Theory TransClosure

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theory TransClosure
imports Main

(*  Title:      HOL/MetisTest/TransClosure.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Testing the metis method
*)

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