Proof Assistants
for One-and-a-Halfth Order Logic
Work in progress
A. Tasistro
Universidad ORT Uruguay
LSFA 2013
IME - USP
Abstract
I will explain:
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The motivations.
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What One-and-a-Halfth Order Logic is.
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What we have done/are doing.
The context
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A Software Engineering program.
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We insist that they learn proof (of programs).
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In particular, formal proof.
Two early courses
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Recursion and induction using (concise) Haskell.
Logic (standard),
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using Natural Deduction.
With a tool for autonomous practice.
Features
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Tree display.
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Backwards/forward derivation.
Paper and pencil simulation
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several simultaneous trees,
drag and drop.
Meta-theorems
∀x(α ∧ β) ` ∀xα ∧ ∀xβ,
not:
∀x(P(x) ∧ Q(x)) ` ∀xP(x) ∧ ∀xQ(x),
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With re-use.
Technical difficulties with meta-theorems in FOL
∀x(α ∨ β) ` α ∨ ∀xβ (x not free in α)
∀x(α → β), α[x := t] ` β[x := t].
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Meta-vars (formulæ, terms).
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Side conditions (restricting dependence on variables).
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Explicit substitution.
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Equality, as in:
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∀xα
∀e
α[x := x]
eq
α.
Matching for re-use of meta-theorems.
Contribution
Lots of didactic proof assistants...
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Pandora
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ADN
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Panda
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Proof Builder
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JAPE
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DC Proof
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Yoda
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...
but none with the feature.
One-and-a-halfth order logic
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The logic of meta-derivations.
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(We don’t want HOL).
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Gabbay and Mathijssen, 2008.
Based upon Nominal Terms.
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atoms,
abstraction,
swapping,
freshness x#α,
α conversion,
unknowns (meta-variables),
explicit substitution.
Judgements Γ `∆ α
∀i
Γ `∆ α
∆ . x#Γ
Γ `∆ ∀xα
What have we done?
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(Joint with J. Pais)
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Full version for propositional logic. (2011, Pais’ Engineering
final project.)
A prototype for FOL (i.e. One-and-a-halfth —2013, Pais’
Master thesis.)
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Adapting original system to ND.
Implementing matching and equality checking (in presence of
explicit substitution).
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We are re-programming the prototype with interface –due to
next semester for use in teaching.
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An LCF-like version (embedded in Haskell) for more
professional use.
Thanks !!!
[email protected]