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refactor(library/data/num): cleanup

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This commit is contained in:
Leonardo de Moura 2014-10-05 13:47:51 -07:00
parent 86591c7272
commit c9e5e40477
1 changed files with 22 additions and 18 deletions
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40
library/data/num.lean
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@ -38,15 +38,15 @@ namespace pos_num
definition size (a : pos_num) : pos_num :=
rec_on a one (λn r, succ r) (λn r, succ r)
theorem succ_not_is_one {a : pos_num} : is_one (succ a) = ff :=
theorem succ_not_is_one (a : pos_num) : is_one (succ a) = ff :=
induction_on a rfl (take n iH, rfl) (take n iH, rfl)
theorem pred_succ {a : pos_num} : pred (succ a) = a :=
theorem pred.succ (a : pos_num) : pred (succ a) = a :=
rec_on a
rfl
(take (n : pos_num) (iH : pred (succ n) = n),
calc
pred (succ (bit1 n)) = cond ff one (bit1 (pred (succ n))) : {succ_not_is_one}
pred (succ (bit1 n)) = cond ff one (bit1 (pred (succ n))) : {!succ_not_is_one}
... = bit1 (pred (succ n)) : rfl
... = bit1 n : {iH})
(take (n : pos_num) (iH : pred (succ n) = n), rfl)
@ -66,32 +66,36 @@ namespace pos_num
infixl `+` := add
theorem add_one_one : one + one = bit0 one :=
section
variables (a b : pos_num)
theorem add.one_one : one + one = bit0 one :=
rfl
theorem add_one_bit0 {a : pos_num} : one + (bit0 a) = bit1 a :=
theorem add.one_bit0 : one + (bit0 a) = bit1 a :=
rfl
theorem add_one_bit1 {a : pos_num} : one + (bit1 a) = succ (bit1 a) :=
theorem add.one_bit1 : one + (bit1 a) = succ (bit1 a) :=
rfl
theorem add_bit0_one {a : pos_num} : (bit0 a) + one = bit1 a :=
theorem add.bit0_one : (bit0 a) + one = bit1 a :=
rfl
theorem add_bit1_one {a : pos_num} : (bit1 a) + one = succ (bit1 a) :=
theorem add.bit1_one : (bit1 a) + one = succ (bit1 a) :=
rfl
theorem add_bit0_bit0 {a b : pos_num} : (bit0 a) + (bit0 b) = bit0 (a + b) :=
theorem add.bit0_bit0 : (bit0 a) + (bit0 b) = bit0 (a + b) :=
rfl
theorem add_bit0_bit1 {a b : pos_num} : (bit0 a) + (bit1 b) = bit1 (a + b) :=
theorem add.bit0_bit1 : (bit0 a) + (bit1 b) = bit1 (a + b) :=
rfl
theorem add_bit1_bit0 {a b : pos_num} : (bit1 a) + (bit0 b) = bit1 (a + b) :=
theorem add.bit1_bit0 : (bit1 a) + (bit0 b) = bit1 (a + b) :=
rfl
theorem add_bit1_bit1 {a b : pos_num} : (bit1 a) + (bit1 b) = succ (bit1 (a + b)) :=
theorem add.bit1_bit1 : (bit1 a) + (bit1 b) = succ (bit1 (a + b)) :=
rfl
end
definition mul (a b : pos_num) : pos_num :=
rec_on a
@ -101,16 +105,16 @@ namespace pos_num
infixl `*` := mul
theorem mul_one_left (a : pos_num) : one * a = a :=
theorem mul.one_left (a : pos_num) : one * a = a :=
rfl
theorem mul_one_right (a : pos_num) : a * one = a :=
theorem mul.one_right (a : pos_num) : a * one = a :=
induction_on a
rfl
(take (n : pos_num) (iH : n * one = n),
calc bit1 n * one = bit0 (n * one) + one : rfl
... = bit0 n + one : {iH}
... = bit1 n : add_bit0_one)
... = bit1 n : !add.bit0_one)
(take (n : pos_num) (iH : n * one = n),
calc bit0 n * one = bit0 (n * one) : rfl
... = bit0 n : {iH})
@ -142,14 +146,14 @@ namespace num
definition size (a : num) : num :=
rec_on a (pos one) (λp, pos (size p))
theorem pred_succ (a : num) : pred (succ a) = a :=
theorem pred.succ (a : num) : pred (succ a) = a :=
rec_on a
rfl
(λp, calc
pred (succ (pos p)) = pred (pos (pos_num.succ p)) : rfl
... = cond ff zero (pos (pos_num.pred (pos_num.succ p))) : {succ_not_is_one}
... = cond ff zero (pos (pos_num.pred (pos_num.succ p))) : {!succ_not_is_one}
... = pos (pos_num.pred (pos_num.succ p)) : !cond.ff
... = pos p : {pos_num.pred_succ})
... = pos p : {!pos_num.pred.succ})
definition add (a b : num) : num :=
rec_on a b (λp_a, rec_on b (pos p_a) (λp_b, pos (pos_num.add p_a p_b)))