259 lines
8.4 KiB
Text
259 lines
8.4 KiB
Text
----------------------------------------------------------------------------------------------------
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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, Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic tools.helper_tactics tools.tactic data.nat.basic
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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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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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cons : T → list T → list T
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namespace list
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infixr `::` := cons
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notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
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variable {T : Type}
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protected theorem induction_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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(Hind : ∀ (x : T) (l : list T), P l → P (x::l)) : P l :=
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rec Hnil Hind l
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protected theorem cases_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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(Hcons : ∀ (x : T) (l : list T), P (x::l)) : P l :=
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induction_on l Hnil (take x l IH, Hcons x l)
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protected definition rec_on {A : Type} {C : list A → Type} (l : list A)
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(H1 : C nil) (H2 : Π (h : A) (t : list A), C t → C (h::t)) : C l :=
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rec H1 H2 l
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-- Concat
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-- ------
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definition append (s t : list T) : list T :=
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rec t (λx l u, x::u) s
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infixl `++` : 65 := append
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theorem append.nil_left (t : list T) : nil ++ t = 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_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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induction_on s rfl (λx l H, H ▸ rfl)
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-- Length
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-- ------
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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.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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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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-- Append
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-- ------
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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.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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-- 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 :=
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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.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.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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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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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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-- Head and tail
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-- -------------
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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.cons (x x' : T) (t : list T) : head x' (x::t) = x
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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)) : {!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.cons (x : T) (l : list T) : tail (x::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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-- List membership
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-- ---------------
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definition mem (x : T) : list T → Prop :=
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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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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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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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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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assume H1 : x ∈ y::s ++ t,
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have H2 : x = y ∨ x ∈ s ++ t, from H1,
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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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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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assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
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assume H : x ∈ y::s ∨ x ∈ t,
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or.elim H
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(assume H1,
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or.elim H1
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(take H2 : x = y, or.inl H2)
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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.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 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 (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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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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(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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decidable.rec_on (H x h)
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(assume Heq : x = h,
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decidable.inl (or.inl Heq))
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(assume Hne : x ≠ h,
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decidable.inl (or.inr Hp)))
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(assume Hn : ¬x ∈ l,
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decidable.rec_on (H x h)
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(assume Heq : x = h,
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decidable.inl (or.inl Heq))
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(assume Hne : x ≠ h,
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have H1 : ¬(x = h ∨ x ∈ l), from
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assume H2 : x = h ∨ x ∈ l, or.elim H2
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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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decidable.inr H2)))
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-- Find
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-- ----
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section
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variable [H : decidable_eq T]
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include H
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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 (x : T) : find x nil = 0
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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 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 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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... = 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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end
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-- nth element
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-- -----------
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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.succ (x : T) (l : list T) (n : nat) : nth x l (succ n) = nth x (tail l) n
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end list
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