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/- Arrays/Slices -/
import Lean
import Lean.Meta.Tactic.Simp
import Init.Data.List.Basic
import Mathlib.Tactic.RunCmd
import Mathlib.Tactic.Linarith
import Base.IList
import Base.Primitives.Scalar
import Base.Primitives.Range
import Base.Primitives.CoreOps
import Base.Arith
import Base.Progress.Base

namespace Primitives

open Result Error core.ops.range

def Array (α : Type u) (n : Usize) := { l : List α // l.length = n.val }

instance (a : Type u) (n : Usize) : Arith.HasIntProp (Array a n) where
  prop_ty := λ v => v.val.len = n.val
  prop := λ  _, l  => by simp[Scalar.max, List.len_eq_length, *]

instance {α : Type u} {n : Usize} (p : Array α n  Prop) : Arith.HasIntProp (Subtype p) where
  prop_ty := λ x => p x
  prop := λ x => x.property

@[simp]
abbrev Array.length {α : Type u} {n : Usize} (v : Array α n) : Int := v.val.len

@[simp]
abbrev Array.v {α : Type u} {n : Usize} (v : Array α n) : List α := v.val

example {α: Type u} {n : Usize} (v : Array α n) : v.length  Scalar.max ScalarTy.Usize := by
  scalar_tac

def Array.make (α : Type u) (n : Usize) (init : List α) (hl : init.len = n.val := by decide) :
  Array α n :=  init, by simp [ List.len_eq_length]; apply hl 

example : Array Int (Usize.ofInt 2) := Array.make Int (Usize.ofInt 2) [0, 1]

@[simp]
abbrev Array.index_s {α : Type u} {n : Usize} [Inhabited α] (v : Array α n) (i : Int) : α :=
  v.val.index i

@[simp]
abbrev Array.slice {α : Type u} {n : Usize} [Inhabited α] (v : Array α n) (i j : Int) : List α :=
  v.val.slice i j

def Array.index_usize (α : Type u) (n : Usize) (v: Array α n) (i: Usize) : Result α :=
  match v.val.indexOpt i.val with
  | none => fail .arrayOutOfBounds
  | some x => ret x

-- For initialization
def Array.repeat (α : Type u) (n : Usize) (x : α) : Array α n :=
   List.ireplicate n.val x, by have h := n.hmin; simp_all [Scalar.min] 

@[pspec]
theorem Array.repeat_spec {α : Type u} (n : Usize) (x : α) :
   a, Array.repeat α n x = a  a.val = List.ireplicate n.val x := by
  simp [Array.repeat]

/- In the theorems below: we don't always need the `∃ ..`, but we use one
   so that `progress` introduces an opaque variable and an equality. This
   helps control the context.
 -/

@[pspec]
theorem Array.index_usize_spec {α : Type u} {n : Usize} [Inhabited α] (v: Array α n) (i: Usize)
  (hbound : i.val < v.length) :
   x, v.index_usize α n i = ret x  x = v.val.index i.val := by
  simp only [index_usize]
  -- TODO: dependent rewrite
  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
  simp [*]

def Array.update_usize (α : Type u) (n : Usize) (v: Array α n) (i: Usize) (x: α) : Result (Array α n) :=
  match v.val.indexOpt i.val with
  | none => fail .arrayOutOfBounds
  | some _ =>
    .ret  v.val.update i.val x, by have := v.property; simp [*] 

@[pspec]
theorem Array.update_usize_spec {α : Type u} {n : Usize} (v: Array α n) (i: Usize) (x : α)
  (hbound : i.val < v.length) :
   nv, v.update_usize α n i x = ret nv 
  nv.val = v.val.update i.val x
  := by
  simp only [update_usize]
  have h := List.indexOpt_bounds v.val i.val
  split
  . simp_all [length]; cases h <;> scalar_tac
  . simp_all

def Array.index_mut_usize (α : Type u) (n : Usize) (v: Array α n) (i: Usize) :
  Result (α × (α -> Result (Array α n))) := do
  let x  index_usize α n v i
  ret (x, update_usize α n v i)

@[pspec]
theorem Array.index_mut_usize_spec {α : Type u} {n : Usize} [Inhabited α] (v: Array α n) (i: Usize)
  (hbound : i.val < v.length) :
   x back, v.index_mut_usize α n i = ret (x, back) 
  x = v.val.index i.val 
  back = update_usize α n v i := by
  simp only [index_mut_usize, Bind.bind, bind]
  have  x, h  := index_usize_spec v i hbound
  simp [h]

def Slice (α : Type u) := { l : List α // l.length  Usize.max }

instance (a : Type u) : Arith.HasIntProp (Slice a) where
  prop_ty := λ v => 0  v.val.len  v.val.len  Scalar.max ScalarTy.Usize
  prop := λ  _, l  => by simp[Scalar.max, List.len_eq_length, *]

instance {α : Type u} (p : Slice α  Prop) : Arith.HasIntProp (Subtype p) where
  prop_ty := λ x => p x
  prop := λ x => x.property

@[simp]
abbrev Slice.length {α : Type u} (v : Slice α) : Int := v.val.len

@[simp]
abbrev Slice.v {α : Type u} (v : Slice α) : List α := v.val

example {a: Type u} (v : Slice a) : v.length  Scalar.max ScalarTy.Usize := by
  scalar_tac

def Slice.new (α : Type u): Slice α :=  [], by apply Scalar.cMax_suffices .Usize; simp; decide 

-- TODO: very annoying that the α is an explicit parameter
def Slice.len (α : Type u) (v : Slice α) : Usize :=
  Usize.ofIntCore v.val.len (by scalar_tac) (by scalar_tac)

@[simp]
theorem Slice.len_val {α : Type u} (v : Slice α) : (Slice.len α v).val = v.length :=
  by rfl

@[simp]
abbrev Slice.index_s {α : Type u} [Inhabited α] (v: Slice α) (i: Int) : α :=
  v.val.index i

@[simp]
abbrev Slice.slice {α : Type u} [Inhabited α] (s : Slice α) (i j : Int) : List α :=
  s.val.slice i j

def Slice.index_usize (α : Type u) (v: Slice α) (i: Usize) : Result α :=
  match v.val.indexOpt i.val with
  | none => fail .arrayOutOfBounds
  | some x => ret x

/- In the theorems below: we don't always need the `∃ ..`, but we use one
   so that `progress` introduces an opaque variable and an equality. This
   helps control the context.
 -/

@[pspec]
theorem Slice.index_usize_spec {α : Type u} [Inhabited α] (v: Slice α) (i: Usize)
  (hbound : i.val < v.length) :
   x, v.index_usize α i = ret x  x = v.val.index i.val := by
  simp only [index_usize]
  -- TODO: dependent rewrite
  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
  simp [*]

def Slice.update_usize (α : Type u) (v: Slice α) (i: Usize) (x: α) : Result (Slice α) :=
  match v.val.indexOpt i.val with
  | none => fail .arrayOutOfBounds
  | some _ =>
    .ret  v.val.update i.val x, by have := v.property; simp [*] 

@[pspec]
theorem Slice.update_usize_spec {α : Type u} (v: Slice α) (i: Usize) (x : α)
  (hbound : i.val < v.length) :
   nv, v.update_usize α i x = ret nv 
  nv.val = v.val.update i.val x
  := by
  simp only [update_usize]
  have h := List.indexOpt_bounds v.val i.val
  split
  . simp_all [length]; cases h <;> scalar_tac
  . simp_all

def Slice.index_mut_usize (α : Type u) (v: Slice α) (i: Usize) :
  Result (α × (α  Result (Slice α))) := do
  let x  Slice.index_usize α v i
  ret (x, Slice.update_usize α v i)

@[pspec]
theorem Slice.index_mut_usize_spec {α : Type u} [Inhabited α] (v: Slice α) (i: Usize)
  (hbound : i.val < v.length) :
   x back, v.index_mut_usize α i = ret (x, back) 
  x = v.val.index i.val 
  back = Slice.update_usize α v i := by
  simp only [index_mut_usize, Bind.bind, bind]
  have  x, h  := Slice.index_usize_spec v i hbound
  simp [h]

/- Array to slice/subslices -/

/- We could make this function not use the `Result` type. By making it monadic, we
   push the user to use the `Array.to_slice_spec` spec theorem below (through the
   `progress` tactic), meaning `Array.to_slice` should be considered as opaque.
   All what the spec theorem reveals is that the "representative" lists are the same. -/
def Array.to_slice (α : Type u) (n : Usize) (v : Array α n) : Result (Slice α) :=
  ret  v.val, by simp [ List.len_eq_length]; scalar_tac 

@[pspec]
theorem Array.to_slice_spec {α : Type u} {n : Usize} (v : Array α n) :
   s, to_slice α n v = ret s  v.val = s.val := by simp [to_slice]

def Array.from_slice (α : Type u) (n : Usize) (_ : Array α n) (s : Slice α) : Result (Array α n) :=
  if h: s.val.len = n.val then
    ret  s.val, by simp [ List.len_eq_length, *] 
  else fail panic

@[pspec]
theorem Array.from_slice_spec {α : Type u} {n : Usize} (a : Array α n) (ns : Slice α) (h : ns.val.len = n.val) :
   na, from_slice α n a ns = ret na  na.val = ns.val
  := by simp [from_slice, *]

def Array.to_slice_mut (α : Type u) (n : Usize) (a : Array α n) :
  Result (Slice α × (Slice α  Result (Array α n))) := do
  let s  Array.to_slice α n a
  ret (s, Array.from_slice α n a)

@[pspec]
theorem Array.to_slice_mut_spec {α : Type u} {n : Usize} (v : Array α n) :
   s back, to_slice_mut α n v = ret (s, back) 
  v.val = s.val 
  back = Array.from_slice α n v
  := by simp [to_slice_mut, to_slice]

def Array.subslice (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) : Result (Slice α) :=
  -- TODO: not completely sure here
  if r.start.val < r.end_.val  r.end_.val  a.val.len then
    ret  a.val.slice r.start.val r.end_.val,
          by
            simp [ List.len_eq_length]
            have := a.val.slice_len_le r.start.val r.end_.val
            scalar_tac 
  else
    fail panic

@[pspec]
theorem Array.subslice_spec {α : Type u} {n : Usize} [Inhabited α] (a : Array α n) (r : Range Usize)
  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val  a.val.len) :
   s, subslice α n a r = ret s 
  s.val = a.val.slice r.start.val r.end_.val 
  ( i, 0  i  i + r.start.val < r.end_.val  s.val.index i = a.val.index (r.start.val + i))
  := by
  simp [subslice, *]
  intro i _ _
  have := List.index_slice r.start.val r.end_.val i a.val (by scalar_tac) (by scalar_tac) (by trivial) (by scalar_tac)
  simp [*]

def Array.update_subslice (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) (s : Slice α) : Result (Array α n) :=
  -- TODO: not completely sure here
  if h: r.start.val < r.end_.val  r.end_.val  a.length  s.val.len = r.end_.val - r.start.val then
    let s_beg := a.val.itake r.start.val
    let s_end := a.val.idrop r.end_.val
    have : s_beg.len = r.start.val := by
      apply List.itake_len
      . simp_all; scalar_tac
      . scalar_tac
    have : s_end.len = a.val.len - r.end_.val := by
      apply List.idrop_len
      . scalar_tac
      . scalar_tac
    let na := s_beg.append (s.val.append s_end)
    have : na.len = a.val.len := by simp [*]
    ret  na, by simp_all [ List.len_eq_length]; scalar_tac 
  else
    fail panic

-- TODO: it is annoying to write `.val` everywhere. We could leverage coercions,
-- but: some symbols like `+` are already overloaded to be notations for monadic
-- operations/
-- We should introduce special symbols for the monadic arithmetic operations
-- (the user will never write those symbols directly).
@[pspec]
theorem Array.update_subslice_spec {α : Type u} {n : Usize} [Inhabited α] (a : Array α n) (r : Range Usize) (s : Slice α)
  (_ : r.start.val < r.end_.val) (_ : r.end_.val  a.length) (_ : s.length = r.end_.val - r.start.val) :
   na, update_subslice α n a r s = ret na 
  ( i, 0  i  i < r.start.val  na.index_s i = a.index_s i) 
  ( i, r.start.val  i  i < r.end_.val  na.index_s i = s.index_s (i - r.start.val)) 
  ( i, r.end_.val  i  i < n.val  na.index_s i = a.index_s i) := by
  simp [update_subslice, *]
  have h := List.replace_slice_index r.start.val r.end_.val a.val s.val
    (by scalar_tac) (by scalar_tac) (by scalar_tac) (by scalar_tac)
  simp [List.replace_slice] at h
  have  h0, h1, h2  := h
  clear h
  split_conjs
  . intro i _ _
    have := h0 i (by int_tac) (by int_tac)
    simp [*]
  . intro i _ _
    have := h1 i (by int_tac) (by int_tac)
    simp [*]
  . intro i _ _
    have := h2 i (by int_tac) (by int_tac)
    simp [*]

def Slice.subslice (α : Type u) (s : Slice α) (r : Range Usize) : Result (Slice α) :=
  -- TODO: not completely sure here
  if r.start.val < r.end_.val  r.end_.val  s.length then
    ret  s.val.slice r.start.val r.end_.val,
          by
            simp [ List.len_eq_length]
            have := s.val.slice_len_le r.start.val r.end_.val
            scalar_tac 
  else
    fail panic

@[pspec]
theorem Slice.subslice_spec {α : Type u} [Inhabited α] (s : Slice α) (r : Range Usize)
  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val  s.val.len) :
   ns, subslice α s r = ret ns 
  ns.val = s.slice r.start.val r.end_.val 
  ( i, 0  i  i + r.start.val < r.end_.val  ns.index_s i = s.index_s (r.start.val + i))
  := by
  simp [subslice, *]
  intro i _ _
  have := List.index_slice r.start.val r.end_.val i s.val (by scalar_tac) (by scalar_tac) (by trivial) (by scalar_tac)
  simp [*]

attribute [pp_dot] List.len List.length List.index -- use the dot notation when printing
set_option pp.coercions false -- do not print coercions with ↑ (this doesn't parse)

def Slice.update_subslice (α : Type u) (s : Slice α) (r : Range Usize) (ss : Slice α) : Result (Slice α) :=
  -- TODO: not completely sure here
  if h: r.start.val < r.end_.val  r.end_.val  s.length  ss.val.len = r.end_.val - r.start.val then
    let s_beg := s.val.itake r.start.val
    let s_end := s.val.idrop r.end_.val
    have : s_beg.len = r.start.val := by
      apply List.itake_len
      . simp_all; scalar_tac
      . scalar_tac
    have : s_end.len = s.val.len - r.end_.val := by
      apply List.idrop_len
      . scalar_tac
      . scalar_tac
    let ns := s_beg.append (ss.val.append s_end)
    have : ns.len = s.val.len := by simp [*]
    ret  ns, by simp_all [ List.len_eq_length]; scalar_tac 
  else
    fail panic

@[pspec]
theorem Slice.update_subslice_spec {α : Type u} [Inhabited α] (a : Slice α) (r : Range Usize) (ss : Slice α)
  (_ : r.start.val < r.end_.val) (_ : r.end_.val  a.length) (_ : ss.length = r.end_.val - r.start.val) :
   na, update_subslice α a r ss = ret na 
  ( i, 0  i  i < r.start.val  na.index_s i = a.index_s i) 
  ( i, r.start.val  i  i < r.end_.val  na.index_s i = ss.index_s (i - r.start.val)) 
  ( i, r.end_.val  i  i < a.length  na.index_s i = a.index_s i) := by
  simp [update_subslice, *]
  have h := List.replace_slice_index r.start.val r.end_.val a.val ss.val
    (by scalar_tac) (by scalar_tac) (by scalar_tac) (by scalar_tac)
  simp [List.replace_slice, *] at h
  have  h0, h1, h2  := h
  clear h
  split_conjs
  . intro i _ _
    have := h0 i (by int_tac) (by int_tac)
    simp [*]
  . intro i _ _
    have := h1 i (by int_tac) (by int_tac)
    simp [*]
  . intro i _ _
    have := h2 i (by int_tac) (by int_tac)
    simp [*]

/- Trait declaration: [core::slice::index::private_slice_index::Sealed] -/
structure core.slice.index.private_slice_index.Sealed (Self : Type) where

/- Trait declaration: [core::slice::index::SliceIndex] -/
structure core.slice.index.SliceIndex (Self T : Type) where
  sealedInst : core.slice.index.private_slice_index.Sealed Self
  Output : Type
  get : Self  T  Result (Option Output)
  get_mut : Self  T  Result (Option Output × (Option Output  Result T))
  get_unchecked : Self  ConstRawPtr T  Result (ConstRawPtr Output)
  get_unchecked_mut : Self  MutRawPtr T  Result (MutRawPtr Output)
  index : Self  T  Result Output
  index_mut : Self  T  Result (Output × (Output  Result T))

/- [core::slice::index::[T]::index]: forward function -/
def core.slice.index.Slice.index
  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T))
  (slice : Slice T) (i : I) : Result inst.Output := do
  let x  inst.get i slice
  match x with
  | none => fail panic
  | some x => ret x

/- [core::slice::index::Range:::get]: forward function -/
def core.slice.index.RangeUsize.get (T : Type) (i : Range Usize) (slice : Slice T) :
  Result (Option (Slice T)) :=
  sorry -- TODO

/- [core::slice::index::Range::get_mut]: forward function -/
def core.slice.index.RangeUsize.get_mut
  (T : Type) : Range Usize  Slice T  Result (Option (Slice T) × (Option (Slice T)  Result (Slice T))) :=
  sorry -- TODO

/- [core::slice::index::Range::get_unchecked]: forward function -/
def core.slice.index.RangeUsize.get_unchecked
  (T : Type) :
  Range Usize  ConstRawPtr (Slice T)  Result (ConstRawPtr (Slice T)) :=
  -- Don't know what the model should be - for now we always fail to make
  -- sure code which uses it fails
  fun _ _ => fail panic

/- [core::slice::index::Range::get_unchecked_mut]: forward function -/
def core.slice.index.RangeUsize.get_unchecked_mut
  (T : Type) :
  Range Usize  MutRawPtr (Slice T)  Result (MutRawPtr (Slice T)) :=
  -- Don't know what the model should be - for now we always fail to make
  -- sure code which uses it fails
  fun _ _ => fail panic

/- [core::slice::index::Range::index]: forward function -/
def core.slice.index.RangeUsize.index
  (T : Type) : Range Usize  Slice T  Result (Slice T) :=
  sorry -- TODO

/- [core::slice::index::Range::index_mut]: forward function -/
def core.slice.index.RangeUsize.index_mut
  (T : Type) : Range Usize  Slice T  Result (Slice T × (Slice T  Result (Slice T))) :=
  sorry -- TODO

/- [core::slice::index::[T]::index_mut]: forward function -/
def core.slice.index.Slice.index_mut
  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T)) :
  Slice T  I  Result (inst.Output × (inst.Output  Result (Slice T))) :=
  sorry -- TODO

/- [core::array::[T; N]::index]: forward function -/
def core.array.Array.index
  (T I : Type) (N : Usize) (inst : core.ops.index.Index (Slice T) I)
  (a : Array T N) (i : I) : Result inst.Output :=
  sorry -- TODO

/- [core::array::[T; N]::index_mut]: forward function -/
def core.array.Array.index_mut
  (T I : Type) (N : Usize) (inst : core.ops.index.IndexMut (Slice T) I)
  (a : Array T N) (i : I) :
  Result (inst.indexInst.Output × (inst.indexInst.Output  Result (Array T N))) :=
  sorry -- TODO

/- Trait implementation: [core::slice::index::private_slice_index::Range] -/
def core.slice.index.private_slice_index.SealedRangeUsizeInst
  : core.slice.index.private_slice_index.Sealed (Range Usize) := {}

/- Trait implementation: [core::slice::index::Range] -/
def core.slice.index.SliceIndexRangeUsizeSliceTInst (T : Type) :
  core.slice.index.SliceIndex (Range Usize) (Slice T) := {
  sealedInst := core.slice.index.private_slice_index.SealedRangeUsizeInst
  Output := Slice T
  get := core.slice.index.RangeUsize.get T
  get_mut := core.slice.index.RangeUsize.get_mut T
  get_unchecked := core.slice.index.RangeUsize.get_unchecked T
  get_unchecked_mut := core.slice.index.RangeUsize.get_unchecked_mut T
  index := core.slice.index.RangeUsize.index T
  index_mut := core.slice.index.RangeUsize.index_mut T
}

/- Trait implementation: [core::slice::index::[T]] -/
def core.ops.index.IndexSliceTIInst (T I : Type)
  (inst : core.slice.index.SliceIndex I (Slice T)) :
  core.ops.index.Index (Slice T) I := {
  Output := inst.Output
  index := core.slice.index.Slice.index T I inst
}

/- Trait implementation: [core::slice::index::[T]] -/
def core.ops.index.IndexMutSliceTIInst (T I : Type)
  (inst : core.slice.index.SliceIndex I (Slice T)) :
  core.ops.index.IndexMut (Slice T) I := {
  indexInst := core.ops.index.IndexSliceTIInst T I inst
  index_mut := core.slice.index.Slice.index_mut T I inst
}

/- Trait implementation: [core::array::[T; N]] -/
def core.ops.index.IndexArrayIInst (T I : Type) (N : Usize)
  (inst : core.ops.index.Index (Slice T) I) :
  core.ops.index.Index (Array T N) I := {
  Output := inst.Output
  index := core.array.Array.index T I N inst
}

/- Trait implementation: [core::array::[T; N]] -/
def core.ops.index.IndexMutArrayIInst (T I : Type) (N : Usize)
  (inst : core.ops.index.IndexMut (Slice T) I) :
  core.ops.index.IndexMut (Array T N) I := {
  indexInst := core.ops.index.IndexArrayIInst T I N inst.indexInst
  index_mut := core.array.Array.index_mut T I N inst
}

/- [core::slice::index::usize::get]: forward function -/
def core.slice.index.Usize.get
  (T : Type) : Usize  Slice T  Result (Option T) :=
  sorry -- TODO

/- [core::slice::index::usize::get_mut]: forward function -/
def core.slice.index.Usize.get_mut
  (T : Type) : Usize  Slice T  Result (Option T × (Option T  Result (Slice T))) :=
  sorry -- TODO

/- [core::slice::index::usize::get_unchecked]: forward function -/
def core.slice.index.Usize.get_unchecked
  (T : Type) : Usize  ConstRawPtr (Slice T)  Result (ConstRawPtr T) :=
  sorry -- TODO

/- [core::slice::index::usize::get_unchecked_mut]: forward function -/
def core.slice.index.Usize.get_unchecked_mut
  (T : Type) : Usize  MutRawPtr (Slice T)  Result (MutRawPtr T) :=
  sorry -- TODO

/- [core::slice::index::usize::index]: forward function -/
def core.slice.index.Usize.index (T : Type) : Usize  Slice T  Result T :=
  sorry -- TODO

/- [core::slice::index::usize::index_mut]: forward function -/
def core.slice.index.Usize.index_mut (T : Type) :
  Usize  Slice T  Result (T × (T  Result (Slice T))) :=
  sorry -- TODO

/- Trait implementation: [core::slice::index::private_slice_index::usize] -/
def core.slice.index.private_slice_index.SealedUsizeInst
  : core.slice.index.private_slice_index.Sealed Usize := {}

/- Trait implementation: [core::slice::index::usize] -/
def core.slice.index.SliceIndexUsizeSliceTInst (T : Type) :
  core.slice.index.SliceIndex Usize (Slice T) := {
  sealedInst := core.slice.index.private_slice_index.SealedUsizeInst
  Output := T
  get := core.slice.index.Usize.get T
  get_mut := core.slice.index.Usize.get_mut T
  get_unchecked := core.slice.index.Usize.get_unchecked T
  get_unchecked_mut := core.slice.index.Usize.get_unchecked_mut T
  index := core.slice.index.Usize.index T
  index_mut := core.slice.index.Usize.index_mut T
}

end Primitives