The Coq Proof Assistant

# Library Coq.ZArith.Zwf

Require Import ZArith_base.
Require Export Wf_nat.
Require Import Lia.
Local Open Scope Z_scope.
Well-founded relations on Z.
We define the following family of relations on Z x Z:
x (Zwf c) y iff x < y & c <= y
Definition Zwf (c x y:Z) := c <= y /\ x < y.
and we prove that (Zwf c) is well founded
Section wf_proof.
Variable c : Z.
The proof of well-foundness is classic: we do the proof by induction on a measure in nat, which is here |x-c|
Let f (z:Z) := Z.abs_nat (z - c).
Lemma Zwf_well_founded : well_founded (Zwf c).
n= 0
inductive case
End wf_proof.
#[global]
Hint Resolve Zwf_well_founded: datatypes.
We also define the other family of relations:
x (Zwf_up c) y iff y < x <= c
Definition Zwf_up (c x y:Z) := y < x <= c.
and we prove that (Zwf_up c) is well founded
Section wf_proof_up.
Variable c : Z.
The proof of well-foundness is classic: we do the proof by induction on a measure in nat, which is here |c-x|
Let f (z:Z) := Z.abs_nat (c - z).
Lemma Zwf_up_well_founded : well_founded (Zwf_up c).
End wf_proof_up.
#[global]
Hint Resolve Zwf_up_well_founded: datatypes.