Proof checking with PVS Book: Chapter 3

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Proof checking with PVSBook Chapter 3
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A Theory

• Name THEORY
• BEGIN
• Definitions (types, variables, constants)
• Axioms
• Lemmas (conjectures, theorems)
• END Name

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Group theory

• (, e), where is the operator and e the unity
  element.
• Associativity (G1) (xy)zx(yz).
• Unity (G2)
• (xe)x

• Right complement (G3)
• ?x ?y xye.
• Want to prove
• ?x ?y yxe.

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Informal proof

• Choose x arbitrarily.
• By G3, there exists y s.t.
• (1) xye.
• By G3, we have z s.t.
• (2) yze.
• yx(yx)e (by G2)
• (yx)(yz) (by (2))

• y(x(yz)) (by G1)
• y((xy)z) (by G1)
• y(ez) (by (1))
• (ye)z (by G1)
• yz (by (G2))
• e (by (2))

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Example groups

• Group THEORY
• BEGIN
• element TYPE
• unit element
• element, element-gt
• element
• lt some axiomsgt

• leftCONJECTURE
• FORALL (x element)
• EXISTS (y element)
• yxunit
• END Group

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Axioms

• associativity AXIOM FORALL (x, y, zelement)
  (xy)zx(yz)
• unity AXIOM FORALL (xelement)
• xunitx
• complement AXIOM FORALL(xelement)
• EXISTS (yelement) xyunity

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Skolemization

• Corresponds to choosing some arbitrary constant
  and proving without loss of generality.
• Want to prove (/\)-gt(\/?x?(x)\/).
• Choose a new constant x. Prove
  (/\)--gt(\/?(x)\/).

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Skolemization

• Corresponds to choosing some unconstrained
  arbitrary constant when one is known to exist.
• Want to prove (/\?x?(x)/\)--gt(\/).
• Choose a new constant x. Prove
  (/\?(x)/\)--gt(\/).

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Skolem in PVS

• (skolem 2 (a1 b2 c7))
• (skolem -3 (a1 _ c7))
• (skolem! -3)invents new constants, e.g., for x
  will invent x!1, x!2, when applied repeatedly.

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Instantiation

• Corresponds to restricting the generality.
• Want to prove (/\?x?(x)/\)--gt(\/).
• Choose a some term t. Prove (/\?(t)/\)--gt(\/)
  .

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Instantiation

• Corresponds to proving the existence of an
  element by showing an evidence.
• Want to prove (/\)--gt(\/??x?(x)\/).
• Choose some term t. Prove (/\)--gt(\/?(t)\/).

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Instantiating in PVS

• (inst -1 xy a bc)
• (inst 2 a _ x)

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Other useful rules

• (replace -1 (-1 2 3))Formula -1 is of the form
  leri. Replace any occurrence of le by ri in
  lines -1, 2, 3.
• (replace -1 (-1 2 3) RL)Similar, but replace ri
  by le instead.
• (assert), (assert -) (assert ) (assert 7) Apply
  algebraic simplification.
• (lemma ltaxiom-namegt) - add axiom as additional
  antecedent.