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/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | set s : Int := gcd_c1 m n | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | set t : Int := gcd_c2 m n | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h4 : s * m + t * n = gcd m n := gcd_lin_comb n m | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | define at h1 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : rel_prime m n
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [h1, Nat.cast_one] at h4 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = ↑(gcd m n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | set x : Int := t * n * a + s * m * b | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h5 : x ≡ a (MOD m) := Lemma_7_4_7_aux h4 a b | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [add_comm] at h4 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : s * ↑m + t * ↑n = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h6 : s * m * b + t * n * a ≡ b (MOD n) :=
Lemma_7_4_7_aux h4 b a | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h7 : s * m * b + t * n * a = x := by ring | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [h7] at h6 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h8 : m * n ≠ 0 := mul_ne_zero (NeZero.ne m) (NeZero.ne n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : m * n ≠ 0
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
⊢ ∃ r < m * n, ↑... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [←neZero_iff] at h8 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : m * n ≠ 0
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : m * n ≠ 0
⊢... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧
x ≡ x % ↑(m * n) (MOD m * n) := mod_cmpl_res (m * n) x | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
⊢ ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h10 : x % ↑(m * n) < ↑(m * n) ∧
x ≡ x % ↑(m * n) (MOD m * n) := h9.right | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | set r : Nat := Int.toNat (x % ↑(m * n)) | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h11 : x % ↑(m * n) = ↑r := (Int.toNat_of_nonneg h9.left).symm | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [h11, Nat.cast_lt] at h10 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | apply Exists.intro r | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | apply And.intro h10.left | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | have h12 : r ≡ x (MOD (m * n)) := congr_symm h10.right | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | rewrite [Lemma_7_4_5 _ _ h1] at h12 | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | apply And.intro | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n) ∧ x ≡ x %... | case left
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n)... | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m *... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | ring | m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
⊢ s * ↑m * ↑b + t * ↑n * ↑a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : s * ↑m * ↑b + t * ↑n * ↑a ≡ ↑b (MOD n)
⊢ s * ↑m * ↑b + t * ↑n * ↑a... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | show r ≡ a (MOD m) from congr_trans h12.left h5 | case left
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : N... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Lemma_7_4_7 | [979, 1] | [1014, 7] | show r ≡ b (MOD n) from congr_trans h12.right h6 | case right
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : NeZero (m * n)
h9 : 0 ≤ x % ↑(m * n) ∧ x % ↑(m * n) < ↑(m * n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
m n : ℕ
inst✝¹ : NeZero m
inst✝ : NeZero n
h1 : gcd m n = 1
a b : ℕ
s : ℤ := gcd_c1 m n
t : ℤ := gcd_c2 m n
h4 : t * ↑n + s * ↑m = 1
x : ℤ := t * ↑n * ↑a + s * ↑m * ↑b
h5 : x ≡ ↑a (MOD m)
h6 : x ≡ ↑b (MOD n)
h7 : s * ↑m * ↑b + t * ↑n * ↑a = x
h8 : ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Set_prod_def | [1016, 1] | [1017, 46] | rfl | U V : Type
A : Set U
B : Set V
a : U
b : V
⊢ (a, b) ∈ A ×ₛ B ↔ a ∈ A ∧ b ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
a : U
b : V
⊢ (a, b) ∈ A ×ₛ B ↔ a ∈ A ∧ b ∈ B
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Rel_prod_def | [1019, 1] | [1021, 53] | rfl | U V W X : Type
R : Rel U V
S : Rel W X
u : U
v : V
w : W
x : X
⊢ (R ×ᵣ S) (u, w) (v, x) ↔ R u v ∧ S w x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V W X : Type
R : Rel U V
S : Rel W X
u : U
v : V
w : W
x : X
⊢ (R ×ᵣ S) (u, w) (v, x) ↔ R u v ∧ S w x
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_1_2_1 | [1029, 1] | [1036, 7] | obtain (R : Rel U V) (h3 : matching R A B) from h1 | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
⊢ A ×ₛ C ∼ B ×ₛ D | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
⊢ A ×ₛ C ∼ B ×ₛ D | Please generate a tactic in lean4 to solve the state.
STATE:
U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
⊢ A ×ₛ C ∼ B ×ₛ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_1_2_1 | [1029, 1] | [1036, 7] | obtain (S : Rel W X) (h4 : matching S C D) from h2 | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
⊢ A ×ₛ C ∼ B ×ₛ D | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ A ×ₛ C ∼ B ×ₛ D | Please generate a tactic in lean4 to solve the state.
STATE:
U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
⊢ A ×ₛ C ∼ B ×ₛ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_1_2_1 | [1029, 1] | [1036, 7] | apply Exists.intro (R ×ᵣ S) | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ A ×ₛ C ∼ B ×ₛ D | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D) | Please generate a tactic in lean4 to solve the state.
STATE:
U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ A ×ₛ C ∼ B ×ₛ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Theorem_8_1_2_1 | [1029, 1] | [1036, 7] | show matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D) from prod_match h3 h4 | U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V W X : Type
A : Set U
B : Set V
C : Set W
D : Set X
h1 : A ∼ B
h2 : C ∼ D
R : Rel U V
h3 : matching R A B
S : Rel W X
h4 : matching S C D
⊢ matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_def | [1038, 1] | [1038, 61] | rfl | n a : ℕ
⊢ qr n a = (a / n, a % n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a : ℕ
⊢ qr n a = (a / n, a % n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | define | n : ℕ
⊢ one_to_one (qr n) | n : ℕ
⊢ ∀ (x1 x2 : ℕ), qr n x1 = qr n x2 → x1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ one_to_one (qr n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | fix a1 : Nat | n : ℕ
⊢ ∀ (x1 x2 : ℕ), qr n x1 = qr n x2 → x1 = x2 | n a1 : ℕ
⊢ ∀ (x2 : ℕ), qr n a1 = qr n x2 → a1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ (x1 x2 : ℕ), qr n x1 = qr n x2 → x1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | fix a2 : Nat | n a1 : ℕ
⊢ ∀ (x2 : ℕ), qr n a1 = qr n x2 → a1 = x2 | n a1 a2 : ℕ
⊢ qr n a1 = qr n a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 : ℕ
⊢ ∀ (x2 : ℕ), qr n a1 = qr n x2 → a1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | assume h1 : qr n a1 = qr n a2 | n a1 a2 : ℕ
⊢ qr n a1 = qr n a2 → a1 = a2 | n a1 a2 : ℕ
h1 : qr n a1 = qr n a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 a2 : ℕ
⊢ qr n a1 = qr n a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | rewrite [qr_def, qr_def] at h1 | n a1 a2 : ℕ
h1 : qr n a1 = qr n a2
⊢ a1 = a2 | n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 a2 : ℕ
h1 : qr n a1 = qr n a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | have h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n := Prod.mk.inj h1 | n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
⊢ a1 = a2 | n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | show a1 = a2 from
calc a1
_ = n * (a1 / n) + a1 % n := (Nat.div_add_mod a1 n).symm
_ = n * (a2 / n) + a2 % n := by rw [h2.left, h2.right]
_ = a2 := Nat.div_add_mod a2 n | n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n
⊢ a1 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.qr_one_one | [1040, 1] | [1051, 7] | rw [h2.left, h2.right] | n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n
⊢ n * (a1 / n) + a1 % n = n * (a2 / n) + a2 % n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n a1 a2 : ℕ
h1 : (a1 / n, a1 % n) = (a2 / n, a2 % n)
h2 : a1 / n = a2 / n ∧ a1 % n = a2 % n
⊢ n * (a1 / n) + a1 % n = n * (a2 / n) + a2 % n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | rewrite [numElts_def] at h1 | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : numElts A m
h2 : numElts B n
⊢ numElts (A ×ₛ B) (m * n) | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : numElts B n
⊢ numElts (A ×ₛ B) (m * n) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : numElts A m
h2 : numElts B n
⊢ numElts (A ×ₛ B) (m * n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | rewrite [numElts_def] at h2 | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : numElts B n
⊢ numElts (A ×ₛ B) (m * n) | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ numElts (A ×ₛ B) (m * n) | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : numElts B n
⊢ numElts (A ×ₛ B) (m * n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | rewrite [numElts_def] | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ numElts (A ×ₛ B) (m * n) | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ I (m * n) ∼ A ×ₛ B | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ numElts (A ×ₛ B) (m * n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | have h3 : I m ×ₛ I n ∼ A ×ₛ B := Theorem_8_1_2_1 h1 h2 | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ I (m * n) ∼ A ×ₛ B | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
⊢ I (m * n) ∼ A ×ₛ B | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
⊢ I (m * n) ∼ A ×ₛ B
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | have h4 : I (m * n) ∼ I m ×ₛ I n := I_prod m n | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
⊢ I (m * n) ∼ A ×ₛ B | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
h4 : I (m * n) ∼ I m ×ₛ I n
⊢ I (m * n) ∼ A ×ₛ B | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
⊢ I (m * n) ∼ A ×ₛ B
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.numElts_prod | [1059, 1] | [1067, 7] | show I (m * n) ∼ A ×ₛ B from Theorem_8_1_3_3 h4 h3 | U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
h4 : I (m * n) ∼ I m ×ₛ I n
⊢ I (m * n) ∼ A ×ₛ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A : Set U
B : Set V
m n : ℕ
h1 : I m ∼ A
h2 : I n ∼ B
h3 : I m ×ₛ I n ∼ A ×ₛ B
h4 : I (m * n) ∼ I m ×ₛ I n
⊢ I (m * n) ∼ A ×ₛ B
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_def | [1069, 1] | [1069, 75] | rfl | m n a : ℕ
⊢ mod_mod m n a = (a % m, a % n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n a : ℕ
⊢ mod_mod m n a = (a % m, a % n)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | define | m n : ℕ
h1 : rel_prime m n
⊢ one_one_on (mod_mod m n) (Set_rp_below (m * n)) | m n : ℕ
h1 : rel_prime m n
⊢ ∀ ⦃x1 x2 : ℕ⦄, x1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n x1 = mod_mod m n x2 → x1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
⊢ one_one_on (mod_mod m n) (Set_rp_below (m * n))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | fix a1 : Nat | m n : ℕ
h1 : rel_prime m n
⊢ ∀ ⦃x1 x2 : ℕ⦄, x1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n x1 = mod_mod m n x2 → x1 = x2 | m n : ℕ
h1 : rel_prime m n
a1 : ℕ
⊢ ∀ ⦃x2 : ℕ⦄, a1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n x2 → a1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
⊢ ∀ ⦃x1 x2 : ℕ⦄, x1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n x1 = mod_mod m n x2 → x1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | fix a2 : Nat | m n : ℕ
h1 : rel_prime m n
a1 : ℕ
⊢ ∀ ⦃x2 : ℕ⦄, a1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n x2 → a1 = x2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
⊢ a1 ∈ Set_rp_below (m * n) → a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 : ℕ
⊢ ∀ ⦃x2 : ℕ⦄, a1 ∈ Set_rp_below (m * n) → x2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n x2 → a1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | assume h2 : a1 ∈ Set_rp_below (m * n) | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
⊢ a1 ∈ Set_rp_below (m * n) → a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
⊢ a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
⊢ a1 ∈ Set_rp_below (m * n) → a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | assume h3 : a2 ∈ Set_rp_below (m * n) | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
⊢ a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
⊢ mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
⊢ a2 ∈ Set_rp_below (m * n) → mod_mod m n a1 = mod_mod m n a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | assume h4 : mod_mod m n a1 = mod_mod m n a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
⊢ mod_mod m n a1 = mod_mod m n a2 → a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
⊢ mod_mod m n a1 = mod_mod m n a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | define at h2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : a1 ∈ Set_rp_below (m * n)
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | define at h3 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : a2 ∈ Set_rp_below (m * n)
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | rewrite [mod_mod_def, mod_mod_def] at h4 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : mod_mod m n a1 = mod_mod m n a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | have h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n := Prod.mk.inj h4 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | have h6 : m * n ≠ 0 := by linarith | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | have h7 : NeZero m := left_NeZero_of_mul h6 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | have h8 : NeZero n := right_NeZero_of_mul h6 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | rewrite [←congr_iff_mod_eq_Nat, ←congr_iff_mod_eq_Nat] at h5 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m) ∧ ↑a1 ≡ ↑a2 (MOD n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | rewrite [←Lemma_7_4_5 _ _ h1] at h5 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m) ∧ ↑a1 ≡ ↑a2 (MOD n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m) ∧ ↑a1 ≡ ↑a2 (MOD n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2
TACTIC:... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | rewrite [congr_iff_mod_eq_Nat] at h5 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % (m * n) = a2 % (m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : ↑a1 ≡ ↑a2 (MOD m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | rewrite [Nat.mod_eq_of_lt h2.right, Nat.mod_eq_of_lt h3.right] at h5 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % (m * n) = a2 % (m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 = a2
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % (m * n) = a2 % (m * n)
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | show a1 = a2 from h5 | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 = a2
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 = a2
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_one_one_on | [1093, 1] | [1112, 7] | linarith | m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
⊢ m * n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
a1 a2 : ℕ
h2 : rel_prime (m * n) a1 ∧ a1 < m * n
h3 : rel_prime (m * n) a2 ∧ a2 < m * n
h4 : (a1 % m, a1 % n) = (a2 % m, a2 % n)
h5 : a1 % m = a2 % m ∧ a1 % n = a2 % n
⊢ m * n ≠ 0
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_elt_Set_rp_below | [1114, 1] | [1120, 7] | define | a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ a % m ∈ Set_rp_below m | a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m (a % m) ∧ a % m < m | Please generate a tactic in lean4 to solve the state.
STATE:
a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ a % m ∈ Set_rp_below m
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_elt_Set_rp_below | [1114, 1] | [1120, 7] | rewrite [rel_prime_mod] | a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m (a % m) ∧ a % m < m | a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m a ∧ a % m < m | Please generate a tactic in lean4 to solve the state.
STATE:
a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m (a % m) ∧ a % m < m
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_elt_Set_rp_below | [1114, 1] | [1120, 7] | show rel_prime m a ∧ a % m < m from
And.intro h1 (mod_nonzero_lt a (NeZero.ne m)) | a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m a ∧ a % m < m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a m : ℕ
inst✝ : NeZero m
h1 : rel_prime m a
⊢ rel_prime m a ∧ a % m < m
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply Set.ext | m n : ℕ
h1 : rel_prime m n
⊢ image (mod_mod m n) (Set_rp_below (m * n)) = Set_rp_below m ×ₛ Set_rp_below n | case h
m n : ℕ
h1 : rel_prime m n
⊢ ∀ (x : ℕ × ℕ), x ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ x ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
⊢ image (mod_mod m n) (Set_rp_below (m * n)) = Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | fix (b, c) : Nat × Nat | case h
m n : ℕ
h1 : rel_prime m n
⊢ ∀ (x : ℕ × ℕ), x ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ x ∈ Set_rp_below m ×ₛ Set_rp_below n | case h
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n : ℕ
h1 : rel_prime m n
⊢ ∀ (x : ℕ × ℕ), x ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ x ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply Iff.intro | case h
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) → (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n → (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) ↔ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | assume h2 : (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) → (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) → (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | define at h2 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | obtain (a : Nat)
(h3 : a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c)) from h2 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [Set_rp_below_def, mod_mod_def] at h3 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h4 : rel_prime (m * n) a := h3.left.left | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime (m * n) a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [Lemma_7_4_6] at h4 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime (m * n) a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime (m * n) a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h5 : a % m = b ∧ a % n = c := Prod.mk.inj h3.right | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | define | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ b ∈ Set_rp_below m ∧ c ∈ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ (b, c) ∈ Set_rp_below m... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [←h5.left, ←h5.right] | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ b ∈ Set_rp_below m ∧ c ∈ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ b ∈ Set_rp_below m ∧ c ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h6 : m * n ≠ 0 := by linarith | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ a % m ∈ Set_rp_below m ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h7 : NeZero m := left_NeZero_of_mul h6 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
⊢ a % m ∈ ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h8 : NeZero n := right_NeZero_of_mul h6 | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_below n | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZer... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply And.intro | case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a % m ∈ Set_rp_below m ∧ a % n ∈ Set_rp_b... | case h.mp.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a % m ∈ Set_rp_below m
case h.mp.ri... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZer... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | linarith | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ m * n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
⊢ m * n ≠ 0
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | show a % m ∈ Set_rp_below m from mod_elt_Set_rp_below h4.left | case h.mp.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a % m ∈ Set_rp_below m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | show a % n ∈ Set_rp_below n from mod_elt_Set_rp_below h4.right | case h.mp.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 : NeZero m
h8 : NeZero n
⊢ a % n ∈ Set_rp_below n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
a : ℕ
h3 : (rel_prime (m * n) a ∧ a < m * n) ∧ (a % m, a % n) = (b, c)
h4 : rel_prime m a ∧ rel_prime n a
h5 : a % m = b ∧ a % n = c
h6 : m * n ≠ 0
h7 :... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | assume h2 : (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n → (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
⊢ (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n → (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [Set_prod_def, Set_rp_below_def, Set_rp_below_def] at h2 | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (b, c) ∈ Set_rp_below m ×ₛ Set_rp_below n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | define | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n)) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ (b, c) ∈ image (mod_mod m n) (Set_rp_below (m * n))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h3 : m ≠ 0 := by linarith | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | have h4 : n ≠ 0 := by linarith | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [←neZero_iff] at h3 | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : m ≠ 0
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | rewrite [←neZero_iff] at h4 | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : n ≠ 0
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | obtain (a : Nat) (h5 : a < m * n ∧ a ≡ b (MOD m) ∧ a ≡ c (MOD n))
from Lemma_7_4_7 h1 b c | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply Exists.intro a | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c) | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ ∃ x ∈ Set_rp_below (m * n), mod_mod m n x = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | apply And.intro | case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c) | case h.mpr.left
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n)
case h.mpr.right
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
h3 : NeZero m
h4 : NeZero n
a : ℕ
h5 : a < m * n ∧ ↑a ≡ ↑b (MOD m) ∧ ↑a ≡ ↑c (MOD n)
⊢ a ∈ Set_rp_below (m * n) ∧ mod_mod m n a = (b, c)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.mod_mod_image | [1122, 1] | [1182, 7] | linarith | m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : ℕ
h1 : rel_prime m n
b c : ℕ
h2 : (rel_prime m b ∧ b < m) ∧ rel_prime n c ∧ c < n
⊢ m ≠ 0
TACTIC:
|