Logical Analysis and Verification of Cryptographic Protocols - Loria

6.5. A DECIDABILITY RESULT 173
Thus:
• If C3 is a ground instance of a clause in S (i.e. C s 3 ∈ S). Then we have
µ(C3) = µ(C1, C2) \ {A} and thus:
µ(C3) ⊆ µ(π) \ {A}
⊆ µ(π) ↓S \{A}
The second inclusion is correct since A is maximal in δS(π, C), and thus in
µ(π).
• Otherwise, by definition of the saturation algorithm the selected resolution
inference applied on C s 1 and C s 2 is redundant and, by definition of re-
dundancy, all its ground instances are redundant and hence the inference
Γ→A,∆ A,Γ ′ →∆ ′
Γ,Γ ′ →∆,∆ ′ is redundant. This implies that the clause C3 = Γ, Γ ′ →
∆, ∆ ′ can be deduced from a finite set S ′ of ground clauses instances of
clauses in S such that
µ(S ′ ) ⊆ µ(C1, C2) ↓S \{A}
⊆ µ(π) ↓S \{A}
Again, the second inclusion holds because A is maximal in δS(π, C), and
thus in µ(π).
By iterating the construction on π we find another proof π ′ such that
µ(π ′ ) ⊆ µ(π) ↓S \{A}
µ(π ′ ) ↓S⊆ (µ(π) ↓S \{A}) ↓S
Since A is a maximal in π for ≺a this implies:
µ(π ′ ) ↓S⊆ ((µ(π)\{A}) ↓S ∪({A} ↓S \{A})) ↓S = (µ(π) ↓S ∪{A} ↓S) ↓S \{A} =
Since A ∈ δS(π, C) we have A /∈ µ(C) ↓S, and thus we have:
δS(π ′ , C) = µ(π ′ ) ↓S \µ(C) ↓S
⊆ ((µ(π) ↓S ∪{A} ↓S) \ {A}) \ µ(C) ↓S
⊆ ((µ(π) ↓S ∪{A} ↓S) \ µ(C) ↓S) \ {A}
< {A}
< δS(π, C)
Thus π is not such that δS(π, C) is minimal. �