Program Semantics and Classical Logic

Corollary 14 (Scott’s Induction Rule) Assume X₁, . . . , X_n are not free
in C. Write C(Q) for [Q/X]C. Then

C ([fail/Xi,..., fail∕Xn]P )⁺
∀X1₁₁.Xn(C(P)^→C∖[Pι∕Xι,₁₁.,Pn∕Xn]Pr AX
C(hXi ^ Pi)i(P))+

8 Hoare’s Calculus

As a simple application we show that our semantics subsumes the Hoare
Calculus, which we shall present in Gentzen form. Sequents Γ 'h C will
consist of a set of asserted programs Γ and an asserted program C. The
derivability relation 'PA between assertions F is given by an axiomatization
of first-order logic plus the first-order Peano axioms. Structural rules of our
Hoare-Gentzen calculus are the following.

The following logical rules embody the Hoare Calculus.

'H {F}B?{F ∧ B}               (Test)

Γi 'h {F1}P1{F2} Γ2 'h {F2}P2{F3}

Γι, Γ2 'h {Fi}Pi; P2{F3}

Γι 'h {F1}P1{F2} Γ2 'h {F1}P2{F2}
Γι, Γ2 'h {F1}P1 ∪ P2{F2}

Γ 'H {F1}[fail∕X1,..., fail∕Xn]P {F2}

Γ, {Fι}P{F2} 'h {F1}[P1∕X1,..., Pn/Xn]P{F2}

Γ 'h {F}lXi ^ Pi)n,_l(P){F2}

Fi 'PA F2 Γ 'h {F2}P{F3} F3 'PA F4
Γ 'h {Fi}P{F4}

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