Proof-Theoretic Conservative Extension of HOL with Ad-hoc Overloading

Gengelbach, Arve; Weber, Tjark

doi:10.1007/978-3-030-64276-1_2

Cited by 2 publications

(4 citation statements)

References 13 publications

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“…We conjecture that the syntactic counterpart of our result holds: if D ′ is an extension of D such that D ′ ⊢ ϕ, where ϕ is a formula whose non-built-in constant instances and types are independent of symbols defined in D ′ \ D, then D ⊢ ϕ. Gengelbach and Weber recently proved this conjecture for constant definition through equality axioms [5]. We leave its study for the more general constant specification…”

Section: Resultsmentioning

confidence: 65%

“…In recent work Gengelbach and Weber [5] prove model-theoretic conservativity of definitional theories [4] for semantics that do not require full function spaces in order to derive their syntactic counterparts. A definitional extension of a theory is proof-theoretically conservative, that is, if a formula's types and constants are unchanged by a theory update, and the formula is derivable after the update, then it is also derivable from the theory before the update.…”

Section: Related Workmentioning

confidence: 99%

“…For Isabelle/HOL conservativity has been studied in an absolute manner, i. e. any definitional theory is a conservative extension of initial HOL [11], the theory of Booleans with Hilbert-choice and infinity axiom. Gengelbach and Weber [5] prove conservativity of any definitional extension above initial HOL, by translating a model-theoretic conservativity for a generalised semantics into its syntactic counterpart. As their semantics are similar to ours, this paper adds to the reliability of their result.…”

Section: Introductionmentioning

confidence: 99%

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Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Gengelbach

Pohjola

Weber

2021

Electron. Proc. Theor. Comput. Sci.

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Definitions of new symbols merely abbreviate expressions in logical frameworks, and no new facts (regarding previously defined symbols) should hold because of a new definition. In Isabelle/HOL, definable symbols are types and constants. The latter may be ad-hoc overloaded, i. e. have different definitions for non-overlapping types. We prove that symbols that are independent of a new definition may keep their interpretation in a model extension. This work revises our earlier notion of modeltheoretic conservative extension and generalises an earlier model construction. We obtain consistency of theories of definitions in higher-order logic (HOL) with ad-hoc overloading as a corollary. Our results are mechanised in the HOL4 theorem prover. Const c ty 1 # Const d ty 2 def = c = d ∨ ty 1 # ty 2. A user may introduce (non-built-in) types and constants by theory extension, as described in Section 2.3. For types and constants we generally say symbols. Built-ins We abbreviate Bool for Tyapp «bool» [ ] x → y for Tyapp «fun» [x; y] Equal ty for Const «=» (ty → ty → Bool) s === t for Comb (Comb (Equal (typeof s)) s) t Any type with any of these type constructors at the top level is built-in, as is the constant Equal. These are the only symbols which are not user-defined. A formula is a term of type Bool.

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Section: Resultsmentioning

confidence: 65%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Gengelbach

Pohjola

Weber

2021

Electron. Proc. Theor. Comput. Sci.

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“…We plan to deploy our framework to formalize various aspects of HOL and Isabelle/HOL's metatheory [44,[60][61][62][63], complementing the work already done in the HOL4 prover on these aspects [3].…”

Section: Future Workmentioning

confidence: 99%

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

Gheri¹,

Popescu²

2021

Preprint

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We have previously published the Isabelle/HOL formalization of a general theory of syntax with bindings. In this companion paper, we instantiate the general theory to the syntax of lambda-calculus and formalize the development leading to several fundamental constructions and results: sound semantic interpretation, the Church-Rosser and standardization theorems, and higher-order abstract syntax encoding. For Church-Rosser and standardization, our work covers both the call-by-name and call-by-value versions of the calculus, following classic papers by Takahashi and Plotkin. During the formalization, we were able to stay focused on the high-level ideas of the development-thanks to the arsenal provided by our general theory: a wealth of basic facts about the substitution, swapping and freshness operators, as well as recursive-definition and reasoning principles, including a specialization to semantic interpretation of syntax.

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Proof-Theoretic Conservative Extension of HOL with Ad-hoc Overloading

Cited by 2 publications

References 13 publications

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

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