## Mutual recursion

Some examples of mutual recursion and corresponding proofs of termination

**Auteurs:** Jean-Christophe Filliâtre

**Catégories:** Tricky termination

**Outils:** Why3

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Some examples of mutual recursion and corresponding proofs of termination

use int.Int

This example is from the book "Program Proofs" by Rustan Leino

let rec f1 (n: int) : int requires { 0 <= n } variant { n, 1 } = if n = 0 then 0 else f2 n + 1 with f2 (n: int) : int requires { 1 <= n } variant { n, 0 } = 2 * f1 (n - 1)

Hofstadter's Female and Male sequences

let rec function f (n: int) : int requires { 0 <= n } variant { n, 1 } ensures { if n = 0 then result = 1 else 1 <= result <= n } = if n = 0 then 1 else n - m (f (n - 1)) with function m (n: int) : int requires { 0 <= n } variant { n, 0 } ensures { if n = 0 then result = 0 else 0 <= result < n } = if n = 0 then 0 else n - f (m (n - 1))

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# Why3 Proof Results for Project "mutual_recursion"

## Theory "mutual_recursion.Top": fully verified

Obligations | Alt-Ergo 2.3.0 | |||

VC for f1 | 0.00 | |||

VC for f2 | 0.01 | |||

VC for f | --- | |||

split_vc | ||||

variant decrease | 0.00 | |||

precondition | 0.00 | |||

variant decrease | 0.00 | |||

precondition | 0.00 | |||

postcondition | --- | |||

split_vc | ||||

postcondition | 0.00 | |||

postcondition | --- | |||

split_vc | ||||

postcondition | 0.00 | |||

postcondition | 0.00 | |||

postcondition | 0.00 | |||

VC for m | --- | |||

split_vc | ||||

variant decrease | 0.00 | |||

precondition | 0.00 | |||

variant decrease | 0.00 | |||

precondition | 0.00 | |||

postcondition | --- | |||

split_vc | ||||

postcondition | --- | |||

split_vc | ||||

postcondition | 0.00 | |||

postcondition | 0.00 | |||

postcondition | 0.00 | |||

postcondition | 0.00 |