Peano.v 13 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 ``````Definition ax12 := ∀∀ (Succ(#1) * #0 = (#1 * #0) + #0). Definition ax13 := ∀∀ (Succ(#1) = Succ(#0) -> #1 = #0). Definition ax14 := ∀ ~(Succ(#0) = Zero). Definition axioms_list := `````` 45 `````` [ ax1; ax2; ax3; ax4; ax5; ax6; ax7; ax8; `````` Pierre Letouzey committed Jul 10, 2019 46 47 `````` 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 ``````Local Open Scope string. `````` 99 100 ``````Local Open Scope formula_scope. `````` Pierre Letouzey committed Jul 10, 2019 101 ``````Definition PeanoTheory := `````` Pierre Letouzey committed Aug 09, 2019 102 `````` {| sign := PeanoSign; `````` Pierre Letouzey committed Jul 10, 2019 103 104 105 `````` IsAxiom := PeanoAx.IsAx; WfAxiom := PeanoAx.WfAx |}. `````` 106 ``````(** Useful lemmas so as to be able to write proofs that take less than 1000 lines. *) `````` Samuel Ben Hamou committed Jun 08, 2020 107 `````` `````` 108 109 ``````Import PeanoAx. `````` Samuel Ben Hamou committed Jun 12, 2020 110 ``````Lemma Symmetry : forall logic A B Γ, BClosed A -> BClosed B -> In ax2 Γ -> Pr logic (Γ ⊢ A = B) -> Pr logic (Γ ⊢ B = A). `````` 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 ``````Proof. intros. apply R_Imp_e with (A := A = B); [ | assumption ]. assert (AX2 : Pr logic (Γ ⊢ ax2)). { apply R_Ax. exact H1. } unfold ax2 in AX2. apply R_All_e with (t := A) in AX2; [ | assumption ]. apply R_All_e with (t := B) in AX2; [ | assumption ]. cbn in AX2. assert (bsubst 0 B (lift A) = A). { assert (lift A = A). { apply lift_nop. exact H. } rewrite H3. apply bclosed_bsubst_id. exact H. } rewrite H3 in AX2. exact AX2. Qed. Lemma Transitivity : forall logic A B C Γ, BClosed A -> BClosed B -> BClosed C -> In ax3 Γ -> Pr logic (Γ ⊢ A = B) -> Pr logic (Γ ⊢ B = C) -> Pr logic (Γ ⊢ A = C). Proof. intros. apply R_Imp_e with (A := A = B /\ B = C); [ | apply R_And_i; assumption ]. assert (AX3 : Pr logic (Γ ⊢ ax3)). { apply R_Ax. exact H2. } unfold ax3 in AX3. apply R_All_e with (t := A) in AX3; [ | assumption ]. apply R_All_e with (t := B) in AX3; [ | assumption ]. apply R_All_e with (t := C) in AX3; [ | assumption ]. cbn in AX3. assert (bsubst 0 C (lift B) = B). { assert (lift B = B). {apply lift_nop. assumption. } rewrite H5. apply bclosed_bsubst_id. assumption. } rewrite H5 in AX3. assert (bsubst 0 C (bsubst 1 (lift B) (lift (lift A))) = A). { assert (lift A = A). { apply lift_nop. assumption. } rewrite H6. rewrite H6. assert (lift B = B). { apply lift_nop. assumption. } rewrite H7. assert (bsubst 1 B A = A). { apply bclosed_bsubst_id. assumption. } rewrite H8. apply bclosed_bsubst_id. assumption. } rewrite H6 in AX3. assumption. Qed. Lemma Hereditarity : forall logic A B Γ, BClosed A -> BClosed B -> In ax4 Γ -> Pr logic (Γ ⊢ A = B) -> Pr logic (Γ ⊢ Succ A = Succ B). Proof. intros. apply R_Imp_e with (A := A = B); [ | assumption ]. assert (AX4 : Pr logic (Γ ⊢ ax4)). { apply R_Ax. assumption. } unfold ax4 in AX4. apply R_All_e with (t := A) in AX4; [ | assumption ]. apply R_All_e with (t := B) in AX4; [ | assumption ]. cbn in AX4. assert (bsubst 0 B (lift A) = A). { assert (lift A = A). { apply lift_nop. assumption. } rewrite H3. apply bclosed_bsubst_id. assumption. } rewrite H3 in AX4. assumption. Qed. Lemma AntiHereditarity : forall logic A B Γ, BClosed A -> BClosed B -> In ax13 Γ -> Pr logic (Γ ⊢ Succ A = Succ B) -> Pr logic (Γ ⊢ A = B). Proof. intros. apply R_Imp_e with (A := Succ A = Succ B); [ | assumption ]. assert (AX13 : Pr logic (Γ ⊢ ax13)). { apply R_Ax. assumption. } unfold ax13 in AX13. apply R_All_e with (t := A) in AX13; [ | assumption ]. apply R_All_e with (t := B) in AX13; [ | assumption ]. cbn in AX13. assert (bsubst 0 B (lift A) = A). { assert (lift A = A). { apply lift_nop. assumption. } rewrite H3. apply bclosed_bsubst_id. assumption. } rewrite H3 in AX13. assumption. Qed. `````` Samuel Ben Hamou committed Jun 16, 2020 183 184 185 186 ``````Ltac axiom ax name := match goal with | |- Pr ?l (?ctx ⊢ _) => assert (name : Pr l (ctx ⊢ ax)); [ apply R_Ax; compute; intuition | ]; unfold ax in name end. `````` Samuel Ben Hamou committed Jun 12, 2020 187 `````` `````` Samuel Ben Hamou committed Jun 12, 2020 188 ``````Ltac app_R_All_i t := apply R_All_i with (x := t); [ rewrite<- Names.mem_spec; now compute | ]. `````` Samuel Ben Hamou committed Jun 16, 2020 189 ``````Ltac eapp_R_All_i := eapply R_All_i; [ rewrite<- Names.mem_spec; now compute | ]. `````` Samuel Ben Hamou committed Jun 12, 2020 190 191 192 `````` Ltac sym := apply Symmetry; [ auto | auto | compute; intuition | ]. `````` Samuel Ben Hamou committed Jun 16, 2020 193 ``````Ltac ahered := apply Hereditarity; [ auto | auto | compute; intuition | ]. `````` Samuel Ben Hamou committed Jun 12, 2020 194 `````` `````` Samuel Ben Hamou committed Jun 16, 2020 195 ``````Ltac hered := apply AntiHereditarity; [ auto | auto | compute; intuition | ]. `````` Samuel Ben Hamou committed Jun 12, 2020 196 `````` `````` Samuel Ben Hamou committed Jun 16, 2020 197 ``````Ltac trans b := apply Transitivity with (B := b); [ auto | auto | auto | compute; intuition | | ]. `````` Samuel Ben Hamou committed Jun 12, 2020 198 199 200 `````` Ltac thm := unfold IsTheorem; split; [ unfold Wf; split; [ auto | split; auto ] | ]. `````` Samuel Ben Hamou committed Jun 16, 2020 201 202 203 204 205 206 207 208 ``````Ltac change_succ := match goal with | |- context[ Fun "S" [?t] ] => change (Fun "S" [t]) with (Succ (t)); change_succ | _ => idtac end. Ltac cbm := cbn; change (Fun "O" []) with Zero; change_succ. `````` Samuel Ben Hamou committed Jul 02, 2020 209 210 211 212 213 214 ``````Ltac parse term := match term with | (_ -> ?queue)%form => parse queue | (∀ ?formula)%form => formula end. `````` Samuel Ben Hamou committed Jun 12, 2020 215 ``````Ltac rec := `````` Samuel Ben Hamou committed Jun 12, 2020 216 `````` match goal with `````` Samuel Ben Hamou committed Jul 02, 2020 217 218 219 `````` | |- exists axs, (Forall (IsAxiom PeanoTheory) axs /\ Pr ?l (axs ⊢ ?A))%type => let form := parse A in exists (induction_schema form :: axioms_list); set (Γ := induction_schema form :: axioms_list); set (rec := induction_schema form); split; `````` Samuel Ben Hamou committed Jun 12, 2020 220 `````` [ apply Forall_forall; intros x H; destruct H; `````` Samuel Ben Hamou committed Jun 16, 2020 221 `````` [ simpl; unfold IsAx; right; exists form ; split; `````` Samuel Ben Hamou committed Jun 12, 2020 222 `````` [ auto | split; [ auto | auto ]] | `````` Samuel Ben Hamou committed Jul 02, 2020 223 224 `````` simpl; unfold IsAx; left; assumption ] | repeat apply R_Imp_i; `````` Samuel Ben Hamou committed Jun 12, 2020 225 `````` eapply R_Imp_e; `````` Samuel Ben Hamou committed Jul 02, 2020 226 `````` [ apply R_Ax; unfold induction_schema; cbm; intuition | simpl; apply R_And_i; cbn ] `````` Samuel Ben Hamou committed Jun 16, 2020 227 228 `````` ]; cbm (* | _ => idtac *) `````` Samuel Ben Hamou committed Jun 12, 2020 229 `````` end. `````` Samuel Ben Hamou committed Jul 02, 2020 230 `````` `````` Samuel Ben Hamou committed Jun 16, 2020 231 `````` `````` 232 233 ``````(** Some basic proofs in Peano arithmetics. *) `````` Samuel Ben Hamou committed Jun 08, 2020 234 ``````Lemma ZeroRight : IsTheorem Intuiti PeanoTheory (∀ (#0 = #0 + Zero)). `````` Samuel Ben Hamou committed Jun 12, 2020 235 236 237 ``````Proof. thm. rec. `````` Samuel Ben Hamou committed Jun 16, 2020 238 `````` + app_R_All_i "y". cbm. `````` Samuel Ben Hamou committed Jun 16, 2020 239 240 241 242 243 244 245 246 247 248 249 250 251 252 `````` sym. axiom ax9 AX9. apply R_All_e with (t := Zero) in AX9; auto. + app_R_All_i "y". cbm. apply R_Imp_i. set (H1 := FVar _ = _). sym. trans (Succ ((FVar "y") + Zero)). - axiom ax10 AX10. apply R_All_e with (t := FVar "y") in AX10; auto. apply R_All_e with (t := Zero) in AX10; auto. - ahered. sym. apply R_Ax. apply in_eq. `````` Samuel Ben Hamou committed Jun 10, 2020 253 ``````Qed. `````` Samuel Ben Hamou committed Jun 08, 2020 254 `````` `````` 255 ``````Lemma SuccRight : IsTheorem Intuiti PeanoTheory (∀∀ (Succ(#1 + #0) = #1 + Succ(#0))). `````` Samuel Ben Hamou committed Jun 08, 2020 256 ``````Proof. `````` Samuel Ben Hamou committed Jun 16, 2020 257 258 `````` thm. rec. `````` Samuel Ben Hamou committed Jun 16, 2020 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 `````` + app_R_All_i "x". app_R_All_i "y". cbm. sym. trans (Succ (FVar "y")). - axiom ax9 AX9. apply R_All_e with (t := Succ (FVar "y")) in AX9; auto. - ahered. sym. axiom ax9 AX9. apply R_All_e with (t := FVar "y") in AX9; auto. + app_R_All_i "x". cbm. apply R_Imp_i. app_R_All_i "y". cbm. set (hyp := ∀ Succ _ = _). trans (Succ (Succ (FVar "x" + FVar "y"))). - ahered. axiom ax10 AX10. apply R_All_e with (t := FVar "x") in AX10; auto. apply R_All_e with (t := FVar "y") in AX10; auto. - trans (Succ (FVar "x" + Succ (FVar "y"))). * ahered. axiom hyp Hyp. apply R_All_e with (t := FVar "y") in Hyp; auto. * axiom ax10 AX10. sym. apply R_All_e with (t := FVar "x") in AX10; auto. apply R_All_e with (t := Succ (FVar "y")) in AX10; auto. `````` Samuel Ben Hamou committed Jun 12, 2020 289 290 ``````Qed. `````` Samuel Ben Hamou committed Jun 30, 2020 291 292 ``````Lemma Comm : IsTheorem Intuiti PeanoTheory `````` Samuel Ben Hamou committed Jul 02, 2020 293 294 `````` ((∀ #0 = #0 + Zero) -> (∀∀ Succ(#1 + #0) = #1 + Succ(#0)) -> (∀∀ #0 + #1 = #1 + #0)). `````` Samuel Ben Hamou committed Jun 30, 2020 295 ``````Proof. `````` Samuel Ben Hamou committed Jul 02, 2020 296 `````` thm. `````` Samuel Ben Hamou committed Jul 03, 2020 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 `````` rec; set (Γ' := _ :: _ :: Γ). + app_R_All_i "x". cbm. app_R_All_i "x". cbm. trans (FVar "x"). - sym. assert (Pr Intuiti (Γ' ⊢ ∀ # 0 = # 0 + Zero)). { apply R_Ax. simpl in *; intuition. } apply R_All_e with (t := FVar "x") in H; auto. - sym. axiom ax9 AX9. apply R_All_e with (t := FVar "x") in AX9; auto. + app_R_All_i "y". cbm. apply R_Imp_i. app_R_All_i "x". cbm. trans (Succ (FVar "x" + FVar "y")). - sym. assert (Pr Intuiti ((∀ # 0 + FVar "y" = FVar "y" + # 0) :: Γ' ⊢ ∀ ∀ Succ (#1 + #0) = #1 + Succ (#0))). { apply R_Ax. simpl in *; intuition. } apply R_All_e with (t := FVar "x") in H; auto. apply R_All_e with (t := FVar "y") in H; auto. - trans (Succ (FVar "y" + FVar "x")). * ahered. assert (Pr Intuiti ((∀ #0 + FVar "y" = FVar "y" + #0) :: Γ' ⊢ ∀ #0 + FVar "y" = FVar "y" + #0)). { apply R_Ax. apply in_eq. } apply R_All_e with (t := FVar "x") in H; auto. * sym. axiom ax10 AX10. apply R_All_e with (t := FVar "y") in AX10; auto. apply R_All_e with (t := FVar "x") in AX10; auto. Qed. `````` Samuel Ben Hamou committed Jun 08, 2020 328 `````` `````` Samuel Ben Hamou committed Jul 03, 2020 329 330 331 332 333 334 335 336 337 ``````Lemma Commutativity : IsTheorem Intuiti PeanoTheory (∀∀ #0 + #1 = #1 + #0). Proof. apply ModusPonens with (A := (∀∀ Succ(#1 + #0) = #1 + Succ(#0))). + apply ModusPonens with (A := ∀ #0 = #0 + Zero). * apply Comm. * apply ZeroRight. + apply SuccRight. Qed. `````` Pierre Letouzey committed Aug 09, 2019 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 ``````(** 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 |}.``````