Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 38,622 Bytes
4365a98 fc5e983 4365a98 fc5e983 4365a98 fc5e983 4365a98 fc5e983 4365a98 fc5e983 4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import data.list.prod_sigma
import data.set.prod
import logic.equiv.fin
import model_theory.language_map
/-!
# Basics on First-Order Syntax
This file defines first-order terms, formulas, sentences, and theories in a style inspired by the
[Flypitch project](https://flypitch.github.io/).
## Main Definitions
* A `first_order.language.term` is defined so that `L.term α` is the type of `L`-terms with free
variables indexed by `α`.
* A `first_order.language.formula` is defined so that `L.formula α` is the type of `L`-formulas with
free variables indexed by `α`.
* A `first_order.language.sentence` is a formula with no free variables.
* A `first_order.language.Theory` is a set of sentences.
* The variables of terms and formulas can be relabelled with `first_order.language.term.relabel`,
`first_order.language.bounded_formula.relabel`, and `first_order.language.formula.relabel`.
* Given an operation on terms and an operation on relations,
`first_order.language.bounded_formula.map_term_rel` gives an operation on formulas.
* `first_order.language.bounded_formula.cast_le` adds more `fin`-indexed variables.
* `first_order.language.bounded_formula.lift_at` raises the indexes of the `fin`-indexed variables
above a particular index.
* `first_order.language.term.subst` and `first_order.language.bounded_formula.subst` substitute
variables with given terms.
* Language maps can act on syntactic objects with functions such as
`first_order.language.Lhom.on_formula`.
* `first_order.language.term.constants_vars_equiv` and
`first_order.language.bounded_formula.constants_vars_equiv` switch terms and formulas between having
constants in the language and having extra variables indexed by the same type.
## Implementation Notes
* Formulas use a modified version of de Bruijn variables. Specifically, a `L.bounded_formula α n`
is a formula with some variables indexed by a type `α`, which cannot be quantified over, and some
indexed by `fin n`, which can. For any `φ : L.bounded_formula α (n + 1)`, we define the formula
`∀' φ : L.bounded_formula α n` by universally quantifying over the variable indexed by
`n : fin (n + 1)`.
## References
For the Flypitch project:
- [J. Han, F. van Doorn, *A formal proof of the independence of the continuum hypothesis*]
[flypitch_cpp]
- [J. Han, F. van Doorn, *A formalization of forcing and the unprovability of
the continuum hypothesis*][flypitch_itp]
-/
universes u v w u' v'
namespace first_order
namespace language
variables (L : language.{u v}) {L' : language}
variables {M : Type w} {N P : Type*} [L.Structure M] [L.Structure N] [L.Structure P]
variables {α : Type u'} {β : Type v'} {γ : Type*}
open_locale first_order
open Structure fin
/-- A term on `α` is either a variable indexed by an element of `α`
or a function symbol applied to simpler terms. -/
inductive term (α : Type u') : Type (max u u')
| var {} : ∀ (a : α), term
| func {} : ∀ {l : ℕ} (f : L.functions l) (ts : fin l → term), term
export term
variable {L}
namespace term
open finset
/-- The `finset` of variables used in a given term. -/
@[simp] def var_finset [decidable_eq α] : L.term α → finset α
| (var i) := {i}
| (func f ts) := univ.bUnion (λ i, (ts i).var_finset)
/-- The `finset` of variables from the left side of a sum used in a given term. -/
@[simp] def var_finset_left [decidable_eq α] : L.term (α ⊕ β) → finset α
| (var (sum.inl i)) := {i}
| (var (sum.inr i)) := ∅
| (func f ts) := univ.bUnion (λ i, (ts i).var_finset_left)
/-- Relabels a term's variables along a particular function. -/
@[simp] def relabel (g : α → β) : L.term α → L.term β
| (var i) := var (g i)
| (func f ts) := func f (λ i, (ts i).relabel)
lemma relabel_id (t : L.term α) :
t.relabel id = t :=
begin
induction t with _ _ _ _ ih,
{ refl, },
{ simp [ih] },
end
@[simp] lemma relabel_id_eq_id :
(term.relabel id : L.term α → L.term α) = id :=
funext relabel_id
@[simp] lemma relabel_relabel (f : α → β) (g : β → γ) (t : L.term α) :
(t.relabel f).relabel g = t.relabel (g ∘ f) :=
begin
induction t with _ _ _ _ ih,
{ refl, },
{ simp [ih] },
end
@[simp] lemma relabel_comp_relabel (f : α → β) (g : β → γ) :
(term.relabel g ∘ term.relabel f : L.term α → L.term γ) = term.relabel (g ∘ f) :=
funext (relabel_relabel f g)
/-- Relabels a term's variables along a bijection. -/
@[simps] def relabel_equiv (g : α ≃ β) : L.term α ≃ L.term β :=
⟨relabel g, relabel g.symm, λ t, by simp, λ t, by simp⟩
/-- Restricts a term to use only a set of the given variables. -/
def restrict_var [decidable_eq α] : Π (t : L.term α) (f : t.var_finset → β), L.term β
| (var a) f := var (f ⟨a, mem_singleton_self a⟩)
| (func F ts) f := func F (λ i, (ts i).restrict_var
(f ∘ set.inclusion (subset_bUnion_of_mem _ (mem_univ i))))
/-- Restricts a term to use only a set of the given variables on the left side of a sum. -/
def restrict_var_left [decidable_eq α] {γ : Type*} :
Π (t : L.term (α ⊕ γ)) (f : t.var_finset_left → β), L.term (β ⊕ γ)
| (var (sum.inl a)) f := var (sum.inl (f ⟨a, mem_singleton_self a⟩))
| (var (sum.inr a)) f := var (sum.inr a)
| (func F ts) f := func F (λ i, (ts i).restrict_var_left
(f ∘ set.inclusion (subset_bUnion_of_mem _ (mem_univ i))))
end term
/-- The representation of a constant symbol as a term. -/
def constants.term (c : L.constants) : (L.term α) :=
func c default
/-- Applies a unary function to a term. -/
def functions.apply₁ (f : L.functions 1) (t : L.term α) : L.term α := func f ![t]
/-- Applies a binary function to two terms. -/
def functions.apply₂ (f : L.functions 2) (t₁ t₂ : L.term α) : L.term α := func f ![t₁, t₂]
namespace term
/-- Sends a term with constants to a term with extra variables. -/
@[simp] def constants_to_vars : L[[γ]].term α → L.term (γ ⊕ α)
| (var a) := var (sum.inr a)
| (@func _ _ 0 f ts) := sum.cases_on f (λ f, func f (λ i, (ts i).constants_to_vars))
(λ c, var (sum.inl c))
| (@func _ _ (n + 1) f ts) := sum.cases_on f (λ f, func f (λ i, (ts i).constants_to_vars))
(λ c, is_empty_elim c)
/-- Sends a term with extra variables to a term with constants. -/
@[simp] def vars_to_constants : L.term (γ ⊕ α) → L[[γ]].term α
| (var (sum.inr a)) := var a
| (var (sum.inl c)) := constants.term (sum.inr c)
| (func f ts) := func (sum.inl f) (λ i, (ts i).vars_to_constants)
/-- A bijection between terms with constants and terms with extra variables. -/
@[simps] def constants_vars_equiv : L[[γ]].term α ≃ L.term (γ ⊕ α) :=
⟨constants_to_vars, vars_to_constants, begin
intro t,
induction t with _ n f _ ih,
{ refl },
{ cases n,
{ cases f,
{ simp [constants_to_vars, vars_to_constants, ih] },
{ simp [constants_to_vars, vars_to_constants, constants.term] } },
{ cases f,
{ simp [constants_to_vars, vars_to_constants, ih] },
{ exact is_empty_elim f } } }
end, begin
intro t,
induction t with x n f _ ih,
{ cases x;
refl },
{ cases n;
{ simp [vars_to_constants, constants_to_vars, ih] } }
end⟩
/-- A bijection between terms with constants and terms with extra variables. -/
def constants_vars_equiv_left : L[[γ]].term (α ⊕ β) ≃ L.term ((γ ⊕ α) ⊕ β) :=
constants_vars_equiv.trans (relabel_equiv (equiv.sum_assoc _ _ _)).symm
@[simp] lemma constants_vars_equiv_left_apply (t : L[[γ]].term (α ⊕ β)) :
constants_vars_equiv_left t = (constants_to_vars t).relabel (equiv.sum_assoc _ _ _).symm :=
rfl
@[simp] lemma constants_vars_equiv_left_symm_apply (t : L.term ((γ ⊕ α) ⊕ β)) :
constants_vars_equiv_left.symm t = vars_to_constants (t.relabel (equiv.sum_assoc _ _ _)) :=
rfl
instance inhabited_of_var [inhabited α] : inhabited (L.term α) :=
⟨var default⟩
instance inhabited_of_constant [inhabited L.constants] : inhabited (L.term α) :=
⟨(default : L.constants).term⟩
/-- Raises all of the `fin`-indexed variables of a term greater than or equal to `m` by `n'`. -/
def lift_at {n : ℕ} (n' m : ℕ) : L.term (α ⊕ fin n) → L.term (α ⊕ fin (n + n')) :=
relabel (sum.map id (λ i, if ↑i < m then fin.cast_add n' i else fin.add_nat n' i))
/-- Substitutes the variables in a given term with terms. -/
@[simp] def subst : L.term α → (α → L.term β) → L.term β
| (var a) tf := tf a
| (func f ts) tf := (func f (λ i, (ts i).subst tf))
end term
localized "prefix `&`:max := first_order.language.term.var ∘ sum.inr" in first_order
namespace Lhom
/-- Maps a term's symbols along a language map. -/
@[simp] def on_term (φ : L →ᴸ L') : L.term α → L'.term α
| (var i) := var i
| (func f ts) := func (φ.on_function f) (λ i, on_term (ts i))
@[simp] lemma id_on_term :
((Lhom.id L).on_term : L.term α → L.term α) = id :=
begin
ext t,
induction t with _ _ _ _ ih,
{ refl },
{ simp_rw [on_term, ih],
refl, },
end
@[simp] lemma comp_on_term {L'' : language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
((φ.comp ψ).on_term : L.term α → L''.term α) = φ.on_term ∘ ψ.on_term :=
begin
ext t,
induction t with _ _ _ _ ih,
{ refl },
{ simp_rw [on_term, ih],
refl, },
end
end Lhom
/-- Maps a term's symbols along a language equivalence. -/
@[simps] def Lequiv.on_term (φ : L ≃ᴸ L') : L.term α ≃ L'.term α :=
{ to_fun := φ.to_Lhom.on_term,
inv_fun := φ.inv_Lhom.on_term,
left_inv := by rw [function.left_inverse_iff_comp, ← Lhom.comp_on_term, φ.left_inv,
Lhom.id_on_term],
right_inv := by rw [function.right_inverse_iff_comp, ← Lhom.comp_on_term, φ.right_inv,
Lhom.id_on_term] }
variables (L) (α)
/-- `bounded_formula α n` is the type of formulas with free variables indexed by `α` and up to `n`
additional free variables. -/
inductive bounded_formula : ℕ → Type (max u v u')
| falsum {} {n} : bounded_formula n
| equal {n} (t₁ t₂ : L.term (α ⊕ fin n)) : bounded_formula n
| rel {n l : ℕ} (R : L.relations l) (ts : fin l → L.term (α ⊕ fin n)) : bounded_formula n
| imp {n} (f₁ f₂ : bounded_formula n) : bounded_formula n
| all {n} (f : bounded_formula (n+1)) : bounded_formula n
/-- `formula α` is the type of formulas with all free variables indexed by `α`. -/
@[reducible] def formula := L.bounded_formula α 0
/-- A sentence is a formula with no free variables. -/
@[reducible] def sentence := L.formula empty
/-- A theory is a set of sentences. -/
@[reducible] def Theory := set L.sentence
variables {L} {α} {n : ℕ}
/-- Applies a relation to terms as a bounded formula. -/
def relations.bounded_formula {l : ℕ} (R : L.relations n) (ts : fin n → L.term (α ⊕ fin l)) :
L.bounded_formula α l := bounded_formula.rel R ts
/-- Applies a unary relation to a term as a bounded formula. -/
def relations.bounded_formula₁ (r : L.relations 1) (t : L.term (α ⊕ fin n)) :
L.bounded_formula α n :=
r.bounded_formula ![t]
/-- Applies a binary relation to two terms as a bounded formula. -/
def relations.bounded_formula₂ (r : L.relations 2) (t₁ t₂ : L.term (α ⊕ fin n)) :
L.bounded_formula α n :=
r.bounded_formula ![t₁, t₂]
/-- The equality of two terms as a bounded formula. -/
def term.bd_equal (t₁ t₂ : L.term (α ⊕ fin n)) : (L.bounded_formula α n) :=
bounded_formula.equal t₁ t₂
/-- Applies a relation to terms as a bounded formula. -/
def relations.formula (R : L.relations n) (ts : fin n → L.term α) :
L.formula α := R.bounded_formula (λ i, (ts i).relabel sum.inl)
/-- Applies a unary relation to a term as a formula. -/
def relations.formula₁ (r : L.relations 1) (t : L.term α) :
L.formula α :=
r.formula ![t]
/-- Applies a binary relation to two terms as a formula. -/
def relations.formula₂ (r : L.relations 2) (t₁ t₂ : L.term α) :
L.formula α :=
r.formula ![t₁, t₂]
/-- The equality of two terms as a first-order formula. -/
def term.equal (t₁ t₂ : L.term α) : (L.formula α) :=
(t₁.relabel sum.inl).bd_equal (t₂.relabel sum.inl)
namespace bounded_formula
instance : inhabited (L.bounded_formula α n) :=
⟨falsum⟩
instance : has_bot (L.bounded_formula α n) := ⟨falsum⟩
/-- The negation of a bounded formula is also a bounded formula. -/
@[pattern] protected def not (φ : L.bounded_formula α n) : L.bounded_formula α n := φ.imp ⊥
/-- Puts an `∃` quantifier on a bounded formula. -/
@[pattern] protected def ex (φ : L.bounded_formula α (n + 1)) : L.bounded_formula α n :=
φ.not.all.not
instance : has_top (L.bounded_formula α n) := ⟨bounded_formula.not ⊥⟩
instance : has_inf (L.bounded_formula α n) := ⟨λ f g, (f.imp g.not).not⟩
instance : has_sup (L.bounded_formula α n) := ⟨λ f g, f.not.imp g⟩
/-- The biimplication between two bounded formulas. -/
protected def iff (φ ψ : L.bounded_formula α n) := φ.imp ψ ⊓ ψ.imp φ
open finset
/-- The `finset` of variables used in a given formula. -/
@[simp] def free_var_finset [decidable_eq α] :
∀ {n}, L.bounded_formula α n → finset α
| n falsum := ∅
| n (equal t₁ t₂) := t₁.var_finset_left ∪ t₂.var_finset_left
| n (rel R ts) := univ.bUnion (λ i, (ts i).var_finset_left)
| n (imp f₁ f₂) := f₁.free_var_finset ∪ f₂.free_var_finset
| n (all f) := f.free_var_finset
/-- Casts `L.bounded_formula α m` as `L.bounded_formula α n`, where `m ≤ n`. -/
@[simp] def cast_le : ∀ {m n : ℕ} (h : m ≤ n), L.bounded_formula α m → L.bounded_formula α n
| m n h falsum := falsum
| m n h (equal t₁ t₂) := equal (t₁.relabel (sum.map id (fin.cast_le h)))
(t₂.relabel (sum.map id (fin.cast_le h)))
| m n h (rel R ts) := rel R (term.relabel (sum.map id (fin.cast_le h)) ∘ ts)
| m n h (imp f₁ f₂) := (f₁.cast_le h).imp (f₂.cast_le h)
| m n h (all f) := (f.cast_le (add_le_add_right h 1)).all
@[simp] lemma cast_le_rfl {n} (h : n ≤ n) (φ : L.bounded_formula α n) :
φ.cast_le h = φ :=
begin
induction φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ simp [fin.cast_le_of_eq], },
{ simp [fin.cast_le_of_eq], },
{ simp [fin.cast_le_of_eq, ih1, ih2], },
{ simp [fin.cast_le_of_eq, ih3], },
end
@[simp] lemma cast_le_cast_le {k m n} (km : k ≤ m) (mn : m ≤ n) (φ : L.bounded_formula α k) :
(φ.cast_le km).cast_le mn = φ.cast_le (km.trans mn) :=
begin
revert m n,
induction φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3;
intros m n km mn,
{ refl },
{ simp },
{ simp only [cast_le, eq_self_iff_true, heq_iff_eq, true_and],
rw [← function.comp.assoc, relabel_comp_relabel],
simp },
{ simp [ih1, ih2] },
{ simp only [cast_le, ih3] }
end
@[simp] lemma cast_le_comp_cast_le {k m n} (km : k ≤ m) (mn : m ≤ n) :
(bounded_formula.cast_le mn ∘ bounded_formula.cast_le km :
L.bounded_formula α k → L.bounded_formula α n) =
bounded_formula.cast_le (km.trans mn) :=
funext (cast_le_cast_le km mn)
/-- Restricts a bounded formula to only use a particular set of free variables. -/
def restrict_free_var [decidable_eq α] : Π {n : ℕ} (φ : L.bounded_formula α n)
(f : φ.free_var_finset → β), L.bounded_formula β n
| n falsum f := falsum
| n (equal t₁ t₂) f := equal
(t₁.restrict_var_left (f ∘ set.inclusion (subset_union_left _ _)))
(t₂.restrict_var_left (f ∘ set.inclusion (subset_union_right _ _)))
| n (rel R ts) f := rel R (λ i, (ts i).restrict_var_left
(f ∘ set.inclusion (subset_bUnion_of_mem _ (mem_univ i))))
| n (imp φ₁ φ₂) f :=
(φ₁.restrict_free_var (f ∘ set.inclusion (subset_union_left _ _))).imp
(φ₂.restrict_free_var (f ∘ set.inclusion (subset_union_right _ _)))
| n (all φ) f := (φ.restrict_free_var f).all
/-- Places universal quantifiers on all extra variables of a bounded formula. -/
def alls : ∀ {n}, L.bounded_formula α n → L.formula α
| 0 φ := φ
| (n + 1) φ := φ.all.alls
/-- Places existential quantifiers on all extra variables of a bounded formula. -/
def exs : ∀ {n}, L.bounded_formula α n → L.formula α
| 0 φ := φ
| (n + 1) φ := φ.ex.exs
/-- Maps bounded formulas along a map of terms and a map of relations. -/
def map_term_rel {g : ℕ → ℕ}
(ft : ∀ n, L.term (α ⊕ fin n) → L'.term (β ⊕ fin (g n)))
(fr : ∀ n, L.relations n → L'.relations n)
(h : ∀ n, L'.bounded_formula β (g (n + 1)) → L'.bounded_formula β (g n + 1)) :
∀ {n}, L.bounded_formula α n → L'.bounded_formula β (g n)
| n falsum := falsum
| n (equal t₁ t₂) := equal (ft _ t₁) (ft _ t₂)
| n (rel R ts) := rel (fr _ R) (λ i, ft _ (ts i))
| n (imp φ₁ φ₂) := φ₁.map_term_rel.imp φ₂.map_term_rel
| n (all φ) := (h n φ.map_term_rel).all
/-- Raises all of the `fin`-indexed variables of a formula greater than or equal to `m` by `n'`. -/
def lift_at : ∀ {n : ℕ} (n' m : ℕ), L.bounded_formula α n → L.bounded_formula α (n + n') :=
λ n n' m φ, φ.map_term_rel (λ k t, t.lift_at n' m) (λ _, id)
(λ _, cast_le (by rw [add_assoc, add_comm 1, add_assoc]))
@[simp] lemma map_term_rel_map_term_rel {L'' : language}
(ft : ∀ n, L.term (α ⊕ fin n) → L'.term (β ⊕ fin n))
(fr : ∀ n, L.relations n → L'.relations n)
(ft' : ∀ n, L'.term (β ⊕ fin n) → L''.term (γ ⊕ fin n))
(fr' : ∀ n, L'.relations n → L''.relations n)
{n} (φ : L.bounded_formula α n) :
(φ.map_term_rel ft fr (λ _, id)).map_term_rel ft' fr' (λ _, id) =
φ.map_term_rel (λ _, (ft' _) ∘ (ft _)) (λ _, (fr' _) ∘ (fr _)) (λ _, id) :=
begin
induction φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ simp [map_term_rel] },
{ simp [map_term_rel] },
{ simp [map_term_rel, ih1, ih2] },
{ simp [map_term_rel, ih3], }
end
@[simp] lemma map_term_rel_id_id_id {n} (φ : L.bounded_formula α n) :
φ.map_term_rel (λ _, id) (λ _, id) (λ _, id) = φ :=
begin
induction φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ simp [map_term_rel] },
{ simp [map_term_rel] },
{ simp [map_term_rel, ih1, ih2] },
{ simp [map_term_rel, ih3], }
end
/-- An equivalence of bounded formulas given by an equivalence of terms and an equivalence of
relations. -/
@[simps] def map_term_rel_equiv (ft : ∀ n, L.term (α ⊕ fin n) ≃ L'.term (β ⊕ fin n))
(fr : ∀ n, L.relations n ≃ L'.relations n) {n} :
L.bounded_formula α n ≃ L'.bounded_formula β n :=
⟨map_term_rel (λ n, ft n) (λ n, fr n) (λ _, id),
map_term_rel (λ n, (ft n).symm) (λ n, (fr n).symm) (λ _, id),
λ φ, by simp, λ φ, by simp⟩
/-- A function to help relabel the variables in bounded formulas. -/
def relabel_aux (g : α → β ⊕ fin n) (k : ℕ) :
α ⊕ fin k → β ⊕ fin (n + k) :=
sum.map id fin_sum_fin_equiv ∘ equiv.sum_assoc _ _ _ ∘ sum.map g id
@[simp] lemma sum_elim_comp_relabel_aux {m : ℕ} {g : α → (β ⊕ fin n)}
{v : β → M} {xs : fin (n + m) → M} :
sum.elim v xs ∘ relabel_aux g m =
sum.elim (sum.elim v (xs ∘ cast_add m) ∘ g) (xs ∘ nat_add n) :=
begin
ext x,
cases x,
{ simp only [bounded_formula.relabel_aux, function.comp_app, sum.map_inl, sum.elim_inl],
cases g x with l r;
simp },
{ simp [bounded_formula.relabel_aux] }
end
@[simp] lemma relabel_aux_sum_inl (k : ℕ) :
relabel_aux (sum.inl : α → α ⊕ fin n) k =
sum.map id (nat_add n) :=
begin
ext x,
cases x;
{ simp [relabel_aux] },
end
/-- Relabels a bounded formula's variables along a particular function. -/
def relabel (g : α → (β ⊕ fin n)) {k} (φ : L.bounded_formula α k) :
L.bounded_formula β (n + k) :=
φ.map_term_rel (λ _ t, t.relabel (relabel_aux g _)) (λ _, id)
(λ _, cast_le (ge_of_eq (add_assoc _ _ _)))
@[simp] lemma relabel_falsum (g : α → (β ⊕ fin n)) {k} :
(falsum : L.bounded_formula α k).relabel g = falsum :=
rfl
@[simp] lemma relabel_bot (g : α → (β ⊕ fin n)) {k} :
(⊥ : L.bounded_formula α k).relabel g = ⊥ :=
rfl
@[simp] lemma relabel_imp (g : α → (β ⊕ fin n)) {k} (φ ψ : L.bounded_formula α k) :
(φ.imp ψ).relabel g = (φ.relabel g).imp (ψ.relabel g) :=
rfl
@[simp] lemma relabel_not (g : α → (β ⊕ fin n)) {k} (φ : L.bounded_formula α k) :
φ.not.relabel g = (φ.relabel g).not :=
by simp [bounded_formula.not]
@[simp] lemma relabel_all (g : α → (β ⊕ fin n)) {k} (φ : L.bounded_formula α (k + 1)) :
φ.all.relabel g = (φ.relabel g).all :=
begin
rw [relabel, map_term_rel, relabel],
simp,
end
@[simp] lemma relabel_ex (g : α → (β ⊕ fin n)) {k} (φ : L.bounded_formula α (k + 1)) :
φ.ex.relabel g = (φ.relabel g).ex :=
by simp [bounded_formula.ex]
@[simp] lemma relabel_sum_inl (φ : L.bounded_formula α n) :
(φ.relabel sum.inl : L.bounded_formula α (0 + n)) =
φ.cast_le (ge_of_eq (zero_add n)) :=
begin
simp only [relabel, relabel_aux_sum_inl],
induction φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ simp [fin.nat_add_zero, cast_le_of_eq, map_term_rel] },
{ simp [fin.nat_add_zero, cast_le_of_eq, map_term_rel] },
{ simp [map_term_rel, ih1, ih2], },
{ simp [map_term_rel, ih3, cast_le], },
end
/-- Substitutes the variables in a given formula with terms. -/
@[simp] def subst {n : ℕ} (φ : L.bounded_formula α n) (f : α → L.term β) : L.bounded_formula β n :=
φ.map_term_rel (λ _ t, t.subst (sum.elim (term.relabel sum.inl ∘ f) (var ∘ sum.inr)))
(λ _, id) (λ _, id)
/-- A bijection sending formulas with constants to formulas with extra variables. -/
def constants_vars_equiv : L[[γ]].bounded_formula α n ≃ L.bounded_formula (γ ⊕ α) n :=
map_term_rel_equiv (λ _, term.constants_vars_equiv_left) (λ _, equiv.sum_empty _ _)
/-- Turns the extra variables of a bounded formula into free variables. -/
@[simp] def to_formula : ∀ {n : ℕ}, L.bounded_formula α n → L.formula (α ⊕ fin n)
| n falsum := falsum
| n (equal t₁ t₂) := t₁.equal t₂
| n (rel R ts) := R.formula ts
| n (imp φ₁ φ₂) := φ₁.to_formula.imp φ₂.to_formula
| n (all φ) := (φ.to_formula.relabel
(sum.elim (sum.inl ∘ sum.inl) (sum.map sum.inr id ∘ fin_sum_fin_equiv.symm))).all
variables {l : ℕ} {φ ψ : L.bounded_formula α l} {θ : L.bounded_formula α l.succ}
variables {v : α → M} {xs : fin l → M}
/-- An atomic formula is either equality or a relation symbol applied to terms.
Note that `⊥` and `⊤` are not considered atomic in this convention. -/
inductive is_atomic : L.bounded_formula α n → Prop
| equal (t₁ t₂ : L.term (α ⊕ fin n)) : is_atomic (bd_equal t₁ t₂)
| rel {l : ℕ} (R : L.relations l) (ts : fin l → L.term (α ⊕ fin n)) :
is_atomic (R.bounded_formula ts)
lemma not_all_is_atomic (φ : L.bounded_formula α (n + 1)) :
¬ φ.all.is_atomic :=
λ con, by cases con
lemma not_ex_is_atomic (φ : L.bounded_formula α (n + 1)) :
¬ φ.ex.is_atomic :=
λ con, by cases con
lemma is_atomic.relabel {m : ℕ} {φ : L.bounded_formula α m} (h : φ.is_atomic)
(f : α → β ⊕ (fin n)) :
(φ.relabel f).is_atomic :=
is_atomic.rec_on h (λ _ _, is_atomic.equal _ _) (λ _ _ _, is_atomic.rel _ _)
lemma is_atomic.lift_at {k m : ℕ} (h : is_atomic φ) : (φ.lift_at k m).is_atomic :=
is_atomic.rec_on h (λ _ _, is_atomic.equal _ _) (λ _ _ _, is_atomic.rel _ _)
lemma is_atomic.cast_le {h : l ≤ n} (hφ : is_atomic φ) :
(φ.cast_le h).is_atomic :=
is_atomic.rec_on hφ (λ _ _, is_atomic.equal _ _) (λ _ _ _, is_atomic.rel _ _)
/-- A quantifier-free formula is a formula defined without quantifiers. These are all equivalent
to boolean combinations of atomic formulas. -/
inductive is_qf : L.bounded_formula α n → Prop
| falsum : is_qf falsum
| of_is_atomic {φ} (h : is_atomic φ) : is_qf φ
| imp {φ₁ φ₂} (h₁ : is_qf φ₁) (h₂ : is_qf φ₂) : is_qf (φ₁.imp φ₂)
lemma is_atomic.is_qf {φ : L.bounded_formula α n} : is_atomic φ → is_qf φ :=
is_qf.of_is_atomic
lemma is_qf_bot : is_qf (⊥ : L.bounded_formula α n) :=
is_qf.falsum
lemma is_qf.not {φ : L.bounded_formula α n} (h : is_qf φ) :
is_qf φ.not :=
h.imp is_qf_bot
lemma is_qf.relabel {m : ℕ} {φ : L.bounded_formula α m} (h : φ.is_qf)
(f : α → β ⊕ (fin n)) :
(φ.relabel f).is_qf :=
is_qf.rec_on h is_qf_bot (λ _ h, (h.relabel f).is_qf) (λ _ _ _ _ h1 h2, h1.imp h2)
lemma is_qf.lift_at {k m : ℕ} (h : is_qf φ) : (φ.lift_at k m).is_qf :=
is_qf.rec_on h is_qf_bot (λ _ ih, ih.lift_at.is_qf) (λ _ _ _ _ ih1 ih2, ih1.imp ih2)
lemma is_qf.cast_le {h : l ≤ n} (hφ : is_qf φ) :
(φ.cast_le h).is_qf :=
is_qf.rec_on hφ is_qf_bot (λ _ ih, ih.cast_le.is_qf) (λ _ _ _ _ ih1 ih2, ih1.imp ih2)
lemma not_all_is_qf (φ : L.bounded_formula α (n + 1)) :
¬ φ.all.is_qf :=
λ con, begin
cases con with _ con,
exact (φ.not_all_is_atomic con),
end
lemma not_ex_is_qf (φ : L.bounded_formula α (n + 1)) :
¬ φ.ex.is_qf :=
λ con, begin
cases con with _ con _ _ con,
{ exact (φ.not_ex_is_atomic con) },
{ exact not_all_is_qf _ con }
end
/-- Indicates that a bounded formula is in prenex normal form - that is, it consists of quantifiers
applied to a quantifier-free formula. -/
inductive is_prenex : ∀ {n}, L.bounded_formula α n → Prop
| of_is_qf {n} {φ : L.bounded_formula α n} (h : is_qf φ) : is_prenex φ
| all {n} {φ : L.bounded_formula α (n + 1)} (h : is_prenex φ) : is_prenex φ.all
| ex {n} {φ : L.bounded_formula α (n + 1)} (h : is_prenex φ) : is_prenex φ.ex
lemma is_qf.is_prenex {φ : L.bounded_formula α n} : is_qf φ → is_prenex φ :=
is_prenex.of_is_qf
lemma is_atomic.is_prenex {φ : L.bounded_formula α n} (h : is_atomic φ) : is_prenex φ :=
h.is_qf.is_prenex
lemma is_prenex.induction_on_all_not {P : ∀ {n}, L.bounded_formula α n → Prop}
{φ : L.bounded_formula α n}
(h : is_prenex φ)
(hq : ∀ {m} {ψ : L.bounded_formula α m}, ψ.is_qf → P ψ)
(ha : ∀ {m} {ψ : L.bounded_formula α (m + 1)}, P ψ → P ψ.all)
(hn : ∀ {m} {ψ : L.bounded_formula α m}, P ψ → P ψ.not) :
P φ :=
is_prenex.rec_on h (λ _ _, hq) (λ _ _ _, ha) (λ _ _ _ ih, hn (ha (hn ih)))
lemma is_prenex.relabel {m : ℕ} {φ : L.bounded_formula α m} (h : φ.is_prenex)
(f : α → β ⊕ (fin n)) :
(φ.relabel f).is_prenex :=
is_prenex.rec_on h
(λ _ _ h, (h.relabel f).is_prenex)
(λ _ _ _ h, by simp [h.all])
(λ _ _ _ h, by simp [h.ex])
lemma is_prenex.cast_le (hφ : is_prenex φ) :
∀ {n} {h : l ≤ n}, (φ.cast_le h).is_prenex :=
is_prenex.rec_on hφ
(λ _ _ ih _ _, ih.cast_le.is_prenex)
(λ _ _ _ ih _ _, ih.all)
(λ _ _ _ ih _ _, ih.ex)
lemma is_prenex.lift_at {k m : ℕ} (h : is_prenex φ) : (φ.lift_at k m).is_prenex :=
is_prenex.rec_on h
(λ _ _ ih, ih.lift_at.is_prenex)
(λ _ _ _ ih, ih.cast_le.all)
(λ _ _ _ ih, ih.cast_le.ex)
/-- An auxiliary operation to `first_order.language.bounded_formula.to_prenex`.
If `φ` is quantifier-free and `ψ` is in prenex normal form, then `φ.to_prenex_imp_right ψ`
is a prenex normal form for `φ.imp ψ`. -/
def to_prenex_imp_right :
∀ {n}, L.bounded_formula α n → L.bounded_formula α n → L.bounded_formula α n
| n φ (bounded_formula.ex ψ) := ((φ.lift_at 1 n).to_prenex_imp_right ψ).ex
| n φ (all ψ) := ((φ.lift_at 1 n).to_prenex_imp_right ψ).all
| n φ ψ := φ.imp ψ
lemma is_qf.to_prenex_imp_right {φ : L.bounded_formula α n} :
Π {ψ : L.bounded_formula α n}, is_qf ψ → (φ.to_prenex_imp_right ψ = φ.imp ψ)
| _ is_qf.falsum := rfl
| _ (is_qf.of_is_atomic (is_atomic.equal _ _)) := rfl
| _ (is_qf.of_is_atomic (is_atomic.rel _ _)) := rfl
| _ (is_qf.imp is_qf.falsum _) := rfl
| _ (is_qf.imp (is_qf.of_is_atomic (is_atomic.equal _ _)) _) := rfl
| _ (is_qf.imp (is_qf.of_is_atomic (is_atomic.rel _ _)) _) := rfl
| _ (is_qf.imp (is_qf.imp _ _) _) := rfl
lemma is_prenex_to_prenex_imp_right {φ ψ : L.bounded_formula α n}
(hφ : is_qf φ) (hψ : is_prenex ψ) :
is_prenex (φ.to_prenex_imp_right ψ) :=
begin
induction hψ with _ _ hψ _ _ _ ih1 _ _ _ ih2,
{ rw hψ.to_prenex_imp_right,
exact (hφ.imp hψ).is_prenex },
{ exact (ih1 hφ.lift_at).all },
{ exact (ih2 hφ.lift_at).ex }
end
/-- An auxiliary operation to `first_order.language.bounded_formula.to_prenex`.
If `φ` and `ψ` are in prenex normal form, then `φ.to_prenex_imp ψ`
is a prenex normal form for `φ.imp ψ`. -/
def to_prenex_imp :
∀ {n}, L.bounded_formula α n → L.bounded_formula α n → L.bounded_formula α n
| n (bounded_formula.ex φ) ψ := (φ.to_prenex_imp (ψ.lift_at 1 n)).all
| n (all φ) ψ := (φ.to_prenex_imp (ψ.lift_at 1 n)).ex
| _ φ ψ := φ.to_prenex_imp_right ψ
lemma is_qf.to_prenex_imp : Π {φ ψ : L.bounded_formula α n}, φ.is_qf →
φ.to_prenex_imp ψ = φ.to_prenex_imp_right ψ
| _ _ is_qf.falsum := rfl
| _ _ (is_qf.of_is_atomic (is_atomic.equal _ _)) := rfl
| _ _ (is_qf.of_is_atomic (is_atomic.rel _ _)) := rfl
| _ _ (is_qf.imp is_qf.falsum _) := rfl
| _ _ (is_qf.imp (is_qf.of_is_atomic (is_atomic.equal _ _)) _) := rfl
| _ _ (is_qf.imp (is_qf.of_is_atomic (is_atomic.rel _ _)) _) := rfl
| _ _ (is_qf.imp (is_qf.imp _ _) _) := rfl
lemma is_prenex_to_prenex_imp {φ ψ : L.bounded_formula α n}
(hφ : is_prenex φ) (hψ : is_prenex ψ) :
is_prenex (φ.to_prenex_imp ψ) :=
begin
induction hφ with _ _ hφ _ _ _ ih1 _ _ _ ih2,
{ rw hφ.to_prenex_imp,
exact is_prenex_to_prenex_imp_right hφ hψ },
{ exact (ih1 hψ.lift_at).ex },
{ exact (ih2 hψ.lift_at).all }
end
/-- For any bounded formula `φ`, `φ.to_prenex` is a semantically-equivalent formula in prenex normal
form. -/
def to_prenex : ∀ {n}, L.bounded_formula α n → L.bounded_formula α n
| _ falsum := ⊥
| _ (equal t₁ t₂) := t₁.bd_equal t₂
| _ (rel R ts) := rel R ts
| _ (imp f₁ f₂) := f₁.to_prenex.to_prenex_imp f₂.to_prenex
| _ (all f) := f.to_prenex.all
lemma to_prenex_is_prenex (φ : L.bounded_formula α n) :
φ.to_prenex.is_prenex :=
bounded_formula.rec_on φ
(λ _, is_qf_bot.is_prenex)
(λ _ _ _, (is_atomic.equal _ _).is_prenex)
(λ _ _ _ _, (is_atomic.rel _ _).is_prenex)
(λ _ _ _ h1 h2, is_prenex_to_prenex_imp h1 h2)
(λ _ _, is_prenex.all)
end bounded_formula
namespace Lhom
open bounded_formula
/-- Maps a bounded formula's symbols along a language map. -/
@[simp] def on_bounded_formula (g : L →ᴸ L') :
∀ {k : ℕ}, L.bounded_formula α k → L'.bounded_formula α k
| k falsum := falsum
| k (equal t₁ t₂) := (g.on_term t₁).bd_equal (g.on_term t₂)
| k (rel R ts) := (g.on_relation R).bounded_formula (g.on_term ∘ ts)
| k (imp f₁ f₂) := (on_bounded_formula f₁).imp (on_bounded_formula f₂)
| k (all f) := (on_bounded_formula f).all
@[simp] lemma id_on_bounded_formula :
((Lhom.id L).on_bounded_formula : L.bounded_formula α n → L.bounded_formula α n) = id :=
begin
ext f,
induction f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ rw [on_bounded_formula, Lhom.id_on_term, id.def, id.def, id.def, bd_equal] },
{ rw [on_bounded_formula, Lhom.id_on_term],
refl, },
{ rw [on_bounded_formula, ih1, ih2, id.def, id.def, id.def] },
{ rw [on_bounded_formula, ih3, id.def, id.def] }
end
@[simp] lemma comp_on_bounded_formula {L'' : language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
((φ.comp ψ).on_bounded_formula : L.bounded_formula α n → L''.bounded_formula α n) =
φ.on_bounded_formula ∘ ψ.on_bounded_formula :=
begin
ext f,
induction f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3,
{ refl },
{ simp only [on_bounded_formula, comp_on_term, function.comp_app],
refl, },
{ simp only [on_bounded_formula, comp_on_relation, comp_on_term, function.comp_app],
refl },
{ simp only [on_bounded_formula, function.comp_app, ih1, ih2, eq_self_iff_true, and_self], },
{ simp only [ih3, on_bounded_formula, function.comp_app] }
end
/-- Maps a formula's symbols along a language map. -/
def on_formula (g : L →ᴸ L') : L.formula α → L'.formula α :=
g.on_bounded_formula
/-- Maps a sentence's symbols along a language map. -/
def on_sentence (g : L →ᴸ L') : L.sentence → L'.sentence :=
g.on_formula
/-- Maps a theory's symbols along a language map. -/
def on_Theory (g : L →ᴸ L') (T : L.Theory) : L'.Theory :=
g.on_sentence '' T
@[simp] lemma mem_on_Theory {g : L →ᴸ L'} {T : L.Theory} {φ : L'.sentence} :
φ ∈ g.on_Theory T ↔ ∃ φ₀, φ₀ ∈ T ∧ g.on_sentence φ₀ = φ :=
set.mem_image _ _ _
end Lhom
namespace Lequiv
/-- Maps a bounded formula's symbols along a language equivalence. -/
@[simps] def on_bounded_formula (φ : L ≃ᴸ L') :
L.bounded_formula α n ≃ L'.bounded_formula α n :=
{ to_fun := φ.to_Lhom.on_bounded_formula,
inv_fun := φ.inv_Lhom.on_bounded_formula,
left_inv := by rw [function.left_inverse_iff_comp, ← Lhom.comp_on_bounded_formula, φ.left_inv,
Lhom.id_on_bounded_formula],
right_inv := by rw [function.right_inverse_iff_comp, ← Lhom.comp_on_bounded_formula, φ.right_inv,
Lhom.id_on_bounded_formula] }
lemma on_bounded_formula_symm (φ : L ≃ᴸ L') :
(φ.on_bounded_formula.symm : L'.bounded_formula α n ≃ L.bounded_formula α n) =
φ.symm.on_bounded_formula :=
rfl
/-- Maps a formula's symbols along a language equivalence. -/
def on_formula (φ : L ≃ᴸ L') :
L.formula α ≃ L'.formula α :=
φ.on_bounded_formula
@[simp] lemma on_formula_apply (φ : L ≃ᴸ L') :
(φ.on_formula : L.formula α → L'.formula α) = φ.to_Lhom.on_formula :=
rfl
@[simp] lemma on_formula_symm (φ : L ≃ᴸ L') :
(φ.on_formula.symm : L'.formula α ≃ L.formula α) = φ.symm.on_formula :=
rfl
/-- Maps a sentence's symbols along a language equivalence. -/
@[simps] def on_sentence (φ : L ≃ᴸ L') :
L.sentence ≃ L'.sentence :=
φ.on_formula
end Lequiv
localized "infix ` =' `:88 := first_order.language.term.bd_equal" in first_order
-- input \~- or \simeq
localized "infixr ` ⟹ `:62 := first_order.language.bounded_formula.imp" in first_order
-- input \==>
localized "prefix `∀'`:110 := first_order.language.bounded_formula.all" in first_order
localized "prefix `∼`:max := first_order.language.bounded_formula.not" in first_order
-- input \~, the ASCII character ~ has too low precedence
localized "infix ` ⇔ `:61 := first_order.language.bounded_formula.iff" in first_order -- input \<=>
localized "prefix `∃'`:110 := first_order.language.bounded_formula.ex" in first_order -- input \ex
namespace formula
/-- Relabels a formula's variables along a particular function. -/
def relabel (g : α → β) : L.formula α → L.formula β :=
@bounded_formula.relabel _ _ _ 0 (sum.inl ∘ g) 0
/-- The graph of a function as a first-order formula. -/
def graph (f : L.functions n) : L.formula (fin (n + 1)) :=
equal (var 0) (func f (λ i, var i.succ))
/-- The negation of a formula. -/
protected def not (φ : L.formula α) : L.formula α := φ.not
/-- The implication between formulas, as a formula. -/
protected def imp : L.formula α → L.formula α → L.formula α := bounded_formula.imp
/-- The biimplication between formulas, as a formula. -/
protected def iff (φ ψ : L.formula α) : L.formula α := φ.iff ψ
lemma is_atomic_graph (f : L.functions n) : (graph f).is_atomic :=
bounded_formula.is_atomic.equal _ _
end formula
namespace relations
variable (r : L.relations 2)
/-- The sentence indicating that a basic relation symbol is reflexive. -/
protected def reflexive : L.sentence := ∀' r.bounded_formula₂ &0 &0
/-- The sentence indicating that a basic relation symbol is irreflexive. -/
protected def irreflexive : L.sentence := ∀' ∼ (r.bounded_formula₂ &0 &0)
/-- The sentence indicating that a basic relation symbol is symmetric. -/
protected def symmetric : L.sentence := ∀' ∀' (r.bounded_formula₂ &0 &1 ⟹ r.bounded_formula₂ &1 &0)
/-- The sentence indicating that a basic relation symbol is antisymmetric. -/
protected def antisymmetric : L.sentence :=
∀' ∀' (r.bounded_formula₂ &0 &1 ⟹ (r.bounded_formula₂ &1 &0 ⟹ term.bd_equal &0 &1))
/-- The sentence indicating that a basic relation symbol is transitive. -/
protected def transitive : L.sentence :=
∀' ∀' ∀' (r.bounded_formula₂ &0 &1 ⟹ r.bounded_formula₂ &1 &2 ⟹ r.bounded_formula₂ &0 &2)
/-- The sentence indicating that a basic relation symbol is total. -/
protected def total : L.sentence :=
∀' ∀' (r.bounded_formula₂ &0 &1 ⊔ r.bounded_formula₂ &1 &0)
end relations
section cardinality
variable (L)
/-- A sentence indicating that a structure has `n` distinct elements. -/
protected def sentence.card_ge (n) : L.sentence :=
(((((list.fin_range n).product (list.fin_range n)).filter (λ ij : _ × _, ij.1 ≠ ij.2)).map
(λ (ij : _ × _), ∼ ((& ij.1).bd_equal (& ij.2)))).foldr (⊓) ⊤).exs
/-- A theory indicating that a structure is infinite. -/
def infinite_theory : L.Theory := set.range (sentence.card_ge L)
/-- A theory that indicates a structure is nonempty. -/
def nonempty_theory : L.Theory := {sentence.card_ge L 1}
/-- A theory indicating that each of a set of constants is distinct. -/
def distinct_constants_theory (s : set α) : L[[α]].Theory :=
(λ ab : α × α, (((L.con ab.1).term.equal (L.con ab.2).term).not)) '' (s ×ˢ s ∩ (set.diagonal α)ᶜ)
variables {L} {α}
open set
lemma monotone_distinct_constants_theory :
monotone (L.distinct_constants_theory : set α → L[[α]].Theory) :=
λ s t st, (image_subset _ (inter_subset_inter_left _ (prod_mono st st)))
lemma directed_distinct_constants_theory :
directed (⊆) (L.distinct_constants_theory : set α → L[[α]].Theory) :=
monotone.directed_le monotone_distinct_constants_theory
lemma distinct_constants_theory_eq_Union (s : set α) :
L.distinct_constants_theory s = ⋃ (t : finset s), L.distinct_constants_theory
(t.map (function.embedding.subtype (λ x, x ∈ s))) :=
begin
classical,
simp only [distinct_constants_theory],
rw [← image_Union, ← Union_inter],
refine congr rfl (congr (congr rfl _) rfl),
ext ⟨i, j⟩,
simp only [prod_mk_mem_set_prod_eq, finset.coe_map, function.embedding.coe_subtype, mem_Union,
mem_image, finset.mem_coe, subtype.exists, subtype.coe_mk, exists_and_distrib_right,
exists_eq_right],
refine ⟨λ h, ⟨{⟨i, h.1⟩, ⟨j, h.2⟩}, ⟨h.1, _⟩, ⟨h.2, _⟩⟩, _⟩,
{ simp },
{ simp },
{ rintros ⟨t, ⟨is, _⟩, ⟨js, _⟩⟩,
exact ⟨is, js⟩ }
end
end cardinality
end language
end first_order
|