117.33/117.58 YES 117.33/117.58 117.33/117.58 Problem 1: 117.33/117.58 117.33/117.58 (VAR v_NonEmpty:S x:S y:S) 117.33/117.58 (RULES 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 ) 117.33/117.58 117.33/117.58 Problem 1: 117.33/117.58 Valid CTRS Processor: 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 -> The system is a deterministic 3-CTRS. 117.33/117.58 117.33/117.58 Problem 1: 117.33/117.58 117.33/117.58 Dependency Pairs Processor: 117.33/117.58 117.33/117.58 Conditional Termination Problem 1: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 117.33/117.58 Conditional Termination Problem 2: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> IMPLIES(implies(x:S,implies(x:S,0)),0) 117.33/117.58 F(x:S) -> IMPLIES(x:S,implies(x:S,0)) 117.33/117.58 F(x:S) -> IMPLIES(x:S,0) 117.33/117.58 IMPLIES(x:S,y:S) -> NOT(x:S) 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 117.33/117.58 117.33/117.58 The problem is decomposed in 2 subproblems. 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 SCC Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 ->Strongly Connected Components: 117.33/117.58 ->->Cycle: 117.33/117.58 ->->-> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 ->->-> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 Simplifying Unifiable Condition Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 ->Transformed Pairs/Rules: 117.33/117.58 ->->Original Pair/Rule: 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 ->-> Transformed Pair/Rule: 117.33/117.58 implies(1,0) -> 0 117.33/117.58 ->->Original Pair/Rule: 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 ->-> Transformed Pair/Rule: 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 SCC Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 ->Strongly Connected Components: 117.33/117.58 ->->Cycle: 117.33/117.58 ->->-> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 ->->-> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 Ohlebusch Transformation Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> F(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 -> New Pairs: 117.33/117.58 F(x:S) -> U17(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 U17(1,x:S) -> F(0) 117.33/117.58 -> New Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 SCC Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> U17(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 U17(1,x:S) -> F(0) 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 ->Strongly Connected Components: 117.33/117.58 ->->Cycle: 117.33/117.58 ->->-> Pairs: 117.33/117.58 F(x:S) -> U17(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 U17(1,x:S) -> F(0) 117.33/117.58 ->->-> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 Reduction Pair Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> U17(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 U17(1,x:S) -> F(0) 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 -> Usable rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 ->Interpretation type: 117.33/117.58 Linear 117.33/117.58 ->Coefficients: 117.33/117.58 Natural Numbers 117.33/117.58 ->Dimension: 117.33/117.58 2 117.33/117.58 ->Bound: 117.33/117.58 1 117.33/117.58 ->Interpretation: 117.33/117.58 117.33/117.58 [and](X1,X2) = [1 0;1 1].X1 + [1 0;0 1].X2 + [1;0] 117.33/117.58 [f](X) = [0 1;1 1].X + [1;1] 117.33/117.58 [implies](X1,X2) = [0 1;1 0].X1 + [0 0;0 1].X2 117.33/117.58 [not](X) = [1 0;1 0].X + [1;0] 117.33/117.58 [or](X1,X2) = [1 0;1 1].X1 + [1 1;1 1].X2 117.33/117.58 [0] = [1;0] 117.33/117.58 [1] = [0;1] 117.33/117.58 [F](X) = [0 1;0 1].X + [1;1] 117.33/117.58 [U15](X1,X2) = [0 0;0 1].X1 + [0 0;1 0].X2 + [1;1] 117.33/117.58 [U16](X1,X2,X3) = [0 0;0 1].X1 + [0 1;0 0].X2 117.33/117.58 [U17](X1,X2) = [0 1;0 0].X1 + [0 0;0 1].X2 + [0;1] 117.33/117.58 117.33/117.58 Problem 1.1: 117.33/117.58 117.33/117.58 SCC Processor: 117.33/117.58 -> Pairs: 117.33/117.58 U17(1,x:S) -> F(0) 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> U15(implies(implies(x:S,implies(x:S,0)),0),x:S) 117.33/117.58 implies(1,0) -> 0 117.33/117.58 implies(x:S,1) -> 1 117.33/117.58 implies(x:S,y:S) -> U16(not(x:S),x:S,y:S) 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 U15(1,x:S) -> f(0) 117.33/117.58 U16(1,x:S,y:S) -> 1 117.33/117.58 ->Strongly Connected Components: 117.33/117.58 There is no strongly connected component 117.33/117.58 117.33/117.58 The problem is finite. 117.33/117.58 117.33/117.58 Problem 1.2: 117.33/117.58 117.33/117.58 SCC Processor: 117.33/117.58 -> Pairs: 117.33/117.58 F(x:S) -> IMPLIES(implies(x:S,implies(x:S,0)),0) 117.33/117.58 F(x:S) -> IMPLIES(x:S,implies(x:S,0)) 117.33/117.58 F(x:S) -> IMPLIES(x:S,0) 117.33/117.58 IMPLIES(x:S,y:S) -> NOT(x:S) 117.33/117.58 -> QPairs: 117.33/117.58 Empty 117.33/117.58 -> Rules: 117.33/117.58 and(not(x:S),x:S) -> 0 117.33/117.58 and(0,x:S) -> 0 117.33/117.58 and(1,x:S) -> x:S 117.33/117.58 and(x:S,not(x:S)) -> 0 117.33/117.58 and(x:S,0) -> 0 117.33/117.58 and(x:S,1) -> x:S 117.33/117.58 f(x:S) -> f(0) | implies(implies(x:S,implies(x:S,0)),0) ->* 1 117.33/117.58 implies(x:S,y:S) -> 0 | x:S ->* 1, y:S ->* 0 117.33/117.58 implies(x:S,y:S) -> 1 | not(x:S) ->* 1 117.33/117.58 implies(x:S,y:S) -> 1 | y:S ->* 1 117.33/117.58 not(0) -> 1 117.33/117.58 not(1) -> 0 117.33/117.58 or(not(x:S),x:S) -> 1 117.33/117.58 or(0,x:S) -> x:S 117.33/117.58 or(1,x:S) -> 1 117.33/117.58 or(x:S,not(x:S)) -> 1 117.33/117.58 or(x:S,0) -> x:S 117.33/117.58 or(x:S,1) -> 1 117.33/117.58 ->Strongly Connected Components: 117.33/117.58 There is no strongly connected component 117.33/117.58 117.33/117.58 The problem is finite. 117.33/117.58 EOF