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core.​init.​data.​int.​comp_lemmas

core.​init.​data.​int.​comp_lemmas

source
int.​bit0_nonneg
int.​bit0_pos
int.​bit1_nonneg
int.​bit1_pos
int.​nat_abs_add_neg
int.​nat_abs_add_nonneg
int.​nat_abs_bit0
int.​nat_abs_bit0_step
int.​nat_abs_bit1_nonneg
int.​nat_abs_bit1_nonneg_step
int.​nat_abs_of_nat_core
int.​nat_abs_of_neg_succ_of_nat
int.​ne_neg_of_ne
int.​ne_neg_of_pos
int.​ne_of_nat_abs_ne_nat_abs_of_nonneg
int.​ne_of_nat_ne_nonneg_case
int.​neg_ne_of_pos
int.​neg_ne_zero_of_ne
int.​neg_succ_of_nat_lt_zero
int.​nonneg_of_pos
int.​of_nat_ge_zero
int.​of_nat_nat_abs_eq_of_nonneg
int.​one_nonneg
int.​one_pos
int.​zero_ne_neg_of_ne
int.​zero_nonneg
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theorem int.​ne_neg_of_ne {a b : ℤ} :
a ≠ b → -a ≠ -b

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theorem int.​neg_ne_zero_of_ne {a : ℤ} :
a ≠ 0 → -a ≠ 0

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theorem int.​zero_ne_neg_of_ne {a : ℤ} :
0 ≠ a → 0 ≠ -a

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theorem int.​neg_ne_of_pos {a b : ℤ} :
a > 0 → b > 0 → -a ≠ b

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theorem int.​ne_neg_of_pos {a b : ℤ} :
a > 0 → b > 0 → a ≠ -b

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theorem int.​one_pos  :
1 > 0

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theorem int.​bit0_pos {a : ℤ} :
a > 0 → bit0 a > 0

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theorem int.​bit1_pos {a : ℤ} :
a ≥ 0 → bit1 a > 0

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theorem int.​zero_nonneg  :
0 ≥ 0

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theorem int.​one_nonneg  :
1 ≥ 0

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theorem int.​bit0_nonneg {a : ℤ} :
a ≥ 0 → bit0 a ≥ 0

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theorem int.​bit1_nonneg {a : ℤ} :
a ≥ 0 → bit1 a ≥ 0

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theorem int.​nonneg_of_pos {a : ℤ} :
a > 0 → a ≥ 0

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theorem int.​neg_succ_of_nat_lt_zero (n : ℕ) :
-[1+ n] < 0

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theorem int.​of_nat_ge_zero (n : ℕ) :
int.of_nat n ≥ 0

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theorem int.​of_nat_nat_abs_eq_of_nonneg {a : ℤ} :
a ≥ 0 → int.of_nat a.nat_abs = a

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theorem int.​ne_of_nat_abs_ne_nat_abs_of_nonneg {a b : ℤ} :
a ≥ 0 → b ≥ 0 → a.nat_abs ≠ b.nat_abs → a ≠ b

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theorem int.​ne_of_nat_ne_nonneg_case {a b : ℤ} {n m : ℕ} :
a ≥ 0 → b ≥ 0 → a.nat_abs = n → b.nat_abs = m → n ≠ m → a ≠ b

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theorem int.​nat_abs_of_nat_core (n : ℕ) :
(int.of_nat n).nat_abs = n

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theorem int.​nat_abs_of_neg_succ_of_nat (n : ℕ) :
-[1+ n].nat_abs = n.succ

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theorem int.​nat_abs_add_nonneg {a b : ℤ} :
a ≥ 0 → b ≥ 0 → (a + b).nat_abs = a.nat_abs + b.nat_abs

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theorem int.​nat_abs_add_neg {a b : ℤ} :
a < 0 → b < 0 → (a + b).nat_abs = a.nat_abs + b.nat_abs

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theorem int.​nat_abs_bit0 (a : ℤ) :
(bit0 a).nat_abs = bit0 a.nat_abs

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theorem int.​nat_abs_bit0_step {a : ℤ} {n : ℕ} :
a.nat_abs = n → (bit0 a).nat_abs = bit0 n

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theorem int.​nat_abs_bit1_nonneg {a : ℤ} :
a ≥ 0 → (bit1 a).nat_abs = bit1 a.nat_abs

source
theorem int.​nat_abs_bit1_nonneg_step {a : ℤ} {n : ℕ} :
a ≥ 0 → a.nat_abs = n → (bit1 a).nat_abs = bit1 n

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