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/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact ⟨hq.1, hq.2, by linarith, m_ne⟩ | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ r / s < ↑q ∧ ↑q < t / u ∧ n ≠ 0 ∧ m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ r / s < ↑q ∧ ↑q < t / u ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast
calc (0:ℝ) ≤ r / s := by positivity
r / s < q := hq.1 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
⊢ 0 < q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
⊢ 0 < q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
⊢ 0 ≤ r / s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
⊢ 0 ≤ r / s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : (nn : ℝ) / m = q := by exact_mod_cast (q.num_den) | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑n / ↑m = ↑q | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑n / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | convert this using 2 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q | case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast Int.toNat_of_nonneg (le_of_lt nn_gt) | case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast (q.num_den) | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑nn / ↑m = ↑q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑nn / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | linarith | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
q : ℚ
hq : r / s < ↑q ∧ ↑q < t / u
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ n ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast m_ne | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ ↑m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ ↑m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast (h m n).2.1 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ ↑m * r < ↑n * s → ↑m * t < ↑n * u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ ↑m * r < ↑n * s → ↑m * t < ↑n * u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | suffices hh : (n:ℝ) * u < m * t | case pos.intro.intro.intro.intro.intro.refine'_2
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ False | case pos.intro.intro.intro.intro.intro.refine'_2
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh✝ : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
hh : ↑n * u < ↑m * t
⊢ False
case hh
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u < ↑m * t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro.intro.intro.refine'_2
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact (lt_self_iff_false ((m : ℝ) * t)).mp (this.trans hh) | case pos.intro.intro.intro.intro.intro.refine'_2
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh✝ : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
hh : ↑n * u < ↑m * t
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro.intro.intro.refine'_2
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh✝ : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
hh : ↑n * u < ↑m * t
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | convert mul_lt_mul hnm_le_tu (by simp : u * m ≤ u * m) (by positivity) (by positivity) using 1 | case hh
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u < ↑m * t | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
case hh
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u < ↑m * t
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | all_goals (field_simp; ring) | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | simp | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ u * ↑m ≤ u * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ u * ↑m ≤ u * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ 0 < u * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ 0 < u * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ 0 ≤ t / u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ 0 ≤ t / u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | (field_simp; ring) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | field_simp | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t * u = t * (u * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t = t / u * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | ring | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t * u = t * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
this : ↑m * t < ↑n * u
⊢ ↑m * t * u = t * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | simp | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ s * ↑m ≤ s * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ s * ↑m ≤ s * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ 0 < s * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ 0 < s * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ 0 ≤ ↑n / ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ 0 ≤ ↑n / ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | (field_simp; ring) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s = ↑n / ↑m * (s * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s = ↑n / ↑m * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | field_simp | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s = ↑n / ↑m * (s * ↑m) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s * ↑m = ↑n * (s * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s = ↑n / ↑m * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | ring | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s * ↑m = ↑n * (s * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : r / s < t / u
n m : ℕ
hrs_le_nm : r / s < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < t / u
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑m * r < ↑n * s → ↑m * t < ↑n * u
⊢ ↑n * s * ↑m = ↑n * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have turs : t / u < r / s := (or_iff_right rstu).mp (lt_or_lt_iff_ne.mpr hh) | case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
⊢ False | case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : ∃ n m : ℕ, t / u < n / m ∧ (n:ℝ) / m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ False | case this
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
this : ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | obtain ⟨n, m, hrs_le_nm, hnm_le_tu, n_ne, m_ne⟩ := this | case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
this : ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
this : ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have m_ne_ℝ : (m : ℝ) ≠ 0 := by exact_mod_cast m_ne | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have mrns : (n : ℝ) * s < m * r → (n : ℝ) * u < m * t := by exact_mod_cast (h m n).2.2 | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : (n : ℝ) * s < m * r := by
convert mul_lt_mul hnm_le_tu (by simp : s * m ≤ s * m) (by positivity) (by positivity) using 1
all_goals (field_simp; ring) | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this : ↑n * s < ↑m * r
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have := mrns this | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this : ↑n * s < ↑m * r
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this : ↑n * s < ↑m * r
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : t * m < n * u := by
convert mul_lt_mul hrs_le_nm (by simp : u * m ≤ u * m) (by positivity) (by positivity) using 1;
all_goals (field_simp; ring) | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ False | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝¹ : ↑n * s < ↑m * r
this✝ : ↑n * u < ↑m * t
this : t * ↑m < ↑n * u
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | linarith | case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝¹ : ↑n * s < ↑m * r
this✝ : ↑n * u < ↑m * t
this : t * ↑m < ↑n * u
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝¹ : ↑n * s < ↑m * r
this✝ : ↑n * u < ↑m * t
this : t * ↑m < ↑n * u
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | obtain ⟨q, hq⟩ := exists_rat_btwn turs | case this
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have q_gt : (0:ℚ) < q := by exact_mod_cast
calc (0 : ℝ) ≤ t / u := by positivity
t / u < q := hq.1 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | let m := q.den | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have m_ne : m ≠ 0 := q.den_nz | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | let nn := q.num | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have nn_gt : 0 < nn := Rat.num_pos_iff_pos.mpr q_gt | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | let n := nn.toNat | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have n_nn : (n:ℤ) = nn := Int.toNat_of_nonneg (le_of_lt nn_gt) | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | use n, m | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ∃ n m, t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : (n : ℝ) / m = q := by
have : (nn : ℝ) / m = q := by exact_mod_cast (q.num_den)
convert this using 2
exact_mod_cast Int.toNat_of_nonneg (le_of_lt nn_gt) | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | rw [this] | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0 | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑q ∧ ↑q < r / s ∧ n ≠ 0 ∧ m ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑n / ↑m ∧ ↑n / ↑m < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact ⟨hq.1, hq.2, by linarith, m_ne⟩ | case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑q ∧ ↑q < r / s ∧ n ≠ 0 ∧ m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.intro
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ t / u < ↑q ∧ ↑q < r / s ∧ n ≠ 0 ∧ m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast
calc (0 : ℝ) ≤ t / u := by positivity
t / u < q := hq.1 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ 0 < q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ 0 < q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ 0 ≤ t / u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
⊢ 0 ≤ t / u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | have : (nn : ℝ) / m = q := by exact_mod_cast (q.num_den) | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑n / ↑m = ↑q | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑n / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | convert this using 2 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q | case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast Int.toNat_of_nonneg (le_of_lt nn_gt) | case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑nn / ↑m = ↑q
⊢ ↑n = ↑nn
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast (q.num_den) | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑nn / ↑m = ↑q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
⊢ ↑nn / ↑m = ↑q
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | linarith | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
q : ℚ
hq : t / u < ↑q ∧ ↑q < r / s
q_gt : 0 < q
m : ℕ := q.den
m_ne : m ≠ 0
nn : ℤ := q.num
nn_gt : 0 < nn
n : ℕ := Int.toNat nn
n_nn : ↑n = nn
this : ↑n / ↑m = ↑q
⊢ n ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast m_ne | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ ↑m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
⊢ ↑m ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | exact_mod_cast (h m n).2.2 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ ↑n * s < ↑m * r → ↑n * u < ↑m * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
⊢ ↑n * s < ↑m * r → ↑n * u < ↑m * t
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | convert mul_lt_mul hnm_le_tu (by simp : s * m ≤ s * m) (by positivity) (by positivity) using 1 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑n * s < ↑m * r | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑n * s = ↑n / ↑m * (s * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑n * s < ↑m * r
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | all_goals (field_simp; ring) | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑n * s = ↑n / ↑m * (s * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑n * s = ↑n / ↑m * (s * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | simp | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ s * ↑m ≤ s * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ s * ↑m ≤ s * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ 0 < s * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ 0 < s * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ 0 ≤ r / s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ 0 ≤ r / s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | (field_simp; ring) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | field_simp | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r * s = r * (s * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r = r / s * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | ring | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r * s = r * (s * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
⊢ ↑m * r * s = r * (s * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | convert mul_lt_mul hrs_le_nm (by simp : u * m ≤ u * m) (by positivity) (by positivity) using 1 | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ t * ↑m < ↑n * u | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ t * ↑m = t / u * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ t * ↑m < ↑n * u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | all_goals (field_simp; ring) | case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ t * ↑m = t / u * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ t * ↑m = t / u * (u * ↑m)
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | simp | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ u * ↑m ≤ u * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ u * ↑m ≤ u * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ 0 < u * ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ 0 < u * ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | positivity | r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ 0 ≤ ↑n / ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ 0 ≤ ↑n / ↑m
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | (field_simp; ring) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | field_simp | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m) | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u * ↑m = ↑n * (u * ↑m) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u = ↑n / ↑m * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | eq_ratio_of_proportion | [34, 1] | [91, 13] | ring | case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u * ↑m = ↑n * (u * ↑m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
r s t u : ℝ
s_ne : s ≠ 0
u_ne : u ≠ 0
hr : 0 ≤ r
hs : 0 ≤ s
ht : 0 ≤ t
hu : 0 ≤ u
h :
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧ (↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
hh : r / s ≠ t / u
rstu : ¬r / s < t / u
turs : t / u < r / s
n m : ℕ
hrs_le_nm : t / u < ↑n / ↑m
hnm_le_tu : ↑n / ↑m < r / s
n_ne : n ≠ 0
m_ne : m ≠ 0
m_ne_ℝ : ↑m ≠ 0
mrns : ↑n * s < ↑m * r → ↑n * u < ↑m * t
this✝ : ↑n * s < ↑m * r
this : ↑n * u < ↑m * t
⊢ ↑n * u * ↑m = ↑n * (u * ↑m)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv | [100, 1] | [106, 88] | dsimp [proportion] at * | r s t u : ℝ
h : proportion r s t u
⊢ proportion s r u t | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ 0 ≤ s ∧
0 ≤ r ∧
0 ≤ u ∧
0 ≤ t ∧
∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧
(↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h : proportion r s t u
⊢ proportion s r u t
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv | [100, 1] | [106, 88] | refine ⟨h.2.1, h.1, h.2.2.2.1, h.2.2.1, ?_⟩ | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ 0 ≤ s ∧
0 ≤ r ∧
0 ≤ u ∧
0 ≤ t ∧
∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧
(↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ ∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ 0 ≤ s ∧
0 ≤ r ∧
0 ≤ u ∧
0 ≤ t ∧
∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧
(↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv | [100, 1] | [106, 88] | intro n m | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ ∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
⊢ ∀ (n m : ℕ),
(↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv | [100, 1] | [106, 88] | have := h.2.2.2.2 m n | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
this : (↑m * r = ↑n * s → ↑m * t = ↑n * u) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t)
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv | [100, 1] | [106, 88] | exact ⟨fun h => (this.1 h.symm).symm, ⟨fun h => (this.2.2 h), fun h => (this.2.1 h)⟩⟩ | r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
this : (↑m * r = ↑n * s → ↑m * t = ↑n * u) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t)
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h :
0 ≤ r ∧
0 ≤ s ∧
0 ≤ t ∧
0 ≤ u ∧
∀ (n m : ℕ),
(↑n * r = ↑m * s → ↑n * t = ↑m * u) ∧
(↑n * r < ↑m * s → ↑n * t < ↑m * u) ∧ (↑m * s < ↑n * r → ↑m * u < ↑n * t)
n m : ℕ
this : (↑m * r = ↑n * s → ↑m * t = ↑n * u) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t)
⊢ (↑n * s = ↑m * r → ↑n * u = ↑m * t) ∧ (↑n * s < ↑m * r → ↑n * u < ↑m * t) ∧ (↑m * r < ↑n * s → ↑m * t < ↑n * u)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv_iff | [109, 1] | [110, 57] | constructor | r s t u : ℝ
⊢ proportion r s t u ↔ proportion s r u t | case mp
r s t u : ℝ
⊢ proportion r s t u → proportion s r u t
case mpr
r s t u : ℝ
⊢ proportion s r u t → proportion r s t u | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
⊢ proportion r s t u ↔ proportion s r u t
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv_iff | [109, 1] | [110, 57] | all_goals exact fun h => proportion_inv h | case mp
r s t u : ℝ
⊢ proportion r s t u → proportion s r u t
case mpr
r s t u : ℝ
⊢ proportion s r u t → proportion r s t u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
r s t u : ℝ
⊢ proportion r s t u → proportion s r u t
case mpr
r s t u : ℝ
⊢ proportion s r u t → proportion r s t u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_inv_iff | [109, 1] | [110, 57] | exact fun h => proportion_inv h | case mpr
r s t u : ℝ
⊢ proportion s r u t → proportion r s t u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
⊢ proportion s r u t → proportion r s t u
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | constructor | r s t u : ℝ
h : proportion r s t u
⊢ s = 0 ↔ u = 0 | case mp
r s t u : ℝ
h : proportion r s t u
⊢ s = 0 → u = 0
case mpr
r s t u : ℝ
h : proportion r s t u
⊢ u = 0 → s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h : proportion r s t u
⊢ s = 0 ↔ u = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | intro s_ne | case mp
r s t u : ℝ
h : proportion r s t u
⊢ s = 0 → u = 0 | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
r s t u : ℝ
h : proportion r s t u
⊢ s = 0 → u = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | have := (h.2.2.2.2 0 1).1 | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ u = 0 | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : ↑0 * r = ↑1 * s → ↑0 * t = ↑1 * u
⊢ u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ u = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | simp at this | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : ↑0 * r = ↑1 * s → ↑0 * t = ↑1 * u
⊢ u = 0 | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : 0 = s → 0 = u
⊢ u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : ↑0 * r = ↑1 * s → ↑0 * t = ↑1 * u
⊢ u = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | exact (this s_ne.symm).symm | case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : 0 = s → 0 = u
⊢ u = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
this : 0 = s → 0 = u
⊢ u = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | intro u_ne | case mpr
r s t u : ℝ
h : proportion r s t u
⊢ u = 0 → s = 0 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
⊢ s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
⊢ u = 0 → s = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | have hh := (h.2.2.2.2 0 1).2.1 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
⊢ s = 0 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : ↑0 * r < ↑1 * s → ↑0 * t < ↑1 * u
⊢ s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
⊢ s = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | simp at hh | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : ↑0 * r < ↑1 * s → ↑0 * t < ↑1 * u
⊢ s = 0 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
⊢ s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : ↑0 * r < ↑1 * s → ↑0 * t < ↑1 * u
⊢ s = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | by_contra s_ne | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
⊢ s = 0 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
⊢ s = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | have : 0 < s := Ne.lt_of_le (fun ss => s_ne ss.symm) h.2.1 | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
⊢ False | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this : 0 < s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | have := hh this | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this : 0 < s
⊢ False | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this✝ : 0 < s
this : 0 < u
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this : 0 < s
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | zero_of_proportion_iff | [113, 1] | [124, 13] | linarith | case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this✝ : 0 < s
this : 0 < u
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
r s t u : ℝ
h : proportion r s t u
u_ne : u = 0
hh : 0 < s → 0 < u
s_ne : ¬s = 0
this✝ : 0 < s
this : 0 < u
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_of_zero | [126, 1] | [126, 70] | simp [proportion] | ⊢ proportion 0 0 0 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ proportion 0 0 0 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | by_cases s_ne : s = 0 | r s t u : ℝ
h : proportion r s t u
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ proportion t u r s
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
r s t u : ℝ
h : proportion r s t u
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | have u_ne := (zero_of_proportion_iff h).mp s_ne | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | have inv := proportion_inv h | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | by_cases r_ne : r = 0 | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
⊢ proportion t u r s
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | have t_ne := (zero_of_proportion_iff inv).mp r_ne | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | rw [s_ne, r_ne, t_ne, u_ne] | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion t u r s | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion 0 0 0 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | exact proportion_of_zero | case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion 0 0 0 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : r = 0
t_ne : t = 0
⊢ proportion 0 0 0 0
TACTIC:
|
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