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add lemma turning 0 < n -> n.+1 #2086

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Sep 12, 2024
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add lemma turning 0 < n -> n.+1 #2086

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25ba748
add lemma turning 0 < n -> n.+1
Alizter Aug 30, 2024
606bb1f
rename and simplify gt_zero_ind
Alizter Sep 12, 2024
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20 changes: 16 additions & 4 deletions theories/Spaces/Nat/Core.v
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Original file line number Diff line number Diff line change
Expand Up @@ -877,6 +877,19 @@ Proof.
rewrite nat_mul_comm; exact _.
Defined.

(** ** Eliminating positive assumptions *)

(** Sometimes we want to prove a predicate which assumes that [0 < x]. In that case, it suffices to prove it for [x.+1] instead. *)
Definition gt_zero_ind (P : nat -> Type)
(H : forall x, P x.+1)
: forall x (l : 0 < x), P x.
Proof.
intros x l.
destruct x.
1: contradiction (lt_irrefl _ l).
apply H.
Defined.

(** Alternative Characterizations of [<=] *)

(** [n <= m] is equivalent to [(n < m) + (n = m)]. This justifies the name "less than or equal to". Note that it is not immediately obvious that the latter type is a hprop. *)
Expand Down Expand Up @@ -1183,14 +1196,13 @@ Defined.
(** Under certain conditions, subtracting a predecessor is the successor of the subtraction. *)
Definition nat_sub_pred_r n m : 0 < m -> m < n -> n - nat_pred m = (n - m).+1.
Proof.
intros H1 H2.
induction m as [|m IHm] in n, H1, H2 |- *.
1: contradiction (not_lt_zero_r _ H1).
revert m; snrapply gt_zero_ind.
intros m H1.
rewrite nat_sub_succ_r.
rewrite nat_succ_pred.
1: reflexivity.
apply equiv_lt_lt_sub.
exact (lt_trans _ H2).
exact (lt_trans _ H1).
Defined.

(** Subtracting from a sum is the sum of subtracting from the second summand. *)
Expand Down
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