1. Proof checking with PVS Book: Chapter 31

2. A Theory Name: THEORY BEGIN Definitions (types, variables, constants)
Axioms
Lemmas (conjectures, theorems)
END Name 2

3. Group theory (*, e), where * is the operator and e the unity element.
Associativity (G1): (x*y)*z=x*(y*z).
Unity (G2):
(x*e)=x
Right complement (G3):
x y x*y=e.
Want to prove:
x y y*x=e. 3

4. Informal proof =y*(x*(y*z)) (by G1) Choose x arbitrarily.
By G3, there exists y s.t.
(1) x*y=e.
By G3, we have z s.t.
(2) y*z=e.
y*x=(y*x)*e (by G2)
=(y*x)*(y*z) (by (2))
=y*(x*(y*z)) (by G1)
=y*((x*y)*z) (by G1)
=y*(e*z) (by (1))
=(y*e)*z (by G1)
=y*z (by (G2))
=e (by (2)) 4

5. Example: groups Group: THEORY BEGIN element: TYPE unit: element
*: [element, element->
element]
< some axioms>
left:CONJECTURE
FORALL (x: element):
EXISTS (y: element):
y*x=unit
END Group 5

6. Axioms associativity: AXIOM FORALL (x, y, z:element): (x*y)*z=x*(y*z)
unity: AXIOM FORALL (x:element):
x*unit=x
complement: AXIOM FORALL(x:element):
EXISTS (y:element): x*y=unity 6

7. Skolemization
Corresponds to choosing some arbitrary constant and proving “without loss of generality”.
Want to prove (…/\…)->(…\/x(x)\/…).
Choose a new constant x’. Prove (…/\…)-->(…\/(x’)\/…). 7

8. Skolemization
Corresponds to choosing some unconstrained arbitrary constant when one is known to exist.
Want to prove (…/\x(x)/\…)-->(…\/…).
Choose a new constant x’. Prove (…/\(x’)/\…)-->(…\/…). 8

9. 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. 9

10. Instantiation Corresponds to restricting the generality.
Want to prove (…/\x(x)/\…)-->(…\/…).
Choose a some term t. Prove (…/\(t)/\…)-->(…\/…). 10

11. Instantiation
Corresponds to proving the existence of an element by showing an evidence.
Want to prove (…/\…)-->(…\/x(x)\/…).
Choose some term t. Prove (…/\…)-->(…\/(t)\/…). 11

12. Instantiating in PVS
(inst -1 “x*y” “a” “b+c”)
(inst 2 “a” “_” “x”) 12

13. Other useful rules
(replace -1 (-1 2 3)) Formula -1 is of the form le=ri. 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 “<axiom-name>”) - add axiom as additional antecedent. 13