-- LeroyCompilerVerificationCourse/Compil.lean

import LeroyCompilerVerificationCourse.Imp

@[grind] inductive instr : Type where
  | Iconst (n : Int)             
  | Ivar (x : ident)             
  | Isetvar (x : ident)          
  | Iadd                         
  | Iopp                         
  | Ibranch (d : Int)            
  | Ibeq (d1 : Int) (d0 : Int)   
  | Ible (d1 : Int) (d0 : Int)   
  | Ihalt                        
    deriving Repr

@[grind] def codelen (c : List instr) : Int := c.length

def stack : Type := List Int

def config : Type := Int × stack × store

@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=
  match C with
  | [] => .none
  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)

@[grind] inductive transition (C : List instr) : config → config → Prop where
  | trans_const : ∀ pc stk s n,
      instr_at C pc = .some (.Iconst n) →
      transition C (pc    , stk     , s)
                   (pc + 1, n :: stk, s)
  | trans_var : ∀ pc stk s x,
      instr_at C pc = .some (.Ivar x) ->
      transition C (pc    , stk       , s)
                   (pc + 1, s x :: stk, s)
  | trans_setvar : ∀ pc stk s x n,
      instr_at C pc = .some (.Isetvar x) ->
      transition C (pc    , n :: stk, s)
                   (pc + 1, stk     , update x n s)
  | trans_add : ∀ pc stk s n1 n2,
      instr_at C pc = .some (.Iadd) ->
      transition C (pc    , n2 :: n1 :: stk , s)
                   (pc + 1, (n1 + n2) :: stk, s)
  | trans_opp : ∀ pc stk s n,
      instr_at C pc = .some (.Iopp) ->
      transition C (pc    , n :: stk    , s)
                   (pc + 1, (- n) :: stk, s)
  | trans_branch : ∀ pc stk s d pc',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      transition C (pc , stk, s)
                   (pc', stk, s)
  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',
      instr_at C pc = .some (.Ibeq d1 d0) ->
      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->
      transition C (pc , n2 :: n1 :: stk, s)
                   (pc', stk            , s)
  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',
      instr_at C pc = .some (.Ible d1 d0) ->
      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->
      transition C (pc , n2 :: n1 :: stk, s)
                   (pc', stk            , s)

@[grind] def transitions (C : List instr) : config → config → Prop :=
  star (transition C)

@[grind] def compile_aexp (a : aexp) : List instr :=
  match a with
  | .CONST n => .Iconst n :: []
  | .VAR x => .Ivar x :: []
  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])
  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])

@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=
  match b with
  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []
  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []
  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []
  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []
  | .NOT b1 => compile_bexp b1 d0 d1
  | .AND b1 b2 =>
      let code2 := compile_bexp b2 d1 d0
      let code1 := compile_bexp b1 0 (codelen code2 + d0)
      code1 ++ code2

@[grind] def compile_com (c : com) : List instr :=
  match c with
  | .SKIP =>
      []
  | .ASSIGN x a =>
      compile_aexp a ++ .Isetvar x :: []
  | .SEQ c1 c2 =>
      compile_com c1 ++ compile_com c2
  | .IFTHENELSE b ifso ifnot =>
      let code_ifso := compile_com ifso
      let code_ifnot := compile_com ifnot
      compile_bexp b 0 (codelen code_ifso + 1)
      ++ code_ifso
      ++ .Ibranch (codelen code_ifnot)
      :: code_ifnot
  | .WHILE b body =>
      let code_body := compile_com body
      let code_test := compile_bexp b 0 (codelen code_body + 1)
      code_test
      ++ code_body
      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []

@[grind] inductive code_at : List instr → Int → List instr → Prop where
  | code_at_intro : ∀ C1 C2 C3 pc,
      pc = codelen C1 ->
      code_at (C1 ++ C2 ++ C3) pc C2

inductive compile_cont (C : List instr) : cont -> Int -> Prop where
  | ccont_stop : ∀ pc,
      instr_at C pc = .some .Ihalt ->
      compile_cont C .Kstop pc
  | ccont_seq : ∀ c k pc pc',
      code_at C pc (compile_com c) ->
      pc' = pc + codelen (compile_com c) ->
      compile_cont C k pc' ->
      compile_cont C (.Kseq c k) pc
  | ccont_while : ∀ b c k pc d pc' pc'',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      code_at C pc' (compile_com (.WHILE b c)) ->
      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->
      compile_cont C k pc'' ->
      compile_cont C (.Kwhile b c k) pc
  | ccont_branch : ∀ d k pc pc',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      compile_cont C k pc' ->
      compile_cont C k pc

inductive match_config (C : List instr) : com × cont × store -> config -> Prop where
  | match_config_intro : ∀ c k st pc,
      code_at C pc (compile_com c) ->
      compile_cont C k (pc + codelen (compile_com c)) ->
      match_config C (c, k, st) (pc, [], st)

def com_size (c : com) : Nat :=
  match c with
  | .SKIP => 1
  | .ASSIGN _ _ => 1
  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)
  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)
  | .WHILE _ c1 => (com_size c1 + 1)

def cont_size (k : cont) : Nat :=
  match k with
  | .Kstop => 0
  | .Kseq c k' => (com_size c + cont_size k')
  | .Kwhile _ _ k' => cont_size k'

def measure' (impconf : com × cont × store) : Nat :=
  match impconf with
  | (c, k, _) => (com_size c + cont_size k)

-- Repository definitions used:
content: notation:10 l:10 " ;; " r:11 => com.SEQ l r

content: inductive com : Type where
  | SKIP                                        
  | ASSIGN (x : ident) (a : aexp)               
  | SEQ (c1 : com) (c2 : com)                   
  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) 
  | WHILE (b : bexp) (c1 : com)                 

content: @[grind] inductive cont where
| Kstop
| Kseq (c : com) (k : cont)
| Kwhile (b : bexp) (c : com) (k : cont)

content: inductive bexp : Type where
  | TRUE                              
  | FALSE                             
  | EQUAL (a1 : aexp) (a2 : aexp)     
  | LESSEQUAL (a1 : aexp) (a2 : aexp) 
  | NOT (b1 : bexp)                   
  | AND (b1 : bexp) (b2 : bexp)       

content: inductive aexp : Type where
  | CONST (n : Int)               
  | VAR (x : ident)               
  | PLUS (a1 : aexp) (a2 : aexp)  
  | MINUS (a1 : aexp) (s2 : aexp) 

content: def ident := String deriving BEq, Repr, Hashable

content: def store : Type := ident → Int

content: @[grind] def update (x : ident) (v : Int) (s : store) : store :=
  fun y => if x == y then v else s y

content: @[grind cases]
inductive plus (R : α → α → Prop) : α → α → Prop where
| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c


grind_pattern plus.plus_left => star R b c, plus R a c

content: @[grind] inductive star (R : α → α → Prop) : α → α → Prop where
  | star_refl : ∀ x : α, star R x x
  | star_step : ∀ {x y z}, R x y → star R y z → star R x z

content: def infseq {α} (R : α → α → Prop) : α → Prop :=
  λ x : α => ∃ y, R x y ∧ infseq R y
  coinductive_fixpoint

content: inductive step : com × cont × store -> com × cont × store -> Prop where
  | step_assign : ∀ x a k s,
      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)
    
  | step_seq : ∀ c1 c2 s k,
      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)
      
  | step_ifthenelse : ∀ b c1 c2 k s,
      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)
      
  | step_while_done : ∀ b c k s,
      beval s b = false ->
      step (.WHILE b c, k, s) (.SKIP, k, s)
      
  | step_while_true : ∀ b c k s,
      beval s b = true ->
      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)
      
  | step_skip_seq : ∀ c k s,
      step (.SKIP, .Kseq c k, s) (c, k, s)
      
  | step_skip_while : ∀ b c k s,
      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)

content: @[grind] def aeval (s : store) (a : aexp) : Int :=
  match a with
  | .CONST n => n
  | .VAR x => s x
  | .PLUS a1 a2 => aeval s a1 + aeval s a2
  | .MINUS a1 a2 => aeval s a1 - aeval s a2

content: @[grind] def beval (s : store) (b : bexp) : Bool :=
  match b with
  | .TRUE => true
  | .FALSE => false
  | .EQUAL a1 a2 => aeval s a1 = aeval s a2
  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2
  | .NOT b1 =>  !(beval s b1)
  | .AND b1 b2 => beval s b1 && beval s b2

content: notation:10 l:10 " ;; " r:11 => com.SEQ l r

-- Repository lemmas used:
content: @[grind] theorem plus_star {a b} (h : plus R a b) : star R a b

content: @[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c

content: theorem plus_right : star R a b -> R b c -> plus R a c

content: theorem star_plus_trans :
  ∀ a b c, star R a b -> plus R b c -> plus R a c

-- Local definitions used:
content: @[grind] inductive instr : Type where
  | Iconst (n : Int)             
  | Ivar (x : ident)             
  | Isetvar (x : ident)          
  | Iadd                         
  | Iopp                         
  | Ibranch (d : Int)            
  | Ibeq (d1 : Int) (d0 : Int)   
  | Ible (d1 : Int) (d0 : Int)   
  | Ihalt                        
    deriving Repr

content: @[grind] def codelen (c : List instr) : Int := c.length

content: def stack : Type := List Int

content: def config : Type := Int × stack × store

content: @[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=
  match C with
  | [] => .none
  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)

content: @[grind] inductive transition (C : List instr) : config → config → Prop where
  | trans_const : ∀ pc stk s n,
      instr_at C pc = .some (.Iconst n) →
      transition C (pc    , stk     , s)
                   (pc + 1, n :: stk, s)
  | trans_var : ∀ pc stk s x,
      instr_at C pc = .some (.Ivar x) ->
      transition C (pc    , stk       , s)
                   (pc + 1, s x :: stk, s)
  | trans_setvar : ∀ pc stk s x n,
      instr_at C pc = .some (.Isetvar x) ->
      transition C (pc    , n :: stk, s)
                   (pc + 1, stk     , update x n s)
  | trans_add : ∀ pc stk s n1 n2,
      instr_at C pc = .some (.Iadd) ->
      transition C (pc    , n2 :: n1 :: stk , s)
                   (pc + 1, (n1 + n2) :: stk, s)
  | trans_opp : ∀ pc stk s n,
      instr_at C pc = .some (.Iopp) ->
      transition C (pc    , n :: stk    , s)
                   (pc + 1, (- n) :: stk, s)
  | trans_branch : ∀ pc stk s d pc',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      transition C (pc , stk, s)
                   (pc', stk, s)
  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',
      instr_at C pc = .some (.Ibeq d1 d0) ->
      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->
      transition C (pc , n2 :: n1 :: stk, s)
                   (pc', stk            , s)
  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',
      instr_at C pc = .some (.Ible d1 d0) ->
      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->
      transition C (pc , n2 :: n1 :: stk, s)
                   (pc', stk            , s)

content: @[grind] def transitions (C : List instr) : config → config → Prop :=
  star (transition C)

content: @[grind] def compile_aexp (a : aexp) : List instr :=
  match a with
  | .CONST n => .Iconst n :: []
  | .VAR x => .Ivar x :: []
  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])
  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])

content: @[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=
  match b with
  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []
  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []
  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []
  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []
  | .NOT b1 => compile_bexp b1 d0 d1
  | .AND b1 b2 =>
      let code2 := compile_bexp b2 d1 d0
      let code1 := compile_bexp b1 0 (codelen code2 + d0)
      code1 ++ code2

content: @[grind] def compile_com (c : com) : List instr :=
  match c with
  | .SKIP =>
      []
  | .ASSIGN x a =>
      compile_aexp a ++ .Isetvar x :: []
  | .SEQ c1 c2 =>
      compile_com c1 ++ compile_com c2
  | .IFTHENELSE b ifso ifnot =>
      let code_ifso := compile_com ifso
      let code_ifnot := compile_com ifnot
      compile_bexp b 0 (codelen code_ifso + 1)
      ++ code_ifso
      ++ .Ibranch (codelen code_ifnot)
      :: code_ifnot
  | .WHILE b body =>
      let code_body := compile_com body
      let code_test := compile_bexp b 0 (codelen code_body + 1)
      code_test
      ++ code_body
      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []

content: @[grind] inductive code_at : List instr → Int → List instr → Prop where
  | code_at_intro : ∀ C1 C2 C3 pc,
      pc = codelen C1 ->
      code_at (C1 ++ C2 ++ C3) pc C2

content: inductive compile_cont (C : List instr) : cont -> Int -> Prop where
  | ccont_stop : ∀ pc,
      instr_at C pc = .some .Ihalt ->
      compile_cont C .Kstop pc
  | ccont_seq : ∀ c k pc pc',
      code_at C pc (compile_com c) ->
      pc' = pc + codelen (compile_com c) ->
      compile_cont C k pc' ->
      compile_cont C (.Kseq c k) pc
  | ccont_while : ∀ b c k pc d pc' pc'',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      code_at C pc' (compile_com (.WHILE b c)) ->
      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->
      compile_cont C k pc'' ->
      compile_cont C (.Kwhile b c k) pc
  | ccont_branch : ∀ d k pc pc',
      instr_at C pc = .some (.Ibranch d) ->
      pc' = pc + 1 + d ->
      compile_cont C k pc' ->
      compile_cont C k pc

content: inductive match_config (C : List instr) : com × cont × store -> config -> Prop where
  | match_config_intro : ∀ c k st pc,
      code_at C pc (compile_com c) ->
      compile_cont C k (pc + codelen (compile_com c)) ->
      match_config C (c, k, st) (pc, [], st)

content: def com_size (c : com) : Nat :=
  match c with
  | .SKIP => 1
  | .ASSIGN _ _ => 1
  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)
  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)
  | .WHILE _ c1 => (com_size c1 + 1)

content: def cont_size (k : cont) : Nat :=
  match k with
  | .Kstop => 0
  | .Kseq c k' => (com_size c + cont_size k')
  | .Kwhile _ _ k' => cont_size k'

content: def measure' (impconf : com × cont × store) : Nat :=
  match impconf with
  | (c, k, _) => (com_size c + cont_size k)

-- Local lemmas used:
content: @[grind =] theorem codelen_cons :
  ∀ i c, codelen (i :: c) = codelen c + 1

content: @[grind =] theorem codelen_app :
  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2

content: @[grind =>] theorem instr_a : ∀ i c2 c1 pc,
  pc = codelen c1 ->
  instr_at (c1 ++ (i :: c2) ) pc = .some i

content: @[grind] theorem code_at_app_right :
  ∀ C pc C1 C2,
  code_at C pc (C1 ++ C2) ->
  code_at C (pc + codelen C1) C2

content: @[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i

content: theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :
  code_at C pc (compile_aexp a) →
  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)

content: theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :
   transitions C
       (pc, stk, s)
       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)

content: theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :
  ∃ pc',
  star (transition C) (pc, [], s) (pc', [], s)
  ∧ code_at C pc' (compile_com c)
  ∧ compile_cont C k (pc' + codelen (compile_com c))

content: theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :
  ∃ pc',
  plus (transition C) (pc, [], s) (pc', [], s)
  ∧ code_at C pc' (compile_com (.WHILE b c))
  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))

content: theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :
 match_config C (.SKIP, k, s) (pc, [], s)

content: theorem simulation_step :
  ∀ C impconf1 impconf2 machconf1,
  step impconf1 impconf2 ->
  match_config C impconf1 machconf1 ->
  ∃ machconf2,
      (plus (transition C) machconf1 machconf2
       \/ (star (transition C) machconf1 machconf2
           /\ (measure' impconf2 < measure' impconf1)))
   /\ match_config C impconf2 machconf2

-- Target theorem and proof:
theorem simulation_infseq_inv :
  ∀ C n impconf1 machconf1,
  infseq step impconf1 -> match_config C impconf1 machconf1 ->
  (measure' impconf1 < n) ->
  ∃ impconf2 machconf2,
      infseq step impconf2
   /\ plus (transition C) machconf1 machconf2
   /\ match_config C impconf2 machconf2
:= by
    intro C n impconf1 h1 h2 h3 h4
    induction n generalizing impconf1 h1
    case zero => contradiction
    case succ n' ih =>
      rw [infseq] at h2
      rcases h2 with ⟨impconf2 , STEP, INFSEQ⟩
      have ⟨ machconf2, h5 , h6 ⟩ := simulation_step C impconf1 impconf2 h1 STEP h3
      cases h5
      next PLUS =>
        exists impconf2
        exists machconf2
      next w =>
        rcases w with ⟨ STAR, MEASURE ⟩
        specialize ih impconf2 machconf2 INFSEQ h6 (by omega)
        rcases ih with ⟨ c1, m1, w⟩
        exists c1
        exists m1
        constructor
        · exact w.1
        · constructor
          · apply star_plus_trans
            · exact STAR
            · exact w.2.1
          · exact w.2.2