--> I) Base case
open INV
  red inv1(init) .
close

--> II) Induction case

--> 1) send1(s)
open ISTEP
-- arbitrary values
-- assumptions
-- successor state
  eq s' = send1(s) .
-- check
  red istep1 .
close

--> 2) rec1(s)
--> c-rec1(s), bit1(s) = b
open ISTEP
-- arbitrary values
  op b : -> Bool .
-- assumptions
  -- eq c-rec1(s) = true .
  eq cell2(s) = c(b) .
  --
  eq bit1(s) = b .
-- successor state
  eq s' = rec1(s) .
-- check
  red istep1 .
close
--> c-rec1(s), ~(bit1(s) = b), bit2(s) = b
open ISTEP
-- arbitrary values
  op b : -> Bool .
-- assumptions
  -- eq c-rec1(s) = true .
  eq cell2(s) = c(b) .
  --
  eq (bit1(s) = b) = false .
  eq bit2(s) = b .
-- successor state
  eq s' = rec1(s) .
-- check
  red eqbool-lemma2(bit1(s),b) implies istep1 .
close
--> c-rec1(s), ~(bit1(s) = b), ~(bit2(s) = b)
open ISTEP
-- arbitrary values
  op b : -> Bool .
-- assumptions
  -- eq c-rec1(s) = true .
  eq cell2(s) = c(b) .
  --
  eq (bit1(s) = b) = false .
  eq (bit2(s) = b) = false .
-- successor state
  eq s' = rec1(s) .
-- check
  red inv2(s) implies istep1 .
close
--> ~c-rec1(s)
open ISTEP
-- arbitrary values
-- assumptions
  eq c-rec1(s) = false .
-- successor state
  eq s' = rec1(s) .
-- check
  red istep1 .
close

--> 3) send2(s)
open ISTEP
-- arbitrary values
-- assumptions
-- successor state
  eq s' = send2(s) .
-- check
  red istep1 .
close

--> 4) rec2(s)
--> c-rec2(s), bit2(s) = fst(p), bit1(s) = fst(p), 
--> pac(next(s)) = snd(p)
open ISTEP
-- arbitrary values
  op p : -> BPPair .
-- assumptions
  -- eq c-rec2(s) = true .
  eq cell1(s) = c(p) .
  --
  eq bit2(s) = fst(p) .
  eq bit1(s) = fst(p) .
  eq pac(next(s)) = snd(p) .
-- successor state
  eq s' = rec2(s) .
-- check
  red eqbool-lemma1(fst(p)) implies istep1 .
close
--> c-rec2(s), bit2(s) = fst(p), bit1(s) = fst(p), 
--> ~(pac(next(s)) = snd(p))
open ISTEP
-- arbitrary values
  op p : -> BPPair .
-- assumptions
  -- eq c-rec2(s) = true .
  eq cell1(s) = c(p) .
  --
  eq bit2(s) = fst(p) .
  eq bit1(s) = fst(p) .
  eq (pac(next(s)) = snd(p)) = false .
-- successor state
  eq s' = rec2(s) .
-- check
  red inv3(s) implies istep1 .
close
--> c-rec2(s), bit2(s) = fst(p), ~(bit1(s) = fst(p))
open ISTEP
-- arbitrary values
  op p : -> BPPair .
-- assumptions
  -- eq c-rec2(s) = true .
  eq cell1(s) = c(p) .
  --
  eq bit2(s) = fst(p) .
  eq (bit1(s) = fst(p)) = false .
-- successor state
  eq s' = rec2(s) .
-- check
  red inv4(s) implies istep1 .
close
--> c-rec2(s), ~(bit2(s) = fst(p))
open ISTEP
-- arbitrary values
  op p : -> BPPair .
-- assumptions
  -- eq c-rec2(s) = true .
  eq cell1(s) = c(p) .
  --
  eq (bit2(s) = fst(p)) = false .
-- successor state
  eq s' = rec2(s) .
-- check
  red istep1 .
close
--> ~c-rec2(s)
open ISTEP
-- arbitrary values
-- assumptions
  eq c-rec2(s) = false .
-- successor state
  eq s' = rec2(s) .
-- check
  red istep1 .
close

--> 5) drop1(s)
--> c-drop1(s)
open ISTEP
-- arbitrary values
  op p : -> BPPair .
-- assumptions
  -- eq c-drop1(s) = true .
  eq cell1(s) = c(p) .
-- successor state
  eq s' = drop1(s) .
-- check
  red istep1 .
close
--> ~c-drop1(s)
open ISTEP
-- arbitrary values
-- assumptions
  eq c-drop1(s) = false .
-- successor state
  eq s' = drop1(s) .
-- check
  red istep1 .
close

--> 6) drop2(s)
--> c-drop2(s)
open ISTEP
-- arbitrary values
  op b : -> Bool .
-- assumptions
  -- eq c-drop2(s) = true .
  eq cell2(s) = c(b) .
-- successor state
  eq s' = drop2(s) .
-- check
  red istep1 .
close
--> ~c-drop2(s)
open ISTEP
-- arbitrary values
-- assumptions
  eq c-drop2(s) = false .
-- successor state
  eq s' = drop2(s) .
-- check
  red istep1 .
close

--> Q.E.D.