diff options
author | Son Ho | 2023-08-04 18:19:37 +0200 |
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committer | Son Ho | 2023-08-04 18:19:37 +0200 |
commit | f7c09787c4a7457568d3d79d38b45caac4af8772 (patch) | |
tree | edac38167e05e76e7e38798e89dadc672df06f3b /backends/lean/Base/IList | |
parent | 931fabe3e8590815548d606b33fc8db31e9f6010 (diff) |
Start adding support for Arrays/Slices in the Lean library
Diffstat (limited to 'backends/lean/Base/IList')
-rw-r--r-- | backends/lean/Base/IList/IList.lean | 238 |
1 files changed, 229 insertions, 9 deletions
diff --git a/backends/lean/Base/IList/IList.lean b/backends/lean/Base/IList/IList.lean index 93047a1b..0b483e90 100644 --- a/backends/lean/Base/IList/IList.lean +++ b/backends/lean/Base/IList/IList.lean @@ -3,6 +3,7 @@ import Std.Data.Int.Lemmas import Base.Arith +import Base.Utils namespace List @@ -87,6 +88,28 @@ def idrop (i : Int) (ls : List α) : List α := | [] => [] | x :: tl => if i = 0 then x :: tl else idrop (i - 1) tl +def itake (i : Int) (ls : List α) : List α := + match ls with + | [] => [] + | hd :: tl => if i = 0 then [] else hd :: itake (i - 1) tl + +def slice (start end_ : Int) (ls : List α) : List α := + (ls.idrop start).itake (end_ - start) + +def replace_slice (start end_ : Int) (l nl : List α) : List α := + let l_beg := l.itake start + let l_end := l.idrop end_ + l_beg ++ nl ++ l_end + +def allP {α : Type u} (l : List α) (p: α → Prop) : Prop := + foldr (fun a r => p a ∧ r) True l + +def pairwise_rel + {α : Type u} (rel : α → α → Prop) (l: List α) : Prop + := match l with + | [] => True + | hd :: tl => allP tl (rel hd) ∧ pairwise_rel rel tl + section Lemmas variable {α : Type u} @@ -99,6 +122,28 @@ variable {α : Type u} @[simp] theorem idrop_zero : idrop 0 (ls : List α) = ls := by cases ls <;> simp [idrop] @[simp] theorem idrop_nzero_cons (hne : i ≠ 0) : idrop i ((x :: tl) : List α) = idrop (i - 1) tl := by simp [*, idrop] +@[simp] theorem itake_nil : itake i ([] : List α) = [] := by simp [itake] +@[simp] theorem itake_zero : itake 0 (ls : List α) = [] := by cases ls <;> simp [itake] +@[simp] theorem itake_nzero_cons (hne : i ≠ 0) : itake i ((x :: tl) : List α) = x :: itake (i - 1) tl := by simp [*, itake] + +@[simp] theorem slice_nil : slice i j ([] : List α) = [] := by simp [slice] +@[simp] theorem slice_zero : slice 0 0 (ls : List α) = [] := by cases ls <;> simp [slice] + +@[simp] +theorem slice_nzero_cons (i j : Int) (x : α) (tl : List α) (hne : i ≠ 0) : slice i j ((x :: tl) : List α) = slice (i - 1) (j - 1) tl := + match tl with + | [] => by simp [slice]; simp [*] + | hd :: tl => + if h: i - 1 = 0 then by + have : i = 1 := by int_tac + simp [*, slice] + else + have := slice_nzero_cons (i - 1) (j - 1) hd tl h + by + conv => lhs; simp [slice, *] + conv at this => lhs; simp [slice, *] + simp [*, slice] + theorem len_eq_length (ls : List α) : ls.len = ls.length := by induction ls . rfl @@ -158,8 +203,33 @@ theorem right_len_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l2.len = l2'.len apply right_length_eq_append_eq assumption +@[simp] +theorem index_append_beg [Inhabited α] (i : Int) (l0 l1 : List α) + (_ : 0 ≤ i) (_ : i < l0.len) : + (l0 ++ l1).index i = l0.index i := + match l0 with + | [] => by simp_all; int_tac + | hd :: tl => + if hi : i = 0 then by simp_all + else by + have := index_append_beg (i - 1) tl l1 (by int_tac) (by simp_all; int_tac) + simp_all + +@[simp] +theorem index_append_end [Inhabited α] (i : Int) (l0 l1 : List α) + (_ : l0.len ≤ i) (_ : i < l0.len + l1.len) : + (l0 ++ l1).index i = l1.index (i - l0.len) := + match l0 with + | [] => by simp_all + | hd :: tl => + have : ¬ i = 0 := by simp_all; int_tac + have := index_append_end (i - 1) tl l1 (by simp_all; int_tac) (by simp_all; int_tac) + -- TODO: canonize arith expressions + have : i - 1 - len tl = i - (1 + len tl) := by int_tac + by simp_all + open Arith in -theorem idrop_eq_nil_of_le (hineq : ls.len ≤ i) : idrop i ls = [] := by +@[simp] theorem idrop_eq_nil_of_le (hineq : ls.len ≤ i) : idrop i ls = [] := by revert i induction ls <;> simp [*] rename_i hd tl hi @@ -175,6 +245,136 @@ theorem idrop_eq_nil_of_le (hineq : ls.len ≤ i) : idrop i ls = [] := by apply hi linarith +theorem idrop_len_le (i : Int) (ls : List α) : (ls.idrop i).len ≤ ls.len := + match ls with + | [] => by simp + | hd :: tl => + if h: i = 0 then by simp [*] + else + have := idrop_len_le (i - 1) tl + by simp [*]; linarith + +@[simp] +theorem idrop_len (i : Int) (ls : List α) (_ : 0 ≤ i) (_ : i ≤ ls.len) : + (ls.idrop i).len = ls.len - i := + match ls with + | [] => by simp_all; linarith + | hd :: tl => + if h: i = 0 then by simp [*] + else + have := idrop_len (i - 1) tl (by int_tac) (by simp at *; int_tac) + by simp [*] at *; int_tac + +theorem itake_len_le (i : Int) (ls : List α) : (ls.itake i).len ≤ ls.len := + match ls with + | [] => by simp + | hd :: tl => + if h: i = 0 then by simp [*]; int_tac + else + have := itake_len_le (i - 1) tl + by simp [*] + +@[simp] +theorem itake_len (i : Int) (ls : List α) (_ : 0 ≤ i) (_ : i ≤ ls.len) : (ls.itake i).len = i := + match ls with + | [] => by simp_all; int_tac + | hd :: tl => + if h: i = 0 then by simp [*] + else + have := itake_len (i - 1) tl (by int_tac) (by simp at *; int_tac) + by simp [*] + +theorem slice_len_le (i j : Int) (ls : List α) : (ls.slice i j).len ≤ ls.len := by + simp [slice] + have := ls.idrop_len_le i + have := (ls.idrop i).itake_len_le (j - i) + int_tac + +@[simp] +theorem index_idrop [Inhabited α] (i : Int) (j : Int) (ls : List α) + (_ : 0 ≤ i) (_ : 0 ≤ j) (_ : i + j < ls.len) : + (ls.idrop i).index j = ls.index (i + j) := + match ls with + | [] => by simp at *; int_tac + | hd :: tl => + if h: i = 0 then by simp [*] + else by + have : ¬ i + j = 0 := by int_tac + simp [*] + -- TODO: rewriting rule: ¬ i = 0 → 0 ≤ i → 0 < i ? + have := index_idrop (i - 1) j tl (by int_tac) (by simp at *; int_tac) (by simp at *; int_tac) + -- TODO: canonize add/subs? + have : i - 1 + j = i + j - 1 := by int_tac + simp [*] + +@[simp] +theorem index_itake [Inhabited α] (i : Int) (j : Int) (ls : List α) + (_ : 0 ≤ j) (_ : j < i) (_ : j < ls.len) : + (ls.itake i).index j = ls.index j := + match ls with + | [] => by simp at * + | hd :: tl => + have : ¬ 0 = i := by int_tac -- TODO: this is slightly annoying + if h: j = 0 then by simp [*] at * + else by + simp [*] + -- TODO: rewriting rule: ¬ i = 0 → 0 ≤ i → 0 < i ? + have := index_itake (i - 1) (j - 1) tl (by simp at *; int_tac) (by simp at *; int_tac) (by simp at *; int_tac) + simp [*] + +@[simp] +theorem index_slice [Inhabited α] (i j k : Int) (ls : List α) + (_ : 0 ≤ i) (_ : j ≤ ls.len) (_ : 0 ≤ k) (_ : i + k < j) : + (ls.slice i j).index k = ls.index (i + k) := + match ls with + | [] => by simp at *; int_tac + | hd :: tl => + if h: i = 0 then by + simp [*, slice] at * + apply index_itake <;> simp_all + int_tac + else by + have : ¬ i + k = 0 := by int_tac + simp [*] + -- TODO: tactics simp_int_tac/simp_scalar_tac? + have := index_slice (i - 1) (j - 1) k tl (by simp at *; int_tac) (by simp at *; int_tac) + (by simp at *; int_tac) (by simp at *; int_tac) + have : (i - 1 + k) = (i + k - 1) := by int_tac -- TODO: canonize add/sub + simp [*] + +@[simp] +theorem index_itake_append_beg [Inhabited α] (i j : Int) (l0 l1 : List α) + (_ : 0 ≤ j) (_ : j < i) (_ : i ≤ l0.len) : + ((l0 ++ l1).itake i).index j = l0.index j := + match l0 with + | [] => by + simp at * + int_tac + | hd :: tl => + have : ¬ i = 0 := by simp at *; int_tac + if hj : j = 0 then by simp [*] + else by + have := index_itake_append_beg (i - 1) (j - 1) tl l1 (by simp_all; int_tac) (by simp_all) (by simp_all; int_tac) + simp [*] + +@[simp] +theorem index_itake_append_end [Inhabited α] (i j : Int) (l0 l1 : List α) + (_ : l0.len ≤ j) (_ : j < i) (_ : i ≤ l0.len + l1.len) : + ((l0 ++ l1).itake i).index j = l1.index (j - l0.len) := + match l0 with + | [] => by + simp at * + have := index_itake i j l1 (by simp_all) (by simp_all) (by simp_all; int_tac) + simp [*] + | hd :: tl => + have : ¬ i = 0 := by simp at *; int_tac + if hj : j = 0 then by simp_all; int_tac -- Contradiction + else by + have := index_itake_append_end (i - 1) (j - 1) tl l1 (by simp_all; int_tac) (by simp_all) (by simp_all; int_tac) + -- TODO: normalization of add/sub + have : j - 1 - len tl = j - (1 + len tl) := by int_tac + simp_all + @[simp] theorem index_ne {α : Type u} [Inhabited α] (l: List α) (i: ℤ) (j: ℤ) (x: α) : @@ -251,8 +451,34 @@ theorem index_map_eq {α : Type u} {β : Type v} [Inhabited α] [Inhabited β] ( by simp [*] -def allP {α : Type u} (l : List α) (p: α → Prop) : Prop := - foldr (fun a r => p a ∧ r) True l +theorem replace_slice_index [Inhabited α] (start end_ : Int) (l nl : List α) + (_ : 0 ≤ start) (_ : start < end_) (_ : end_ ≤ l.len) (_ : nl.len = end_ - start) : + let l1 := l.replace_slice start end_ nl + (∀ i, 0 ≤ i → i < start → l1.index i = l.index i) ∧ + (∀ i, start ≤ i → i < end_ → l1.index i = nl.index (i - start)) ∧ + (∀ i, end_ ≤ i → i < l.len → l1.index i = l.index i) + := by + -- let s_end := s.val ++ a.val.idrop r.end_.val + -- We need those assumptions everywhere + -- have : 0 ≤ start := by scalar_tac + have : start ≤ l.len := by int_tac + simp only [replace_slice] + split_conjs + . intro i _ _ + -- Introducing exactly the assumptions we need to make the rewriting work + have : i < l.len := by int_tac + simp_all + . intro i _ _ + have : (List.itake start l).len ≤ i := by simp_all + have : i < (List.itake start l).len + (nl ++ List.idrop end_ l).len := by + simp_all; int_tac + simp_all + . intro i _ _ + have : 0 ≤ end_ := by scalar_tac + have : end_ ≤ l.len := by int_tac + have : (List.itake start l).len ≤ i := by simp_all; int_tac + have : i < (List.itake start l).len + (nl ++ List.idrop end_ l).len := by simp_all + simp_all @[simp] theorem allP_nil {α : Type u} (p: α → Prop) : allP [] p := @@ -263,12 +489,6 @@ theorem allP_cons {α : Type u} (hd: α) (tl : List α) (p: α → Prop) : allP (hd :: tl) p ↔ p hd ∧ allP tl p := by simp [allP, foldr] -def pairwise_rel - {α : Type u} (rel : α → α → Prop) (l: List α) : Prop - := match l with - | [] => True - | hd :: tl => allP tl (rel hd) ∧ pairwise_rel rel tl - @[simp] theorem pairwise_rel_nil {α : Type u} (rel : α → α → Prop) : pairwise_rel rel [] |