diff options
Diffstat (limited to 'theories/PArith/BinPos.v')
| -rw-r--r-- | theories/PArith/BinPos.v | 382 |
1 files changed, 206 insertions, 176 deletions
diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v index 7baf102a..000d895e 100644 --- a/theories/PArith/BinPos.v +++ b/theories/PArith/BinPos.v @@ -1,10 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Export BinNums. @@ -813,6 +815,34 @@ Proof. rewrite compare_cont_spec. unfold ge. destruct (p ?= q); easy'. Qed. +Lemma compare_cont_Lt_not_Lt p q : + compare_cont Lt p q <> Lt <-> p > q. +Proof. + rewrite compare_cont_Lt_Lt. + unfold gt, le, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + +Lemma compare_cont_Lt_not_Gt p q : + compare_cont Lt p q <> Gt <-> p <= q. +Proof. + apply not_iff_compat, compare_cont_Lt_Gt. +Qed. + +Lemma compare_cont_Gt_not_Lt p q : + compare_cont Gt p q <> Lt <-> p >= q. +Proof. + apply not_iff_compat, compare_cont_Gt_Lt. +Qed. + +Lemma compare_cont_Gt_not_Gt p q : + compare_cont Gt p q <> Gt <-> p < q. +Proof. + rewrite compare_cont_Gt_Gt. + unfold ge, lt, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + (** We can express recursive equations for [compare] *) Lemma compare_xO_xO p q : (p~0 ?= q~0) = (p ?= q). @@ -1625,7 +1655,7 @@ Qed. Lemma sqrtrem_spec p : SqrtSpec (sqrtrem p) p. Proof. -revert p. fix 1. +revert p. fix sqrtrem_spec 1. destruct p; try destruct p; try (constructor; easy); apply sqrtrem_step_spec; auto. Qed. @@ -1875,180 +1905,180 @@ Notation IsNul := Pos.IsNul (only parsing). Notation IsPos := Pos.IsPos (only parsing). Notation IsNeg := Pos.IsNeg (only parsing). -Notation Psucc := Pos.succ (compat "8.3"). -Notation Pplus := Pos.add (compat "8.3"). -Notation Pplus_carry := Pos.add_carry (compat "8.3"). -Notation Ppred := Pos.pred (compat "8.3"). -Notation Piter_op := Pos.iter_op (compat "8.3"). -Notation Piter_op_succ := Pos.iter_op_succ (compat "8.3"). -Notation Pmult_nat := (Pos.iter_op plus) (compat "8.3"). -Notation nat_of_P := Pos.to_nat (compat "8.3"). -Notation P_of_succ_nat := Pos.of_succ_nat (compat "8.3"). -Notation Pdouble_minus_one := Pos.pred_double (compat "8.3"). -Notation positive_mask := Pos.mask (compat "8.3"). -Notation positive_mask_rect := Pos.mask_rect (compat "8.3"). -Notation positive_mask_ind := Pos.mask_ind (compat "8.3"). -Notation positive_mask_rec := Pos.mask_rec (compat "8.3"). -Notation Pdouble_plus_one_mask := Pos.succ_double_mask (compat "8.3"). -Notation Pdouble_mask := Pos.double_mask (compat "8.3"). -Notation Pdouble_minus_two := Pos.double_pred_mask (compat "8.3"). -Notation Pminus_mask := Pos.sub_mask (compat "8.3"). -Notation Pminus_mask_carry := Pos.sub_mask_carry (compat "8.3"). -Notation Pminus := Pos.sub (compat "8.3"). -Notation Pmult := Pos.mul (compat "8.3"). -Notation iter_pos := @Pos.iter (compat "8.3"). -Notation Ppow := Pos.pow (compat "8.3"). -Notation Pdiv2 := Pos.div2 (compat "8.3"). -Notation Pdiv2_up := Pos.div2_up (compat "8.3"). -Notation Psize := Pos.size_nat (compat "8.3"). -Notation Psize_pos := Pos.size (compat "8.3"). -Notation Pcompare x y m := (Pos.compare_cont m x y) (compat "8.3"). -Notation Plt := Pos.lt (compat "8.3"). -Notation Pgt := Pos.gt (compat "8.3"). -Notation Ple := Pos.le (compat "8.3"). -Notation Pge := Pos.ge (compat "8.3"). -Notation Pmin := Pos.min (compat "8.3"). -Notation Pmax := Pos.max (compat "8.3"). -Notation Peqb := Pos.eqb (compat "8.3"). -Notation positive_eq_dec := Pos.eq_dec (compat "8.3"). -Notation xI_succ_xO := Pos.xI_succ_xO (compat "8.3"). -Notation Psucc_discr := Pos.succ_discr (compat "8.3"). +Notation Psucc := Pos.succ (compat "8.6"). +Notation Pplus := Pos.add (only parsing). +Notation Pplus_carry := Pos.add_carry (only parsing). +Notation Ppred := Pos.pred (compat "8.6"). +Notation Piter_op := Pos.iter_op (compat "8.6"). +Notation Piter_op_succ := Pos.iter_op_succ (compat "8.6"). +Notation Pmult_nat := (Pos.iter_op plus) (only parsing). +Notation nat_of_P := Pos.to_nat (only parsing). +Notation P_of_succ_nat := Pos.of_succ_nat (only parsing). +Notation Pdouble_minus_one := Pos.pred_double (only parsing). +Notation positive_mask := Pos.mask (only parsing). +Notation positive_mask_rect := Pos.mask_rect (only parsing). +Notation positive_mask_ind := Pos.mask_ind (only parsing). +Notation positive_mask_rec := Pos.mask_rec (only parsing). +Notation Pdouble_plus_one_mask := Pos.succ_double_mask (only parsing). +Notation Pdouble_mask := Pos.double_mask (compat "8.6"). +Notation Pdouble_minus_two := Pos.double_pred_mask (only parsing). +Notation Pminus_mask := Pos.sub_mask (only parsing). +Notation Pminus_mask_carry := Pos.sub_mask_carry (only parsing). +Notation Pminus := Pos.sub (only parsing). +Notation Pmult := Pos.mul (only parsing). +Notation iter_pos := @Pos.iter (only parsing). +Notation Ppow := Pos.pow (compat "8.6"). +Notation Pdiv2 := Pos.div2 (compat "8.6"). +Notation Pdiv2_up := Pos.div2_up (compat "8.6"). +Notation Psize := Pos.size_nat (only parsing). +Notation Psize_pos := Pos.size (only parsing). +Notation Pcompare x y m := (Pos.compare_cont m x y) (only parsing). +Notation Plt := Pos.lt (compat "8.6"). +Notation Pgt := Pos.gt (compat "8.6"). +Notation Ple := Pos.le (compat "8.6"). +Notation Pge := Pos.ge (compat "8.6"). +Notation Pmin := Pos.min (compat "8.6"). +Notation Pmax := Pos.max (compat "8.6"). +Notation Peqb := Pos.eqb (compat "8.6"). +Notation positive_eq_dec := Pos.eq_dec (only parsing). +Notation xI_succ_xO := Pos.xI_succ_xO (only parsing). +Notation Psucc_discr := Pos.succ_discr (compat "8.6"). Notation Psucc_o_double_minus_one_eq_xO := - Pos.succ_pred_double (compat "8.3"). + Pos.succ_pred_double (only parsing). Notation Pdouble_minus_one_o_succ_eq_xI := - Pos.pred_double_succ (compat "8.3"). -Notation xO_succ_permute := Pos.double_succ (compat "8.3"). + Pos.pred_double_succ (only parsing). +Notation xO_succ_permute := Pos.double_succ (only parsing). Notation double_moins_un_xO_discr := - Pos.pred_double_xO_discr (compat "8.3"). -Notation Psucc_not_one := Pos.succ_not_1 (compat "8.3"). -Notation Ppred_succ := Pos.pred_succ (compat "8.3"). -Notation Psucc_pred := Pos.succ_pred_or (compat "8.3"). -Notation Psucc_inj := Pos.succ_inj (compat "8.3"). -Notation Pplus_carry_spec := Pos.add_carry_spec (compat "8.3"). -Notation Pplus_comm := Pos.add_comm (compat "8.3"). -Notation Pplus_succ_permute_r := Pos.add_succ_r (compat "8.3"). -Notation Pplus_succ_permute_l := Pos.add_succ_l (compat "8.3"). -Notation Pplus_no_neutral := Pos.add_no_neutral (compat "8.3"). -Notation Pplus_carry_plus := Pos.add_carry_add (compat "8.3"). -Notation Pplus_reg_r := Pos.add_reg_r (compat "8.3"). -Notation Pplus_reg_l := Pos.add_reg_l (compat "8.3"). -Notation Pplus_carry_reg_r := Pos.add_carry_reg_r (compat "8.3"). -Notation Pplus_carry_reg_l := Pos.add_carry_reg_l (compat "8.3"). -Notation Pplus_assoc := Pos.add_assoc (compat "8.3"). -Notation Pplus_xO := Pos.add_xO (compat "8.3"). -Notation Pplus_xI_double_minus_one := Pos.add_xI_pred_double (compat "8.3"). -Notation Pplus_xO_double_minus_one := Pos.add_xO_pred_double (compat "8.3"). -Notation Pplus_diag := Pos.add_diag (compat "8.3"). -Notation PeanoView := Pos.PeanoView (compat "8.3"). -Notation PeanoOne := Pos.PeanoOne (compat "8.3"). -Notation PeanoSucc := Pos.PeanoSucc (compat "8.3"). -Notation PeanoView_rect := Pos.PeanoView_rect (compat "8.3"). -Notation PeanoView_ind := Pos.PeanoView_ind (compat "8.3"). -Notation PeanoView_rec := Pos.PeanoView_rec (compat "8.3"). -Notation peanoView_xO := Pos.peanoView_xO (compat "8.3"). -Notation peanoView_xI := Pos.peanoView_xI (compat "8.3"). -Notation peanoView := Pos.peanoView (compat "8.3"). -Notation PeanoView_iter := Pos.PeanoView_iter (compat "8.3"). -Notation eq_dep_eq_positive := Pos.eq_dep_eq_positive (compat "8.3"). -Notation PeanoViewUnique := Pos.PeanoViewUnique (compat "8.3"). -Notation Prect := Pos.peano_rect (compat "8.3"). -Notation Prect_succ := Pos.peano_rect_succ (compat "8.3"). -Notation Prect_base := Pos.peano_rect_base (compat "8.3"). -Notation Prec := Pos.peano_rec (compat "8.3"). -Notation Pind := Pos.peano_ind (compat "8.3"). -Notation Pcase := Pos.peano_case (compat "8.3"). -Notation Pmult_1_r := Pos.mul_1_r (compat "8.3"). -Notation Pmult_Sn_m := Pos.mul_succ_l (compat "8.3"). -Notation Pmult_xO_permute_r := Pos.mul_xO_r (compat "8.3"). -Notation Pmult_xI_permute_r := Pos.mul_xI_r (compat "8.3"). -Notation Pmult_comm := Pos.mul_comm (compat "8.3"). -Notation Pmult_plus_distr_l := Pos.mul_add_distr_l (compat "8.3"). -Notation Pmult_plus_distr_r := Pos.mul_add_distr_r (compat "8.3"). -Notation Pmult_assoc := Pos.mul_assoc (compat "8.3"). -Notation Pmult_xI_mult_xO_discr := Pos.mul_xI_mul_xO_discr (compat "8.3"). -Notation Pmult_xO_discr := Pos.mul_xO_discr (compat "8.3"). -Notation Pmult_reg_r := Pos.mul_reg_r (compat "8.3"). -Notation Pmult_reg_l := Pos.mul_reg_l (compat "8.3"). -Notation Pmult_1_inversion_l := Pos.mul_eq_1_l (compat "8.3"). -Notation Psquare_xO := Pos.square_xO (compat "8.3"). -Notation Psquare_xI := Pos.square_xI (compat "8.3"). -Notation iter_pos_swap_gen := Pos.iter_swap_gen (compat "8.3"). -Notation iter_pos_swap := Pos.iter_swap (compat "8.3"). -Notation iter_pos_succ := Pos.iter_succ (compat "8.3"). -Notation iter_pos_plus := Pos.iter_add (compat "8.3"). -Notation iter_pos_invariant := Pos.iter_invariant (compat "8.3"). -Notation Ppow_1_r := Pos.pow_1_r (compat "8.3"). -Notation Ppow_succ_r := Pos.pow_succ_r (compat "8.3"). -Notation Peqb_refl := Pos.eqb_refl (compat "8.3"). -Notation Peqb_eq := Pos.eqb_eq (compat "8.3"). -Notation Pcompare_refl_id := Pos.compare_cont_refl (compat "8.3"). -Notation Pcompare_eq_iff := Pos.compare_eq_iff (compat "8.3"). -Notation Pcompare_Gt_Lt := Pos.compare_cont_Gt_Lt (compat "8.3"). -Notation Pcompare_eq_Lt := Pos.compare_lt_iff (compat "8.3"). -Notation Pcompare_Lt_Gt := Pos.compare_cont_Lt_Gt (compat "8.3"). - -Notation Pcompare_antisym := Pos.compare_cont_antisym (compat "8.3"). -Notation ZC1 := Pos.gt_lt (compat "8.3"). -Notation ZC2 := Pos.lt_gt (compat "8.3"). -Notation Pcompare_spec := Pos.compare_spec (compat "8.3"). -Notation Pcompare_p_Sp := Pos.lt_succ_diag_r (compat "8.3"). -Notation Pcompare_succ_succ := Pos.compare_succ_succ (compat "8.3"). -Notation Pcompare_1 := Pos.nlt_1_r (compat "8.3"). -Notation Plt_1 := Pos.nlt_1_r (compat "8.3"). -Notation Plt_1_succ := Pos.lt_1_succ (compat "8.3"). -Notation Plt_lt_succ := Pos.lt_lt_succ (compat "8.3"). -Notation Plt_irrefl := Pos.lt_irrefl (compat "8.3"). -Notation Plt_trans := Pos.lt_trans (compat "8.3"). -Notation Plt_ind := Pos.lt_ind (compat "8.3"). -Notation Ple_lteq := Pos.le_lteq (compat "8.3"). -Notation Ple_refl := Pos.le_refl (compat "8.3"). -Notation Ple_lt_trans := Pos.le_lt_trans (compat "8.3"). -Notation Plt_le_trans := Pos.lt_le_trans (compat "8.3"). -Notation Ple_trans := Pos.le_trans (compat "8.3"). -Notation Plt_succ_r := Pos.lt_succ_r (compat "8.3"). -Notation Ple_succ_l := Pos.le_succ_l (compat "8.3"). -Notation Pplus_compare_mono_l := Pos.add_compare_mono_l (compat "8.3"). -Notation Pplus_compare_mono_r := Pos.add_compare_mono_r (compat "8.3"). -Notation Pplus_lt_mono_l := Pos.add_lt_mono_l (compat "8.3"). -Notation Pplus_lt_mono_r := Pos.add_lt_mono_r (compat "8.3"). -Notation Pplus_lt_mono := Pos.add_lt_mono (compat "8.3"). -Notation Pplus_le_mono_l := Pos.add_le_mono_l (compat "8.3"). -Notation Pplus_le_mono_r := Pos.add_le_mono_r (compat "8.3"). -Notation Pplus_le_mono := Pos.add_le_mono (compat "8.3"). -Notation Pmult_compare_mono_l := Pos.mul_compare_mono_l (compat "8.3"). -Notation Pmult_compare_mono_r := Pos.mul_compare_mono_r (compat "8.3"). -Notation Pmult_lt_mono_l := Pos.mul_lt_mono_l (compat "8.3"). -Notation Pmult_lt_mono_r := Pos.mul_lt_mono_r (compat "8.3"). -Notation Pmult_lt_mono := Pos.mul_lt_mono (compat "8.3"). -Notation Pmult_le_mono_l := Pos.mul_le_mono_l (compat "8.3"). -Notation Pmult_le_mono_r := Pos.mul_le_mono_r (compat "8.3"). -Notation Pmult_le_mono := Pos.mul_le_mono (compat "8.3"). -Notation Plt_plus_r := Pos.lt_add_r (compat "8.3"). -Notation Plt_not_plus_l := Pos.lt_not_add_l (compat "8.3"). -Notation Ppow_gt_1 := Pos.pow_gt_1 (compat "8.3"). -Notation Ppred_mask := Pos.pred_mask (compat "8.3"). -Notation Pminus_mask_succ_r := Pos.sub_mask_succ_r (compat "8.3"). -Notation Pminus_mask_carry_spec := Pos.sub_mask_carry_spec (compat "8.3"). -Notation Pminus_succ_r := Pos.sub_succ_r (compat "8.3"). -Notation Pminus_mask_diag := Pos.sub_mask_diag (compat "8.3"). - -Notation Pplus_minus_eq := Pos.add_sub (compat "8.3"). -Notation Pmult_minus_distr_l := Pos.mul_sub_distr_l (compat "8.3"). -Notation Pminus_lt_mono_l := Pos.sub_lt_mono_l (compat "8.3"). -Notation Pminus_compare_mono_l := Pos.sub_compare_mono_l (compat "8.3"). -Notation Pminus_compare_mono_r := Pos.sub_compare_mono_r (compat "8.3"). -Notation Pminus_lt_mono_r := Pos.sub_lt_mono_r (compat "8.3"). -Notation Pminus_decr := Pos.sub_decr (compat "8.3"). -Notation Pminus_xI_xI := Pos.sub_xI_xI (compat "8.3"). -Notation Pplus_minus_assoc := Pos.add_sub_assoc (compat "8.3"). -Notation Pminus_plus_distr := Pos.sub_add_distr (compat "8.3"). -Notation Pminus_minus_distr := Pos.sub_sub_distr (compat "8.3"). -Notation Pminus_mask_Lt := Pos.sub_mask_neg (compat "8.3"). -Notation Pminus_Lt := Pos.sub_lt (compat "8.3"). -Notation Pminus_Eq := Pos.sub_diag (compat "8.3"). -Notation Psize_monotone := Pos.size_nat_monotone (compat "8.3"). -Notation Psize_pos_gt := Pos.size_gt (compat "8.3"). -Notation Psize_pos_le := Pos.size_le (compat "8.3"). + Pos.pred_double_xO_discr (only parsing). +Notation Psucc_not_one := Pos.succ_not_1 (only parsing). +Notation Ppred_succ := Pos.pred_succ (compat "8.6"). +Notation Psucc_pred := Pos.succ_pred_or (only parsing). +Notation Psucc_inj := Pos.succ_inj (compat "8.6"). +Notation Pplus_carry_spec := Pos.add_carry_spec (only parsing). +Notation Pplus_comm := Pos.add_comm (only parsing). +Notation Pplus_succ_permute_r := Pos.add_succ_r (only parsing). +Notation Pplus_succ_permute_l := Pos.add_succ_l (only parsing). +Notation Pplus_no_neutral := Pos.add_no_neutral (only parsing). +Notation Pplus_carry_plus := Pos.add_carry_add (only parsing). +Notation Pplus_reg_r := Pos.add_reg_r (only parsing). +Notation Pplus_reg_l := Pos.add_reg_l (only parsing). +Notation Pplus_carry_reg_r := Pos.add_carry_reg_r (only parsing). +Notation Pplus_carry_reg_l := Pos.add_carry_reg_l (only parsing). +Notation Pplus_assoc := Pos.add_assoc (only parsing). +Notation Pplus_xO := Pos.add_xO (only parsing). +Notation Pplus_xI_double_minus_one := Pos.add_xI_pred_double (only parsing). +Notation Pplus_xO_double_minus_one := Pos.add_xO_pred_double (only parsing). +Notation Pplus_diag := Pos.add_diag (only parsing). +Notation PeanoView := Pos.PeanoView (only parsing). +Notation PeanoOne := Pos.PeanoOne (only parsing). +Notation PeanoSucc := Pos.PeanoSucc (only parsing). +Notation PeanoView_rect := Pos.PeanoView_rect (only parsing). +Notation PeanoView_ind := Pos.PeanoView_ind (only parsing). +Notation PeanoView_rec := Pos.PeanoView_rec (only parsing). +Notation peanoView_xO := Pos.peanoView_xO (only parsing). +Notation peanoView_xI := Pos.peanoView_xI (only parsing). +Notation peanoView := Pos.peanoView (only parsing). +Notation PeanoView_iter := Pos.PeanoView_iter (only parsing). +Notation eq_dep_eq_positive := Pos.eq_dep_eq_positive (only parsing). +Notation PeanoViewUnique := Pos.PeanoViewUnique (only parsing). +Notation Prect := Pos.peano_rect (only parsing). +Notation Prect_succ := Pos.peano_rect_succ (only parsing). +Notation Prect_base := Pos.peano_rect_base (only parsing). +Notation Prec := Pos.peano_rec (only parsing). +Notation Pind := Pos.peano_ind (only parsing). +Notation Pcase := Pos.peano_case (only parsing). +Notation Pmult_1_r := Pos.mul_1_r (only parsing). +Notation Pmult_Sn_m := Pos.mul_succ_l (only parsing). +Notation Pmult_xO_permute_r := Pos.mul_xO_r (only parsing). +Notation Pmult_xI_permute_r := Pos.mul_xI_r (only parsing). +Notation Pmult_comm := Pos.mul_comm (only parsing). +Notation Pmult_plus_distr_l := Pos.mul_add_distr_l (only parsing). +Notation Pmult_plus_distr_r := Pos.mul_add_distr_r (only parsing). +Notation Pmult_assoc := Pos.mul_assoc (only parsing). +Notation Pmult_xI_mult_xO_discr := Pos.mul_xI_mul_xO_discr (only parsing). +Notation Pmult_xO_discr := Pos.mul_xO_discr (only parsing). +Notation Pmult_reg_r := Pos.mul_reg_r (only parsing). +Notation Pmult_reg_l := Pos.mul_reg_l (only parsing). +Notation Pmult_1_inversion_l := Pos.mul_eq_1_l (only parsing). +Notation Psquare_xO := Pos.square_xO (compat "8.6"). +Notation Psquare_xI := Pos.square_xI (compat "8.6"). +Notation iter_pos_swap_gen := Pos.iter_swap_gen (only parsing). +Notation iter_pos_swap := Pos.iter_swap (only parsing). +Notation iter_pos_succ := Pos.iter_succ (only parsing). +Notation iter_pos_plus := Pos.iter_add (only parsing). +Notation iter_pos_invariant := Pos.iter_invariant (only parsing). +Notation Ppow_1_r := Pos.pow_1_r (compat "8.6"). +Notation Ppow_succ_r := Pos.pow_succ_r (compat "8.6"). +Notation Peqb_refl := Pos.eqb_refl (compat "8.6"). +Notation Peqb_eq := Pos.eqb_eq (compat "8.6"). +Notation Pcompare_refl_id := Pos.compare_cont_refl (only parsing). +Notation Pcompare_eq_iff := Pos.compare_eq_iff (only parsing). +Notation Pcompare_Gt_Lt := Pos.compare_cont_Gt_Lt (only parsing). +Notation Pcompare_eq_Lt := Pos.compare_lt_iff (only parsing). +Notation Pcompare_Lt_Gt := Pos.compare_cont_Lt_Gt (only parsing). + +Notation Pcompare_antisym := Pos.compare_cont_antisym (only parsing). +Notation ZC1 := Pos.gt_lt (only parsing). +Notation ZC2 := Pos.lt_gt (only parsing). +Notation Pcompare_spec := Pos.compare_spec (compat "8.6"). +Notation Pcompare_p_Sp := Pos.lt_succ_diag_r (only parsing). +Notation Pcompare_succ_succ := Pos.compare_succ_succ (compat "8.6"). +Notation Pcompare_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1 := Pos.nlt_1_r (only parsing). +Notation Plt_1_succ := Pos.lt_1_succ (compat "8.6"). +Notation Plt_lt_succ := Pos.lt_lt_succ (compat "8.6"). +Notation Plt_irrefl := Pos.lt_irrefl (compat "8.6"). +Notation Plt_trans := Pos.lt_trans (compat "8.6"). +Notation Plt_ind := Pos.lt_ind (compat "8.6"). +Notation Ple_lteq := Pos.le_lteq (compat "8.6"). +Notation Ple_refl := Pos.le_refl (compat "8.6"). +Notation Ple_lt_trans := Pos.le_lt_trans (compat "8.6"). +Notation Plt_le_trans := Pos.lt_le_trans (compat "8.6"). +Notation Ple_trans := Pos.le_trans (compat "8.6"). +Notation Plt_succ_r := Pos.lt_succ_r (compat "8.6"). +Notation Ple_succ_l := Pos.le_succ_l (compat "8.6"). +Notation Pplus_compare_mono_l := Pos.add_compare_mono_l (only parsing). +Notation Pplus_compare_mono_r := Pos.add_compare_mono_r (only parsing). +Notation Pplus_lt_mono_l := Pos.add_lt_mono_l (only parsing). +Notation Pplus_lt_mono_r := Pos.add_lt_mono_r (only parsing). +Notation Pplus_lt_mono := Pos.add_lt_mono (only parsing). +Notation Pplus_le_mono_l := Pos.add_le_mono_l (only parsing). +Notation Pplus_le_mono_r := Pos.add_le_mono_r (only parsing). +Notation Pplus_le_mono := Pos.add_le_mono (only parsing). +Notation Pmult_compare_mono_l := Pos.mul_compare_mono_l (only parsing). +Notation Pmult_compare_mono_r := Pos.mul_compare_mono_r (only parsing). +Notation Pmult_lt_mono_l := Pos.mul_lt_mono_l (only parsing). +Notation Pmult_lt_mono_r := Pos.mul_lt_mono_r (only parsing). +Notation Pmult_lt_mono := Pos.mul_lt_mono (only parsing). +Notation Pmult_le_mono_l := Pos.mul_le_mono_l (only parsing). +Notation Pmult_le_mono_r := Pos.mul_le_mono_r (only parsing). +Notation Pmult_le_mono := Pos.mul_le_mono (only parsing). +Notation Plt_plus_r := Pos.lt_add_r (only parsing). +Notation Plt_not_plus_l := Pos.lt_not_add_l (only parsing). +Notation Ppow_gt_1 := Pos.pow_gt_1 (compat "8.6"). +Notation Ppred_mask := Pos.pred_mask (compat "8.6"). +Notation Pminus_mask_succ_r := Pos.sub_mask_succ_r (only parsing). +Notation Pminus_mask_carry_spec := Pos.sub_mask_carry_spec (only parsing). +Notation Pminus_succ_r := Pos.sub_succ_r (only parsing). +Notation Pminus_mask_diag := Pos.sub_mask_diag (only parsing). + +Notation Pplus_minus_eq := Pos.add_sub (only parsing). +Notation Pmult_minus_distr_l := Pos.mul_sub_distr_l (only parsing). +Notation Pminus_lt_mono_l := Pos.sub_lt_mono_l (only parsing). +Notation Pminus_compare_mono_l := Pos.sub_compare_mono_l (only parsing). +Notation Pminus_compare_mono_r := Pos.sub_compare_mono_r (only parsing). +Notation Pminus_lt_mono_r := Pos.sub_lt_mono_r (only parsing). +Notation Pminus_decr := Pos.sub_decr (only parsing). +Notation Pminus_xI_xI := Pos.sub_xI_xI (only parsing). +Notation Pplus_minus_assoc := Pos.add_sub_assoc (only parsing). +Notation Pminus_plus_distr := Pos.sub_add_distr (only parsing). +Notation Pminus_minus_distr := Pos.sub_sub_distr (only parsing). +Notation Pminus_mask_Lt := Pos.sub_mask_neg (only parsing). +Notation Pminus_Lt := Pos.sub_lt (only parsing). +Notation Pminus_Eq := Pos.sub_diag (only parsing). +Notation Psize_monotone := Pos.size_nat_monotone (only parsing). +Notation Psize_pos_gt := Pos.size_gt (only parsing). +Notation Psize_pos_le := Pos.size_le (only parsing). (** More complex compatibility facts, expressed as lemmas (to preserve scopes for instance) *) |