-- I) Base case

open INV
  red inv2(init,i) .
close

-- II) Inductive cases

--> 1) want(s,k)
-- 1.1) c-want(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-want(s,k) = true .
  eq pc(s,k) = l1 .
--
  eq i = k .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-want(s,k) = true .
  eq pc(s,k) = l1 .
--
  eq (i = k) = false .
  eq pc(s,i) = cs .
  eq top(queue(s)) = i .
  eq empty?(queue(s)) = true .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red inv3(s,i) implies istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-want(s,k) = true .
  eq pc(s,k) = l1 .
--
  eq (i = k) = false .
  eq pc(s,i) = cs .
  eq top(queue(s)) = i .
  eq empty?(queue(s)) = false .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-want(s,k) = true .
  eq pc(s,k) = l1 .
--
  eq (i = k) = false .
  eq pc(s,i) = cs .
  eq (top(queue(s)) = i) = false .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-want(s,k) = true .
  eq pc(s,k) = l1 .
--
  eq (i = k) = false .
  eq (pc(s,i) = cs) = false .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
-- 1.2) not c-want(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  eq c-want(s,k) = false .
-- successor state
  eq s' = want(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close

--> 2) try(s,k)
-- 2.1) c-try(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-try(s,k) = true .
  eq pc(s,k) = l2 .
  eq top(queue(s)) = k .
--
  eq i = k .
-- successor state
  eq s' = try(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-try(s,k) = true .
  eq pc(s,k) = l2 .
  eq top(queue(s)) = k .
--
  eq (i = k) = false .
-- successor state
  eq s' = try(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
-- 2.2) not c-try(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  eq c-try(s,k) = false .
-- successor state
  eq s' = try(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close

--> 3) exit(s,k)
-- 3.1) c-exit(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-exit(s,k) = true .
  eq pc(s,k) = cs .
--
  eq i = k .
-- successor state
  eq s' = exit(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-exit(s,k) = true .
  eq pc(s,k) = cs .
--
  eq (i = k) = false .
  eq pc(s,i) = cs .
-- successor state
  eq s' = exit(s,k) .
-- check if the predicate is true.
  red inv1(s,i,k) implies istep2(i) .
close
--
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  -- eq c-exit(s,k) = true .
  eq pc(s,k) = cs .
--
  eq (i = k) = false .
  eq (pc(s,i) = cs) = false .
-- successor state
  eq s' = exit(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close
--
-- 3.2) not c-exit(s,k)
open ISTEP
-- arbitrary objects
  op k : -> Pid .
-- assumptions
  eq c-exit(s,k) = false .
-- successor state
  eq s' = exit(s,k) .
-- check if the predicate is true.
  red istep2(i) .
close

--> Q.E.D