1 (*********************************************************************************************************************************)
2 (* NaturalDeduction: structurally explicit proofs in Coq *)
3 (*********************************************************************************************************************************)
5 Generalizable All Variables.
6 Require Import Preamble.
7 Require Import General.
8 Require Import Coq.Strings.Ascii.
9 Require Import Coq.Strings.String.
14 * Unlike most formalizations, this library offers TWO different ways
15 * to represent a natural deduction proof. To demonstrate this,
16 * consider the signature of the propositional calculus:
18 * Variable PropositionalVariable : Type.
20 * Inductive Formula : Prop :=
21 * | formula_var : PropositionalVariable -> Formula (* every propositional variable is a formula *)
22 * | formula_and : Formula -> Formula -> Formula (* the conjunction of any two formulae is a formula *)
23 * | formula_or : Formula -> Formula -> Formula (* the disjunction of any two formulae is a formula *)
25 * And couple this with the theory of conjunction and disjunction:
26 * φ\/ψ is true if either φ is true or ψ is true, and φ/\ψ is true
27 * if both φ and ψ are true.
29 * 1) Structurally implicit proofs
31 * This is what you would call the "usual" representation –- it's
32 * what most people learn when they first start programming in Coq:
34 * Inductive IsTrue : Formula -> Prop :=
35 * | IsTrue_or1 : forall f1 f2, IsTrue f1 -> IsTrue (formula_or f1 f2)
36 * | IsTrue_or2 : forall f1 f2, IsTrue f2 -> IsTrue (formula_or f1 f2)
37 * | IsTrue_and : forall f1 f2, IsTrue f2 -> IsTrue f2 -> IsTrue (formula_and f1 f2)
39 * Here each judgment (such as "φ is true") is represented by a Coq
42 * 1. A proof of a judgment is any inhabitant of that Coq type.
44 * 2. A proof of a judgment "J2" from hypothesis judgment "J1"
45 * is any Coq function from the Coq type for J1 to the Coq
46 * type for J2; Composition of (hypothetical) proofs is
47 * represented by composition of Coq functions.
49 * 3. A pair of judgments is represented by their product (Coq
50 * type [prod]) in Coq; a pair of proofs is represented by
51 * their pair (Coq function [pair]) in Coq.
53 * 4. Duplication of hypotheses is represented by the Coq
54 * function (fun x => (x,x)). Dereliction of hypotheses is
55 * represented by the coq function (fun (x,y) => x) or (fun
56 * (x,y) => y). Exchange of the order of hypotheses is
57 * represented by the Coq function (fun (x,y) => (y,x)).
59 * The above can be done using lists instead of tuples.
61 * The advantage of this approach is that it requires a minimum
62 * amount of syntax, and takes maximum advantage of Coq's
63 * automation facilities.
65 * The disadvantage is that one cannot reason about proof-theoretic
66 * properties *generically* across different signatures and
67 * theories. Each signature has its own type of judgments, and
68 * each theory has its own type of proofs. In the present
69 * development we will want to prove –– in this generic manner --
70 * that various classes of natural deduction calculi form various
71 * kinds of categories. So we will need this ability to reason
72 * about proofs independently of the type used to represent
73 * judgments and (more importantly) of the set of basic inference
76 * 2) Structurally explicit proofs
78 * Structurally explicit proofs are formalized in this file
79 * (NaturalDeduction.v) and are designed specifically in order to
80 * circumvent the problem in the previous paragraph.
85 * REGARDING LISTS versus TREES:
87 * You'll notice that this formalization uses (Tree (option A)) in a
88 * lot of places where you might find (list A) more natural. If this
89 * bothers you, see the end of the file for the technical reasons why.
90 * In short, it lets us avoid having to mess about with JMEq or EqDep,
91 * which are not as well-supported by the Coq core as the theory of
95 Section Natural_Deduction.
97 (* any Coq Type may be used as the set of judgments *)
98 Context {Judgment : Type}.
100 (* any Coq Type –- indexed by the hypothesis and conclusion judgments -- may be used as the set of basic inference rules *)
101 Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}.
104 * This type represents a valid Natural Deduction proof from a list
105 * (tree) of hypotheses; the notation H/⋯⋯/C is meant to look like
106 * a proof tree with the middle missing if you tilt your head to
107 * the left (yeah, I know it's a stretch). Note also that this
108 * type is capable of representing proofs with multiple
109 * conclusions, whereas a Rule may have only one conclusion.
112 forall hypotheses:Tree ??Judgment,
113 forall conclusions:Tree ??Judgment,
116 (* natural deduction: you may infer anything from itself -- "identity proof" *)
117 | nd_id0 : [ ] /⋯⋯/ [ ]
118 | nd_id1 : forall h, [ h ] /⋯⋯/ [ h ]
120 (* natural deduction: you may discard conclusions *)
121 | nd_weak : forall h, [ h ] /⋯⋯/ [ ]
123 (* natural deduction: you may duplicate conclusions *)
124 | nd_copy : forall h, h /⋯⋯/ (h,,h)
126 (* natural deduction: you may write two proof trees side by side on a piece of paper -- "proof product" *)
127 | nd_prod : forall {h1 h2 c1 c2}
130 ( h1 ,, h2 /⋯⋯/ c1 ,, c2)
132 (* natural deduction: given a proof of every hypothesis, you may discharge them -- "proof composition" *)
139 (* structural rules on lists of judgments *)
140 | nd_cancell : forall {a}, [] ,, a /⋯⋯/ a
141 | nd_cancelr : forall {a}, a ,, [] /⋯⋯/ a
142 | nd_llecnac : forall {a}, a /⋯⋯/ [] ,, a
143 | nd_rlecnac : forall {a}, a /⋯⋯/ a ,, []
144 | nd_assoc : forall {a b c}, (a,,b),,c /⋯⋯/ a,,(b,,c)
145 | nd_cossa : forall {a b c}, a,,(b,,c) /⋯⋯/ (a,,b),,c
147 (* any Rule by itself counts as a proof *)
148 | nd_rule : forall {h c} (r:Rule h c), h /⋯⋯/ c
150 where "H /⋯⋯/ C" := (ND H C).
152 Notation "H /⋯⋯/ C" := (ND H C) : pf_scope.
153 Notation "a ;; b" := (nd_comp a b) : nd_scope.
154 Notation "a ** b" := (nd_prod a b) : nd_scope.
158 (* a proof is "structural" iff it does not contain any invocations of nd_rule *)
159 Inductive Structural : forall {h c}, h /⋯⋯/ c -> Prop :=
160 | nd_structural_id0 : Structural nd_id0
161 | nd_structural_id1 : forall h, Structural (nd_id1 h)
162 | nd_structural_weak : forall h, Structural (nd_weak h)
163 | nd_structural_copy : forall h, Structural (nd_copy h)
164 | nd_structural_prod : forall `(pf1:h1/⋯⋯/c1)`(pf2:h2/⋯⋯/c2), Structural pf1 -> Structural pf2 -> Structural (pf1**pf2)
165 | nd_structural_comp : forall `(pf1:h1/⋯⋯/x) `(pf2: x/⋯⋯/c2), Structural pf1 -> Structural pf2 -> Structural (pf1;;pf2)
166 | nd_structural_cancell : forall {a}, Structural (@nd_cancell a)
167 | nd_structural_cancelr : forall {a}, Structural (@nd_cancelr a)
168 | nd_structural_llecnac : forall {a}, Structural (@nd_llecnac a)
169 | nd_structural_rlecnac : forall {a}, Structural (@nd_rlecnac a)
170 | nd_structural_assoc : forall {a b c}, Structural (@nd_assoc a b c)
171 | nd_structural_cossa : forall {a b c}, Structural (@nd_cossa a b c)
174 (* multi-judgment generalization of nd_id0 and nd_id1; making nd_id0/nd_id1 primitive and deriving all other *)
175 Fixpoint nd_id (sl:Tree ??Judgment) : sl /⋯⋯/ sl :=
177 | T_Leaf None => nd_id0
178 | T_Leaf (Some x) => nd_id1 x
179 | T_Branch a b => nd_prod (nd_id a) (nd_id b)
182 Hint Constructors Structural.
183 Lemma nd_id_structural : forall sl, Structural (nd_id sl).
185 induction sl; simpl; auto.
189 (* An equivalence relation on proofs which is sensitive only to the logical content of the proof -- insensitive to
190 * structural variations *)
192 { ndr_eqv : forall {h c }, h /⋯⋯/ c -> h /⋯⋯/ c -> Prop where "pf1 === pf2" := (@ndr_eqv _ _ pf1 pf2)
193 ; ndr_eqv_equivalence : forall h c, Equivalence (@ndr_eqv h c)
195 (* the relation must respect composition, be associative wrt composition, and be left and right neutral wrt the identity proof *)
196 ; ndr_comp_respects : forall {a b c}(f f':a/⋯⋯/b)(g g':b/⋯⋯/c), f === f' -> g === g' -> f;;g === f';;g'
197 ; ndr_comp_associativity : forall `(f:a/⋯⋯/b)`(g:b/⋯⋯/c)`(h:c/⋯⋯/d), (f;;g);;h === f;;(g;;h)
198 ; ndr_comp_left_identity : forall `(f:a/⋯⋯/c), nd_id _ ;; f === f
199 ; ndr_comp_right_identity : forall `(f:a/⋯⋯/c), f ;; nd_id _ === f
201 (* the relation must respect products, be associative wrt products, and be left and right neutral wrt the identity proof *)
202 ; ndr_prod_respects : forall {a b c d}(f f':a/⋯⋯/b)(g g':c/⋯⋯/d), f===f' -> g===g' -> f**g === f'**g'
203 ; ndr_prod_associativity : forall `(f:a/⋯⋯/a')`(g:b/⋯⋯/b')`(h:c/⋯⋯/c'), (f**g)**h === nd_assoc ;; f**(g**h) ;; nd_cossa
204 ; ndr_prod_left_identity : forall `(f:a/⋯⋯/b), (nd_id0 ** f ) === nd_cancell ;; f ;; nd_llecnac
205 ; ndr_prod_right_identity : forall `(f:a/⋯⋯/b), (f ** nd_id0) === nd_cancelr ;; f ;; nd_rlecnac
207 (* products and composition must distribute over each other *)
208 ; ndr_prod_preserves_comp : forall `(f:a/⋯⋯/b)`(f':a'/⋯⋯/b')`(g:b/⋯⋯/c)`(g':b'/⋯⋯/c'), (f;;g)**(f';;g') === (f**f');;(g**g')
210 (* any two _structural_ proofs with the same hypotheses/conclusions must be considered equal *)
211 ; ndr_structural_indistinguishable : forall `(f:a/⋯⋯/b)(g:a/⋯⋯/b), Structural f -> Structural g -> f===g
215 * Single-conclusion proofs; this is an alternate representation
216 * where each inference has only a single conclusion. These have
217 * worse compositionality properties than ND's, but are easier to
218 * emit as LaTeX code.
220 Inductive SCND : Tree ??Judgment -> Tree ??Judgment -> Type :=
221 | scnd_axiom : forall c , SCND [] [c]
222 | scnd_comp : forall ht ct c , SCND ht ct -> Rule ct [c] -> SCND ht [c]
223 | scnd_weak : forall c , SCND c []
224 | scnd_leaf : forall ht c , SCND ht [c] -> SCND ht [c]
225 | scnd_branch : forall ht c1 c2, SCND ht c1 -> SCND ht c2 -> SCND ht (c1,,c2)
227 Hint Constructors SCND.
229 Definition mkSCNDAxioms (h:Tree ??Judgment) : SCND [] h.
232 apply scnd_leaf. apply scnd_axiom.
234 apply scnd_branch; auto.
237 (* Any ND whose primitive Rules have at most one conclusion (note that nd_prod is allowed!) can be turned into an SCND. *)
238 Definition mkSCND (all_rules_one_conclusion : forall h c1 c2, Rule h (c1,,c2) -> False)
239 : forall h x c, ND x c -> SCND h x -> SCND h c.
241 induction nd; intro k.
245 eapply scnd_branch; apply k.
247 apply (scnd_branch _ _ _ (IHnd1 X) (IHnd2 X0)).
251 inversion k; subst; auto.
252 inversion k; subst; auto.
253 apply scnd_branch; auto.
254 apply scnd_branch; auto.
255 inversion k; subst; inversion X; subst; auto.
256 inversion k; subst; inversion X0; subst; auto.
259 apply scnd_leaf. eapply scnd_comp. apply k. apply r.
261 set (all_rules_one_conclusion _ _ _ r) as bogus.
265 (* a "ClosedND" is a proof with no open hypotheses and no multi-conclusion rules *)
266 Inductive ClosedND : Tree ??Judgment -> Type :=
267 | cnd_weak : ClosedND []
268 | cnd_rule : forall h c , ClosedND h -> Rule h c -> ClosedND c
269 | cnd_branch : forall c1 c2, ClosedND c1 -> ClosedND c2 -> ClosedND (c1,,c2)
273 (* we can turn an SCND without hypotheses into a ClosedND *)
274 Definition closedFromSCND h c (pn2:SCND h c)(cnd:ClosedND h) : ClosedND c.
275 refine ((fix closedFromPnodes h c (pn2:SCND h c)(cnd:ClosedND h) {struct pn2} :=
276 (match pn2 in SCND H C return H=h -> C=c -> _ with
277 | scnd_weak c => let case_weak := tt in _
278 | scnd_leaf ht z pn' => let case_leaf := tt in let qq := closedFromPnodes _ _ pn' in _
279 | scnd_axiom c => let case_axiom := tt in _
280 | scnd_comp ht ct c pn' rule => let case_comp := tt in let qq := closedFromPnodes _ _ pn' in _
281 | scnd_branch ht c1 c2 pn' pn'' => let case_branch := tt in
282 let q1 := closedFromPnodes _ _ pn' in
283 let q2 := closedFromPnodes _ _ pn'' in _
285 end (refl_equal _) (refl_equal _))) h c pn2 cnd).
310 destruct case_branch.
313 apply q1. subst. apply cnd0.
314 apply q2. subst. apply cnd0.
318 Close Scope nd_scope.
321 End Natural_Deduction.
323 Implicit Arguments ND [ Judgment ].
324 Hint Constructors Structural.
325 Hint Extern 1 => apply nd_id_structural.
326 Hint Extern 1 => apply ndr_structural_indistinguishable.
328 (* This first notation gets its own scope because it can be confusing when we're working with multiple different kinds
329 * of proofs. When only one kind of proof is in use, it's quite helpful though. *)
330 Notation "H /⋯⋯/ C" := (@ND _ _ H C) : pf_scope.
331 Notation "a ;; b" := (nd_comp a b) : nd_scope.
332 Notation "a ** b" := (nd_prod a b) : nd_scope.
333 Notation "[# a #]" := (nd_rule a) : nd_scope.
334 Notation "a === b" := (@ndr_eqv _ _ _ _ _ a b) : nd_scope.
336 (* enable setoid rewriting *)
340 Add Parametric Relation {jt rt ndr h c} : (h/⋯⋯/c) (@ndr_eqv jt rt ndr h c)
341 reflexivity proved by (@Equivalence_Reflexive _ _ (ndr_eqv_equivalence h c))
342 symmetry proved by (@Equivalence_Symmetric _ _ (ndr_eqv_equivalence h c))
343 transitivity proved by (@Equivalence_Transitive _ _ (ndr_eqv_equivalence h c))
344 as parametric_relation_ndr_eqv.
345 Add Parametric Morphism {jt rt ndr h x c} : (@nd_comp jt rt h x c)
346 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
347 as parametric_morphism_nd_comp.
348 intros; apply ndr_comp_respects; auto.
350 Add Parametric Morphism {jt rt ndr a b c d} : (@nd_prod jt rt a b c d)
351 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
352 as parametric_morphism_nd_prod.
353 intros; apply ndr_prod_respects; auto.
356 (* a generalization of the procedure used to build (nd_id n) from nd_id0 and nd_id1 *)
357 Definition nd_replicate
363 (forall (o:Ob), @ND Judgment Rule [h o] [c o]) ->
364 @ND Judgment Rule (mapOptionTree h j) (mapOptionTree c j).
373 (* "map" over natural deduction proofs, where the result proof has the same judgments (but different rules) *)
376 {Judgment}{Rule0}{Rule1}
377 (r:forall h c, Rule0 h c -> @ND Judgment Rule1 h c)
379 (pf:@ND Judgment Rule0 h c)
381 @ND Judgment Rule1 h c.
382 intros Judgment Rule0 Rule1 r.
384 refine ((fix nd_map h c pf {struct pf} :=
388 @ND Judgment Rule1 H C
390 | nd_id0 => let case_nd_id0 := tt in _
391 | nd_id1 h => let case_nd_id1 := tt in _
392 | nd_weak h => let case_nd_weak := tt in _
393 | nd_copy h => let case_nd_copy := tt in _
394 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
395 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
396 | nd_rule _ _ rule => let case_nd_rule := tt in _
397 | nd_cancell _ => let case_nd_cancell := tt in _
398 | nd_cancelr _ => let case_nd_cancelr := tt in _
399 | nd_llecnac _ => let case_nd_llecnac := tt in _
400 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
401 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
402 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
403 end))) ); simpl in *.
405 destruct case_nd_id0. apply nd_id0.
406 destruct case_nd_id1. apply nd_id1.
407 destruct case_nd_weak. apply nd_weak.
408 destruct case_nd_copy. apply nd_copy.
409 destruct case_nd_prod. apply (nd_prod (nd_map _ _ lpf) (nd_map _ _ rpf)).
410 destruct case_nd_comp. apply (nd_comp (nd_map _ _ top) (nd_map _ _ bot)).
411 destruct case_nd_cancell. apply nd_cancell.
412 destruct case_nd_cancelr. apply nd_cancelr.
413 destruct case_nd_llecnac. apply nd_llecnac.
414 destruct case_nd_rlecnac. apply nd_rlecnac.
415 destruct case_nd_assoc. apply nd_assoc.
416 destruct case_nd_cossa. apply nd_cossa.
420 (* "map" over natural deduction proofs, where the result proof has different judgments *)
423 {Judgment0}{Rule0}{Judgment1}{Rule1}
424 (f:Judgment0->Judgment1)
425 (r:forall h c, Rule0 h c -> @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c))
427 (pf:@ND Judgment0 Rule0 h c)
429 @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c).
430 intros Judgment0 Rule0 Judgment1 Rule1 f r.
432 refine ((fix nd_map' h c pf {struct pf} :=
436 @ND Judgment1 Rule1 (mapOptionTree f H) (mapOptionTree f C)
438 | nd_id0 => let case_nd_id0 := tt in _
439 | nd_id1 h => let case_nd_id1 := tt in _
440 | nd_weak h => let case_nd_weak := tt in _
441 | nd_copy h => let case_nd_copy := tt in _
442 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
443 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
444 | nd_rule _ _ rule => let case_nd_rule := tt in _
445 | nd_cancell _ => let case_nd_cancell := tt in _
446 | nd_cancelr _ => let case_nd_cancelr := tt in _
447 | nd_llecnac _ => let case_nd_llecnac := tt in _
448 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
449 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
450 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
451 end))) ); simpl in *.
453 destruct case_nd_id0. apply nd_id0.
454 destruct case_nd_id1. apply nd_id1.
455 destruct case_nd_weak. apply nd_weak.
456 destruct case_nd_copy. apply nd_copy.
457 destruct case_nd_prod. apply (nd_prod (nd_map' _ _ lpf) (nd_map' _ _ rpf)).
458 destruct case_nd_comp. apply (nd_comp (nd_map' _ _ top) (nd_map' _ _ bot)).
459 destruct case_nd_cancell. apply nd_cancell.
460 destruct case_nd_cancelr. apply nd_cancelr.
461 destruct case_nd_llecnac. apply nd_llecnac.
462 destruct case_nd_rlecnac. apply nd_rlecnac.
463 destruct case_nd_assoc. apply nd_assoc.
464 destruct case_nd_cossa. apply nd_cossa.
468 (* witnesses the fact that every Rule in a particular proof satisfies the given predicate *)
469 Inductive nd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h}{c}, @ND Judgment Rule h c -> Prop :=
470 | nd_property_structural : forall h c pf, Structural pf -> @nd_property _ _ P h c pf
471 | nd_property_prod : forall h0 c0 pf0 h1 c1 pf1,
472 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P h1 c1 pf1 -> @nd_property _ _ P _ _ (nd_prod pf0 pf1)
473 | nd_property_comp : forall h0 c0 pf0 c1 pf1,
474 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P c0 c1 pf1 -> @nd_property _ _ P _ _ (nd_comp pf0 pf1)
475 | nd_property_rule : forall h c r, P h c r -> @nd_property _ _ P h c (nd_rule r).
476 Hint Constructors nd_property.
478 Close Scope pf_scope.
479 Close Scope nd_scope.