--> ****************************************************************
-->           PROOF score for the initial states condition 
--> ****************************************************************

--> ----------------------------------------------------------------
--> require qlock-prop
require qlock-prop

--> ----------------------------------------------------------------
--> INITcheck: module for checking initial condition 
--> ----------------------------------------------------------------
mod INITcheck {
inc(INIT + MX + HQ=C)
pred check-init_ : State .
eq (check-init S:State) = (init S) implies ((mx S) and (hq=c S)) .
-- fresh constants
op aq : -> Aq .
ops as-r as-w as-c : -> As .
-- initial condition proposition
op initCheck : -> Bool .
eq initCheck = check-init [aq r as-r w as-w c as-c] .
}

--> ================================================================
--> check that (init implies q=wc) (1)
--> ----------------------------------------------------------------
select INITcheck .
:goal{eq initCheck = true .}
:def csp-q = :csp{eq aq = empQ . eq (aq =aq empQ) = false .}
:def csp-r = :csp{eq as-r = empS . eq (as-r =as empS) = false .}
:def csp-w = :csp{eq as-w = empS . eq (as-w =as empS) = false .}
:def csp-c = :csp{eq as-c = empS . eq (as-c =as empS) = false .}
:apply(csp-q rd- csp-r rd- csp-w rd- csp-c rd-) 
--> QED
--> ================================================================

#|
INITcheck(X.STATE)> :show proof
root*
[csp-q] 1*
[csp-r] 1-1*
[csp-r] 1-2*
[csp-w] 1-2-1*
[csp-c] 1-2-1-1*
[csp-c] 1-2-1-2*
[csp-w] 1-2-2*
[csp-q] 2*

INITcheck(X.STATE)> :describe proof
==> root*
    -- context module: #Goal-root
    -- targeted sentence:
      eq initCheck = true .
[csp-q] 1*
    -- context module: #Goal-1
    -- assumption
      eq [csp-q]: aq = empQ .
    -- targeted sentence:
      eq initCheck = true .
[csp-r] 1-1*
    -- context module: #Goal-1-1
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r = empS .
    -- discharged sentence:
      eq [RD-]: initCheck = true .
[csp-r] 1-2*
    -- context module: #Goal-1-2
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r =as empS = false .
    -- targeted sentence:
      eq initCheck = true .
[csp-w] 1-2-1*
    -- context module: #Goal-1-2-1
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r =as empS = false .
      eq [csp-w]: as-w = empS .
    -- targeted sentence:
      eq initCheck = true .
[csp-c] 1-2-1-1*
    -- context module: #Goal-1-2-1-1
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r =as empS = false .
      eq [csp-w]: as-w = empS .
      eq [csp-c]: as-c = empS .
    -- discharged sentence:
      eq [RD-]: initCheck = true .
[csp-c] 1-2-1-2*
    -- context module: #Goal-1-2-1-2
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r =as empS = false .
      eq [csp-w]: as-w = empS .
      eq [csp-c]: as-c =as empS = false .
    -- discharged sentence:
      eq [RD-]: initCheck = true .
[csp-w] 1-2-2*
    -- context module: #Goal-1-2-2
    -- assumptions
      eq [csp-q]: aq = empQ .
      eq [csp-r]: as-r =as empS = false .
      eq [csp-w]: as-w =as empS = false .
    -- discharged sentence:
      eq [RD-]: initCheck = true .
[csp-q] 2*
    -- context module: #Goal-2
    -- assumption
      eq [csp-q]: aq =aq empQ = false .
    -- discharged sentence:
      eq [RD-]: initCheck = true .
|#

--> ================================================================
--> check that (init implies q=wc) (2)
--> ----------------------------------------------------------------
--> 1-1
open INITcheck .
eq aq = empQ . 
eq as-r = empS .
red initCheck .
close
--> 1-2-1-1
open INITcheck .
eq aq = empQ . 
eq (as-r =as empS) = false .
eq as-w = empS .
eq as-c = empS .
red initCheck .
close
--> 1-2-1-2
open INITcheck .
eq aq = empQ . 
eq (as-r =as empS) = false .
eq as-w = empS .
eq (as-c =as empS) = false .
red initCheck .
close
--> 1-2-2
open INITcheck .
eq aq = empQ . 
eq (as-r =as empS) = false .
eq (as-w =as empS) = false .
red initCheck .
close
--> 2
open INITcheck .
eq (aq =aq empQ) = false .
red initCheck .
close
--> QED

--> ****************************************************************
--> end of file
eof
--> ****************************************************************