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**1 - 6**of**6**### Church–Rosser Made Easy

"... The Church–Rosser theorem states that the λ-calculus is confluent under α- and β-reductions. The standard proof of this result is due to Tait and Martin-Löf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give confluence ..."

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The Church–Rosser theorem states that the λ-calculus is confluent under α- and β-reductions. The standard proof of this result is due to Tait and Martin-Löf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give confluence of the βη-calculus. 1

### 1.1 Booleans

, 2010

"... Even though the pure λ-calculus consists only of λ-terms, we can represent and manipulate common data ..."

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Even though the pure λ-calculus consists only of λ-terms, we can represent and manipulate common data

### DOI 10.3233/FI-2010-306 IOS Press Church–Rosser Made Easy

"... Abstract. The Church–Rosser theorem states that the λ-calculus is confluent under β-reductions. The standard proof of this result is due to Tait and Martin-Löf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give conflue ..."

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Abstract. The Church–Rosser theorem states that the λ-calculus is confluent under β-reductions. The standard proof of this result is due to Tait and Martin-Löf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give confluence of the βη-calculus. Keywords: lambda-calculus, confluence, Church–Rosser theorem

### unknown title

"... In this supplementary lecture we prove that the λ-calculus is confluent. This is result is due to Alonzo Church (1903– 1995) and J. Barkley Rosser (1907–1989) and is known as the Church–Rosser theorem. The proof given here takes an alternative approach to the standard proof due to Tait and Martin-Lö ..."

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In this supplementary lecture we prove that the λ-calculus is confluent. This is result is due to Alonzo Church (1903– 1995) and J. Barkley Rosser (1907–1989) and is known as the Church–Rosser theorem. The proof given here takes an alternative approach to the standard proof due to Tait and Martin-Löf as given for example in Barendregt [1] (see also [2, 3]). 1 Terms as Labeled Trees In the proof of confluence, we will need to talk about occurrences of subterms in a term as designated by a path from the root. Thus we need to develop notation for terms viewed as labeled trees. We will refer to this view of terms as the coalgebraic view, although the rationale for this terminology will only become clear much later in the course. 1.1 Algebraic View of Terms A ranked alphabet is a set Σ together with an arity function arity: Σ → N. The letters f ∈ Σ are viewed as operator symbols, and arity f ∈ N denotes the arity of f, or the number of inputs of f. The symbol f is called unary, binary, ternary, or n-ary according as its arity is 1, 2, 3, or n, respectively. It is called a constant if its arity is 0. Finite terms over Σ can be defined by induction: • If t0,..., tn−1 are terms and f ∈ Σ with arity f = n, then f t0..., tn−1 is a term. Note the base case n = 0 is included; the first premise is vacuous in that case. This defines terms in prefix notation,

### Author manuscript, published in "Second international conference on Certified Programs and Proofs (2012)" Proof Pearl: Abella Formalization of λ-Calculus Cube Property

, 2013

"... Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λ-calculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We re-interp ..."

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Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λ-calculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We re-interpret his work in Abella, a recent proof assistant based on higher-order abstract syntax and provided with a nominal quantifier. By revisiting Huet’s approach and exploiting the features of Abella, we get a strikingly compact and natural development, which makes Huet’s idea really shine. 1

### unknown title

"... In this supplementary lecture we prove that the λ-calculus is confluent. This is result is due to Alonzo Church (1903–1995) and J. Barkley Rosser (1907–1989) and is known as the Church–Rosser theorem. The proof given here takes an alternative approach to the standard proof due to Tait and Martin-Löf ..."

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In this supplementary lecture we prove that the λ-calculus is confluent. This is result is due to Alonzo Church (1903–1995) and J. Barkley Rosser (1907–1989) and is known as the Church–Rosser theorem. The proof given here takes an alternative approach to the standard proof due to Tait and Martin-Löf as given for example in Barendregt [Bar84] (see also [Pol95, Tak95]). 1 Terms as Labeled Trees In the proof of confluence, we will need to talk about occurrences of subterms in a term as designated by a path from the root. Thus we need to develop notation for terms viewed as labeled trees. We will refer to this view of terms as the coalgebraic view, although the rationale for this terminology will only become clear much later in the course. 1.1 Algebraic View of Terms A ranked alphabet is a set Σ together with an arity function arity: Σ → N. The letters f ∈ Σ are viewed as operator symbols, and arity f ∈ N denotes the arity of f, or the number of inputs of f. The symbol f is called unary, binary, ternary, or n-ary according as its arity is 1, 2, 3, or n, respectively. It is called a constant if its arity is 0. Finite terms over Σ can be defined by induction: • If t0,..., tn−1 are terms and f ∈ Σ with arity f = n, then f t0..., tn−1 is a term. Note the base case n = 0 is included; the first premise is vacuous in that case. This defines terms in prefix notation, but really we are interested in the abstract syntax. 1.2 Coalgebraic View of Terms Alternatively, we can define terms as labeled trees. Let N ∗ denote the set of finite-length strings of natural numbers. A subset t ⊆ N ∗ is a tree if it is nonempty and prefix-closed (that is, if αβ ∈ t, then α ∈ t). Any tree must contain the empty string, which is the root of the tree. A term is then a partial function e: N ∗ ⇀ Σ such that • dom e is a tree; • if α ∈ dom e, then α i ∈ dom e iff i < arity e(α). The second condition says that a node α of the tree has exactly n children if the arity of its label is n. A term is finite if its domain is a finite set. Thus one advantage of this definition is that it admits infinite terms. 1 1 Infinite terms exist in OCaml. Type the following at the interpreter and see what happens: type x = C of x;; let rec x = C x;;