diff options
Diffstat (limited to 'plugins/setoid_ring/InitialRing.v')
| -rw-r--r-- | plugins/setoid_ring/InitialRing.v | 84 |
1 files changed, 56 insertions, 28 deletions
diff --git a/plugins/setoid_ring/InitialRing.v b/plugins/setoid_ring/InitialRing.v index 9c690e2b..f5db2754 100644 --- a/plugins/setoid_ring/InitialRing.v +++ b/plugins/setoid_ring/InitialRing.v @@ -1,9 +1,11 @@ (************************************************************************) -(* 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 Import Zbool. @@ -48,12 +50,19 @@ Section ZMORPHISM. Notation "x - y " := (rsub x y). Notation "- x" := (ropp x). Notation "x == y" := (req x y). Variable Rsth : Setoid_Theory R req. - Add Setoid R req Rsth as R_setoid3. + Add Parametric Relation : R req + reflexivity proved by Rsth.(@Equivalence_Reflexive _ _) + symmetry proved by Rsth.(@Equivalence_Symmetric _ _) + transitivity proved by Rsth.(@Equivalence_Transitive _ _) + as R_setoid3. Ltac rrefl := gen_reflexivity Rsth. Variable Reqe : ring_eq_ext radd rmul ropp req. - Add Morphism radd : radd_ext3. exact (Radd_ext Reqe). Qed. - Add Morphism rmul : rmul_ext3. exact (Rmul_ext Reqe). Qed. - Add Morphism ropp : ropp_ext3. exact (Ropp_ext Reqe). Qed. + Add Morphism radd with signature (req ==> req ==> req) as radd_ext3. + Proof. exact (Radd_ext Reqe). Qed. + Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext3. + Proof. exact (Rmul_ext Reqe). Qed. + Add Morphism ropp with signature (req ==> req) as ropp_ext3. + Proof. exact (Ropp_ext Reqe). Qed. Fixpoint gen_phiPOS1 (p:positive) : R := match p with @@ -103,7 +112,8 @@ Section ZMORPHISM. Section ALMOST_RING. Variable ARth : almost_ring_theory 0 1 radd rmul rsub ropp req. - Add Morphism rsub : rsub_ext3. exact (ARsub_ext Rsth Reqe ARth). Qed. + Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext3. + Proof. exact (ARsub_ext Rsth Reqe ARth). Qed. Ltac norm := gen_srewrite Rsth Reqe ARth. Ltac add_push := gen_add_push radd Rsth Reqe ARth. @@ -151,7 +161,8 @@ Section ZMORPHISM. Variable Rth : ring_theory 0 1 radd rmul rsub ropp req. Let ARth := Rth_ARth Rsth Reqe Rth. - Add Morphism rsub : rsub_ext4. exact (ARsub_ext Rsth Reqe ARth). Qed. + Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext4. + Proof. exact (ARsub_ext Rsth Reqe ARth). Qed. Ltac norm := gen_srewrite Rsth Reqe ARth. Ltac add_push := gen_add_push radd Rsth Reqe ARth. @@ -255,7 +266,11 @@ Section NMORPHISM. Notation "0" := rO. Notation "1" := rI. Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y). Variable Rsth : Setoid_Theory R req. - Add Setoid R req Rsth as R_setoid4. + Add Parametric Relation : R req + reflexivity proved by Rsth.(@Equivalence_Reflexive _ _) + symmetry proved by Rsth.(@Equivalence_Symmetric _ _) + transitivity proved by Rsth.(@Equivalence_Transitive _ _) + as R_setoid4. Ltac rrefl := gen_reflexivity Rsth. Variable SReqe : sring_eq_ext radd rmul req. Variable SRth : semi_ring_theory 0 1 radd rmul req. @@ -265,8 +280,10 @@ Section NMORPHISM. Let rsub := (@SRsub R radd). Notation "x - y " := (rsub x y). Notation "- x" := (ropp x). Notation "x == y" := (req x y). - Add Morphism radd : radd_ext4. exact (Radd_ext Reqe). Qed. - Add Morphism rmul : rmul_ext4. exact (Rmul_ext Reqe). Qed. + Add Morphism radd with signature (req ==> req ==> req) as radd_ext4. + Proof. exact (Radd_ext Reqe). Qed. + Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext4. + Proof. exact (Rmul_ext Reqe). Qed. Ltac norm := gen_srewrite_sr Rsth Reqe ARth. Definition gen_phiN1 x := @@ -374,15 +391,23 @@ Section NWORDMORPHISM. Notation "x - y " := (rsub x y). Notation "- x" := (ropp x). Notation "x == y" := (req x y). Variable Rsth : Setoid_Theory R req. - Add Setoid R req Rsth as R_setoid5. + Add Parametric Relation : R req + reflexivity proved by Rsth.(@Equivalence_Reflexive _ _) + symmetry proved by Rsth.(@Equivalence_Symmetric _ _) + transitivity proved by Rsth.(@Equivalence_Transitive _ _) + as R_setoid5. Ltac rrefl := gen_reflexivity Rsth. Variable Reqe : ring_eq_ext radd rmul ropp req. - Add Morphism radd : radd_ext5. exact (Radd_ext Reqe). Qed. - Add Morphism rmul : rmul_ext5. exact (Rmul_ext Reqe). Qed. - Add Morphism ropp : ropp_ext5. exact (Ropp_ext Reqe). Qed. + Add Morphism radd with signature (req ==> req ==> req) as radd_ext5. + Proof. exact (Radd_ext Reqe). Qed. + Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext5. + Proof. exact (Rmul_ext Reqe). Qed. + Add Morphism ropp with signature (req ==> req) as ropp_ext5. + Proof. exact (Ropp_ext Reqe). Qed. Variable ARth : almost_ring_theory 0 1 radd rmul rsub ropp req. - Add Morphism rsub : rsub_ext7. exact (ARsub_ext Rsth Reqe ARth). Qed. + Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext7. + Proof. exact (ARsub_ext Rsth Reqe ARth). Qed. Ltac norm := gen_srewrite Rsth Reqe ARth. Ltac add_push := gen_add_push radd Rsth Reqe ARth. @@ -555,12 +580,20 @@ Section GEN_DIV. Variable morph : ring_morph rO rI radd rmul rsub ropp req cO cI cadd cmul csub copp ceqb phi. (* Useful tactics *) - Add Setoid R req Rsth as R_set1. + Add Parametric Relation : R req + reflexivity proved by Rsth.(@Equivalence_Reflexive _ _) + symmetry proved by Rsth.(@Equivalence_Symmetric _ _) + transitivity proved by Rsth.(@Equivalence_Transitive _ _) + as R_set1. Ltac rrefl := gen_reflexivity Rsth. - Add Morphism radd : radd_ext. exact (Radd_ext Reqe). Qed. - Add Morphism rmul : rmul_ext. exact (Rmul_ext Reqe). Qed. - Add Morphism ropp : ropp_ext. exact (Ropp_ext Reqe). Qed. - Add Morphism rsub : rsub_ext. exact (ARsub_ext Rsth Reqe ARth). Qed. + Add Morphism radd with signature (req ==> req ==> req) as radd_ext. + Proof. exact (Radd_ext Reqe). Qed. + Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext. + Proof. exact (Rmul_ext Reqe). Qed. + Add Morphism ropp with signature (req ==> req) as ropp_ext. + Proof. exact (Ropp_ext Reqe). Qed. + Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext. + Proof. exact (ARsub_ext Rsth Reqe ARth). Qed. Ltac rsimpl := gen_srewrite Rsth Reqe ARth. Definition triv_div x y := @@ -859,8 +892,3 @@ Ltac isZcst t := (* *) | _ => constr:(false) end. - - - - - |