Library mathcomp.ssreflect.ssrnat

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype.
Require Import BinNat.
Require BinPos Ndec.
Require Export Ring.

A version of arithmetic on nat (natural numbers) that is better suited to small scale reflection than the Coq Arith library. It contains an extensive equational theory (including, e.g., the AGM inequality), as well as support for the ring tactic, and congruence tactics. The following operations and notations are provided: successor and predecessor n.+1, n.+2, n.+3, n.+4 and n.-1, n.-2 this frees the names "S" and "pred" basic arithmetic m + n, m - n, m * n Important: m - n denotes TRUNCATED subtraction: m - n = 0 if m <= n. The definitions use the nosimpl tag to prevent undesirable computation computation during simplification, but remain compatible with the ones provided in the Coq.Init.Peano prelude. For computation, a module NatTrec rebinds all arithmetic notations to less convenient but also less inefficient tail-recursive functions; the auxiliary functions used by these versions are flagged with %Nrec. Also, there is support for input and output of large nat values. Num 3 082 241 inputs the number 3082241 [Num of n] outputs the value n There are coercions num >-> BinNat.N >-> nat; ssrnat rebinds the scope delimiter for BinNat.N to %num, as it uses the shorter %N for its own notations (Peano notations are flagged with %coq_nat). doubling, halving, and parity n.*2, n./2, odd n, uphalf n, with uphalf n = n.+1./2 bool coerces to nat so we can write, e.g., n = odd n + n./2.*2. iteration iter n f x0 == f ( .. (f x0)) iteri n g x0 == g n.-1 (g ... (g 0 x0)) iterop n op x x0 == op x (... op x x) (n x's) or x0 if n = 0 exponentiation, factorial m ^ n, n`! m ^ 1 is convertible to m, and m ^ 2 to m * m comparison m <= n, m < n, m >= n, m > n, m == n, m <= n <= p, etc., comparisons are BOOLEAN operators, and m == n is the generic eqType operation. Most compatibility lemmas are stated as boolean equalities; this keeps the size of the library down. All the inequalities refer to the same constant "leq"; in particular m < n is identical to m.+1 <= n.

> patterns for contextual rewriting: leqLHS := (X in (X <= _)%N)%pattern leqRHS := (X in (_ <= X)%N)%pattern ltnLHS := (X in (X < _)%N)%pattern ltnRHS := (X in (_ < X)%N)%pattern conditionally strict inequality `leqif' m <= n ?= iff condition == (m <= n) and ((m == n) = condition) This is actually a pair of boolean equalities, so rewriting with an `leqif' lemma can affect several kinds of comparison. The transitivity lemma for leqif aggregates the conditions, allowing for arguments of the form ``m <= n <= p <= m, so equality holds throughout''. maximum and minimum maxn m n, minn m n Note that maxn m n = m + (n - m), due to the truncating subtraction. Absolute difference (linear distance) between nats is defined in the int library (in the int.IntDist sublibrary), with the syntax `|m - n|. The '-' in this notation is the signed integer difference. countable choice ex_minn : forall P : pred nat, (exists n, P n) -> nat This returns the smallest n such that P n holds. ex_maxn : forall (P : pred nat) m, (exists n, P n) -> (forall n, P n -> n <= m) -> nat This returns the largest n such that P n holds (given an explicit upper bound). This file adds the following suffix conventions to those documented in

ssrbool.v and eqtype.v: A (infix) -- conjunction, as in ltn_neqAle : (m < n) = (m != n) && (m <= n). B -- subtraction, as in subBn : (m - n) - p = m - (n + p). D -- addition, as in mulnDl : (m + n) * p = m * p + n * p. M -- multiplication, as in expnMn : (m * n) ^ p = m ^ p * n ^ p. p (prefix) -- positive, as in eqn_pmul2l : m > 0 -> (m * n1 == m * n2) = (n1 == n2). P -- greater than 1, as in ltn_Pmull : 1 < n -> 0 < m -> m < n * m. S -- successor, as in addSn : n.+1 + m = (n + m).+1. V (infix) -- disjunction, as in leq_eqVlt : (m <= n) = (m == n) || (m < n). X - exponentiation, as in lognX : logn p (m ^ n) = logn p m * n in file prime.v (the suffix is not used in this file). Suffixes that abbreviate operations (D, B, M and X) are used to abbreviate second-rank operations in equational lemma names that describe left-hand sides (e.g., mulnDl); they are not used to abbreviate the main operation of relational lemmas (e.g., leq_add2l). For the asymmetrical exponentiation operator expn (m ^ n) a right suffix indicates an operation on the exponent, e.g., expnM : m ^ (n1 * n2) = ...; a trailing "n" is used to indicate the left operand, e.g., expnMn : (m1 * m2) ^ n = ... The operands of other operators are selected using the l/r suffixes.

Set Implicit Arguments.

Declare Scope coq_nat_scope.
Declare Scope nat_rec_scope.

Disable Coq prelude hints to improve proof script robustness.

#[global] Remove Hints plus_n_O plus_n_Sm mult_n_O mult_n_Sm : core.

Declare legacy Arith operators in new scope.

Delimit Scope coq_nat_scope with coq_nat.

Notation "m + n" := (plus m n) : coq_nat_scope.
Notation "m - n" := (minus m n) : coq_nat_scope.
Notation "m * n" := (mult m n) : coq_nat_scope.
Notation "m <= n" := (le m n) : coq_nat_scope.
Notation "m < n" := (lt m n) : coq_nat_scope.
Notation "m >= n" := (ge m n) : coq_nat_scope.
Notation "m > n" := (gt m n) : coq_nat_scope.

Rebind scope delimiters, reserving a scope for the "recursive", i.e., unprotected version of operators.

Delimit Scope N_scope with num.

use # [warning="-hiding-delimiting-key" ] attribute once we require Coq 8.18

Delimit Scope nat_scope with N.
Delimit Scope nat_rec_scope with Nrec.

Postfix notation for the successor and predecessor functions. SSreflect uses "pred" for the generic predicate type, and S as a local bound variable.

Notation succn := Datatypes.S.
Notation predn := Peano.pred.

Notation "n .+1" := (succn n) (at level 2, left associativity,
  format "n .+1") : nat_scope.
Notation "n .+2" := n.+1.+1 (at level 2, left associativity,
  format "n .+2") : nat_scope.
Notation "n .+3" := n.+2 .+1 (at level 2, left associativity,
  format "n .+3") : nat_scope.
Notation "n .+4" := n.+2.+2 (at level 2, left associativity,
  format "n .+4") : nat_scope.

Notation "n .-1" := (predn n) (at level 2, left associativity,
  format "n .-1") : nat_scope.
Notation "n .-2" := n.-1.-1 (at level 2, left associativity,
  format "n .-2") : nat_scope.

Lemma succnK : cancel succn predn.
Lemma succn_inj : injective succn.

Predeclare postfix doubling/halving operators.

Reserved Notation "n .*2" (at level 2, left associativity, format "n .*2").
Reserved Notation "n ./2" (at level 2, left associativity, format "n ./2").

Canonical comparison and eqType for nat.

Fixpoint eqn m n {struct m} :=
  match m, n with
  | 0, 0 ⇒ true
  | m'.+1, n'.+1 ⇒ eqn m' n'
  | _, _ ⇒ false
  end.

Lemma eqnP : Equality.axiom eqn.

Canonical nat_eqMixin := EqMixin eqnP.
Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin.

Arguments eqn !m !n.
Arguments eqnP {x y}.

Lemma eqnE : eqn = eq_op.

Lemma eqSS m n : (m .+1 == n .+1 ) = (m == n ).

Lemma nat_irrelevance (x y : nat) (E E' : x = y) : E = E'.

Protected addition, with a more systematic set of lemmas.

Definition addn_rec := plus.
Notation "m + n" := (addn_rec m n) : nat_rec_scope.

Definition addn := nosimpl addn_rec.
Notation "m + n" := (addn m n) : nat_scope.

Lemma addnE : addn = addn_rec.

Lemma plusE : plus = addn.

Lemma add0n : left_id 0 addn.
Lemma addSn m n : m .+1 + n = (m + n ).+1.
Lemma add1n n : 1 + n = n .+1.

Lemma addn0 : right_id 0 addn.

Lemma addnS m n : m + n .+1 = (m + n ).+1.

Lemma addSnnS m n : m .+1 + n = m + n .+1.

Lemma addnCA : left_commutative addn.

Lemma addnC : commutative addn.

Lemma addn1 n : n + 1 = n .+1.

Lemma addnA : associative addn.

Lemma addnAC : right_commutative addn.

Lemma addnCAC m n p : m + n + p = p + n + m.

Lemma addnACl m n p: m + n + p = n + (p + m ).

Lemma addnACA : interchange addn addn.

Lemma addn_eq0 m n : (m + n == 0) = (m == 0) && (n == 0).

Lemma addn_eq1 m n :
(m + n == 1) = ((m == 1) && (n == 0)) || ((m == 0) && (n == 1)).

Lemma eqn_add2l p m n : (p + m == p + n ) = (m == n ).

Lemma eqn_add2r p m n : (m + p == n + p ) = (m == n ).

Lemma addnI : right_injective addn.

Lemma addIn : left_injective addn.

Lemma addn2 m : m + 2 = m .+2.
Lemma add2n m : 2 + m = m .+2.
Lemma addn3 m : m + 3 = m .+3.
Lemma add3n m : 3 + m = m .+3.
Lemma addn4 m : m + 4 = m .+4.
Lemma add4n m : 4 + m = m .+4.

Protected, structurally decreasing subtraction, and basic lemmas. Further properties depend on ordering conditions.

Definition subn_rec := minus.
Arguments subn_rec : simpl nomatch.
Notation "m - n" := (subn_rec m n) : nat_rec_scope.

Definition subn := nosimpl subn_rec.
Notation "m - n" := (subn m n) : nat_scope.

Lemma subnE : subn = subn_rec.
Lemma minusE : minus = subn.

Lemma sub0n : left_zero 0 subn.
Lemma subn0 : right_id 0 subn.
Lemma subnn : self_inverse 0 subn.

Lemma subSS n m : m .+1 - n .+1 = m - n.
Lemma subn1 n : n - 1 = n .-1.
Lemma subn2 n : (n - 2)%N = n .-2.

Lemma subnDl p m n : (p + m ) - (p + n ) = m - n.

Lemma subnDr p m n : (m + p ) - (n + p ) = m - n.

Lemma addnK n : cancel (addn ^~ n) (subn ^~ n).

Lemma addKn n : cancel (addn n) (subn ^~ n).

Lemma subSnn n : n .+1 - n = 1.

Lemma subnDA m n p : n - (m + p ) = (n - m ) - p.

Lemma subnAC : right_commutative subn.

Lemma subnS m n : m - n .+1 = (m - n ).-1.

Lemma subSKn m n : (m .+1 - n ).-1 = m - n.

Integer ordering, and its interaction with the other operations.

Definition leq m n := m - n == 0.

Notation "m <= n" := (leq m n) : nat_scope.
Notation "m < n" := (m.+1 ≤ n) : nat_scope.
Notation "m >= n" := (n ≤ m) (only parsing) : nat_scope.
Notation "m > n" := (n < m) (only parsing) : nat_scope.

For sorting, etc.

Definition geq := [rel m n | m ≥ n ].
Definition ltn := [rel m n | m < n ].
Definition gtn := [rel m n | m > n ].

Notation "m <= n <= p" := ((m ≤ n) && (n ≤ p)) : nat_scope.
Notation "m < n <= p" := ((m < n) && (n ≤ p)) : nat_scope.
Notation "m <= n < p" := ((m ≤ n) && (n < p)) : nat_scope.
Notation "m < n < p" := ((m < n) && (n < p)) : nat_scope.

Lemma ltnS m n : (m < n .+1 ) = (m ≤ n ).
Lemma leq0n n : 0 ≤ n.
Lemma ltn0Sn n : 0 < n .+1.
Lemma ltn0 n : n < 0 = false.
Lemma leqnn n : n ≤ n.
#[global] Hint Resolve leqnn : core.
Lemma ltnSn n : n < n .+1.
Lemma eq_leq m n : m = n → m ≤ n.
Lemma leqnSn n : n ≤ n .+1.
#[global] Hint Resolve leqnSn : core.
Lemma leq_pred n : n .-1 ≤ n.
Lemma leqSpred n : n ≤ n .-1 .+1.

Lemma ltn_predL n : (n .-1 < n ) = (0 < n ).

Lemma ltn_predRL m n : (m < n .-1 ) = (m .+1 < n ).

Lemma ltn_predK m n : m < n → n .-1 .+1 = n.

Lemma prednK n : 0 < n → n .-1 .+1 = n.

Lemma leqNgt m n : (m ≤ n ) = ~~ (n < m ).

Lemma leqVgt m n : (m ≤ n ) || (n < m ).

Lemma ltnNge m n : (m < n ) = ~~ (n ≤ m ).

Lemma ltnn n : n < n = false.

Lemma leqn0 n : (n ≤ 0) = (n == 0).
Lemma lt0n n : (0 < n ) = (n != 0).
Lemma lt0n_neq0 n : 0 < n → n != 0.
Lemma eqn0Ngt n : (n == 0) = ~~ (n > 0).
Lemma neq0_lt0n n : (n == 0) = false → 0 < n.
#[global] Hint Resolve lt0n_neq0 neq0_lt0n : core.

Lemma eqn_leq m n : (m == n ) = (m ≤ n ≤ m ).

Lemma anti_leq : antisymmetric leq.

Lemma neq_ltn m n : (m != n ) = (m < n ) || (n < m ).

Lemma gtn_eqF m n : m < n → n == m = false.

Lemma ltn_eqF m n : m < n → m == n = false.

Lemma ltn_geF m n : m < n → m ≥ n = false.

Lemma leq_gtF m n : m ≤ n → m > n = false.

Lemma leq_eqVlt m n : (m ≤ n ) = (m == n ) || (m < n ).

Lemma ltn_neqAle m n : (m < n ) = (m != n ) && (m ≤ n ).

Lemma leq_trans n m p : m ≤ n → n ≤ p → m ≤ p.

Lemma leq_ltn_trans n m p : m ≤ n → n < p → m < p.

Lemma ltnW m n : m < n → m ≤ n.
#[global] Hint Resolve ltnW : core.

Lemma leqW m n : m ≤ n → m ≤ n .+1.

Lemma ltn_trans n m p : m < n → n < p → m < p.

Lemma leq_total m n : (m ≤ n ) || (m ≥ n ).

Helper lemmas to support generalized induction over a nat measure. The idiom for a proof by induction over a measure Mxy : nat involving variables x, y, ... (e.g., size x + size y) is have [n leMn] := ubnP Mxy; elim: n => // n IHn in x y ... leMn ... *. after which the current goal (possibly modified by generalizations in the in ... part) can be proven with the extra context assumptions n : nat IHn : forall x y ..., Mxy < n -> ... -> the_initial_goal leMn : Mxy < n.+1 This is preferable to the legacy idiom relying on numerical occurrence selection, which is fragile if there can be multiple occurrences of x, y, ... in the measure expression Mxy (e.g., in #|y| with x : finType and y : {set x}). The leMn statement is convertible to Mxy <= n; if it is necessary to have _exactly_ leMn : Mxy <= n, the ltnSE helper lemma may be used as follows have [n] := ubnP Mxy; elim: n => // n IHn in x y ... * => /ltnSE-leMn. We also provide alternative helper lemmas for proofs where the upper bound appears in the goal, and we assume nonstrict (in)equality. In either case the proof will have to dispatch an Mxy = 0 case. have [n defM] := ubnPleq Mxy; elim: n => [|n IHn] in x y ... defM ... *. yields two subgoals, in which Mxy has been replaced by 0 and n.+1, with the extra assumption defM : Mxy <= 0 / Mxy <= n.+1, respectively. The second goal also has the inductive assumption IHn : forall x y ..., Mxy <= n -> ... -> the_initial_goal[n / Mxy]. Using ubnPgeq or ubnPeq instead of ubnPleq yields assumptions with Mxy >= 0/n.+1 or Mxy == 0/n.+1 instead of Mxy <= 0/n.+1, respectively. These introduce a different kind of induction; for example ubnPgeq M lets us remember that n < M throughout the induction. Finally, the ltn_ind lemma provides a generalized induction view for a property of a single integer (i.e., the case Mxy := x).

Lemma ubnP m : {n | m < n }.
Lemma ltnSE m n : m < n .+1 → m ≤ n.
Variant ubn_leq_spec m : nat → Type := UbnLeq n of m ≤ n : ubn_leq_spec m n.
Variant ubn_geq_spec m : nat → Type := UbnGeq n of m ≥ n : ubn_geq_spec m n.
Variant ubn_eq_spec m : nat → Type := UbnEq n of m == n : ubn_eq_spec m n.
Lemma ubnPleq m : ubn_leq_spec m m.
Lemma ubnPgeq m : ubn_geq_spec m m.
Lemma ubnPeq m : ubn_eq_spec m m.
Lemma ltn_ind P : (∀ n, (∀ m, m < n → P m ) → P n ) → ∀ n, P n.

Link to the legacy comparison predicates.

Lemma leP m n : reflect (m ≤ n)%coq_nat (m ≤ n).
Arguments leP {m n}.

Lemma le_irrelevance m n le_mn1 le_mn2 : le_mn1 = le_mn2 :> (m ≤ n)%coq_nat.

Lemma ltP m n : reflect (m < n)%coq_nat (m < n).
Arguments ltP {m n}.

Lemma lt_irrelevance m n lt_mn1 lt_mn2 : lt_mn1 = lt_mn2 :> (m < n)%coq_nat.

Monotonicity lemmas

Lemma leq_add2l p m n : (p + m ≤ p + n ) = (m ≤ n ).

Lemma ltn_add2l p m n : (p + m < p + n ) = (m < n ).

Lemma leq_add2r p m n : (m + p ≤ n + p ) = (m ≤ n ).

Lemma ltn_add2r p m n : (m + p < n + p ) = (m < n ).

Lemma leq_add m1 m2 n1 n2 : m1 ≤ n1 → m2 ≤ n2 → m1 + m2 ≤ n1 + n2.

Lemma leq_addl m n : n ≤ m + n.

Lemma leq_addr m n : n ≤ n + m.

Lemma ltn_addl m n p : m < n → m < p + n.

Lemma ltn_addr m n p : m < n → m < n + p.

Lemma addn_gt0 m n : (0 < m + n ) = (0 < m ) || (0 < n ).

Lemma subn_gt0 m n : (0 < n - m ) = (m < n ).

Lemma subn_eq0 m n : (m - n == 0) = (m ≤ n ).

Lemma leq_subLR m n p : (m - n ≤ p ) = (m ≤ n + p ).

Lemma leq_subr m n : n - m ≤ n.

Lemma ltn_subrR m n : (n < n - m ) = false.

Lemma leq_subrR m n : (n ≤ n - m ) = (m == 0) || (n == 0).

Lemma ltn_subrL m n : (n - m < n ) = (0 < m ) && (0 < n ).

Lemma subnKC m n : m ≤ n → m + (n - m ) = n.

Lemma addnBn m n : m + (n - m ) = m - n + n.

Lemma subnK m n : m ≤ n → (n - m ) + m = n.

Lemma addnBA m n p : p ≤ n → m + (n - p ) = m + n - p.

Lemma addnBAC m n p : n ≤ m → m - n + p = m + p - n.

Lemma addnBCA m n p : p ≤ m → p ≤ n → m + (n - p ) = n + (m - p ).

Lemma addnABC m n p : p ≤ m → p ≤ n → m + (n - p ) = m - p + n.

Lemma subnBA m n p : p ≤ n → m - (n - p ) = m + p - n.

Lemma subnA m n p : p ≤ n → n ≤ m → m - (n - p ) = m - n + p.

Lemma subKn m n : m ≤ n → n - (n - m ) = m.

Lemma subSn m n : m ≤ n → n .+1 - m = (n - m ).+1.

Lemma subnSK m n : m < n → (n - m .+1).+1 = n - m.

Lemma addnCBA m n p : p ≤ n → m + (n - p ) = n + m - p.

Lemma addnBr_leq n p m : n ≤ p → m + (n - p ) = m.

Lemma addnBl_leq m n p : m ≤ n → m - n + p = p.

Lemma subnDAC m n p : m - (n + p ) = m - p - n.

Lemma subnCBA m n p : p ≤ n → m - (n - p ) = p + m - n.

Lemma subnBr_leq n p m : n ≤ p → m - (n - p ) = m.

Lemma subnBl_leq m n p : m ≤ n → (m - n ) - p = 0.

Lemma subnBAC m n p : p ≤ n → n ≤ m → m - (n - p ) = p + (m - n ).

Lemma subDnAC m n p : p ≤ n → m + n - p = n - p + m.

Lemma subDnCA m n p : p ≤ m → m + n - p = n + (m - p ).

Lemma subDnCAC m n p : m ≤ p → m + n - p = n - (p - m ).

Lemma addnBC m n : m - n + n = n - m + m.

Lemma addnCB m n : m - n + n = m + (n - m ).

Lemma addBnAC m n p : n ≤ m → m - n + p = p + m - n.

Lemma addBnCAC m n p : n ≤ m → n ≤ p → m - n + p = p - n + m.

Lemma addBnA m n p : n ≤ m → p ≤ n → m - n + p = m - (n - p ).

Lemma subBnAC m n p : m - n - p = m - (p + n ).

Lemma predn_sub m n : (m - n ).-1 = (m .-1 - n ).

Lemma leq_sub2r p m n : m ≤ n → m - p ≤ n - p.

Lemma leq_sub2l p m n : m ≤ n → p - n ≤ p - m.

Lemma leq_sub m1 m2 n1 n2 : m1 ≤ m2 → n2 ≤ n1 → m1 - n1 ≤ m2 - n2.

Lemma ltn_sub2r p m n : p < n → m < n → m - p < n - p.

Lemma ltn_sub2l p m n : m < p → m < n → p - n < p - m.

Lemma ltn_subRL m n p : (n < p - m ) = (m + n < p ).

Lemma leq_psubRL m n p : 0 < n → (n ≤ p - m ) = (m + n ≤ p ).

Lemma ltn_psubLR m n p : 0 < p → (m - n < p ) = (m < n + p ).

Lemma leq_subRL m n p : m ≤ p → (n ≤ p - m ) = (m + n ≤ p ).

Lemma ltn_subLR m n p : n ≤ m → (m - n < p ) = (m < n + p ).

Lemma leq_subCl m n p : (m - n ≤ p ) = (m - p ≤ n ).

Lemma ltn_subCr m n p : (p < m - n ) = (n < m - p ).

Lemma leq_psubCr m n p : 0 < p → 0 < n → (p ≤ m - n ) = (n ≤ m - p ).

Lemma ltn_psubCl m n p : 0 < p → 0 < n → (m - n < p ) = (m - p < n ).

Lemma leq_subCr m n p : n ≤ m → p ≤ m → (p ≤ m - n ) = (n ≤ m - p ).

Lemma ltn_subCl m n p : n ≤ m → p ≤ m → (m - n < p ) = (m - p < n ).

Lemma leq_sub2rE p m n : p ≤ n → (m - p ≤ n - p ) = (m ≤ n ).

Lemma leq_sub2lE m n p : n ≤ m → (m - p ≤ m - n ) = (n ≤ p ).

Lemma ltn_sub2rE p m n : p ≤ m → (m - p < n - p ) = (m < n ).

Lemma ltn_sub2lE m n p : p ≤ m → (m - p < m - n ) = (n < p ).

Lemma eqn_sub2rE p m n : p ≤ m → p ≤ n → (m - p == n - p ) = (m == n ).

Lemma eqn_sub2lE m n p : p ≤ m → n ≤ m → (m - p == m - n ) = (p == n ).

Max and min.

Definition maxn m n := if m < n then n else m.

Definition minn m n := if m < n then m else n.

Lemma max0n : left_id 0 maxn.
Lemma maxn0 : right_id 0 maxn.

Lemma maxnC : commutative maxn.

Lemma maxnE m n : maxn m n = m + (n - m ).

Lemma maxnAC : right_commutative maxn.

Lemma maxnA : associative maxn.

Lemma maxnCA : left_commutative maxn.

Lemma maxnACA : interchange maxn maxn.

Lemma maxn_idPl {m n} : reflect (maxn m n = m) (m ≥ n).

Lemma maxn_idPr {m n} : reflect (maxn m n = n) (m ≤ n).

Lemma maxnn : idempotent maxn.

Lemma leq_max m n1 n2 : (m ≤ maxn n1 n2 ) = (m ≤ n1 ) || (m ≤ n2 ).
Lemma leq_maxl m n : m ≤ maxn m n.
Lemma leq_maxr m n : n ≤ maxn m n.

Lemma gtn_max m n1 n2 : (m > maxn n1 n2 ) = (m > n1 ) && (m > n2 ).

Lemma geq_max m n1 n2 : (m ≥ maxn n1 n2 ) = (m ≥ n1 ) && (m ≥ n2 ).

Lemma maxnSS m n : maxn m .+1 n .+1 = (maxn m n ).+1.

Lemma addn_maxl : left_distributive addn maxn.

Lemma addn_maxr : right_distributive addn maxn.

Lemma subn_maxl : left_distributive subn maxn.

Lemma min0n : left_zero 0 minn.
Lemma minn0 : right_zero 0 minn.

Lemma minnC : commutative minn.

Lemma addn_min_max m n : minn m n + maxn m n = m + n.

Lemma minnE m n : minn m n = m - (m - n ).

Lemma minnAC : right_commutative minn.

Lemma minnA : associative minn.

Lemma minnCA : left_commutative minn.

Lemma minnACA : interchange minn minn.

Lemma minn_idPl {m n} : reflect (minn m n = m) (m ≤ n).

Lemma minn_idPr {m n} : reflect (minn m n = n) (m ≥ n).

Lemma minnn : idempotent minn.

Lemma leq_min m n1 n2 : (m ≤ minn n1 n2 ) = (m ≤ n1 ) && (m ≤ n2 ).

Lemma gtn_min m n1 n2 : (m > minn n1 n2 ) = (m > n1 ) || (m > n2 ).

Lemma geq_min m n1 n2 : (m ≥ minn n1 n2 ) = (m ≥ n1 ) || (m ≥ n2 ).

Lemma geq_minl m n : minn m n ≤ m.
Lemma geq_minr m n : minn m n ≤ n.

Lemma addn_minr : right_distributive addn minn.

Lemma addn_minl : left_distributive addn minn.

Lemma subn_minl : left_distributive subn minn.

Lemma minnSS m n : minn m .+1 n .+1 = (minn m n ).+1.

Quasi-cancellation (really, absorption) lemmas

Lemma maxnK m n : minn (maxn m n) m = m.

Lemma maxKn m n : minn n (maxn m n) = n.

Lemma minnK m n : maxn (minn m n) m = m.

Lemma minKn m n : maxn n (minn m n) = n.

Distributivity.

Lemma maxn_minl : left_distributive maxn minn.

Lemma maxn_minr : right_distributive maxn minn.

Lemma minn_maxl : left_distributive minn maxn.

Lemma minn_maxr : right_distributive minn maxn.

Comparison predicates.

Variant leq_xor_gtn m n : nat → nat → nat → nat → bool → bool → Set :=
  | LeqNotGtn of m ≤ n : leq_xor_gtn m n m m n n true false
  | GtnNotLeq of n < m : leq_xor_gtn m n n n m m false true.

Lemma leqP m n : leq_xor_gtn m n (minn n m) (minn m n) (maxn n m) (maxn m n)
                                 (m ≤ n) (n < m).

Variant ltn_xor_geq m n : nat → nat → nat → nat → bool → bool → Set :=
  | LtnNotGeq of m < n : ltn_xor_geq m n m m n n false true
  | GeqNotLtn of n ≤ m : ltn_xor_geq m n n n m m true false.

Lemma ltnP m n : ltn_xor_geq m n (minn n m) (minn m n) (maxn n m) (maxn m n)
                                 (n ≤ m) (m < n).

Variant eqn0_xor_gt0 n : bool → bool → Set :=
  | Eq0NotPos of n = 0 : eqn0_xor_gt0 n true false
  | PosNotEq0 of n > 0 : eqn0_xor_gt0 n false true.

Lemma posnP n : eqn0_xor_gt0 n (n == 0) (0 < n).

Variant compare_nat m n : nat → nat → nat → nat →
                          bool → bool → bool → bool → bool → bool → Set :=
  | CompareNatLt of m < n :
      compare_nat m n m m n n false false false true false true
  | CompareNatGt of m > n :
      compare_nat m n n n m m false false true false true false
  | CompareNatEq of m = n :
      compare_nat m n m m m m true true true true false false.

Lemma ltngtP m n :
  compare_nat m n (minn n m) (minn m n) (maxn n m) (maxn m n)
                  (n == m) (m == n) (n ≤ m) (m ≤ n) (n < m) (m < n).

Eliminating the idiom for structurally decreasing compare and subtract.

Lemma subn_if_gt T m n F (E : T) :
(if m .+1 - n is m'.+1 then F m' else E ) = (if n ≤ m then F (m - n) else E ).

Notation leqLHS := (X in (X ≤ _)%N)%pattern.
Notation leqRHS := (X in (_ ≤ X)%N)%pattern.
Notation ltnLHS := (X in (X < _)%N)%pattern.
Notation ltnRHS := (X in (_ < X)%N)%pattern.

Getting a concrete value from an abstract existence proof.

Section ExMinn.

Variable P : pred nat.
Hypothesis exP : ∃ n , P n.

Inductive acc_nat i : Prop := AccNat0 of P i | AccNatS of acc_nat i .+1.

Lemma find_ex_minn : {m | P m & ∀ n, P n → n ≥ m }.

Definition ex_minn := s2val find_ex_minn.

Inductive ex_minn_spec : nat → Type :=
  ExMinnSpec m of P m & (∀ n, P n → n ≥ m) : ex_minn_spec m.

Lemma ex_minnP : ex_minn_spec ex_minn.

End ExMinn.

Section ExMaxn.

Variables (P : pred nat) (m : nat).
Hypotheses (exP : ∃ i , P i) (ubP : ∀ i, P i → i ≤ m).

Lemma ex_maxn_subproof : ∃ i , P (m - i).

Definition ex_maxn := m - ex_minn ex_maxn_subproof.

Variant ex_maxn_spec : nat → Type :=
  ExMaxnSpec i of P i & (∀ j, P j → j ≤ i) : ex_maxn_spec i.

Lemma ex_maxnP : ex_maxn_spec ex_maxn.

End ExMaxn.

Lemma eq_ex_minn P Q exP exQ : P =1 Q → @ex_minn P exP = @ex_minn Q exQ.

Lemma eq_ex_maxn (P Q : pred nat) m n exP ubP exQ ubQ :
  P =1 Q → @ex_maxn P m exP ubP = @ex_maxn Q n exQ ubQ.

Section Iteration.

Variable T : Type.
Implicit Types m n : nat.
Implicit Types x y : T.
Implicit Types S : {pred T }.

Definition iter n f x :=
  let fix loop m := if m is i.+1 then f (loop i) else x in loop n.

Definition iteri n f x :=
  let fix loop m := if m is i.+1 then f i (loop i) else x in loop n.

Definition iterop n op x :=
  let f i y := if i is 0 then x else op x y in iteri n f.

Lemma iterSr n f x : iter n .+1 f x = iter n f (f x).

Lemma iterS n f x : iter n .+1 f x = f (iter n f x).

Lemma iterD n m f x : iter (n + m) f x = iter n f (iter m f x).

Lemma iteriS n f x : iteri n .+1 f x = f n (iteri n f x).

Lemma iteropS idx n op x : iterop n .+1 op x idx = iter n (op x) x.

Lemma eq_iter f f' : f =1 f' → ∀ n, iter n f =1 iter n f'.

Lemma iter_fix n f x : f x = x → iter n f x = x.

Lemma eq_iteri f f' : f =2 f' → ∀ n, iteri n f =1 iteri n f'.

Lemma eq_iterop n op op' : op =2 op' → iterop n op =2 iterop n op'.

Lemma iter_in f S i : {homo f : x / x \in S } → {homo iter i f : x / x \in S }.

End Iteration.

Lemma iter_succn m n : iter n succn m = m + n.

Lemma iter_succn_0 n : iter n succn 0 = n.

Lemma iter_predn m n : iter n predn m = m - n.

Multiplication.

Definition muln_rec := mult.
Notation "m * n" := (muln_rec m n) : nat_rec_scope.

Definition muln := nosimpl muln_rec.
Notation "m * n" := (muln m n) : nat_scope.

Lemma multE : mult = muln.
Lemma mulnE : muln = muln_rec.

Lemma mul0n : left_zero 0 muln.
Lemma muln0 : right_zero 0 muln.
Lemma mul1n : left_id 1 muln.
Lemma mulSn m n : m .+1 × n = n + m × n.
Lemma mulSnr m n : m .+1 × n = m × n + n.

Lemma mulnS m n : m × n .+1 = m + m × n.
Lemma mulnSr m n : m × n .+1 = m × n + m.

Lemma iter_addn m n p : iter n (addn m) p = m × n + p.

Lemma iter_addn_0 m n : iter n (addn m) 0 = m × n.

Lemma muln1 : right_id 1 muln.

Lemma mulnC : commutative muln.

Lemma mulnDl : left_distributive muln addn.

Lemma mulnDr : right_distributive muln addn.

Lemma mulnBl : left_distributive muln subn.

Lemma mulnBr : right_distributive muln subn.

Lemma mulnA : associative muln.

Lemma mulnCA : left_commutative muln.

Lemma mulnAC : right_commutative muln.

Lemma mulnACA : interchange muln muln.

Lemma muln_eq0 m n : (m × n == 0) = (m == 0) || (n == 0).

Lemma muln_eq1 m n : (m × n == 1) = (m == 1) && (n == 1).

Lemma muln_gt0 m n : (0 < m × n ) = (0 < m ) && (0 < n ).

Lemma leq_pmull m n : n > 0 → m ≤ n × m.

Lemma leq_pmulr m n : n > 0 → m ≤ m × n.

Lemma leq_mul2l m n1 n2 : (m × n1 ≤ m × n2 ) = (m == 0) || (n1 ≤ n2 ).

Lemma leq_mul2r m n1 n2 : (n1 × m ≤ n2 × m ) = (m == 0) || (n1 ≤ n2 ).

Lemma leq_mul m1 m2 n1 n2 : m1 ≤ n1 → m2 ≤ n2 → m1 × m2 ≤ n1 × n2.

Lemma eqn_mul2l m n1 n2 : (m × n1 == m × n2 ) = (m == 0) || (n1 == n2 ).

Lemma eqn_mul2r m n1 n2 : (n1 × m == n2 × m ) = (m == 0) || (n1 == n2 ).

Lemma leq_pmul2l m n1 n2 : 0 < m → (m × n1 ≤ m × n2 ) = (n1 ≤ n2 ).
Arguments leq_pmul2l [m n1 n2].

Lemma leq_pmul2r m n1 n2 : 0 < m → (n1 × m ≤ n2 × m ) = (n1 ≤ n2 ).
Arguments leq_pmul2r [m n1 n2].

Lemma eqn_pmul2l m n1 n2 : 0 < m → (m × n1 == m × n2 ) = (n1 == n2 ).
Arguments eqn_pmul2l [m n1 n2].

Lemma eqn_pmul2r m n1 n2 : 0 < m → (n1 × m == n2 × m ) = (n1 == n2 ).
Arguments eqn_pmul2r [m n1 n2].

Lemma ltn_mul2l m n1 n2 : (m × n1 < m × n2 ) = (0 < m ) && (n1 < n2 ).

Lemma ltn_mul2r m n1 n2 : (n1 × m < n2 × m ) = (0 < m ) && (n1 < n2 ).

Lemma ltn_pmul2l m n1 n2 : 0 < m → (m × n1 < m × n2 ) = (n1 < n2 ).
Arguments ltn_pmul2l [m n1 n2].

Lemma ltn_pmul2r m n1 n2 : 0 < m → (n1 × m < n2 × m ) = (n1 < n2 ).
Arguments ltn_pmul2r [m n1 n2].

Lemma ltn_Pmull m n : 1 < n → 0 < m → m < n × m.

Lemma ltn_Pmulr m n : 1 < n → 0 < m → m < m × n.

Lemma ltn_mul m1 m2 n1 n2 : m1 < n1 → m2 < n2 → m1 × m2 < n1 × n2.

Lemma maxnMr : right_distributive muln maxn.

Lemma maxnMl : left_distributive muln maxn.

Lemma minnMr : right_distributive muln minn.

Lemma minnMl : left_distributive muln minn.

Lemma iterM (T : Type) (n m : nat) (f : T → T) :
iter (n × m) f =1 iter n (iter m f).

Exponentiation.

Definition expn_rec m n := iterop n muln m 1.
Notation "m ^ n" := (expn_rec m n) : nat_rec_scope.
Definition expn := nosimpl expn_rec.
Notation "m ^ n" := (expn m n) : nat_scope.

Lemma expnE : expn = expn_rec.

Lemma expn0 m : m ^ 0 = 1.
Lemma expn1 m : m ^ 1 = m.
Lemma expnS m n : m ^ n .+1 = m × m ^ n.
Lemma expnSr m n : m ^ n .+1 = m ^ n × m.

Lemma iter_muln m n p : iter n (muln m) p = m ^ n × p.

Lemma iter_muln_1 m n : iter n (muln m) 1 = m ^ n.

Lemma exp0n n : 0 < n → 0 ^ n = 0.

Lemma exp1n n : 1 ^ n = 1.

Lemma expnD m n1 n2 : m ^ (n1 + n2 ) = m ^ n1 × m ^ n2.

Lemma expnMn m1 m2 n : (m1 × m2 ) ^ n = m1 ^ n × m2 ^ n.

Lemma expnM m n1 n2 : m ^ (n1 × n2 ) = (m ^ n1 ) ^ n2.

Lemma expnAC m n1 n2 : (m ^ n1 ) ^ n2 = (m ^ n2 ) ^ n1.

Lemma expn_gt0 m n : (0 < m ^ n ) = (0 < m ) || (n == 0).

Lemma expn_eq0 m e : (m ^ e == 0) = (m == 0) && (e > 0).

Lemma ltn_expl m n : 1 < m → n < m ^ n.

Lemma leq_exp2l m n1 n2 : 1 < m → (m ^ n1 ≤ m ^ n2 ) = (n1 ≤ n2 ).

Lemma ltn_exp2l m n1 n2 : 1 < m → (m ^ n1 < m ^ n2 ) = (n1 < n2 ).

Lemma eqn_exp2l m n1 n2 : 1 < m → (m ^ n1 == m ^ n2 ) = (n1 == n2 ).

Lemma expnI m : 1 < m → injective (expn m).

Lemma leq_pexp2l m n1 n2 : 0 < m → n1 ≤ n2 → m ^ n1 ≤ m ^ n2.

Lemma ltn_pexp2l m n1 n2 : 0 < m → m ^ n1 < m ^ n2 → n1 < n2.

Lemma ltn_exp2r m n e : e > 0 → (m ^ e < n ^ e ) = (m < n ).

Lemma leq_exp2r m n e : e > 0 → (m ^ e ≤ n ^ e ) = (m ≤ n ).

Lemma eqn_exp2r m n e : e > 0 → (m ^ e == n ^ e ) = (m == n ).

Lemma expIn e : e > 0 → injective (expn ^~ e).

Lemma iterX (T : Type) (n m : nat) (f : T → T) :
iter (n ^ m) f =1 iter m (iter n) f.

Factorial.

Fixpoint fact_rec n := if n is n'.+1 then n × fact_rec n' else 1.

Definition factorial := nosimpl fact_rec.

Notation "n `!" := (factorial n) (at level 2, format "n `!") : nat_scope.

Lemma factE : factorial = fact_rec.

Lemma fact0 : 0`! = 1.

Lemma factS n : (n .+1 )`! = n .+1 × n `!.

Lemma fact_gt0 n : n `! > 0.

Lemma fact_geq n : n ≤ n `!.

Lemma ltn_fact m n : 0 < m → m < n → m `! < n `!.

Parity and bits.

Coercion nat_of_bool (b : bool) := if b then 1 else 0.

Lemma leq_b1 (b : bool) : b ≤ 1.

Lemma addn_negb (b : bool) : ~~ b + b = 1.

Lemma eqb0 (b : bool) : (b == 0 :> nat ) = ~~ b.

Lemma eqb1 (b : bool) : (b == 1 :> nat ) = b.

Lemma lt0b (b : bool) : (b > 0) = b.

Lemma sub1b (b : bool) : 1 - b = ~~ b.

Lemma mulnb (b1 b2 : bool) : b1 × b2 = b1 && b2.

Lemma mulnbl (b : bool) n : b × n = (if b then n else 0).

Lemma mulnbr (b : bool) n : n × b = (if b then n else 0).

Fixpoint odd n := if n is n'.+1 then ~~ odd n' else false.

Lemma oddS n : odd n .+1 = ~~ odd n.

Lemma oddb (b : bool) : odd b = b.

Lemma oddD m n : odd (m + n) = odd m (+) odd n.

Lemma oddB m n : n ≤ m → odd (m - n) = odd m (+) odd n.

Lemma oddN i m : odd m = false → i ≤ m → odd (m - i) = odd i.

Lemma oddM m n : odd (m × n) = odd m && odd n.

Lemma oddX m n : odd (m ^ n) = (n == 0) || odd m.

Doubling.

Fixpoint double_rec n := if n is n'.+1 then n'.*2%Nrec.+2 else 0
where "n .*2" := (double_rec n) : nat_rec_scope.

Definition double := nosimpl double_rec.
Notation "n .*2" := (double n) : nat_scope.

Lemma doubleE : double = double_rec.

Lemma double0 : 0.*2 = 0.

Lemma doubleS n : n .+1 .*2 = n .*2 .+2.

Lemma double_pred n : n .-1 .*2 = n .*2 .-2.

Lemma addnn n : n + n = n .*2.

Lemma mul2n m : 2 × m = m .*2.

Lemma muln2 m : m × 2 = m .*2.

Lemma doubleD m n : (m + n ).*2 = m .*2 + n .*2.

Lemma doubleB m n : (m - n ).*2 = m .*2 - n .*2.

Lemma leq_double m n : (m .*2 ≤ n .*2 ) = (m ≤ n ).

Lemma ltn_double m n : (m .*2 < n .*2 ) = (m < n ).

Lemma ltn_Sdouble m n : (m .*2 .+1 < n .*2 ) = (m < n ).

Lemma leq_Sdouble m n : (m .*2 ≤ n .*2 .+1 ) = (m ≤ n ).

Lemma odd_double n : odd n .*2 = false.

Lemma double_gt0 n : (0 < n .*2 ) = (0 < n ).

Lemma double_eq0 n : (n .*2 == 0) = (n == 0).

Lemma doubleMl m n : (m × n ).*2 = m .*2 × n.

Lemma doubleMr m n : (m × n ).*2 = m × n .*2.

Halving.

Fixpoint half (n : nat) : nat := if n is n'.+1 then uphalf n' else n
with uphalf (n : nat) : nat := if n is n'.+1 then n'./2 .+1 else n
where "n ./2" := (half n) : nat_scope.

Lemma uphalfE n : uphalf n = n .+1 ./2.

Lemma doubleK : cancel double half.

Definition half_double := doubleK.
Definition double_inj := can_inj doubleK.

Lemma uphalf_double n : uphalf n .*2 = n.

Lemma uphalf_half n : uphalf n = odd n + n ./2.

Lemma odd_double_half n : odd n + n ./2 .*2 = n.

Lemma halfK n : n ./2 .*2 = n - odd n.

Lemma uphalfK n : (uphalf n ).*2 = odd n + n.

Lemma odd_halfK n : odd n → n ./2 .*2 = n .-1.

Lemma even_halfK n : ~~ odd n → n ./2 .*2 = n.

Lemma odd_uphalfK n : odd n → (uphalf n ).*2 = n .+1.

Lemma even_uphalfK n : ~~ odd n → (uphalf n ).*2 = n.

Lemma half_bit_double n (b : bool) : (b + n .*2 )./2 = n.

Lemma halfD m n : (m + n )./2 = (odd m && odd n ) + (m ./2 + n ./2 ).

Lemma half_leq m n : m ≤ n → m ./2 ≤ n ./2.

Lemma geq_half_double m n : (m ≤ n ./2 ) = (m .*2 ≤ n ).

Lemma ltn_half_double m n : (m ./2 < n ) = (m < n .*2 ).

Lemma leq_half_double m n : (m ./2 ≤ n ) = (m ≤ n .*2 .+1 ).

Lemma gtn_half_double m n : (n < m ./2 ) = (n .*2 .+1 < m ).

Lemma half_gt0 n : (0 < n ./2 ) = (1 < n ).

Lemma uphalf_leq m n : m ≤ n → uphalf m ≤ uphalf n.

Lemma leq_uphalf_double m n : (uphalf m ≤ n ) = (m ≤ n .*2 ).

Lemma geq_uphalf_double m n : (m ≤ uphalf n ) = (m .*2 ≤ n .+1 ).

Lemma gtn_uphalf_double m n : (n < uphalf m ) = (n .*2 < m ).

Lemma ltn_uphalf_double m n : (uphalf m < n ) = (m .+1 < n .*2 ).

Lemma uphalf_gt0 n : (0 < uphalf n ) = (0 < n ).

Lemma odd_geq m n : odd n → (m ≤ n ) = (m ./2 .*2 ≤ n ).

Lemma odd_ltn m n : odd n → (n < m ) = (n < m ./2 .*2 ).

Lemma odd_gt0 n : odd n → n > 0.

Lemma odd_gt2 n : odd n → n > 1 → n > 2.

Squares and square identities.

Lemma mulnn m : m × m = m ^ 2.

Lemma sqrnD m n : (m + n ) ^ 2 = m ^ 2 + n ^ 2 + 2 × (m × n ).

Lemma sqrnB m n : n ≤ m → (m - n ) ^ 2 = m ^ 2 + n ^ 2 - 2 × (m × n ).

Lemma sqrnD_sub m n : n ≤ m → (m + n ) ^ 2 - 4 × (m × n ) = (m - n ) ^ 2.

Lemma subn_sqr m n : m ^ 2 - n ^ 2 = (m - n ) × (m + n ).

Lemma ltn_sqr m n : (m ^ 2 < n ^ 2) = (m < n ).

Lemma leq_sqr m n : (m ^ 2 ≤ n ^ 2) = (m ≤ n ).

Lemma sqrn_gt0 n : (0 < n ^ 2) = (0 < n ).

Lemma eqn_sqr m n : (m ^ 2 == n ^ 2) = (m == n ).

Lemma sqrn_inj : injective (expn ^~ 2).

Almost strict inequality: an inequality that is strict unless some specific condition holds, such as the Cauchy-Schwartz or the AGM inequality (we only prove the order-2 AGM here; the general one requires sequences). We formalize the concept as a rewrite multirule, that can be used both to rewrite the non-strict inequality to true, and the equality to the specific condition (for strict inequalities use the ltn_neqAle lemma); in addition, the conditional equality also coerces to a non-strict one.

Definition leqif m n C := ((m ≤ n ) × ((m == n ) = C ))%type.

Notation "m <= n ?= 'iff' C" := (leqif m n C) : nat_scope.

Coercion leq_of_leqif m n C (H : m ≤ n ?= iff C) := H .1 : m ≤ n.

Lemma leqifP m n C : reflect (m ≤ n ?= iff C) (if C then m == n else m < n).

Lemma leqif_refl m C : reflect (m ≤ m ?= iff C) C.

Lemma leqif_trans m1 m2 m3 C12 C23 :
  m1 ≤ m2 ?= iff C12 → m2 ≤ m3 ?= iff C23 → m1 ≤ m3 ?= iff C12 && C23.

Lemma mono_leqif f : {mono f : m n / m ≤ n } →
  ∀ m n C, (f m ≤ f n ?= iff C ) = (m ≤ n ?= iff C ).

Lemma leqif_geq m n : m ≤ n → m ≤ n ?= iff (m ≥ n ).

Lemma leqif_eq m n : m ≤ n → m ≤ n ?= iff (m == n ).

Lemma geq_leqif a b C : a ≤ b ?= iff C → (b ≤ a ) = C.

Lemma ltn_leqif a b C : a ≤ b ?= iff C → (a < b ) = ~~ C.

Lemma ltnNleqif x y C : x ≤ y ?= iff ~~ C → (x < y ) = C.

Lemma eq_leqif x y C : x ≤ y ?= iff C → (x == y ) = C.

Lemma eqTleqif x y C : x ≤ y ?= iff C → C → x = y.

Lemma leqif_add m1 n1 C1 m2 n2 C2 :
    m1 ≤ n1 ?= iff C1 → m2 ≤ n2 ?= iff C2 →
  m1 + m2 ≤ n1 + n2 ?= iff C1 && C2.

Lemma leqif_mul m1 n1 C1 m2 n2 C2 :
    m1 ≤ n1 ?= iff C1 → m2 ≤ n2 ?= iff C2 →
  m1 × m2 ≤ n1 × n2 ?= iff (n1 × n2 == 0) || (C1 && C2 ).

Lemma nat_Cauchy m n : 2 × (m × n ) ≤ m ^ 2 + n ^ 2 ?= iff (m == n ).

Lemma nat_AGM2 m n : 4 × (m × n ) ≤ (m + n ) ^ 2 ?= iff (m == n ).

Section ContraLeq.
Implicit Types (b : bool) (m n : nat) (P : Prop).

Lemma contraTleq b m n : (n < m → ~~ b ) → (b → m ≤ n ).

Lemma contraTltn b m n : (n ≤ m → ~~ b ) → (b → m < n ).

Lemma contraPleq P m n : (n < m → ¬ P ) → (P → m ≤ n ).

Lemma contraPltn P m n : (n ≤ m → ¬ P ) → (P → m < n ).

Lemma contraNleq b m n : (n < m → b ) → (~~ b → m ≤ n ).

Lemma contraNltn b m n : (n ≤ m → b ) → (~~ b → m < n ).

Lemma contra_not_leq P m n : (n < m → P ) → (¬ P → m ≤ n ).

Lemma contra_not_ltn P m n : (n ≤ m → P ) → (¬ P → m < n ).

Lemma contraFleq b m n : (n < m → b ) → (b = false → m ≤ n ).

Lemma contraFltn b m n : (n ≤ m → b ) → (b = false → m < n ).

Lemma contra_leqT b m n : (~~ b → m < n ) → (n ≤ m → b ).

Lemma contra_ltnT b m n : (~~ b → m ≤ n ) → (n < m → b ).

Lemma contra_leqN b m n : (b → m < n ) → (n ≤ m → ~~ b ).

Lemma contra_ltnN b m n : (b → m ≤ n ) → (n < m → ~~ b ).

Lemma contra_leq_not P m n : (P → m < n ) → (n ≤ m → ¬ P ).

Lemma contra_ltn_not P m n : (P → m ≤ n ) → (n < m → ¬ P ).

Lemma contra_leqF b m n : (b → m < n ) → (n ≤ m → b = false ).

Lemma contra_ltnF b m n : (b → m ≤ n ) → (n < m → b = false ).

Lemma contra_leq m n p q : (q < p → n < m ) → (m ≤ n → p ≤ q ).

Lemma contra_leq_ltn m n p q : (q ≤ p → n < m ) → (m ≤ n → p < q ).

Lemma contra_ltn_leq m n p q : (q < p → n ≤ m ) → (m < n → p ≤ q ).

Lemma contra_ltn m n p q : (q ≤ p → n ≤ m ) → (m < n → p < q ).

End ContraLeq.

Section Monotonicity.
Variable T : Type.

Lemma homo_ltn_in (D : {pred nat }) (f : nat → T) (r : T → T → Prop) :
  (∀ y x z, r x y → r y z → r x z ) →
  {in D &, ∀ i j k, i < k < j → k \in D } →
  {in D , ∀ i, i .+1 \in D → r (f i) (f i .+1)} →
  {in D &, {homo f : i j / i < j >-> r i j }}.

Lemma homo_ltn (f : nat → T) (r : T → T → Prop) :
  (∀ y x z, r x y → r y z → r x z ) →
  (∀ i, r (f i) (f i .+1)) → {homo f : i j / i < j >-> r i j }.

Lemma homo_leq_in (D : {pred nat }) (f : nat → T) (r : T → T → Prop) :
  (∀ x, r x x ) → (∀ y x z, r x y → r y z → r x z ) →
  {in D &, ∀ i j k, i < k < j → k \in D } →
  {in D , ∀ i, i .+1 \in D → r (f i) (f i .+1)} →
  {in D &, {homo f : i j / i ≤ j >-> r i j }}.

Lemma homo_leq (f : nat → T) (r : T → T → Prop) :
   (∀ x, r x x ) → (∀ y x z, r x y → r y z → r x z ) →
  (∀ i, r (f i) (f i .+1)) → {homo f : i j / i ≤ j >-> r i j }.

Section NatToNat.
Variable (f : nat → nat).

This listing of "Let"s factor out the required premises for the subsequent lemmas, putting them in the context so that "done" solves the goals quickly

Support for larger integers. The normal definitions of +, - and even IO are unsuitable for Peano integers larger than 2000 or so because they are not tail-recursive. We provide a workaround module, along with a rewrite multirule to change the tailrec operators to the normal ones. We handle IO via the NatBin module, but provide our own (more efficient) conversion functions.

Module NatTrec.

Usage: Import NatTrec. in section defining functions, rebinds all non-tail recursive operators. rewrite !trecE. in the correctness proof, restores operators

Fixpoint add m n := if m is m'.+1 then m' + n .+1 else n
where "n + m" := (add n m) : nat_scope.

Fixpoint add_mul m n s := if m is m'.+1 then add_mul m' n (n + s) else s.

Definition mul m n := if m is m'.+1 then add_mul m' n n else 0.

Notation "n * m" := (mul n m) : nat_scope.

Fixpoint mul_exp m n p := if n is n'.+1 then mul_exp m n' (m × p) else p.

Definition exp m n := if n is n'.+1 then mul_exp m n' m else 1.

Notation "n ^ m" := (exp n m) : nat_scope.

Local Notation oddn := odd.
Fixpoint odd n := if n is n'.+2 then odd n' else eqn n 1.

Local Notation doublen := double.
Definition double n := if n is n'.+1 then n' + n .+1 else 0.
Notation "n .*2" := (double n) : nat_scope.

Lemma addE : add =2 addn.

Lemma doubleE : double =1 doublen.

Lemma add_mulE n m s : add_mul n m s = addn (muln n m) s.

Lemma mulE : mul =2 muln.

Lemma mul_expE m n p : mul_exp m n p = muln (expn m n) p.

Lemma expE : exp =2 expn.

Lemma oddE : odd =1 oddn.

Definition trecE := (addE , (doubleE , oddE ), (mulE , add_mulE , (expE , mul_expE ))).

End NatTrec.

Notation natTrecE := NatTrec.trecE.

Lemma eq_binP : Equality.axiom N.eqb.

Canonical bin_nat_eqMixin := EqMixin eq_binP.
Canonical bin_nat_eqType := Eval hnf in EqType N bin_nat_eqMixin.

Arguments N.eqb !n !m.

Section NumberInterpretation.

use # [warning="-hiding-delimiting-key" ] attribute once we require Coq 8.18

Import BinPos.

Section Trec.

Import NatTrec.

Fixpoint nat_of_pos p0 :=
  match p0 with
  | xO p ⇒ (nat_of_pos p).*2
  | xI p ⇒ (nat_of_pos p).*2 .+1
  | xH ⇒ 1
  end.

End Trec.

Local Coercion nat_of_pos : positive >-> nat.

Coercion nat_of_bin b := if b is Npos p then p : nat else 0.

Fixpoint pos_of_nat n0 m0 :=
  match n0, m0 with
  | n.+1, m.+2 ⇒ pos_of_nat n m
  | n.+1, 1 ⇒ xO (pos_of_nat n n)
  | n.+1, 0 ⇒ xI (pos_of_nat n n)
  | 0, _ ⇒ xH
  end.

Definition bin_of_nat n0 := if n0 is n.+1 then Npos (pos_of_nat n n) else 0%num.

Lemma bin_of_natK : cancel bin_of_nat nat_of_bin.

Lemma nat_of_binK : cancel nat_of_bin bin_of_nat.

Lemma nat_of_succ_pos p : Pos.succ p = p .+1 :> nat.

Lemma nat_of_add_pos p q : (p + q)%positive = p + q :> nat.

Lemma nat_of_mul_pos p q : (p × q)%positive = p × q :> nat.

Lemma nat_of_add_bin b1 b2 : (b1 + b2)%num = b1 + b2 :> nat.

Lemma nat_of_mul_bin b1 b2 : (b1 × b2)%num = b1 × b2 :> nat.

Lemma nat_of_exp_bin n (b : N) : n ^ b = pow_N 1 muln n b.

End NumberInterpretation.

Big(ger) nat IO; usage: Num 1 072 399 to create large numbers for test cases Eval compute in [Num of some expression] to display the result of an expression that returns a larger integer.

Record number : Type := Num {bin_of_number :> N}.

Definition extend_number (nn : number) m := Num (nn × 1000 + bin_of_nat m).

Coercion extend_number : number >-> Funclass.

Canonical number_subType := [newType for bin_of_number ].
Definition number_eqMixin := Eval hnf in [eqMixin of number by <:].
Canonical number_eqType := Eval hnf in EqType number number_eqMixin.

Notation "[ 'Num' 'of' e ]" := (Num (bin_of_nat e))
(at level 0, format "[ 'Num' 'of' e ]") : nat_scope.

Interface to ring/ring_simplify tactics

Lemma nat_semi_ring : semi_ring_theory 0 1 addn muln (@eq _).

Lemma nat_semi_morph :
semi_morph 0 1 addn muln (@eq _) 0%num 1%num Nplus Nmult pred1 nat_of_bin.

Lemma nat_power_theory : power_theory 1 muln (@eq _) nat_of_bin expn.

Interface to the ring tactic machinery.

Fixpoint pop_succn e := if e is e'.+1 then fun n ⇒ pop_succn e' n .+1 else id.

Ltac pop_succn e := eval lazy beta iota delta [pop_succn] in (pop_succn e 1).

Ltac nat_litteral e :=
  match pop_succn e with
  | ?n.+1 ⇒ constr: (bin_of_nat n)
  | _ ⇒ NotConstant
  end.

Ltac succn_to_add :=
  match goal with
  | |- context G [?e.+1] ⇒
    let x := fresh "NatLit0" in
    match pop_succn e with
    | ?n.+1 ⇒ pose x := n.+1; let G' := context G [x] in change G'
    | _ ?e' ?n ⇒ pose x := n; let G' := context G [x + e'] in change G'
    end; succn_to_add; rewrite {}/x
  | _ ⇒ idtac
  end.

Add Ring nat_ring_ssr : nat_semi_ring (morphism nat_semi_morph,
   constants [nat_litteral], preprocess [succn_to_add],
   power_tac nat_power_theory [nat_litteral]).

A congruence tactic, similar to the boolean one, along with an .+1/+ normalization tactic.

Ltac nat_norm :=
  succn_to_add; rewrite ?add0n ?addn0 -?addnA ?(addSn , addnS , add0n , addn0 ).

Ltac nat_congr := first
[ apply: (congr1 succn _)
| apply: (congr1 predn _)
| apply: (congr1 (addn _) _)
| apply: (congr1 (subn _) _)
| apply: (congr1 (addn ^~ _) _)
| match goal with |- (?X1 + ?X2 = ?X3) ⇒
     symmetry;
     rewrite -1?(addnC X1) -?(addnCA X1);
     apply: (congr1 (addn X1) _);
     symmetry
   end ].