refactor(data/list): use '!' operator, and new name convention for declaration names
This commit is contained in:
Leonardo de Moura
parent
efaeeb0726
commit
15779c5d1e
1 changed files with 60 additions and 65 deletions
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@ -1,21 +1,15 @@
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----------------------------------------------------------------------------------------------------
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--- Copyright (c) 2014 Parikshit Khanna. All rights reserved.
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--- Released under Apache 2.0 license as described in the file LICENSE.
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--- Authors: Parikshit Khanna, Jeremy Avigad
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--- Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic tools.helper_tactics tools.tactic data.nat
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-- Theory list
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-- ===========
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--
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-- Basic properties of lists.
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import tools.tactic
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import data.nat
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import logic tools.helper_tactics
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open nat
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open eq.ops
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open helper_tactics
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open eq.ops helper_tactics nat
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inductive list (T : Type) : Type :=
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nil {} : list T,
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@ -25,7 +19,6 @@ namespace list
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infix `::` := cons
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section
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parameter {T : Type}
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protected theorem induction_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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@ -51,14 +44,14 @@ rec t (λx l u, x::u) s
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infixl `++` : 65 := append
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theorem nil_append {t : list T} : nil ++ t = t
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theorem append.nil_left (t : list T) : nil ++ t = t
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theorem cons_append {x : T} {s t : list T} : x::s ++ t = x::(s ++ t)
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theorem append.cons (x : T) (s t : list T) : x::s ++ t = x::(s ++ t)
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theorem append_nil {t : list T} : t ++ nil = t :=
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theorem append.nil_right (t : list T) : t ++ nil = t :=
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induction_on t rfl (λx l H, H ▸ rfl)
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theorem append_assoc {s t u : list T} : s ++ t ++ u = s ++ (t ++ u) :=
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theorem append.assoc (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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induction_on s rfl (λx l H, H ▸ rfl)
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-- Length
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@ -67,11 +60,11 @@ induction_on s rfl (λx l H, H ▸ rfl)
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definition length : list T → nat :=
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rec 0 (λx l m, succ m)
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theorem length_nil : length (@nil T) = 0
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theorem length.nil : length (@nil T) = 0
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theorem length_cons {x : T} {t : list T} : length (x::t) = succ (length t)
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theorem length.cons (x : T) (t : list T) : length (x::t) = succ (length t)
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theorem length_append {s t : list T} : length (s ++ t) = length s + length t :=
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theorem length.append (s t : list T) : length (s ++ t) = length s + length t :=
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induction_on s (!add.zero_left⁻¹) (λx s H, !add.succ_left⁻¹ ▸ H ▸ rfl)
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-- add_rewrite length_nil length_cons
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@ -82,11 +75,11 @@ induction_on s (!add.zero_left⁻¹) (λx s H, !add.succ_left⁻¹ ▸ H ▸ rfl
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definition concat (x : T) : list T → list T :=
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rec [x] (λy l l', y::l')
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theorem concat_nil {x : T} : concat x nil = [x]
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theorem concat.nil (x : T) : concat x nil = [x]
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theorem concat_cons {x y : T} {l : list T} : concat x (y::l) = y::(concat x l)
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theorem concat.cons (x y : T) (l : list T) : concat x (y::l) = y::(concat x l)
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theorem concat_eq_append {x : T} {l : list T} : concat x l = l ++ [x]
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theorem concat.eq_append (x : T) (l : list T) : concat x l = l ++ [x]
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-- add_rewrite append_nil append_cons
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@ -96,26 +89,26 @@ theorem concat_eq_append {x : T} {l : list T} : concat x l = l ++ [x]
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definition reverse : list T → list T :=
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rec nil (λx l r, r ++ [x])
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theorem reverse_nil : reverse (@nil T) = nil
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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) = concat x (reverse l)
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theorem reverse.cons (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
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theorem reverse_singleton {x : T} : reverse [x] = [x]
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theorem reverse.singleton (x : T) : reverse [x] = [x]
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theorem reverse_append {s t : list T} : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
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induction_on s (append_nil⁻¹)
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theorem reverse.append (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
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induction_on s (!append.nil_right⁻¹)
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(λx s H, calc
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reverse (x::s ++ t) = reverse t ++ reverse s ++ [x] : {H}
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... = reverse t ++ (reverse s ++ [x]) : append_assoc)
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... = reverse t ++ (reverse s ++ [x]) : !append.assoc)
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theorem reverse_reverse {l : list T} : reverse (reverse l) = l :=
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induction_on l rfl (λx l' H, H ▸ reverse_append)
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theorem reverse.reverse (l : list T) : reverse (reverse l) = l :=
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induction_on l rfl (λx l' H, H ▸ !reverse.append)
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theorem concat_eq_reverse_cons {x : T} {l : list T} : concat x l = reverse (x :: reverse l) :=
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theorem concat.eq_reverse_cons (x : T) (l : list T) : concat x l = reverse (x :: reverse l) :=
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induction_on l rfl
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(λy l' H, calc
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concat x (y::l') = (y::l') ++ [x] : concat_eq_append
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... = reverse (reverse (y::l')) ++ [x] : {reverse_reverse⁻¹})
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concat x (y::l') = (y::l') ++ [x] : !concat.eq_append
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... = reverse (reverse (y::l')) ++ [x] : {!reverse.reverse⁻¹})
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-- Head and tail
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-- -------------
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@ -123,27 +116,27 @@ induction_on l rfl
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definition head (x : T) : list T → T :=
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rec x (λx l h, x)
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theorem head_nil {x : T} : head x nil = x
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theorem head.nil (x : T) : head x nil = x
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theorem head_cons {x x' : T} {t : list T} : head x' (x::t) = x
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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} {x : T} : s ≠ nil → (head x (s ++ t) = head x s) :=
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theorem head.concat {s : list T} (t : list T) (x : T) : s ≠ nil → (head x (s ++ t) = head x s) :=
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cases_on s
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(take H : nil ≠ nil, absurd rfl H)
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(take x s, take H : x::s ≠ nil,
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calc
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head x (x::s ++ t) = head x (x::(s ++ t)) : {cons_append}
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... = x : {head_cons}
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... = head x (x::s) : {head_cons⁻¹})
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head x (x::s ++ t) = head x (x::(s ++ t)) : {!append.cons}
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... = x : !head.cons
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... = head x (x::s) : !head.cons⁻¹)
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definition tail : list T → list T :=
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rec nil (λx l b, l)
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theorem tail_nil : tail (@nil T) = nil
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theorem tail.nil : tail (@nil T) = nil
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theorem tail_cons {x : T} {l : list T} : tail (x::l) = l
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theorem tail.cons (x : T) (l : list T) : tail (x::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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theorem cons_head_tail {l : list T} (x : T) : l ≠ nil → (head x l)::(tail l) = l :=
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cases_on l
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(assume H : nil ≠ nil, absurd rfl H)
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(take x l, assume H : x::l ≠ nil, rfl)
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@ -156,13 +149,13 @@ rec false (λy l H, x = y ∨ H)
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infix `∈` := mem
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theorem mem_nil {x : T} : x ∈ nil ↔ false :=
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theorem mem.nil (x : T) : x ∈ nil ↔ false :=
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iff.rfl
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theorem mem_cons {x y : T} {l : list T} : x ∈ y::l ↔ (x = y ∨ x ∈ l) :=
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theorem mem.cons (x y : T) (l : list T) : x ∈ y::l ↔ (x = y ∨ x ∈ l) :=
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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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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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@ -171,7 +164,7 @@ 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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induction_on s
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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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@ -184,29 +177,29 @@ 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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induction_on l
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(take H : x ∈ nil, false_elim (iff.elim_left mem_nil 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 (exists_intro l (H1 ▸ rfl)))
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exists_intro nil (!exists_intro (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 H3 ▸ rfl,
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exists_intro _ (exists_intro _ H4)))
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!exists_intro (!exists_intro H4)))
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definition mem_is_decidable [instance] {H : decidable_eq T} {x : T} {l : list T} : decidable (x ∈ l) :=
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definition mem.is_decidable [instance] {H : decidable_eq T} (x : T) (l : list T) : decidable (x ∈ l) :=
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rec_on l
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(decidable.inr (iff.false_elim mem_nil))
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(λ (h : T) (l : list T) (iH : decidable (x ∈ l)),
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(decidable.inr (iff.false_elim !mem.nil))
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(take (h : T) (l : list T) (iH : decidable (x ∈ l)),
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show decidable (x ∈ h::l), from
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decidable.rec_on iH
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(assume Hp : x ∈ l,
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@ -225,34 +218,36 @@ rec_on l
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(assume Heq, absurd Heq Hne)
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(assume Hp, absurd Hp Hn),
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have H2 : ¬x ∈ h::l, from
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iff.elim_right (iff.flip_sign mem_cons) H1,
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iff.elim_right (iff.flip_sign !mem.cons) H1,
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decidable.inr H2)))
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-- Find
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-- ----
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section
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parameter {H : decidable_eq T}
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include H
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definition find {H : decidable_eq T} (x : T) : list T → nat :=
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definition find (x : T) : list T → nat :=
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rec 0 (λy l b, if x = y then 0 else succ b)
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theorem find_nil {H : decidable_eq T} {f : T} : find f nil = 0
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theorem find.nil (x : T) : find x nil = 0
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theorem find_cons {H : decidable_eq T} {x y : T} {l : list T} :
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find x (y::l) = if x = y then 0 else succ (find x l)
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theorem find.cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
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theorem not_mem_find {H : decidable_eq T} {l : list T} {x : T} :
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¬x ∈ l → find x l = length l :=
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theorem find.not_mem {l : list T} {x : T} : ¬x ∈ l → find x l = length l :=
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rec_on l
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(assume P₁ : ¬x ∈ nil, rfl)
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(take y l,
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assume iH : ¬x ∈ l → find x l = length l,
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assume P₁ : ¬x ∈ y::l,
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have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (iff.flip_sign mem_cons) P₁,
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have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (iff.flip_sign !mem.cons) P₁,
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have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or P₂),
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calc
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find x (y::l) = if x = y then 0 else succ (find x l) : find_cons
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find x (y::l) = if x = y then 0 else succ (find x l) : !find.cons
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... = succ (find x l) : if_neg (and.elim_left P₃)
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... = succ (length l) : {iH (and.elim_right P₃)}
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... = length (y::l) : length_cons⁻¹)
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... = length (y::l) : !length.cons⁻¹)
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end
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-- nth element
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-- -----------
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@ -260,8 +255,8 @@ rec_on l
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definition nth (x : T) (l : list T) (n : nat) : 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 {x : T} {l : list T} : nth x l 0 = head x l
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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 {x : T} {l : list T} {n : nat} : nth x l (succ n) = nth x (tail l) n
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theorem nth.succ (x : T) (l : list T) (n : nat) : nth x l (succ n) = nth x (tail l) n
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end
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end list
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