb : hpo ∣ b
hdvdhpo : b ∣ hpo
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [dvd_def] at hdvdb | case right
hpo b c : ℕplus
hdvdb : hpo ∣ b
hdvdhpo : b ∣ hpo
⊢ hpo = b | case right
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdhpo : b ∣ hpo
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [dvd_def] at hdvdhpo | case right
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdhpo : b ∣ hpo
⊢ hpo = b | case right
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdhpo : ∃ c, hpo = b * c
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rcases hdvdb with ⟨k, hk⟩ | case right
hpo b c : ℕplus
hdvdb : ∃ c, b = hpo * c
hdvdhpo : ∃ c, hpo = b * c
⊢ hpo = b | case right.intro
hpo b c : ℕplus
hdvdhpo : ∃ c, hpo = b * c
k : ℕplus
hk : b = hpo * k
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rcases hdvdhpo with ⟨l, hl⟩ | case right.intro
hpo b c : ℕplus
hdvdhpo : ∃ c, hpo = b * c
k : ℕplus
hk : b = hpo * k
⊢ hpo = b | case right.intro.intro
hpo b c k : ℕplus
hk : b = hpo * k
l : ℕplus
hl : hpo = b * l
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | rw [Subtype.ext_iff, ℕplus.mul_def] at hk | case right.intro.intro
hpo b c k : ℕplus
hk : b = hpo * k
l : ℕplus
hl : hpo = b * l
⊢ hpo = b | case right.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑{ val := ↑hpo * ↑k, property := (_ : ↑hpo * ↑k ≠ 0) }
l : ℕplus
hl : hpo = b * l
⊢ hpo = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex6.lean | r3_partial_order | [362, 1] | [398, 13] | simp at hk | case right.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑{ val := ↑hpo * ↑k, property := (_ : ↑hpo * ↑k ≠ 0) }
l : ℕplus
hl : hpo = b * l
⊢ hpo = b | case right.intro.intro
hpo b c k : ℕplus
hk : ↑b = ↑hpo * ↑k
l : ℕplus
hl : hpo = b * l
⊢ hpo = b |
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