-- I) Base case

open INV
  red inv200(init,pair1,pair2,pair3,pfifo1,pfifo2) .
close

-- II) Inductive ceses

--> 1) send1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq (put(fifo1(s),< tag1(s) , next(s) >) = pfifo1 @ (pair1 | pair2 | pfifo2)) = false .
-- successor state
  eq s' = send1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  eq put(fifo1(s),< tag1(s) , next(s) >) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo2 = empty .
-- facts
  eq pair2 = < tag1(s) , next(s) > .
-- successor state
  eq s' = send1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
  op pair10 : -> Pair .
  op pfifo10 : -> PFifo .
-- assumptions
  eq put(fifo1(s),< tag1(s) , next(s) >) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo2 = pair10 | pfifo10 .
  eq pair3 \in (del(pair10 | pfifo10)) = true .
-- facts
  eq fifo1(s) = del(pfifo1 @ (pair1 | pair2 | pfifo2)) .
-- successor state
  eq s' = send1(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,pfifo1,del(pair10 | pfifo10)) 
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
  op pair10 : -> Pair .
  op pfifo10 : -> PFifo .
-- assumptions
  eq put(fifo1(s),< tag1(s) , next(s) >) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo2 = pair10 | pfifo10 .
  eq pair3 \in (del(pair10 | pfifo10)) = false .
  eq pair3 \in (pair10 | pfifo10) = false .
-- facts
  eq fifo1(s) = del(pfifo1 @ (pair1 | pair2 | pfifo2)) .
-- successor state
  eq s' = send1(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,pfifo1,del(pair10 | pfifo10)) 
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
  op pair10 : -> Pair .
  op pfifo10 : -> PFifo .
-- assumptions
  eq put(fifo1(s),< tag1(s) , next(s) >) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo2 = pair10 | pfifo10 .
  eq pair3 \in (del(pair10 | pfifo10)) = false . -- (1)
  eq pair3 \in (pair10 | pfifo10) = true .       -- (2)
-- facts
  eq fifo1(s) = del(pfifo1 @ (pair1 | pair2 | pfifo2)) .
  eq pair3 = < tag1(s) , next(s) > . -- from (1) and (2)
-- successor state
  eq s' = send1(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,pfifo1,del(pair10 | pfifo10)) 
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 2) rec1(s)
-- 2.1) c-rec1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec1(s) = true .
  eq empty?(fifo2(s)) = false .
--
  eq tag1(s) = top(fifo2(s)) .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec1(s) = true .
  eq empty?(fifo2(s)) = false .
--
  eq (tag1(s) = top(fifo2(s))) = false .
  eq (fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2)) = false .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec1(s) = true .
  eq empty?(fifo2(s)) = false .
--
  eq (tag1(s) = top(fifo2(s))) = false .
  eq fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq (pair1 = < tag1(s) , next(s) >) = false .
-- facts
  eq pair1 \in (pfifo1 @ (pair1 | pair2 | pfifo2)) = true .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red inv220(s,pair1) implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec1(s) = true .
  eq empty?(fifo2(s)) = false .
--
  eq (tag1(s) = top(fifo2(s))) = false .
  eq fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq (pair2 = < tag1(s) , next(s) >) = false .
-- facts
  eq pair2 \in (pfifo1 @ (pair1 | pair2 | pfifo2)) = true .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red inv220(s,pair2) implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec1(s) = true .
  eq empty?(fifo2(s)) = false .
--
  eq (tag1(s) = top(fifo2(s))) = false .
  eq fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pair1 = < tag1(s) , next(s) > .
  eq pair2 = < tag1(s) , next(s) > .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 2.2) not c-rec1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-rec1(s) = false .
-- successor state
  eq s' = rec1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 3) send2(s)
open ISTEP
-- arbitrary objects
-- assumptions
-- successor state
  eq s' = send2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 4) rec2(s)
-- 4.1) c-rec2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec2(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq (get(fifo1(s)) = pfifo1 @ (pair1 | pair2 | pfifo2)) = false .
-- successor state
  eq s' = rec2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-rec2(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq get(fifo1(s)) = pfifo1 @ (pair1 | pair2 | pfifo2) .
-- facts
  eq (fifo1(s) = top(fifo1(s)) | (pfifo1 @ (pair1 | pair2 | pfifo2))) = true .
-- successor state
  eq s' = rec2(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,(top(fifo1(s)) | pfifo1),pfifo2)
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 4.2) c-rec2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-rec2(s) = false .
-- successor state
  eq s' = rec2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 5) drop1(s)
-- 5.1) c-drop1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-drop1(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq (get(fifo1(s)) = pfifo1 @ (pair1 | pair2 | pfifo2)) = false .
-- successor state
  eq s' = drop1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-drop1(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq get(fifo1(s)) = pfifo1 @ (pair1 | pair2 | pfifo2) .
-- facts
  eq (fifo1(s) = top(fifo1(s)) | (pfifo1 @ (pair1 | pair2 | pfifo2))) = true .
-- successor state
  eq s' = drop1(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,(top(fifo1(s)) | pfifo1),pfifo2)
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 5.2) not c-drop1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-drop1(s) = false .
-- successor state
  eq s' = drop1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 6) dup1(s)
-- 6.1) c-dup1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-dup1(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq (top(fifo1(s)) | fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2)) = false .
-- successor state
  eq s' = dup1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-dup1(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq top(fifo1(s)) | fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo1 = empty .
-- facts
  eq pair1 = pair2 .
-- successor state
  eq s' = dup1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
open ISTEP
-- arbitrary objects
  op pair10 : -> Pair .
  op pfifo10 : -> PFifo .
-- assumptions
  -- eq c-dup1(s) = true .
  eq empty?(fifo1(s)) = false .
--
  eq top(fifo1(s)) | fifo1(s) = pfifo1 @ (pair1 | pair2 | pfifo2) .
  eq pfifo1 = pair10 | pfifo10 .
-- facts
  eq fifo1(s) = pfifo10 @ (pair1 | pair2 | pfifo2) .
-- successor state
  eq s' = dup1(s) .
-- check if the predicate is true.
  red inv200(s,pair1,pair2,pair3,pfifo10,pfifo2)
      implies istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 6.2) not c-dup1(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-dup1(s) = false .
-- successor state
  eq s' = dup1(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 7) drop2(s)
-- 7.1) c-drop2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-drop2(s) = true .
  eq empty?(fifo2(s)) = false .
-- successor state
  eq s' = drop2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 7.2) not c-drop2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-drop2(s) = false .
-- successor state
  eq s' = drop2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> 8) dup2(s)
-- 8.1) c-dup2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  -- eq c-dup2(s) = true .
  eq empty?(fifo2(s)) = false .
-- successor state
  eq s' = dup2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close
--
-- 8.2) not c-dup2(s)
open ISTEP
-- arbitrary objects
-- assumptions
  eq c-dup2(s) = false .
-- successor state
  eq s' = dup2(s) .
-- check if the predicate is true.
  red istep200(pair1,pair2,pair3,pfifo1,pfifo2) .
close

--> Q.E.D.