Peano.v 5 KB
 Pierre Letouzey committed Jul 10, 2019 1 `````` `````` Pierre Letouzey committed Aug 09, 2019 2 3 4 5 6 7 ``````(** * The Theory of Peano Arithmetic and its Coq model *) (** The NatDed development, Pierre Letouzey, 2019. This file is released under the CC0 License, see the LICENSE file *) Require Import Defs NameProofs Mix Meta Theories PreModels Models. `````` Pierre Letouzey committed Jul 10, 2019 8 9 10 11 ``````Import ListNotations. Local Open Scope bool_scope. Local Open Scope eqb_scope. `````` Pierre Letouzey committed Aug 09, 2019 12 13 14 15 ``````(** The Peano axioms *) Definition PeanoSign := Finite.to_infinite peano_sign. `````` Pierre Letouzey committed Jul 10, 2019 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 ``````Definition Zero := Fun "O" []. Definition Succ x := Fun "S" [x]. Notation "x = y" := (Pred "=" [x;y]) : formula_scope. Notation "x + y" := (Fun "+" [x;y]) : formula_scope. Notation "x * y" := (Fun "*" [x;y]) : formula_scope. Module PeanoAx. Local Open Scope formula_scope. Definition ax1 := ∀ (#0 = #0). Definition ax2 := ∀∀ (#1 = #0 -> #0 = #1). Definition ax3 := ∀∀∀ (#2 = #1 /\ #1 = #0 -> #2 = #0). Definition ax4 := ∀∀ (#1 = #0 -> Succ (#1) = Succ (#0)). Definition ax5 := ∀∀∀ (#2 = #1 -> #2 + #0 = #1 + #0). Definition ax6 := ∀∀∀ (#1 = #0 -> #2 + #1 = #2 + #0). Definition ax7 := ∀∀∀ (#2 = #1 -> #2 * #0 = #1 * #0). Definition ax8 := ∀∀∀ (#1 = #0 -> #2 * #1 = #2 * #0). Definition ax9 := ∀ (Zero + #0 = #0). Definition ax10 := ∀∀ (Succ(#1) + #0 = Succ(#1 + #0)). `````` Pierre Letouzey committed Aug 09, 2019 38 ``````Definition ax11 := ∀ (Zero * #0 = Zero). `````` Pierre Letouzey committed Jul 10, 2019 39 40 41 42 43 44 45 46 47 ``````Definition ax12 := ∀∀ (Succ(#1) * #0 = (#1 * #0) + #0). Definition ax13 := ∀∀ (Succ(#1) = Succ(#0) -> #1 = #0). Definition ax14 := ∀ ~(Succ(#0) = Zero). Definition axioms_list := [ ax1; ax2; ax3; ax4; ax5; ax6; ax7; ax8; ax9; ax10; ax11; ax12; ax13; ax14 ]. `````` Pierre Letouzey committed Aug 09, 2019 48 49 50 ``````(** Beware, [bsubst] is ok below for turning [#0] into [Succ #0], but only since it contains now a [lift] of substituted terms inside quantifiers. `````` Pierre Letouzey committed Jul 10, 2019 51 52 53 `````` And the unconventional [∀] before [A[0]] is to get the right bounded vars (Hack !). *) `````` Pierre Letouzey committed Aug 09, 2019 54 55 56 ``````Definition induction_schema A_x := let A_0 := bsubst 0 Zero A_x in let A_Sx := bsubst 0 (Succ(#0)) A_x in `````` Pierre Letouzey committed Jul 10, 2019 57 `````` nForall `````` Pierre Letouzey committed Aug 09, 2019 58 59 `````` (Nat.pred (level A_x)) (((∀ A_0) /\ (∀ (A_x -> A_Sx))) -> ∀ A_x). `````` Pierre Letouzey committed Jul 10, 2019 60 61 62 63 64 65 `````` Local Close Scope formula_scope. Definition IsAx A := List.In A axioms_list \/ exists B, A = induction_schema B /\ `````` Pierre Letouzey committed Aug 09, 2019 66 `````` check PeanoSign B = true /\ `````` Pierre Letouzey committed Jul 10, 2019 67 68 `````` FClosed B. `````` Pierre Letouzey committed Aug 09, 2019 69 ``````Lemma WfAx A : IsAx A -> Wf PeanoSign A. `````` Pierre Letouzey committed Jul 10, 2019 70 71 72 73 74 75 76 ``````Proof. intros [ IN | (B & -> & HB & HB')]. - apply Wf_iff. unfold axioms_list in IN. simpl in IN. intuition; subst; reflexivity. - repeat split; unfold induction_schema; cbn. + rewrite nForall_check. cbn. `````` Pierre Letouzey committed Aug 09, 2019 77 `````` rewrite !check_bsubst, HB; auto. `````` Pierre Letouzey committed Jul 10, 2019 78 79 80 `````` + red. rewrite nForall_level. cbn. assert (level (bsubst 0 Zero B) <= level B). { apply level_bsubst'. auto. } `````` Pierre Letouzey committed Aug 09, 2019 81 82 `````` assert (level (bsubst 0 (Succ(BVar 0)) B) <= level B). { apply level_bsubst'. auto. } `````` Pierre Letouzey committed Jul 10, 2019 83 84 85 86 87 `````` omega with *. + apply nForall_fclosed. red. cbn. assert (FClosed (bsubst 0 Zero B)). { red. rewrite bsubst_fvars. intro x. rewrite Names.union_spec. cbn. red in HB'. intuition. } `````` Pierre Letouzey committed Aug 09, 2019 88 89 90 91 92 `````` assert (FClosed (bsubst 0 (Succ(BVar 0)) B)). { red. rewrite bsubst_fvars. intro x. rewrite Names.union_spec. cbn - [Names.union]. rewrite Names.union_spec. generalize (HB' x) (@Names.empty_spec x). intuition. } `````` Pierre Letouzey committed Jul 10, 2019 93 94 95 96 97 `````` unfold FClosed in *. intuition. Qed. End PeanoAx. `````` Pierre Letouzey committed Aug 09, 2019 98 99 ``````Local Open Scope string. `````` Pierre Letouzey committed Jul 10, 2019 100 ``````Definition PeanoTheory := `````` Pierre Letouzey committed Aug 09, 2019 101 `````` {| sign := PeanoSign; `````` Pierre Letouzey committed Jul 10, 2019 102 103 104 `````` IsAxiom := PeanoAx.IsAx; WfAxiom := PeanoAx.WfAx |}. `````` Pierre Letouzey committed Aug 09, 2019 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 ``````(** A Coq model of this Peano theory, based on the [nat] type *) Definition PeanoFuns : modfuns nat := fun f => if f =? "O" then Some (existT _ 0 0) else if f =? "S" then Some (existT _ 1 S) else if f =? "+" then Some (existT _ 2 Nat.add) else if f =? "*" then Some (existT _ 2 Nat.mul) else None. Definition PeanoPreds : modpreds nat := fun p => if p =? "=" then Some (existT _ 2 (@Logic.eq nat)) else None. Lemma PeanoFuns_ok s : funsymbs PeanoSign s = get_arity (PeanoFuns s). Proof. unfold PeanoSign, peano_sign, PeanoFuns. simpl. unfold eqb, eqb_inst_string. repeat (case string_eqb; auto). Qed. Lemma PeanoPreds_ok s : predsymbs PeanoSign s = get_arity (PeanoPreds s). Proof. unfold PeanoSign, peano_sign, PeanoPreds. simpl. unfold eqb, eqb_inst_string. case string_eqb; auto. Qed. Definition PeanoPreModel : PreModel nat PeanoTheory := {| someone := 0; funs := PeanoFuns; preds := PeanoPreds; funsOk := PeanoFuns_ok; predsOk := PeanoPreds_ok |}. Lemma PeanoAxOk A : IsAxiom PeanoTheory A -> forall genv, interp_form PeanoPreModel genv [] A. Proof. unfold PeanoTheory. simpl. unfold PeanoAx.IsAx. intros [IN|(B & -> & CK & CL)]. - compute in IN. intuition; subst; cbn; intros; subst; omega. - intros genv. unfold PeanoAx.induction_schema. apply interp_nforall. intros stk Len. rewrite app_nil_r. cbn. intros (Base,Step). (* The Peano induction emulated by a Coq induction :-) *) induction m. + specialize (Base 0). apply -> interp_form_bsubst_gen in Base; simpl; eauto. + apply Step in IHm. apply -> interp_form_bsubst_gen in IHm; simpl; eauto. now intros [|k]. Qed. Definition PeanoModel : Model nat PeanoTheory := {| pre := PeanoPreModel; AxOk := PeanoAxOk |}.``````