theory TransClosure
imports Main
begin
types addr = nat
datatype val
= Unit -- "dummy result value of void expressions"
| Null -- "null reference"
| Bool bool -- "Boolean value"
| Intg int -- "integer value"
| Addr addr -- "addresses of objects in the heap"
consts R::"(addr × addr) set"
consts f:: "addr => val"
ML {*AtpWrapper.problem_name := "TransClosure__test"*}
lemma "[| f c = Intg x; ∀ y. f b = Intg y --> y ≠ x; (a,b) ∈ R*; (b,c) ∈ R* |]
==> ∃ c. (b,c) ∈ R ∧ (a,c) ∈ R*"
by (metis Transitive_Closure.rtrancl_into_rtrancl converse_rtranclE trancl_reflcl)
lemma "[| f c = Intg x; ∀ y. f b = Intg y --> y ≠ x; (a,b) ∈ R*; (b,c) ∈ R* |]
==> ∃ c. (b,c) ∈ R ∧ (a,c) ∈ R*"
proof (neg_clausify)
assume 0: "f c = Intg x"
assume 1: "(a, b) ∈ R*"
assume 2: "(b, c) ∈ R*"
assume 3: "f b ≠ Intg x"
assume 4: "!!A. (b, A) ∉ R ∨ (a, A) ∉ R*"
have 5: "b = c ∨ b ∈ Domain R"
by (metis Not_Domain_rtrancl 2)
have 6: "!!X1. (a, X1) ∈ R* ∨ (b, X1) ∉ R"
by (metis Transitive_Closure.rtrancl_into_rtrancl 1)
have 7: "!!X1. (b, X1) ∉ R"
by (metis 6 4)
have 8: "b ∉ Domain R"
by (metis 7 DomainE)
have 9: "c = b"
by (metis 5 8)
have 10: "f b = Intg x"
by (metis 0 9)
show "False"
by (metis 10 3)
qed
ML {*AtpWrapper.problem_name := "TransClosure__test_simpler"*}
lemma "[| f c = Intg x; ∀ y. f b = Intg y --> y ≠ x; (a,b) ∈ R*; (b,c) ∈ R* |]
==> ∃ c. (b,c) ∈ R ∧ (a,c) ∈ R*"
apply (erule_tac x="b" in converse_rtranclE)
apply (metis rel_pow_0_E rel_pow_0_I)
apply (metis DomainE Domain_iff Transitive_Closure.rtrancl_into_rtrancl)
done
end