feat(library/standard/bit): add theorems and notation
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura
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438a42d010
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1 changed files with 118 additions and 15 deletions
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@ -6,8 +6,11 @@ namespace bit
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inductive bit : Type :=
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| b0 : bit
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| b1 : bit
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notation `'0` := b0
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notation `'1` := b1
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notation `'0`:max := b0
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notation `'1`:max := b1
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theorem induction_on {p : bit → Bool} (b : bit) (H0 : p '0) (H1 : p '1) : p b
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:= bit_rec H0 H1 b
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theorem inhabited_bit [instance] : inhabited bit
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:= inhabited_intro b0
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@ -15,24 +18,124 @@ theorem inhabited_bit [instance] : inhabited bit
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definition cond {A : Type} (b : bit) (t e : A)
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:= bit_rec e t b
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theorem cond_b0 {A : Type} (t e : A) : cond b0 t e = e
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:= refl (cond b0 t e)
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theorem dichotomy (b : bit) : b = '0 ∨ b = '1
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:= induction_on b (or_intro_left _ (refl '0)) (or_intro_right _ (refl '1))
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theorem cond_b1 {A : Type} (t e : A) : cond b1 t e = t
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:= refl (cond b1 t e)
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theorem cond_b0 {A : Type} (t e : A) : cond '0 t e = e
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:= refl (cond '0 t e)
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theorem cond_b1 {A : Type} (t e : A) : cond '1 t e = t
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:= refl (cond '1 t e)
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theorem b0_ne_b1 : ¬ '0 = '1
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:= not_intro (assume H : '0 = '1, absurd
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(calc true = cond '1 true false : symm (cond_b1 _ _)
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... = cond '0 true false : {symm H}
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... = false : cond_b0 _ _)
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true_ne_false)
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definition bor (a b : bit)
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:= bit_rec (bit_rec b0 b1 b) b1 a
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:= bit_rec (bit_rec '0 '1 b) '1 a
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theorem bor_b1_left (a : bit) : bor b1 a = b1
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:= refl (bor b1 a)
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theorem bor_b1_left (a : bit) : bor '1 a = '1
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:= refl (bor '1 a)
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theorem bor_b1_right (a : bit) : bor a b1 = b1
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:= bit_rec (refl (bor b0 b1)) (refl (bor b1 b1)) a
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infixl `||`:65 := bor
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theorem bor_b0_left (a : bit) : bor b0 a = a
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:= bit_rec (refl (bor b0 b0)) (refl (bor b0 b1)) a
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theorem bor_b1_right (a : bit) : a || '1 = '1
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:= induction_on a (refl ('0 || '1)) (refl ('1 || '1))
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theorem bor_b0_right (a : bit) : bor a b0 = a
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:= bit_rec (refl (bor b0 b0)) (refl (bor b1 b0)) a
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theorem bor_b0_left (a : bit) : '0 || a = a
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:= induction_on a (refl ('0 || '0)) (refl ('0 || '1))
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theorem bor_b0_right (a : bit) : a || '0 = a
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:= induction_on a (refl ('0 || '0)) (refl ('1 || '0))
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theorem bor_id (a : bit) : a || a = a
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:= induction_on a (refl ('0 || '0)) (refl ('1 || '1))
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theorem bor_swap (a b : bit) : a || b = b || a
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:= induction_on a
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(induction_on b (refl ('0 || '0)) (refl ('0 || '1)))
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(induction_on b (refl ('1 || '0)) (refl ('1 || '1)))
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definition band (a b : bit)
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:= bit_rec '0 (bit_rec '0 '1 b) a
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infixl `&&`:75 := band
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theorem band_b0_left (a : bit) : '0 && a = '0
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:= refl ('0 && a)
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theorem band_b1_left (a : bit) : '1 && a = a
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:= induction_on a (refl ('1 && '0)) (refl ('1 && '1))
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theorem band_b0_right (a : bit) : a && '0 = '0
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:= induction_on a (refl ('0 && '0)) (refl ('1 && '0))
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theorem band_b1_right (a : bit) : a && '1 = a
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:= induction_on a (refl ('0 && '1)) (refl ('1 && '1))
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theorem band_id (a : bit) : a && a = a
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:= induction_on a (refl ('0 && '0)) (refl ('1 && '1))
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theorem band_swap (a b : bit) : a && b = b && a
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:= induction_on a
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(induction_on b (refl ('0 && '0)) (refl ('0 && '1)))
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(induction_on b (refl ('1 && '0)) (refl ('1 && '1)))
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theorem band_eq_b1_elim_left {a b : bit} (H : a && b = '1) : a = '1
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:= or_elim (dichotomy a)
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(assume H0 : a = '0,
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absurd_elim (a = '1)
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(calc '0 = '0 && b : symm (band_b0_left _)
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... = a && b : {symm H0}
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... = '1 : H)
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b0_ne_b1)
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(assume H1 : a = '1, H1)
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theorem band_eq_b1_elim_right {a b : bit} (H : a && b = '1) : b = '1
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:= band_eq_b1_elim_left (trans (band_swap b a) H)
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definition bnot (a : bit) := bit_rec '1 '0 a
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prefix `!`:85 := bnot
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theorem bnot_bnot (a : bit) : !!a = a
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:= induction_on a (refl (!!'0)) (refl (!!'1))
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theorem bnot_false : !'0 = '1
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:= refl _
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theorem bnot_true : !'1 = '0
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:= refl _
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definition beq (a b : bit) : bit
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:= bit_rec (bit_rec '1 '0 b) (bit_rec '0 '1 b) a
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infix `==`:50 := beq
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theorem beq_refl (a : bit) : (a == a) = '1
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:= induction_on a (refl ('0 == '0)) (refl ('1 == '1))
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theorem beq_b1_left (a : bit) : ('1 == a) = a
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:= induction_on a (refl ('1 == '0)) (refl ('1 == '1))
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theorem beq_b1_right (a : bit) : (a == '1) = a
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:= induction_on a (refl ('0 == '1)) (refl ('1 == '1))
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theorem beq_symm (a b : bit) : (a == b) = (b == a)
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:= induction_on a
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(induction_on b (refl ('0 == '0)) (refl ('0 == '1)))
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(induction_on b (refl ('1 == '0)) (refl ('1 == '1)))
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theorem to_eq {a b : bit} : a == b = '1 → a = b
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:= induction_on a
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(induction_on b (assume H, refl '0) (assume H, absurd_elim ('0 = '1) (trans (symm (beq_b1_right '0)) H) b0_ne_b1))
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(induction_on b (assume H, absurd_elim ('1 = '0) (trans (symm (beq_b1_left '0)) H) b0_ne_b1) (assume H, refl '1))
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theorem beq_eq (a b : bit) : (a == b) = '1 ↔ a = b
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:= iff_intro
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(assume H, to_eq H)
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(assume H, subst H (beq_refl a))
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end
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