Semantics of a Small Imperative Programming Language

*** the data types for the programming language
    
obj ZZ is pr INT .
  op  _is_ : Int Int -> Bool .
   
  var I J K L : Int .
  eq  I is I  =  true .
  eq  (I + J) is (K + J)  =  I is K .
  eq  (I - J) is (K - J)  =  I is K .
  cq  I is J  =  false   if  (I < J)  or  (J < I) .
  eq  I + - I  =  0 .
  eq  -(I + J)  =  - I + - J .
  eq  0 * I  =  0 .
  eq  - I * J  =  -(I * J) .
  eq  I - J  =  I + - J .
  eq  I * (J + K)  =  (I * J) + (I * K) .
  cq  I * J  =  I + (I * (J - 1))   if  0 < J .
  eq  (I + J) * K  =  (I * K) + (J * K) .
    
  eq  not(I <= J)  =  J < I .
  eq  not(I < J)  =  J <= I .
  eq  I + 1 <= J  =  I < J .
  eq  I < J + 1  =  I <= J .
  eq  I <= J + -1  =  I < J .
  eq  I <= J + - K  =  I + K <= J .
  eq  I < J + - K  =  I + K < J .
  eq  I + -1 < J  =  I <= J .
  eq  I <= I  =  true .
  eq  I < I  =  false .
  cq  I < I + J  =  true   if  0 < J .
  eq  I + -1 < I  =  true .
  cq  I + J < I  =  true   if  J < 0 .
  cq  I <= J  =  true   if  I < J .
  cq  I <= J + 1  =  true  if  I <= J .
  cq  I <= J + K  =  true  if  (I <= J)  and  (I <= K) .
  cq  I + J <= K + L  =  true  if  (I <= K)  and  (J <= L) .
endo
    
*** the programming language: expressions ***
obj EXP is pr ZZ .
  dfn Var is QID .
  sorts  Exp Arvar Arcomp .
  subsorts  Var Int Arcomp < Exp .
  ops  a b c : -> Arvar .
  op  _+_  : Exp Exp -> Exp [prec 10] .
  op  _*_  : Exp Exp -> Exp [prec 8] .
  op   -_  : Exp -> Exp [prec 1] .
  op  _-_  : Exp Exp -> Exp [prec 10] .
  op  _[_] : Arvar Exp -> Arcomp [prec 1] .
endo 

*** the programming language: tests ***
obj TST is pr EXP .
  sort Tst .
  subsort  Bool < Tst .
  op  _<_ : Exp Exp -> Tst [prec 15] .
  op  _<=_ : Exp Exp -> Tst [prec 15] .
  op  _is_ : Exp Exp -> Tst [prec 15] .
  op  not_ : Tst -> Tst [prec 1] .
  op  _and_ : Tst Tst -> Tst [prec 20] .
  op  _or_ : Tst Tst -> Tst [prec 25] .
endo  

*** the programming language: basic programs ***
obj BPGM is pr TST .
  sort BPgm .  
  op  _:=_ : Var Exp -> BPgm [prec 20] .
  op  _:=_ : Arcomp Exp -> BPgm [prec 20] .
endo
    
*** semantics of basic programs ***
th STORE is pr BPGM .
            pr ARRAY .
  sort Store .
  op initial : -> Store .
  op  _[[_]] : Store Exp -> Int [prec 65] .
  op  _[[_]] : Store Tst -> Bool [prec 65] .
  op  _[[_]] : Store Arvar -> Array [prec 65] .
  op     _;_ : Store BPgm -> Store [prec 60] .
  var  S : Store .
  vars X1 X2 : Var .
  var  I : Int .
  vars E1 E2 : Exp .
  vars T1 T2 : Tst .
  var  B : Bool .
  vars AV AV' : Arvar .
    
  eq  initial [[X1]]  =  0 . 
  
  eq  S [[I]]  =  I .
  eq  S [[- E1]]  =  -(S[[E1]]) .
  eq  S [[E1 - E2]]  =  (S[[E1]]) - (S[[E2]]) .
  eq  S [[E1 + E2]]  =  (S[[E1]]) + (S[[E2]]) .
  eq  S [[E1 * E2]]  =  (S[[E1]]) * (S[[E2]]) .
  eq  S [[ AV[E1] ]]  =  (S[[AV]])[ S[[E1]] ] .
   
  eq  S [[B]]  =  B .
  eq  S [[E1 is E2]]  =  (S [[E1]]) is (S [[E2]]) .
  eq  S [[E1 <= E2]]  =  (S [[E1]]) <= (S [[E2]]) .
  eq  S [[E1 < E2]]  =  (S [[E1]]) < (S [[E2]]) .
  eq  S [[not T1]]  =  not(S [[T1]]) .
  eq  S [[T1 and T2]]  =  (S [[T1]]) and (S [[T2]]) .
  eq  S [[T1 or T2]]  =  (S [[T1]]) or (S [[T2]]) .
 
  eq  S ; X1 := E1 [[X1]]  =  S [[E1]] .
  cq  S ; X1 := E1 [[X2]]  =  S [[X2]]   if  X1 =/= X2 .
  eq  S ; X1 := E1 [[AV]]  =  S [[AV]] .
    
  eq  S ; AV[E1] := E2 [[AV]]  =
        (S[[AV]])[ S[[E1]] <- S[[E2]] ] .
  cq  S ; AV[E1] := E2 [[AV']]  =  S [[AV']]    if  AV =/= AV' .
  eq  S ; AV[E1] := E2 [[X1]]  =  S [[X1]] .
endth
    
*** extended programming language ***
obj PGM is pr BPGM .
  sort  Pgm .
  subsort  BPgm < Pgm .
  op  skip  : -> Pgm . 
  op  _;_   : Pgm Pgm -> Pgm [assoc prec 50] . 
  op  if_then_else_fi : Tst Pgm Pgm -> Pgm [prec 40] .
  op  while_do_od     : Tst Pgm -> Pgm [prec 40] .
endo  
   
obj SEM is pr PGM .
           pr STORE .
  sort EStore .
  subsort Store < EStore .
  op  _;_ : EStore Pgm -> EStore [prec 60] .
  var S : Store .
  var T : Tst . 
  var P1 P2 : Pgm . 
  eq  S ; skip  =  S .
  eq  S ; (P1 ; P2)  =  (S ; P1) ; P2 .
  cq  S ; if T then P1 else P2 fi  =  S ; P1
    if  S[[T]] .
  cq  S ; if T then P1 else P2 fi  =  S ; P2
    if  not(S[[T]]) .
  cq  S ; while T do P1 od  =  (S ; P1) ; while T do P1 od 
    if  S[[T]] .
  cq  S ; while T do P1 od  =  S
    if  not(S[[T]]) .
endo