refactor(library/data/list): avoid placeholders '_', make first argument of false_elim implicit
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura
parent
aace5c37cd
commit
8dec18018c
9 changed files with 84 additions and 86 deletions
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@ -11,15 +11,15 @@
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import tools.tactic
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import data.nat
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import logic
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import logic tools.helper_tactics
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-- import if -- for find
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using nat
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using eq_ops
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using helper_tactics
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namespace list
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-- Type
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-- ----
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@ -52,24 +52,23 @@ list_rec t (fun x : T, fun l : list T, fun u : list T, cons x u) s
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infixl `++` : 65 := concat
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theorem nil_concat (t : list T) : nil ++ t = t := refl _
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theorem nil_concat {t : list T} : nil ++ t = t
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theorem cons_concat (x : T) (s t : list T) : (x :: s) ++ t = x :: (s ++ t) := refl _
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theorem cons_concat {x : T} {s t : list T} : (x :: s) ++ t = x :: (s ++ t)
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theorem concat_nil (t : list T) : t ++ nil = t :=
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list_induction_on t (refl _)
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theorem concat_nil {t : list T} : t ++ nil = t :=
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list_induction_on t rfl
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(take (x : T) (l : list T) (H : concat l nil = l),
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show concat (cons x l) nil = cons x l, from H ▸ refl _)
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show concat (cons x l) nil = cons x l, from H ▸ rfl)
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theorem concat_assoc (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s (refl _)
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theorem concat_assoc {s t u : list T} : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s rfl
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(take x l,
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assume H : concat (concat l t) u = concat l (concat t u),
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calc
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concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
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... = cons x (concat l (concat t u)) : { H }
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... = concat (cons x l) (concat t u) : refl _)
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concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : rfl
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... = cons x (concat l (concat t u)) : {H}
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... = concat (cons x l) (concat t u) : rfl)
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-- Length
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-- ------
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@ -78,9 +77,9 @@ definition length : list T → ℕ := list_rec 0 (fun x l m, succ m)
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theorem length_nil : length (@nil T) = 0 := rfl
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theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := rfl
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theorem length_cons {x : T} {t : list T} : length (x :: t) = succ (length t)
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theorem length_concat (s t : list T) : length (s ++ t) = length s + length t :=
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theorem length_concat {s t : list T} : length (s ++ t) = length s + length t :=
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list_induction_on s
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(calc
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length (concat nil t) = length t : rfl
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@ -90,99 +89,95 @@ list_induction_on s
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assume H : length (concat s t) = length s + length t,
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calc
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length (concat (cons x s) t ) = succ (length (concat s t)) : rfl
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... = succ (length s + length t) : { H }
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... = succ (length s + length t) : {H}
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... = succ (length s) + length t : {add_succ_left⁻¹}
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... = length (cons x s) + length t : rfl)
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-- add_rewrite length_nil length_cons
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-- Append
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-- ------
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definition append (x : T) : list T → list T := list_rec [x] (fun y l l', y :: l')
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theorem append_nil (x : T) : append x nil = [x] := refl _
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theorem append_nil {x : T} : append x nil = [x]
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theorem append_cons (x : T) (y : T) (l : list T) : append x (y :: l) = y :: (append x l) := refl _
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theorem append_cons {x y : T} {l : list T} : append x (y :: l) = y :: (append x l)
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theorem append_eq_concat (x : T) (l : list T) : append x l = l ++ [x] := refl _
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theorem append_eq_concat {x : T} {l : list T} : append x l = l ++ [x]
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-- add_rewrite append_nil append_cons
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-- Reverse
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-- -------
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definition reverse : list T → list T := list_rec nil (fun x l r, r ++ [x])
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theorem reverse_nil : reverse (@nil T) = nil := refl _
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theorem reverse_nil : reverse (@nil T) = nil
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theorem reverse_cons (x : T) (l : list T) : reverse (x :: l) = append x (reverse l) := refl _
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theorem reverse_cons {x : T} {l : list T} : reverse (x :: l) = append x (reverse l)
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theorem reverse_singleton (x : T) : reverse [x] = [x] := refl _
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theorem reverse_singleton {x : T} : reverse [x] = [x]
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theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
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list_induction_on s (symm (concat_nil _))
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theorem reverse_concat {s t : list T} : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
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list_induction_on s (concat_nil⁻¹)
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(take x s,
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assume IH : reverse (s ++ t) = concat (reverse t) (reverse s),
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calc
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reverse ((x :: s) ++ t) = reverse (s ++ t) ++ [x] : refl _
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reverse ((x :: s) ++ t) = reverse (s ++ t) ++ [x] : rfl
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... = reverse t ++ reverse s ++ [x] : {IH}
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... = reverse t ++ (reverse s ++ [x]) : concat_assoc _ _ _
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... = reverse t ++ (reverse (x :: s)) : refl _)
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... = reverse t ++ (reverse s ++ [x]) : concat_assoc
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... = reverse t ++ (reverse (x :: s)) : rfl)
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theorem reverse_reverse (l : list T) : reverse (reverse l) = l :=
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list_induction_on l (refl _)
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theorem reverse_reverse {l : list T} : reverse (reverse l) = l :=
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list_induction_on l rfl
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(take x l',
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assume H: reverse (reverse l') = l',
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show reverse (reverse (x :: l')) = x :: l', from
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calc
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reverse (reverse (x :: l')) = reverse (reverse l' ++ [x]) : refl _
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... = reverse [x] ++ reverse (reverse l') : reverse_concat _ _
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... = [x] ++ l' : { H }
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... = x :: l' : refl _)
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reverse (reverse (x :: l')) = reverse (reverse l' ++ [x]) : rfl
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... = reverse [x] ++ reverse (reverse l') : reverse_concat
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... = [x] ++ l' : {H}
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... = x :: l' : rfl)
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theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
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list_induction_on l (refl _)
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theorem append_eq_reverse_cons {x : T} {l : list T} : append x l = reverse (x :: reverse l) :=
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list_induction_on l rfl
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(take y l',
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assume H : append x l' = reverse (x :: reverse l'),
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calc
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append x (y :: l') = (y :: l') ++ [ x ] : append_eq_concat _ _
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... = concat (reverse (reverse (y :: l'))) [ x ] : {symm (reverse_reverse _)}
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... = reverse (x :: (reverse (y :: l'))) : refl _)
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append x (y :: l') = (y :: l') ++ [ x ] : append_eq_concat
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... = concat (reverse (reverse (y :: l'))) [ x ] : {reverse_reverse⁻¹}
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... = reverse (x :: (reverse (y :: l'))) : rfl)
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-- Head and tail
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-- -------------
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definition head (x0 : T) : list T → T := list_rec x0 (fun x l h, x)
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definition head (x : T) : list T → T := list_rec x (fun x l h, x)
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theorem head_nil (x0 : T) : head x0 (@nil T) = x0 := refl _
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theorem head_nil {x : T} : head x (@nil T) = x
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theorem head_cons (x : T) (x0 : T) (t : list T) : head x0 (x :: t) = x := refl _
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theorem head_cons {x x' : T} {t : list T} : head x' (x :: t) = x
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theorem head_concat (s t : list T) (x0 : T) : s ≠ nil → (head x0 (s ++ t) = head x0 s) :=
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theorem head_concat {s t : list T} {x : T} : s ≠ nil → (head x (s ++ t) = head x s) :=
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list_cases_on s
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(take H : nil ≠ nil, absurd (refl nil) H)
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(take x s,
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take H : cons x s ≠ nil,
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(take H : nil ≠ nil, absurd rfl H)
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(take x s, take H : cons x s ≠ nil,
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calc
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head x0 (concat (cons x s) t) = head x0 (cons x (concat s t)) : {cons_concat _ _ _}
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... = x : {head_cons _ _ _}
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... = head x0 (cons x s) : {symm ( head_cons x x0 s)})
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head x (concat (cons x s) t) = head x (cons x (concat s t)) : {cons_concat}
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... = x : {head_cons}
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... = head x (cons x s) : {head_cons⁻¹})
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definition tail : list T → list T := list_rec nil (fun x l b, l)
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theorem tail_nil : tail (@nil T) = nil := refl _
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theorem tail_nil : tail (@nil T) = nil
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theorem tail_cons (x : T) (l : list T) : tail (cons x l) = l := refl _
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theorem tail_cons {x : T} {l : list T} : tail (cons x l) = l
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theorem cons_head_tail (x0 : T) (l : list T) : l ≠ nil → (head x0 l) :: (tail l) = l :=
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theorem cons_head_tail {x : T} {l : list T} : l ≠ nil → (head x l) :: (tail l) = l :=
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list_cases_on l
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(assume H : nil ≠ nil, absurd (refl _) H)
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(take x l, assume H : cons x l ≠ nil, refl _)
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(assume H : nil ≠ nil, absurd rfl H)
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(take x l, assume H : cons x l ≠ nil, rfl)
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-- List membership
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-- ---------------
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@ -192,11 +187,11 @@ definition mem (x : T) : list T → Prop := list_rec false (fun y l H, x = y ∨
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infix `∈` := mem
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-- TODO: constructively, equality is stronger. Use that?
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theorem mem_nil (x : T) : x ∈ nil ↔ false := iff_refl _
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theorem mem_nil {x : T} : x ∈ nil ↔ false := iff_rfl
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theorem mem_cons (x : T) (y : T) (l : list T) : mem x (cons y l) ↔ (x = y ∨ mem x l) := iff_refl _
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theorem mem_cons {x y : T} {l : list T} : mem x (cons y l) ↔ (x = y ∨ mem x l) := iff_rfl
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theorem mem_concat_imp_or (x : T) (s t : list T) : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
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theorem mem_concat_imp_or {x : T} {s t : list T} : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
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list_induction_on s or_inr
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(take y s,
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assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
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@ -205,9 +200,9 @@ list_induction_on s or_inr
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have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_imp_or_right H2 IH,
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iff_elim_right or_assoc H3)
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theorem mem_or_imp_concat (x : T) (s t : list T) : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
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theorem mem_or_imp_concat {x : T} {s t : list T} : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
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list_induction_on s
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(take H, or_elim H (false_elim _) (assume H, H))
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(take H, or_elim H false_elim (assume H, H))
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(take y s,
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assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
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assume H : x ∈ y :: s ∨ x ∈ t,
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@ -218,24 +213,24 @@ list_induction_on s
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(take H2 : x ∈ s, or_inr (IH (or_inl H2))))
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(assume H1 : x ∈ t, or_inr (IH (or_inr H1))))
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theorem mem_concat (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t
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:= iff_intro (mem_concat_imp_or _ _ _) (mem_or_imp_concat _ _ _)
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theorem mem_concat {x : T} {s t : list T} : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t
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:= iff_intro mem_concat_imp_or mem_or_imp_concat
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theorem mem_split (x : T) (l : list T) : x ∈ l → ∃s t : list T, l = s ++ (x :: t) :=
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theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x :: t) :=
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list_induction_on l
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(take H : x ∈ nil, false_elim _ (iff_elim_left (mem_nil x) H))
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(take H : x ∈ nil, false_elim (iff_elim_left mem_nil H))
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(take y l,
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assume IH : x ∈ l → ∃s t : list T, l = s ++ (x :: t),
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assume H : x ∈ y :: l,
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or_elim H
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(assume H1 : x = y,
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exists_intro nil
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(exists_intro l (subst H1 (refl _))))
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(exists_intro l (subst H1 rfl)))
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(assume H1 : x ∈ l,
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obtain s (H2 : ∃t : list T, l = s ++ (x :: t)), from IH H1,
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obtain t (H3 : l = s ++ (x :: t)), from H2,
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have H4 : y :: l = (y :: s) ++ (x :: t),
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from subst H3 (refl (y :: l)),
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from subst H3 rfl,
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exists_intro _ (exists_intro _ H4)))
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-- Find
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@ -276,12 +271,12 @@ list_induction_on l
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-- nth element
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-- -----------
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definition nth (x0 : T) (l : list T) (n : ℕ) : T :=
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nat_rec (λl, head x0 l) (λm f l, f (tail l)) n l
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definition nth (x : T) (l : list T) (n : ℕ) : T :=
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nat_rec (λl, head x l) (λm f l, f (tail l)) n l
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theorem nth_zero (x0 : T) (l : list T) : nth x0 l 0 = head x0 l := refl _
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theorem nth_zero {x : T} {l : list T} : nth x l 0 = head x l
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theorem nth_succ (x0 : T) (l : list T) (n : ℕ) : nth x0 l (succ n) = nth x0 (tail l) n := refl _
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theorem nth_succ {x : T} {l : list T} {n : ℕ} : nth x l (succ n) = nth x (tail l) n
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end
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@ -28,7 +28,7 @@ section
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sigma_rec H p
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theorem dpair_ext (p : sigma B) : dpair (dpr1 p) (dpr2 p) = p :=
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sigma_destruct p (take a b, refl _)
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sigma_destruct p (take a b, rfl)
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-- Note that we give the general statment explicitly, to help the unifier
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theorem dpair_eq {a1 a2 : A} {b1 : B a1} {b2 : B a2} (H1 : a1 = a2) (H2 : eq_rec_on H1 b1 = b2) :
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@ -71,13 +71,13 @@ rec_on s1
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(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
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(take b2,
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have H3 : (inl B a1 = inr A b2) ↔ false,
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from iff_intro inl_neq_inr (assume H4, false_elim _ H4),
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from iff_intro inl_neq_inr (assume H4, false_elim H4),
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show decidable (inl B a1 = inr A b2), from decidable_iff_equiv _ (iff_symm H3)))
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(take b1, show decidable (inr A b1 = s2), from
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rec_on s2
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(take a2,
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have H3 : (inr A b1 = inl B a2) ↔ false,
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from iff_intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim _ H4),
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from iff_intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim H4),
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show decidable (inr A b1 = inl B a2), from decidable_iff_equiv _ (iff_symm H3))
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(take b2, show decidable (inr A b1 = inr A b2), from H2 b1 b2))
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@ -16,6 +16,9 @@ notation `⋆`:max := star
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theorem unit_eq (a b : unit) : a = b :=
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unit_rec (unit_rec (refl ⋆) b) a
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theorem unit_eq_star (a : unit) : a = star :=
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unit_eq a star
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theorem unit_inhabited [instance] : inhabited unit :=
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inhabited_mk ⋆
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@ -35,9 +35,9 @@ theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
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or_elim (prop_complete a)
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(assume Hat, or_elim (prop_complete b)
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(assume Hbt, Hat ⬝ Hbt⁻¹)
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(assume Hbf, false_elim (a = b) (Hbf ▸ (Hab (eq_true_elim Hat)))))
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(assume Hbf, false_elim (Hbf ▸ (Hab (eq_true_elim Hat)))))
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(assume Haf, or_elim (prop_complete b)
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(assume Hbt, false_elim (a = b) (Haf ▸ (Hba (eq_true_elim Hbt))))
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(assume Hbt, false_elim (Haf ▸ (Hba (eq_true_elim Hbt))))
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(assume Hbf, Haf ⬝ Hbf⁻¹))
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theorem iff_to_eq {a b : Prop} (H : a ↔ b) : a = b :=
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@ -44,7 +44,7 @@ or_elim (em a) (assume Ha, Hab Ha) (assume Hna, Hnab Hna)
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theorem by_contradiction {p : Prop} {Hp : decidable p} (H : ¬p → false) : p :=
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or_elim (em p)
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(assume H1 : p, H1)
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(assume H1 : ¬p, false_elim p (H H1))
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(assume H1 : ¬p, false_elim (H H1))
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theorem and_decidable [instance] {a b : Prop} (Ha : decidable a) (Hb : decidable b) :
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decidable (a ∧ b) :=
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@ -47,7 +47,7 @@ theorem not_not_elim {a : Prop} {D : decidable a} (H : ¬¬a) : a :=
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iff_mp not_not_iff H
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theorem not_true : (¬true) ↔ false :=
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iff_intro (assume H, H trivial) (false_elim _)
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iff_intro (assume H, H trivial) false_elim
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theorem not_false : (¬false) ↔ true :=
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iff_intro (assume H, trivial) (assume H H', H')
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@ -117,12 +117,12 @@ iff_intro
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theorem iff_false_intro {a : Prop} (H : ¬a) : a ↔ false :=
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iff_intro
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(assume H1 : a, absurd H1 H)
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(assume H2 : false, false_elim a H2)
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(assume H2 : false, false_elim H2)
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theorem a_neq_a {A : Type} (a : A) : (a ≠ a) ↔ false :=
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iff_intro
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(assume H, a_neq_a_elim H)
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(assume H, false_elim (a ≠ a) H)
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(assume H, false_elim H)
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theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
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iff_true_intro (refl a)
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@ -135,7 +135,7 @@ iff_intro
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(assume H,
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have H' : ¬a, from assume Ha, (H ▸ Ha) Ha,
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H' (H⁻¹ ▸ H'))
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(assume H, false_elim (a ↔ ¬a) H)
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(assume H, false_elim H)
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theorem true_eq_false : (true ↔ false) ↔ false :=
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not_true ▸ (a_iff_not_a true)
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|
|
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@ -18,7 +18,7 @@ abbreviation imp (a b : Prop) : Prop := a → b
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inductive false : Prop
|
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theorem false_elim (c : Prop) (H : false) : c :=
|
||||
theorem false_elim {c : Prop} (H : false) : c :=
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false_rec c H
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inductive true : Prop :=
|
||||
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|
@ -36,7 +36,7 @@ theorem not_intro {a : Prop} (H : a → false) : ¬a := H
|
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theorem not_elim {a : Prop} (H1 : ¬a) (H2 : a) : false := H1 H2
|
||||
|
||||
theorem absurd {a : Prop} {b : Prop} (H1 : a) (H2 : ¬a) : b :=
|
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false_elim b (H2 H1)
|
||||
false_elim (H2 H1)
|
||||
|
||||
theorem not_not_intro {a : Prop} (Ha : a) : ¬¬a :=
|
||||
assume Hna : ¬a, absurd Ha Hna
|
||||
|
|
|
|||
|
|
@ -57,13 +57,13 @@ rec_on s1
|
|||
(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
|
||||
(take b2,
|
||||
have H3 : (inl B a1 = inr A b2) ↔ false,
|
||||
from iff_intro inl_neq_inr (assume H4, false_elim _ H4),
|
||||
from iff_intro inl_neq_inr (assume H4, false_elim H4),
|
||||
show decidable (inl B a1 = inr A b2), from decidable_iff_equiv _ (iff_symm H3)))
|
||||
(take b1, show decidable (inr A b1 = s2), from
|
||||
rec_on s2
|
||||
(take a2,
|
||||
have H3 : (inr A b1 = inl B a2) ↔ false,
|
||||
from iff_intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim _ H4),
|
||||
from iff_intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim H4),
|
||||
show decidable (inr A b1 = inl B a2), from decidable_iff_equiv _ (iff_symm H3))
|
||||
(take b2, show decidable (inr A b1 = inr A b2), from H2 b1 b2))
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue