-- CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean

import Auto

import Lean

import Mathlib.Algebra.BigOperators.Intervals

import Mathlib.Algebra.Ring.Int.Defs

import Loom.MonadAlgebras.NonDetT.Extract

import Loom.MonadAlgebras.WP.Tactic

import Loom.MonadAlgebras.WP.DoNames'

import CaseStudies.Velvet.Std

section SpMV

structure SpV (valTyp : Type) where
  ind: Array Nat
  val: Array valTyp
  size: ℕ
  size_eq: ind.size = size ∧ val.size = size
  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!

def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=
  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then
    0
  else
    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then
      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)
    else
      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then
        spv_dot spv1 spv2 (pnt1 + 1) pnt2
      else
        spv_dot spv1 spv2 pnt1 (pnt2 + 1)
  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)


method SpVSpV
  (mut out: Array Int)
  (spv1: SpV Int)
  (spv2: SpV Int) return (u: Unit)
  ensures out.size = 1
  ensures out[0]! = spv_dot spv1 spv2 0 0
  do
    out := Array.replicate 1 0
    let mut pnt1 := 0
    let mut pnt2 := 0
    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size
    invariant out.size = 1
    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size
    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0
    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size
    do
      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then
        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!
        pnt1 := pnt1 + 1
        pnt2 := pnt2 + 1
      else
        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then
          pnt1 := pnt1 + 1
        else
          pnt2 := pnt2 + 1
    return


method SpMSpV
  (mut out: Array Int)
  (spm: Array (SpV Int))
  (spv: SpV Int) return (u: Unit)
  ensures out.size = spm.size
  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0
  do
    out := Array.replicate spm.size 0
    let mut spmInd := Array.replicate spm.size 0
    let mut spvInd := Array.replicate spm.size 0
    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size
    invariant spvInd.size = spm.size
    invariant spmInd.size = spm.size
    invariant out.size = spm.size
    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size
    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size
    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0
    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size
    do
      let ind_m := spmInd[i]!
      let ind_v := spvInd[i]!
      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then
        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!
        spmInd[i] += 1
        spvInd[i] += 1
      else
        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then
          spmInd[i] += 1
        else
          spvInd[i] += 1
    return

-- Repository definitions used:
content: syntax "method" ident leafny_binder* "return" "(" ident ":" term ")"
  (require_caluse )*
  (ensures_caluse)* "do" doSeq
  Termination.suffix : command

syntax "ensures" termBeforeReqEnsDo : ensures_caluse

syntax "while_some" term ":|" termBeforeDo "do" doSeq : doElem

syntax "while_some" term ":|" term
  (invariantClause)+
  (doneWith)?
  "do" doSeq : doElem

syntax "let" term ":|" term : doElem

syntax "done_with" termBeforeDo : doneWith

syntax "invariant" termBeforeDo linebreak : invariantClause

syntax "while" term
  (invariantClause)*
  (doneWith)?
  (decreasingTerm)?
  "do" doSeq : doElem

syntax "(mut" ident ":" term ")" : leafny_binder

content: macro_rules
  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))

content: macro_rules
  | `(doElem| while $t do $seq:doSeq) => do
    let decr <- withRef (<- getRef) `(decreasing none)
    let invs <- withRef (<- getRef) `(invariants [])
    `(doElem|
      for _ in Lean.Loop.mk do
        $invs:term
        onDoneGadget (with_name_prefix `done ¬$t:term)
        $decr:term
        if $t then
          $seq:doSeq
        else break)
  | `(doElem| while $t
              $[invariant $inv:term
              ]*
              $[done_with $inv_done]?
              $[decreasing $measure]?
              do $seq:doSeq) => do
      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])
      let invd_some ← match inv_done with
      | some invd_some => withRef invd_some ``($invd_some)
      | none => ``(¬$t:term)
      match measure with
      | some measure_some =>
        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)
        `(doElem|
          for _ in Lean.Loop.mk do
            $invs:term
            onDoneGadget (with_name_prefix `done $invd_some:term)
            $decr:term
            if $t then
              $seq:doSeq
            else break)
      | none => do
        let decr <- withRef (<- getRef) `(decreasing none)
        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])
        `(doElem|
          for _ in Lean.Loop.mk do
            $invs:term
            onDoneGadget (with_name_prefix `done $invd_some:term)
            $decr:term
            if $t then
              $seq:doSeq
            else break)
  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>
    match seq with
    | `(doSeq| $[$seq:doElem]*)
    | `(doSeq| $[$seq:doElem;]*)
    | `(doSeq| { $[$seq:doElem]* }) =>
      `(doElem|
        while ∃ $x:ident, $t do
          let $x :| $t
          $[$seq:doElem]*)
    | _ => Lean.Macro.throwError "while_some expects a sequence of do-elements"
  | `(doElem| while_some $x:ident :| $t
              $[invariant $inv:term
              ]*
              $[done_with $inv_done]? do
                $seq:doSeq) => do
    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])
    let invd_some ← match inv_done with
    | some invd_some => withRef invd_some ``($invd_some)
    | none => ``(¬$t:term)
    match seq with
    | `(doSeq| $[$seq:doElem]*)
    | `(doSeq| $[$seq:doElem;]*)
    | `(doSeq| { $[$seq:doElem]* }) =>
      let decr <- withRef (<- getRef) `(decreasing none)
      `(doElem|
        for _ in Lean.Loop.mk do
          $invs:term
          onDoneGadget (with_name_prefix `done $invd_some:term)
          $decr:term
          if ∃ $x:ident, $t then
            let $x :| $t
            $[$seq:doElem]*
          else break)
    | _ => Lean.Macro.throwError "while_some expects a sequence of do-elements"
  | `(doElem| for $x:ident in $t
            $[invariant $inv:term
            ]*
            do $seq:doSeq) => do
      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])
      match seq with
      | `(doSeq| $[$seq:doElem]*)
      | `(doSeq| $[$seq:doElem;]*)
      | `(doSeq| { $[$seq:doElem]* }) =>
        `(doElem|
          for $x:ident in $t do
            $invs:term
            $[$seq:doElem]*)
      | _ => Lean.Macro.throwError "for expects a sequence of do-elements"

content: macro_rules
  | `(tactic|loom_solver) =>
    `(tactic|(
      try simp at *
      try aesop))

content: macro_rules
  | `(doElem| while $t
              $[invariant $inv:term
              ]*
              $[done_with $inv_done]?
              $[decreasing $measure]?
              do $seq:doSeq) => do
      let balance := mkIdent `balance_name
      let balanceType <- `(term| Bal)
      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)
      let invd_some <- match inv_done with
      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)
      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)
      match measure with
      | some measure_some =>
        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))
        do
        `(doElem|
          for _ in Lean.Loop.mk do
            invariantGadget [ $[$inv:term],* ]
            onDoneGadget ($invd_some:term)
            decreasingGadget ($measure_some:term)
            if $t then
              $seq:doSeq
            else break)
      | none => do
        `(doElem|
          for _ in Lean.Loop.mk do
            invariantGadget [ $[$inv:term],* ]
            onDoneGadget ($invd_some:term)
            decreasingGadget none
            if $t then
              $seq:doSeq
            else break)

content: macro_rules
| `(doElem|balance_set $t) => do
  let balId := mkIdent `balance
  `(doElem|do
    $balId:ident := $t
    set $balId:ident
    $balId:ident ← get)

content: macro_rules
  | `(doElem|$id:ident[$idx:term] := $val:term) =>
    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))
  | `(doElem|$id:ident[$idx:term] += $val:term) =>
    `(doElem| $id:term := ($id:term).modify $idx (· + $val))

-- Repository lemmas used:
content: theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]

-- Local definitions used:
content: structure SpV (valTyp : Type) where
  ind: Array Nat
  val: Array valTyp
  size: ℕ
  size_eq: ind.size = size ∧ val.size = size
  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!

content: def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=
  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then
    0
  else
    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then
      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)
    else
      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then
        spv_dot spv1 spv2 (pnt1 + 1) pnt2
      else
        spv_dot spv1 spv2 pnt1 (pnt2 + 1)
  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)


method SpVSpV
  (mut out: Array Int)
  (spv1: SpV Int)
  (spv2: SpV Int) return (u: Unit)
  ensures out.size = 1
  ensures out[0]! = spv_dot spv1 spv2 0 0
  do
    out := Array.replicate 1 0
    let mut pnt1 := 0
    let mut pnt2 := 0
    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size
    invariant out.size = 1
    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size
    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0
    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size
    do
      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then
        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!
        pnt1 := pnt1 + 1
        pnt2 := pnt2 + 1
      else
        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then
          pnt1 := pnt1 + 1
        else
          pnt2 := pnt2 + 1
    return


method SpMSpV
  (mut out: Array Int)
  (spm: Array (SpV Int))
  (spv: SpV Int) return (u: Unit)
  ensures out.size = spm.size
  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0
  do
    out := Array.replicate spm.size 0
    let mut spmInd := Array.replicate spm.size 0
    let mut spvInd := Array.replicate spm.size 0
    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size
    invariant spvInd.size = spm.size
    invariant spmInd.size = spm.size
    invariant out.size = spm.size
    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size
    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size
    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0
    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size
    do
      let ind_m := spmInd[i]!
      let ind_v := spvInd[i]!
      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then
        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!
        spmInd[i] += 1
        spvInd[i] += 1
      else
        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then
          spmInd[i] += 1
        else
          spvInd[i] += 1
    return

-- Local lemmas used:
content: theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!

content: theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0

content: theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)
  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)
  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):
  spv_dot spv1 spv2 pnt1 pnt2 =
    ∑ i ∈ Finset.range n,
      if max
        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)
        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then
        spv1[i] * spv2[i]
      else
        0

-- Target theorem and proof:
theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)
  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):
    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i]
:= by
      simp [spv_dot_pure_gen spv1 spv2 n 0 0 sz1 sz2]
      apply Finset.sum_congr
      { rfl }
      intro x hx
      simp at hx
      by_cases em1: spv1.size = 0 <;> simp [em1]
      { simp [lt_iff_le_not_ge.mp hx]
        simp [getValSpV_empty spv1 x (by intro i hi; omega)] }
      by_cases em2: spv2.size = 0 <;> simp [em2]
      { intros
        simp [getValSpV_empty spv2 x (by intro i hi; omega)] }
      intro zer_ineq
      have zer_lemma (spv: SpV Int) (i: ℕ) (hsz: spv.size ≠ 0): i < spv.ind[0]! → spv[i] = 0 := by
        intro lt
        have all_none: ∀ j < spv.size, spv.ind[j]! ≠ i := by
          intro i1 hi1
          by_cases i10 : i1 = 0
          { simp [←i10] at lt
            omega }
          have contra := spv.inc 0 i1 (by omega) (by omega) (by omega)
          simp at lt
          omega
        simp [getValSpV_empty spv i all_none]
      by_cases sm1: spv1.ind[0]! ≤ x
      { simp [sm1] at zer_ineq
        simp [zer_lemma spv2 x em2 zer_ineq] }
      simp at sm1
      simp [zer_lemma spv1 x em1 sm1]