Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 2. BACKGROUND AND RELATED WORK 31
1 theory MinLogic = Pure :
2 types
3 o
4 arities
5 o :: logic
6 consts
7 Trueprop :: "o ⇒ prop" (" " 5)
8 "−→ ":: "[o,o] ⇒ o" (infixr 10)
9 False :: "o"
10 axioms
11 impI : "(P =⇒ Q) =⇒ P −→ Q"
12 impE : " P −→ Q; P =⇒ Q"
13 falseE : "False =⇒ P"
14 end
HOL system.
Figure 2.5: A minimal Isabelle logic
2.5.3 Unification, Resolution and Proof
Unification [70] is equation solving, for example the solution of solving f(?x, c) ≡ f(d, ?y) is
?x = d and ?y = c. Isabelle uses higher-order unification, which operates on typed λ-terms
as an equation solving mechanism to support the application of rules to goals. Higher-order
unification also handles function unknowns so must guess the unknown function ?f in order
to solve the equation ?f(t) = g(u1, . . . , uk). Isabelle denotes unknowns for unification (called
schematic variables) by prefixing their name with a question mark. Logically schematic
variables are similar to free variables however while ordinary variables remain fixed unknowns
may be instantiated by unification.
Resolution is used to combine two Isabelle theorems, renaming variables and instantiating
unknowns as necessary to allow rules to be unified with the current proof state to create the
next state. Resolution can be used for both forward and backward proof, although backward
proof tends to be the preferred proof mechanism in Isabelle. A meta-level theorem such as
A ; B =⇒ C can be regarded as an inference rule with premises A and B and conclusion
C or can equally be viewed as a proof state with subgoals A and B and main goal C.
In backward proof a goal is unified with the conclusion of a rule and the premises are created
as new subgoals. For example, consider the trivial proof of the theorem A −→ A in the logic