A Coinductive Confluence Proof for Infinitary Lambda-Calculus
A Coinductive Confluence Proof for Infinitary Lambda-Calculus Łukasz Czajka Institute of Informatics Faculty of Mathematics, Informatics and Mechanics University of Warsaw
14 July 2014
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A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
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A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
1. Coinduction.
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A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
1. Coinduction. 2. Infinitary lambda-calculus.
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Coinduction Coinductive definitions
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Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T)
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Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T) The set T consists of all finite and infinite terms built up from variables and the constructors A and B.
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Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T) The set T consists of all finite and infinite terms built up from variables and the constructors A and B. The set of all possibly infinite labelled trees with labels specified by the grammar.
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T.
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T. substtx (x) substtx (A(s)) substtx (y ) t substx (B(s1 , s2 ))
= = = =
t A(substtx (s)) y if y 6= x B(substtx (s1 ), substtx (s2 ))
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T. substtx (x) substtx (A(s)) substtx (y ) t substx (B(s1 , s2 ))
= = = =
t A(substtx (s)) y if y 6= x B(substtx (s1 ), substtx (s2 ))
Each (co)recursive call of substtx occurs directly inside a constructor for T.
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
The relation → is the greatest fixpoint νF of a function F : P(T × T) → P(T × T) defined as follows. F (R) = {ht1 , t2 i | (t1 ≡ t2 ≡ x)∨ ∃t, t 0 (t1 ≡ A(t) ∧ t2 ≡ A(t 0 ) ∧ R(t, t 0 )) ∨ ∃s, t, s 0 , t 0 (t1 ≡ B(s, t) ∧ t2 ≡ B(s 0 , t 0 )∧ R(s, s 0 ) ∧ R(t, t 0 ))∨ 0 ∃t, t (t1 ≡ A(t) ∧ t2 ≡ B(t 0 , t 0 ) ∧ R(t, t 0 ))}
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
The relation → is the greatest fixpoint νF of a function F : P(T × T) → P(T × T) defined as follows. F (R) = {ht1 , t2 i | (t1 ≡ t2 ≡ x)∨ ∃t, t 0 (t1 ≡ A(t) ∧ t2 ≡ A(t 0 ) ∧ R(t, t 0 )) ∨ ∃s, t, s 0 , t 0 (t1 ≡ B(s, t) ∧ t2 ≡ B(s 0 , t 0 )∧ R(s, s 0 ) ∧ R(t, t 0 ))∨ 0 ∃t, t (t1 ≡ A(t) ∧ t2 ≡ B(t 0 , t 0 ) ∧ R(t, t 0 ))} F is monotone, i.e., F (R) ⊆ F (S) for R ⊆ S. 5 / 30
Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t.
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2). If t ≡ B(t1 , t2 ) then t1 → t1 and t2 → t2 by the coinductive hypothesis, so t → t by rule (3).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2). If t ≡ B(t1 , t2 ) then t1 → t1 and t2 → t2 by the coinductive hypothesis, so t → t by rule (3). 2
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A.
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A. By the Knaster-Tarski fixpoint theorem: \ µF = {X ∈ P(A) | F (X ) ⊆ X } νF =
[ {X ∈ P(A) | X ⊆ F (X )}.
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A. By the Knaster-Tarski fixpoint theorem: \ µF = {X ∈ P(A) | F (X ) ⊆ X } νF =
[ {X ∈ P(A) | X ⊆ F (X )}.
This yields the following proof principles X ⊆ F (X ) F (X ) ⊆ X (IND) (COIND) µF ⊆ X X ⊆ νF where X ∈ P(A).
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Coinduction Alternative characterisation of νF
Monotone F : P(A) → P(A) for some set A.
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Coinduction Alternative characterisation of νF
Monotone F : P(A) → P(A) for some set A. I
ν 0 F = A,
I
ν α+1 F = F (ν α F ), T ν α F = β
14 July 2014
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A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
2 / 30
A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
1. Coinduction.
2 / 30
A Coinductive Confluence Proof for Infinitary Lamba-calculus Presentation plan
1. Coinduction. 2. Infinitary lambda-calculus.
2 / 30
Coinduction Coinductive definitions
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Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T)
3 / 30
Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T) The set T consists of all finite and infinite terms built up from variables and the constructors A and B.
3 / 30
Coinduction Coinductive definitions
T ::= V k A(T) k B(T, T) The set T consists of all finite and infinite terms built up from variables and the constructors A and B. The set of all possibly infinite labelled trees with labels specified by the grammar.
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T.
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T. substtx (x) substtx (A(s)) substtx (y ) t substx (B(s1 , s2 ))
= = = =
t A(substtx (s)) y if y 6= x B(substtx (s1 ), substtx (s2 ))
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Coinduction Guarded corecursion
For t ∈ T, x ∈ V , substtx : T → T. substtx (x) substtx (A(s)) substtx (y ) t substx (B(s1 , s2 ))
= = = =
t A(substtx (s)) y if y 6= x B(substtx (s1 ), substtx (s2 ))
Each (co)recursive call of substtx occurs directly inside a constructor for T.
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
The relation → is the greatest fixpoint νF of a function F : P(T × T) → P(T × T) defined as follows. F (R) = {ht1 , t2 i | (t1 ≡ t2 ≡ x)∨ ∃t, t 0 (t1 ≡ A(t) ∧ t2 ≡ A(t 0 ) ∧ R(t, t 0 )) ∨ ∃s, t, s 0 , t 0 (t1 ≡ B(s, t) ∧ t2 ≡ B(s 0 , t 0 )∧ R(s, s 0 ) ∧ R(t, t 0 ))∨ 0 ∃t, t (t1 ≡ A(t) ∧ t2 ≡ B(t 0 , t 0 ) ∧ R(t, t 0 ))}
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Coinduction Coinductive definitions of relations
x → x (1)
t → t0 (2) A(t) → A(t 0 )
s → s0 t → t0 (3) B(s, t) → B(s 0 , t 0 )
t → t0 (4) A(t) → B(t 0 , t 0 )
The relation → is the greatest fixpoint νF of a function F : P(T × T) → P(T × T) defined as follows. F (R) = {ht1 , t2 i | (t1 ≡ t2 ≡ x)∨ ∃t, t 0 (t1 ≡ A(t) ∧ t2 ≡ A(t 0 ) ∧ R(t, t 0 )) ∨ ∃s, t, s 0 , t 0 (t1 ≡ B(s, t) ∧ t2 ≡ B(s 0 , t 0 )∧ R(s, s 0 ) ∧ R(t, t 0 ))∨ 0 ∃t, t (t1 ≡ A(t) ∧ t2 ≡ B(t 0 , t 0 ) ∧ R(t, t 0 ))} F is monotone, i.e., F (R) ⊆ F (S) for R ⊆ S. 5 / 30
Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t.
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2). If t ≡ B(t1 , t2 ) then t1 → t1 and t2 → t2 by the coinductive hypothesis, so t → t by rule (3).
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Coinduction Sample coinductive proof
We show: for all t ∈ T, t → t. If t ≡ x then this follows by rule (1). If t ≡ A(t 0 ) then t 0 → t 0 by the coinductive hypothesis, so t ≡ A(t 0 ) → A(t 0 ) ≡ t by rule (2). If t ≡ B(t1 , t2 ) then t1 → t1 and t2 → t2 by the coinductive hypothesis, so t → t by rule (3). 2
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A.
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A. By the Knaster-Tarski fixpoint theorem: \ µF = {X ∈ P(A) | F (X ) ⊆ X } νF =
[ {X ∈ P(A) | X ⊆ F (X )}.
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Coinduction Usual coinduction principle
Monotone F : P(A) → P(A) for some set A. By the Knaster-Tarski fixpoint theorem: \ µF = {X ∈ P(A) | F (X ) ⊆ X } νF =
[ {X ∈ P(A) | X ⊆ F (X )}.
This yields the following proof principles X ⊆ F (X ) F (X ) ⊆ X (IND) (COIND) µF ⊆ X X ⊆ νF where X ∈ P(A).
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Coinduction Alternative characterisation of νF
Monotone F : P(A) → P(A) for some set A.
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Coinduction Alternative characterisation of νF
Monotone F : P(A) → P(A) for some set A. I
ν 0 F = A,
I
ν α+1 F = F (ν α F ), T ν α F = β
