New generalised induction idiom (#434)

Replaced the legacy generalised induction idiom with a more robust one
that does not rely on the `{-2}` numerical occurrence selector, using
either new helper lemmas `ubnP` and `ltnSE` or a specific `nat`
induction principle `ltn_ind`.

Added (non-strict in)equality induction helper lemmas

  Added `ubnP[lg]?eq` helper lemmas that abstract an integer expression
along with some (in)equality, in preparation for some generalised
induction. Note that while `ubnPleq` is very similar to `ubnP` (indeed
`ubnP M` is basically `ubnPleq M.+1`), `ubnPgeq` is used to remember
that the inductive value remains below the initial one.
  Used the change log to give notice to users to update the generalised
induction idioms in their proofs to one of the new forms before
Mathcomp 1.11.

1 files changed, 2 insertions, 2 deletions

diff --git a/mathcomp/solvable/pgroup.v b/mathcomp/solvable/pgroup.v
index 5c572b7..2ed68f0 100644
--- a/mathcomp/solvable/pgroup.v
+++ b/mathcomp/solvable/pgroup.v
@@ -623,8 +623,8 @@ Proof. by rewrite /psubgroup sub1G pgroup1. Qed.
 
 Lemma Cauchy p G : prime p -> p %| #|G| -> {x | x \in G & #[x] = p}.
 Proof.
-move=> p_pr; elim: {G}_.+1 {-2}G (ltnSn #|G|) => // n IHn G.
-rewrite ltnS => leGn pG; pose xpG := [pred x in G | #[x] == p].
+move=> p_pr; have [n] := ubnP #|G|; elim: n G => // n IHn G /ltnSE-leGn pG.
+pose xpG := [pred x in G | #[x] == p].
 have [x /andP[Gx /eqP] | no_x] := pickP xpG; first by exists x.
 have{pG n leGn IHn} pZ: p %| #|'C_G(G)|.
   suffices /dvdn_addl <-:  p %| #|G :\: 'C(G)| by rewrite cardsID.