1 (*********************************************************************************************************************************)
2 (* NaturalDeduction: *)
4 (* Structurally explicit natural deduction proofs. *)
6 (*********************************************************************************************************************************)
8 Generalizable All Variables.
9 Require Import Preamble.
10 Require Import General.
11 Require Import Coq.Strings.Ascii.
12 Require Import Coq.Strings.String.
15 * Unlike most formalizations, this library offers two different ways
16 * to represent a natural deduction proof. To demonstrate this,
17 * consider the signature of the propositional calculus:
19 * Variable PropositionalVariable : Type.
21 * Inductive Formula : Prop :=
22 * | formula_var : PropositionalVariable -> Formula (* every propositional variable is a formula *)
23 * | formula_and : Formula -> Formula -> Formula (* the conjunction of any two formulae is a formula *)
24 * | formula_or : Formula -> Formula -> Formula (* the disjunction of any two formulae is a formula *)
26 * And couple this with the theory of conjunction and disjunction:
27 * φ\/ψ is true if either φ is true or ψ is true, and φ/\ψ is true
28 * if both φ and ψ are true.
30 * 1) Structurally implicit proofs
32 * This is what you would call the "usual" representation –- it's
33 * what most people learn when they first start programming in Coq:
35 * Inductive IsTrue : Formula -> Prop :=
36 * | IsTrue_or1 : forall f1 f2, IsTrue f1 -> IsTrue (formula_or f1 f2)
37 * | IsTrue_or2 : forall f1 f2, IsTrue f2 -> IsTrue (formula_or f1 f2)
38 * | IsTrue_and : forall f1 f2, IsTrue f2 -> IsTrue f2 -> IsTrue (formula_and f1 f2)
40 * Here each judgment (such as "φ is true") is represented by a Coq
43 * 1. A proof of a judgment is any inhabitant of that Coq type.
45 * 2. A proof of a judgment "J2" from hypothesis judgment "J1"
46 * is any Coq function from the Coq type for J1 to the Coq
47 * type for J2; Composition of (hypothetical) proofs is
48 * represented by composition of Coq functions.
50 * 3. A pair of judgments is represented by their product (Coq
51 * type [prod]) in Coq; a pair of proofs is represented by
52 * their pair (Coq function [pair]) in Coq.
54 * 4. Duplication of hypotheses is represented by the Coq
55 * function (fun x => (x,x)). Dereliction of hypotheses is
56 * represented by the coq function (fun (x,y) => x) or (fun
57 * (x,y) => y). Exchange of the order of hypotheses is
58 * represented by the Coq function (fun (x,y) => (y,x)).
60 * The above can be done using lists instead of tuples.
62 * The advantage of this approach is that it requires a minimum
63 * amount of syntax, and takes maximum advantage of Coq's
64 * automation facilities.
66 * The disadvantage is that one cannot reason about proof-theoretic
67 * properties *generically* across different signatures and
68 * theories. Each signature has its own type of judgments, and
69 * each theory has its own type of proofs. In the present
70 * development we will want to prove –– in this generic manner --
71 * that various classes of natural deduction calculi form various
72 * kinds of categories. So we will need this ability to reason
73 * about proofs independently of the type used to represent
74 * judgments and (more importantly) of the set of basic inference
77 * 2) Structurally explicit proofs
79 * Structurally explicit proofs are formalized in this file
80 * (NaturalDeduction.v) and are designed specifically in order to
81 * circumvent the problem in the previous paragraph.
83 * These proofs are actually structurally explicit on (potentially)
84 * two different levels. The beginning of this file formalizes
85 * natural deduction proofs with explicit structural operations for
86 * manipulating lists of judgments – for example, the open
87 * hypotheses of an incomplete proof. The class
88 * TreeStructuralRules further down in the file instantiates ND
89 * such that Judgments is actually a pair of trees of propositions,
90 * and there will be a whole *other* set of rules for manipulating
91 * the structure of a tree of propositions *within* a single
94 * The flattening functor ends up mapping the first kind of
95 * structural operation (moving around judgments) onto the second
96 * kind (moving around propositions/types). That's why everything
97 * is so laboriously explicit - there's important information in
98 * those structural operations.
102 * REGARDING LISTS versus TREES:
104 * You'll notice that this formalization uses (Tree (option A)) in a
105 * lot of places where you might find (list A) more natural. If this
106 * bothers you, see the end of the file for the technical reasons why.
107 * In short, it lets us avoid having to mess about with JMEq or EqDep,
108 * which are not as well-supported by the Coq core as the theory of
112 Section Natural_Deduction.
114 (* any Coq Type may be used as the set of judgments *)
115 Context {Judgment : Type}.
117 (* any Coq Type –- indexed by the hypothesis and conclusion judgments -- may be used as the set of basic inference rules *)
118 Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}.
121 * This type represents a valid Natural Deduction proof from a list
122 * (tree) of hypotheses; the notation H/⋯⋯/C is meant to look like
123 * a proof tree with the middle missing if you tilt your head to
124 * the left (yeah, I know it's a stretch). Note also that this
125 * type is capable of representing proofs with multiple
126 * conclusions, whereas a Rule may have only one conclusion.
129 forall hypotheses:Tree ??Judgment,
130 forall conclusions:Tree ??Judgment,
133 (* natural deduction: you may infer nothing from nothing *)
134 | nd_id0 : [ ] /⋯⋯/ [ ]
136 (* natural deduction: you may infer anything from itself -- "identity proof" *)
137 | nd_id1 : forall h, [ h ] /⋯⋯/ [ h ]
139 (* natural deduction: you may discard conclusions *)
140 | nd_weak1 : forall h, [ h ] /⋯⋯/ [ ]
142 (* natural deduction: you may duplicate conclusions *)
143 | nd_copy : forall h, h /⋯⋯/ (h,,h)
145 (* natural deduction: you may write two proof trees side by side on a piece of paper -- "proof product" *)
146 | nd_prod : forall {h1 h2 c1 c2}
149 ( h1 ,, h2 /⋯⋯/ c1 ,, c2)
151 (* natural deduction: given a proof of every hypothesis, you may discharge them -- "proof composition" *)
158 (* Structural rules on lists of judgments - note that this is completely separate from the structural
159 * rules for *contexts* within a sequent. The rules below manipulate lists of *judgments* rather than
160 * lists of *propositions*. *)
161 | nd_cancell : forall {a}, [] ,, a /⋯⋯/ a
162 | nd_cancelr : forall {a}, a ,, [] /⋯⋯/ a
163 | nd_llecnac : forall {a}, a /⋯⋯/ [] ,, a
164 | nd_rlecnac : forall {a}, a /⋯⋯/ a ,, []
165 | nd_assoc : forall {a b c}, (a,,b),,c /⋯⋯/ a,,(b,,c)
166 | nd_cossa : forall {a b c}, a,,(b,,c) /⋯⋯/ (a,,b),,c
168 (* any Rule by itself counts as a proof *)
169 | nd_rule : forall {h c} (r:Rule h c), h /⋯⋯/ c
171 where "H /⋯⋯/ C" := (ND H C).
173 Notation "H /⋯⋯/ C" := (ND H C) : pf_scope.
174 Notation "a ;; b" := (nd_comp a b) : nd_scope.
175 Notation "a ** b" := (nd_prod a b) : nd_scope.
179 (* a predicate on proofs *)
180 Definition NDPredicate := forall h c, h /⋯⋯/ c -> Prop.
182 (* the structural inference rules are those which do not change, add, remove, or re-order the judgments *)
183 Inductive Structural : forall {h c}, h /⋯⋯/ c -> Prop :=
184 | nd_structural_id0 : Structural nd_id0
185 | nd_structural_id1 : forall h, Structural (nd_id1 h)
186 | nd_structural_cancell : forall {a}, Structural (@nd_cancell a)
187 | nd_structural_cancelr : forall {a}, Structural (@nd_cancelr a)
188 | nd_structural_llecnac : forall {a}, Structural (@nd_llecnac a)
189 | nd_structural_rlecnac : forall {a}, Structural (@nd_rlecnac a)
190 | nd_structural_assoc : forall {a b c}, Structural (@nd_assoc a b c)
191 | nd_structural_cossa : forall {a b c}, Structural (@nd_cossa a b c)
194 (* the closure of an NDPredicate under nd_comp and nd_prod *)
195 Inductive NDPredicateClosure (P:NDPredicate) : forall {h c}, h /⋯⋯/ c -> Prop :=
196 | ndpc_p : forall h c f, P h c f -> NDPredicateClosure P f
197 | ndpc_prod : forall `(pf1:h1/⋯⋯/c1)`(pf2:h2/⋯⋯/c2),
198 NDPredicateClosure P pf1 -> NDPredicateClosure P pf2 -> NDPredicateClosure P (pf1**pf2)
199 | ndpc_comp : forall `(pf1:h1/⋯⋯/x) `(pf2: x/⋯⋯/c2),
200 NDPredicateClosure P pf1 -> NDPredicateClosure P pf2 -> NDPredicateClosure P (pf1;;pf2).
202 (* proofs built up from structural rules via comp and prod *)
203 Definition StructuralND {h}{c} f := @NDPredicateClosure (@Structural) h c f.
205 (* The Predicate (BuiltFrom f P h) asserts that "h" was built from a single occurrence of "f" and proofs which satisfy P *)
206 Inductive BuiltFrom {h'}{c'}(f:h'/⋯⋯/c')(P:NDPredicate) : forall {h c}, h/⋯⋯/c -> Prop :=
207 | builtfrom_refl : BuiltFrom f P f
208 | builtfrom_P : forall h c g, @P h c g -> BuiltFrom f P g
209 | builtfrom_prod1 : forall h1 c1 f1 h2 c2 f2, P h1 c1 f1 -> @BuiltFrom _ _ f P h2 c2 f2 -> BuiltFrom f P (f1 ** f2)
210 | builtfrom_prod2 : forall h1 c1 f1 h2 c2 f2, P h1 c1 f1 -> @BuiltFrom _ _ f P h2 c2 f2 -> BuiltFrom f P (f2 ** f1)
211 | builtfrom_comp1 : forall h x c f1 f2, P h x f1 -> @BuiltFrom _ _ f P x c f2 -> BuiltFrom f P (f1 ;; f2)
212 | builtfrom_comp2 : forall h x c f1 f2, P x c f1 -> @BuiltFrom _ _ f P h x f2 -> BuiltFrom f P (f2 ;; f1).
214 (* multi-judgment generalization of nd_id0 and nd_id1; making nd_id0/nd_id1 primitive and deriving all other *)
215 Fixpoint nd_id (sl:Tree ??Judgment) : sl /⋯⋯/ sl :=
217 | T_Leaf None => nd_id0
218 | T_Leaf (Some x) => nd_id1 x
219 | T_Branch a b => nd_prod (nd_id a) (nd_id b)
222 Fixpoint nd_weak (sl:Tree ??Judgment) : sl /⋯⋯/ [] :=
223 match sl as SL return SL /⋯⋯/ [] with
224 | T_Leaf None => nd_id0
225 | T_Leaf (Some x) => nd_weak1 x
226 | T_Branch a b => nd_prod (nd_weak a) (nd_weak b) ;; nd_cancelr
229 Hint Constructors Structural.
230 Hint Constructors BuiltFrom.
231 Hint Constructors NDPredicateClosure.
233 Hint Extern 1 => apply nd_structural_id0.
234 Hint Extern 1 => apply nd_structural_id1.
235 Hint Extern 1 => apply nd_structural_cancell.
236 Hint Extern 1 => apply nd_structural_cancelr.
237 Hint Extern 1 => apply nd_structural_llecnac.
238 Hint Extern 1 => apply nd_structural_rlecnac.
239 Hint Extern 1 => apply nd_structural_assoc.
240 Hint Extern 1 => apply nd_structural_cossa.
241 Hint Extern 1 => apply ndpc_p.
242 Hint Extern 1 => apply ndpc_prod.
243 Hint Extern 1 => apply ndpc_comp.
245 Lemma nd_id_structural : forall sl, StructuralND (nd_id sl).
247 induction sl; simpl; auto.
251 (* An equivalence relation on proofs which is sensitive only to the logical content of the proof -- insensitive to
252 * structural variations *)
254 { ndr_eqv : forall {h c }, h /⋯⋯/ c -> h /⋯⋯/ c -> Prop where "pf1 === pf2" := (@ndr_eqv _ _ pf1 pf2)
255 ; ndr_eqv_equivalence : forall h c, Equivalence (@ndr_eqv h c)
257 (* the relation must respect composition, be associative wrt composition, and be left and right neutral wrt the identity proof *)
258 ; ndr_comp_respects : forall {a b c}(f f':a/⋯⋯/b)(g g':b/⋯⋯/c), f === f' -> g === g' -> f;;g === f';;g'
259 ; ndr_comp_associativity : forall `(f:a/⋯⋯/b)`(g:b/⋯⋯/c)`(h:c/⋯⋯/d), (f;;g);;h === f;;(g;;h)
261 (* the relation must respect products, be associative wrt products, and be left and right neutral wrt the identity proof *)
262 ; ndr_prod_respects : forall {a b c d}(f f':a/⋯⋯/b)(g g':c/⋯⋯/d), f===f' -> g===g' -> f**g === f'**g'
263 ; ndr_prod_associativity : forall `(f:a/⋯⋯/a')`(g:b/⋯⋯/b')`(h:c/⋯⋯/c'), (f**g)**h === nd_assoc ;; f**(g**h) ;; nd_cossa
265 (* products and composition must distribute over each other *)
266 ; 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')
268 (* Given a proof f, any two proofs built from it using only structural rules are indistinguishable. Keep in mind that
269 * nd_weak and nd_copy aren't considered structural, so the hypotheses and conclusions of such proofs will be an identical
270 * list, differing only in the "parenthesization" and addition or removal of empty leaves. *)
271 ; ndr_builtfrom_structural : forall `(f:a/⋯⋯/b){a' b'}(g1 g2:a'/⋯⋯/b'),
272 BuiltFrom f (@StructuralND) g1 ->
273 BuiltFrom f (@StructuralND) g2 ->
276 (* proofs of nothing are not distinguished from each other *)
277 ; ndr_void_proofs_irrelevant : forall `(f:a/⋯⋯/[])(g:a/⋯⋯/[]), f === g
279 (* products and duplication must distribute over each other *)
280 ; ndr_prod_preserves_copy : forall `(f:a/⋯⋯/b), nd_copy a;; f**f === f ;; nd_copy b
282 (* duplicating a hypothesis and discarding it is irrelevant *)
283 ; ndr_copy_then_weak_left : forall a, nd_copy a;; (nd_weak _ ** nd_id _) ;; nd_cancell === nd_id _
284 ; ndr_copy_then_weak_right : forall a, nd_copy a;; (nd_id _ ** nd_weak _) ;; nd_cancelr === nd_id _
288 * Natural Deduction proofs which are Structurally Implicit on the
289 * level of judgments. These proofs have poor compositionality
290 * properties (vertically, they look more like lists than trees) but
291 * are easier to do induction over.
293 Inductive SIND : Tree ??Judgment -> Tree ??Judgment -> Type :=
294 | scnd_weak : forall c , SIND c []
295 | scnd_comp : forall ht ct c , SIND ht ct -> Rule ct [c] -> SIND ht [c]
296 | scnd_branch : forall ht c1 c2, SIND ht c1 -> SIND ht c2 -> SIND ht (c1,,c2)
298 Hint Constructors SIND.
300 (* Any ND whose primitive Rules have at most one conclusion (note that nd_prod is allowed!) can be turned into an SIND. *)
301 Definition mkSIND (all_rules_one_conclusion : forall h c1 c2, Rule h (c1,,c2) -> False)
302 : forall h x c, ND x c -> SIND h x -> SIND h c.
304 induction nd; intro k.
308 eapply scnd_branch; apply k.
310 apply (scnd_branch _ _ _ (IHnd1 X) (IHnd2 X0)).
314 inversion k; subst; auto.
315 inversion k; subst; auto.
316 apply scnd_branch; auto.
317 apply scnd_branch; auto.
318 inversion k; subst; inversion X; subst; auto.
319 inversion k; subst; inversion X0; subst; auto.
322 eapply scnd_comp. apply k. apply r.
324 set (all_rules_one_conclusion _ _ _ r) as bogus.
328 (* Natural Deduction systems whose judgments happen to be pairs of the same type *)
330 Context {S:Type}. (* type of sequent components *)
331 Context {sequent:S->S->Judgment}. (* pairing operation which forms a sequent from its halves *)
332 Notation "a |= b" := (sequent a b).
334 (* a SequentND is a natural deduction whose judgments are sequents, has initial sequents, and has a cut rule *)
336 { snd_initial : forall a, [ ] /⋯⋯/ [ a |= a ]
337 ; snd_cut : forall a b c, [ a |= b ] ,, [ b |= c ] /⋯⋯/ [ a |= c ]
340 Context (sequentND:SequentND).
341 Context (ndr:ND_Relation).
344 * A predicate singling out structural rules, initial sequents,
347 * Proofs using only structural rules cannot add or remove
348 * judgments - their hypothesis and conclusion judgment-trees will
349 * differ only in "parenthesization" and the presence/absence of
350 * empty leaves. This means that a proof involving only
351 * structural rules, cut, and initial sequents can ADD new
352 * non-empty judgment-leaves only via snd_initial, and can only
353 * REMOVE non-empty judgment-leaves only via snd_cut. Since the
354 * initial sequent is a left and right identity for cut, and cut
355 * is associative, any two proofs (with the same hypotheses and
356 * conclusions) using only structural rules, cut, and initial
357 * sequents are logically indistinguishable - their differences
358 * are logically insignificant.
360 * Note that it is important that nd_weak and nd_copy aren't
361 * considered to be "structural".
363 Inductive SequentND_Inert : forall h c, h/⋯⋯/c -> Prop :=
364 | snd_inert_initial : forall a, SequentND_Inert _ _ (snd_initial a)
365 | snd_inert_cut : forall a b c, SequentND_Inert _ _ (snd_cut a b c)
366 | snd_inert_structural: forall a b f, Structural f -> SequentND_Inert a b f
369 (* An ND_Relation for a sequent deduction should not distinguish between two proofs having the same hypotheses and conclusions
370 * if those proofs use only initial sequents, cut, and structural rules (see comment above) *)
371 Class SequentND_Relation :=
373 ; sndr_inert : forall a b (f g:a/⋯⋯/b),
374 NDPredicateClosure SequentND_Inert f ->
375 NDPredicateClosure SequentND_Inert g ->
380 (* Deductions on sequents whose antecedent is a tree of propositions (i.e. a context) *)
382 Context {P:Type}{sequent:Tree ??P -> Tree ??P -> Judgment}.
383 Context {snd:SequentND(sequent:=sequent)}.
384 Notation "a |= b" := (sequent a b).
386 (* Note that these rules mirror nd_{cancell,cancelr,rlecnac,llecnac,assoc,cossa} but are completely separate from them *)
388 { cnd_ant_assoc : forall x a b c, ND [((a,,b),,c) |= x] [(a,,(b,,c)) |= x ]
389 ; cnd_ant_cossa : forall x a b c, ND [(a,,(b,,c)) |= x] [((a,,b),,c) |= x ]
390 ; cnd_ant_cancell : forall x a , ND [ [],,a |= x] [ a |= x ]
391 ; cnd_ant_cancelr : forall x a , ND [a,,[] |= x] [ a |= x ]
392 ; cnd_ant_llecnac : forall x a , ND [ a |= x] [ [],,a |= x ]
393 ; cnd_ant_rlecnac : forall x a , ND [ a |= x] [ a,,[] |= x ]
394 ; cnd_expand_left : forall a b c , ND [ a |= b] [ c,,a |= c,,b]
395 ; cnd_expand_right : forall a b c , ND [ a |= b] [ a,,c |= b,,c]
399 Context `(ContextND).
402 * A predicate singling out initial sequents, cuts, context expansion,
403 * and structural rules.
405 * Any two proofs (with the same hypotheses and conclusions) whose
406 * non-structural rules do nothing other than expand contexts,
407 * re-arrange contexts, or introduce additional initial-sequent
408 * conclusions are indistinguishable. One important consequence
409 * is that asking for a small initial sequent and then expanding
410 * it using cnd_expand_{right,left} is no different from simply
411 * asking for the larger initial sequent in the first place.
414 Inductive ContextND_Inert : forall h c, h/⋯⋯/c -> Prop :=
415 | cnd_inert_initial : forall a, ContextND_Inert _ _ (snd_initial a)
416 | cnd_inert_cut : forall a b c, ContextND_Inert _ _ (snd_cut a b c)
417 | cnd_inert_structural : forall a b f, Structural f -> ContextND_Inert a b f
418 | cnd_inert_cnd_ant_assoc : forall x a b c, ContextND_Inert _ _ (cnd_ant_assoc x a b c)
419 | cnd_inert_cnd_ant_cossa : forall x a b c, ContextND_Inert _ _ (cnd_ant_cossa x a b c)
420 | cnd_inert_cnd_ant_cancell : forall x a , ContextND_Inert _ _ (cnd_ant_cancell x a)
421 | cnd_inert_cnd_ant_cancelr : forall x a , ContextND_Inert _ _ (cnd_ant_cancelr x a)
422 | cnd_inert_cnd_ant_llecnac : forall x a , ContextND_Inert _ _ (cnd_ant_llecnac x a)
423 | cnd_inert_cnd_ant_rlecnac : forall x a , ContextND_Inert _ _ (cnd_ant_rlecnac x a)
424 | cnd_inert_se_expand_left : forall t g s , ContextND_Inert _ _ (@cnd_expand_left _ t g s)
425 | cnd_inert_se_expand_right : forall t g s , ContextND_Inert _ _ (@cnd_expand_right _ t g s).
427 Class ContextND_Relation {ndr}{sndr:SequentND_Relation _ ndr} :=
428 { cndr_inert : forall {a}{b}(f g:a/⋯⋯/b),
429 NDPredicateClosure ContextND_Inert f ->
430 NDPredicateClosure ContextND_Inert g ->
435 (* a proof is Analytic if it does not use cut *)
437 Definition Analytic_Rule : NDPredicate :=
438 fun h c f => forall c, not (snd_cut _ _ c = f).
439 Definition AnalyticND := NDPredicateClosure Analytic_Rule.
441 (* a proof system has cut elimination if, for every proof, there is an analytic proof with the same conclusion *)
442 Class CutElimination :=
443 { ce_eliminate : forall h c, h/⋯⋯/c -> h/⋯⋯/c
444 ; ce_analytic : forall h c f, AnalyticND (ce_eliminate h c f)
447 (* cut elimination is strong if the analytic proof is furthermore equivalent to the original proof *)
448 Class StrongCutElimination :=
449 { sce_ce <: CutElimination
450 ; ce_strong : forall h c f, f === ce_eliminate h c f
456 Close Scope nd_scope.
459 End Natural_Deduction.
461 Coercion snd_cut : SequentND >-> Funclass.
462 Coercion cnd_snd : ContextND >-> SequentND.
463 Coercion sndr_ndr : SequentND_Relation >-> ND_Relation.
464 Coercion cndr_sndr : ContextND_Relation >-> SequentND_Relation.
466 Implicit Arguments ND [ Judgment ].
467 Hint Constructors Structural.
468 Hint Extern 1 => apply nd_id_structural.
469 Hint Extern 1 => apply ndr_builtfrom_structural.
470 Hint Extern 1 => apply nd_structural_id0.
471 Hint Extern 1 => apply nd_structural_id1.
472 Hint Extern 1 => apply nd_structural_cancell.
473 Hint Extern 1 => apply nd_structural_cancelr.
474 Hint Extern 1 => apply nd_structural_llecnac.
475 Hint Extern 1 => apply nd_structural_rlecnac.
476 Hint Extern 1 => apply nd_structural_assoc.
477 Hint Extern 1 => apply nd_structural_cossa.
478 Hint Extern 1 => apply ndpc_p.
479 Hint Extern 1 => apply ndpc_prod.
480 Hint Extern 1 => apply ndpc_comp.
481 Hint Extern 1 => apply builtfrom_refl.
482 Hint Extern 1 => apply builtfrom_prod1.
483 Hint Extern 1 => apply builtfrom_prod2.
484 Hint Extern 1 => apply builtfrom_comp1.
485 Hint Extern 1 => apply builtfrom_comp2.
486 Hint Extern 1 => apply builtfrom_P.
488 Hint Extern 1 => apply snd_inert_initial.
489 Hint Extern 1 => apply snd_inert_cut.
490 Hint Extern 1 => apply snd_inert_structural.
492 Hint Extern 1 => apply cnd_inert_initial.
493 Hint Extern 1 => apply cnd_inert_cut.
494 Hint Extern 1 => apply cnd_inert_structural.
495 Hint Extern 1 => apply cnd_inert_cnd_ant_assoc.
496 Hint Extern 1 => apply cnd_inert_cnd_ant_cossa.
497 Hint Extern 1 => apply cnd_inert_cnd_ant_cancell.
498 Hint Extern 1 => apply cnd_inert_cnd_ant_cancelr.
499 Hint Extern 1 => apply cnd_inert_cnd_ant_llecnac.
500 Hint Extern 1 => apply cnd_inert_cnd_ant_rlecnac.
501 Hint Extern 1 => apply cnd_inert_se_expand_left.
502 Hint Extern 1 => apply cnd_inert_se_expand_right.
504 (* This first notation gets its own scope because it can be confusing when we're working with multiple different kinds
505 * of proofs. When only one kind of proof is in use, it's quite helpful though. *)
506 Notation "H /⋯⋯/ C" := (@ND _ _ H C) : pf_scope.
507 Notation "a ;; b" := (nd_comp a b) : nd_scope.
508 Notation "a ** b" := (nd_prod a b) : nd_scope.
509 Notation "[# a #]" := (nd_rule a) : nd_scope.
510 Notation "a === b" := (@ndr_eqv _ _ _ _ _ a b) : nd_scope.
512 (* enable setoid rewriting *)
516 Add Parametric Relation {jt rt ndr h c} : (h/⋯⋯/c) (@ndr_eqv jt rt ndr h c)
517 reflexivity proved by (@Equivalence_Reflexive _ _ (ndr_eqv_equivalence h c))
518 symmetry proved by (@Equivalence_Symmetric _ _ (ndr_eqv_equivalence h c))
519 transitivity proved by (@Equivalence_Transitive _ _ (ndr_eqv_equivalence h c))
520 as parametric_relation_ndr_eqv.
521 Add Parametric Morphism {jt rt ndr h x c} : (@nd_comp jt rt h x c)
522 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
523 as parametric_morphism_nd_comp.
524 intros; apply ndr_comp_respects; auto.
526 Add Parametric Morphism {jt rt ndr a b c d} : (@nd_prod jt rt a b c d)
527 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
528 as parametric_morphism_nd_prod.
529 intros; apply ndr_prod_respects; auto.
532 Section ND_Relation_Facts.
533 Context `{ND_Relation}.
536 Lemma ndr_comp_right_identity : forall h c (f:h/⋯⋯/c), ndr_eqv (f ;; nd_id c) f.
537 intros; apply (ndr_builtfrom_structural f); auto.
541 Lemma ndr_comp_left_identity : forall h c (f:h/⋯⋯/c), ndr_eqv (nd_id h ;; f) f.
542 intros; apply (ndr_builtfrom_structural f); auto.
545 Ltac nd_prod_preserves_comp_ltac P EQV :=
547 [ |- context [ (?A ** ?B) ;; (?C ** ?D) ] ] =>
548 set (@ndr_prod_preserves_comp _ _ EQV _ _ A _ _ B _ C _ D) as P
551 Lemma nd_swap A B C D (f:ND _ A B) (g:ND _ C D) :
552 (f ** nd_id C) ;; (nd_id B ** g) ===
553 (nd_id A ** g) ;; (f ** nd_id D).
554 setoid_rewrite <- ndr_prod_preserves_comp.
555 setoid_rewrite ndr_comp_left_identity.
556 setoid_rewrite ndr_comp_right_identity.
560 (* this tactical searches the environment; setoid_rewrite doesn't seem to be able to do that properly sometimes *)
561 Ltac nd_swap_ltac P EQV :=
563 [ |- context [ (?F ** nd_id _) ;; (nd_id _ ** ?G) ] ] =>
564 set (@nd_swap _ _ EQV _ _ _ _ F G) as P
567 Lemma nd_prod_split_left A B C D (f:ND _ A B) (g:ND _ B C) :
568 nd_id D ** (f ;; g) ===
569 (nd_id D ** f) ;; (nd_id D ** g).
570 setoid_rewrite <- ndr_prod_preserves_comp.
571 setoid_rewrite ndr_comp_left_identity.
575 Lemma nd_prod_split_right A B C D (f:ND _ A B) (g:ND _ B C) :
576 (f ;; g) ** nd_id D ===
577 (f ** nd_id D) ;; (g ** nd_id D).
578 setoid_rewrite <- ndr_prod_preserves_comp.
579 setoid_rewrite ndr_comp_left_identity.
583 End ND_Relation_Facts.
585 (* a generalization of the procedure used to build (nd_id n) from nd_id0 and nd_id1 *)
586 Definition nd_replicate
592 (forall (o:Ob), @ND Judgment Rule [h o] [c o]) ->
593 @ND Judgment Rule (mapOptionTree h j) (mapOptionTree c j).
602 (* "map" over natural deduction proofs, where the result proof has the same judgments (but different rules) *)
605 {Judgment}{Rule0}{Rule1}
606 (r:forall h c, Rule0 h c -> @ND Judgment Rule1 h c)
608 (pf:@ND Judgment Rule0 h c)
610 @ND Judgment Rule1 h c.
611 intros Judgment Rule0 Rule1 r.
613 refine ((fix nd_map h c pf {struct pf} :=
617 @ND Judgment Rule1 H C
619 | nd_id0 => let case_nd_id0 := tt in _
620 | nd_id1 h => let case_nd_id1 := tt in _
621 | nd_weak1 h => let case_nd_weak := tt in _
622 | nd_copy h => let case_nd_copy := tt in _
623 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
624 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
625 | nd_rule _ _ rule => let case_nd_rule := tt in _
626 | nd_cancell _ => let case_nd_cancell := tt in _
627 | nd_cancelr _ => let case_nd_cancelr := tt in _
628 | nd_llecnac _ => let case_nd_llecnac := tt in _
629 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
630 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
631 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
632 end))) ); simpl in *.
634 destruct case_nd_id0. apply nd_id0.
635 destruct case_nd_id1. apply nd_id1.
636 destruct case_nd_weak. apply nd_weak.
637 destruct case_nd_copy. apply nd_copy.
638 destruct case_nd_prod. apply (nd_prod (nd_map _ _ lpf) (nd_map _ _ rpf)).
639 destruct case_nd_comp. apply (nd_comp (nd_map _ _ top) (nd_map _ _ bot)).
640 destruct case_nd_cancell. apply nd_cancell.
641 destruct case_nd_cancelr. apply nd_cancelr.
642 destruct case_nd_llecnac. apply nd_llecnac.
643 destruct case_nd_rlecnac. apply nd_rlecnac.
644 destruct case_nd_assoc. apply nd_assoc.
645 destruct case_nd_cossa. apply nd_cossa.
649 (* "map" over natural deduction proofs, where the result proof has different judgments *)
652 {Judgment0}{Rule0}{Judgment1}{Rule1}
653 (f:Judgment0->Judgment1)
654 (r:forall h c, Rule0 h c -> @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c))
656 (pf:@ND Judgment0 Rule0 h c)
658 @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c).
659 intros Judgment0 Rule0 Judgment1 Rule1 f r.
661 refine ((fix nd_map' h c pf {struct pf} :=
665 @ND Judgment1 Rule1 (mapOptionTree f H) (mapOptionTree f C)
667 | nd_id0 => let case_nd_id0 := tt in _
668 | nd_id1 h => let case_nd_id1 := tt in _
669 | nd_weak1 h => let case_nd_weak := tt in _
670 | nd_copy h => let case_nd_copy := tt in _
671 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
672 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
673 | nd_rule _ _ rule => let case_nd_rule := tt in _
674 | nd_cancell _ => let case_nd_cancell := tt in _
675 | nd_cancelr _ => let case_nd_cancelr := tt in _
676 | nd_llecnac _ => let case_nd_llecnac := tt in _
677 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
678 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
679 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
680 end))) ); simpl in *.
682 destruct case_nd_id0. apply nd_id0.
683 destruct case_nd_id1. apply nd_id1.
684 destruct case_nd_weak. apply nd_weak.
685 destruct case_nd_copy. apply nd_copy.
686 destruct case_nd_prod. apply (nd_prod (nd_map' _ _ lpf) (nd_map' _ _ rpf)).
687 destruct case_nd_comp. apply (nd_comp (nd_map' _ _ top) (nd_map' _ _ bot)).
688 destruct case_nd_cancell. apply nd_cancell.
689 destruct case_nd_cancelr. apply nd_cancelr.
690 destruct case_nd_llecnac. apply nd_llecnac.
691 destruct case_nd_rlecnac. apply nd_rlecnac.
692 destruct case_nd_assoc. apply nd_assoc.
693 destruct case_nd_cossa. apply nd_cossa.
697 (* witnesses the fact that every Rule in a particular proof satisfies the given predicate *)
698 Inductive nd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h}{c}, @ND Judgment Rule h c -> Prop :=
699 | nd_property_structural : forall h c pf, Structural pf -> @nd_property _ _ P h c pf
700 | nd_property_prod : forall h0 c0 pf0 h1 c1 pf1,
701 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P h1 c1 pf1 -> @nd_property _ _ P _ _ (nd_prod pf0 pf1)
702 | nd_property_comp : forall h0 c0 pf0 c1 pf1,
703 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P c0 c1 pf1 -> @nd_property _ _ P _ _ (nd_comp pf0 pf1)
704 | nd_property_rule : forall h c r, P h c r -> @nd_property _ _ P h c (nd_rule r).
705 Hint Constructors nd_property.
707 (* witnesses the fact that every Rule in a particular proof satisfies the given predicate (for SIND) *)
708 Inductive scnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h c}, @SIND Judgment Rule h c -> Prop :=
709 | scnd_property_weak : forall c, @scnd_property _ _ P _ _ (scnd_weak c)
710 | scnd_property_comp : forall h x c r cnd',
712 @scnd_property _ _ P h x cnd' ->
713 @scnd_property _ _ P h _ (scnd_comp _ _ _ cnd' r)
714 | scnd_property_branch :
715 forall x c1 c2 cnd1 cnd2,
716 @scnd_property _ _ P x c1 cnd1 ->
717 @scnd_property _ _ P x c2 cnd2 ->
718 @scnd_property _ _ P x _ (scnd_branch _ _ _ cnd1 cnd2).
720 (* renders a proof as LaTeX code *)
723 Context {Judgment : Type}.
724 Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}.
725 Context {JudgmentToLatexMath : ToLatexMath Judgment}.
726 Context {RuleToLatexMath : forall h c, ToLatexMath (Rule h c)}.
728 Open Scope string_scope.
730 Definition judgments2latex (j:Tree ??Judgment) := treeToLatexMath (mapOptionTree toLatexMath j).
732 Definition eolL : LatexMath := rawLatexMath eol.
734 (* invariant: each proof shall emit its hypotheses visibly, except nd_id0 *)
735 Section SIND_toLatex.
737 (* indicates which rules should be hidden (omitted) from the rendered proof; useful for structural operations *)
738 Context (hideRule : forall h c (r:Rule h c), bool).
740 Fixpoint SIND_toLatexMath {h}{c}(pns:SIND(Rule:=Rule) h c) : LatexMath :=
742 | scnd_branch ht c1 c2 pns1 pns2 => SIND_toLatexMath pns1 +++ rawLatexMath " \hspace{1cm} " +++ SIND_toLatexMath pns2
743 | scnd_weak c => rawLatexMath ""
744 | scnd_comp ht ct c pns rule => if hideRule _ _ rule
745 then SIND_toLatexMath pns
746 else rawLatexMath "\trfrac["+++ toLatexMath rule +++ rawLatexMath "]{" +++ eolL +++
747 SIND_toLatexMath pns +++ rawLatexMath "}{" +++ eolL +++
748 toLatexMath c +++ rawLatexMath "}" +++ eolL
752 (* this is a work-in-progress; please use SIND_toLatexMath for now *)
753 Fixpoint nd_toLatexMath {h}{c}(nd:@ND _ Rule h c)(indent:string) :=
755 | nd_id0 => rawLatexMath indent +++
756 rawLatexMath "% nd_id0 " +++ eolL
757 | nd_id1 h' => rawLatexMath indent +++
758 rawLatexMath "% nd_id1 "+++ judgments2latex h +++ eolL
759 | nd_weak1 h' => rawLatexMath indent +++
760 rawLatexMath "\inferrule*[Left=ndWeak]{" +++ toLatexMath h' +++ rawLatexMath "}{ }" +++ eolL
761 | nd_copy h' => rawLatexMath indent +++
762 rawLatexMath "\inferrule*[Left=ndCopy]{"+++judgments2latex h+++
763 rawLatexMath "}{"+++judgments2latex c+++rawLatexMath "}" +++ eolL
764 | nd_prod h1 h2 c1 c2 pf1 pf2 => rawLatexMath indent +++
765 rawLatexMath "% prod " +++ eolL +++
766 rawLatexMath indent +++
767 rawLatexMath "\begin{array}{c c}" +++ eolL +++
768 (nd_toLatexMath pf1 (" "+++indent)) +++
769 rawLatexMath indent +++
770 rawLatexMath " & " +++ eolL +++
771 (nd_toLatexMath pf2 (" "+++indent)) +++
772 rawLatexMath indent +++
773 rawLatexMath "\end{array}"
774 | nd_comp h m c pf1 pf2 => rawLatexMath indent +++
775 rawLatexMath "% comp " +++ eolL +++
776 rawLatexMath indent +++
777 rawLatexMath "\begin{array}{c}" +++ eolL +++
778 (nd_toLatexMath pf1 (" "+++indent)) +++
779 rawLatexMath indent +++
780 rawLatexMath " \\ " +++ eolL +++
781 (nd_toLatexMath pf2 (" "+++indent)) +++
782 rawLatexMath indent +++
783 rawLatexMath "\end{array}"
784 | nd_cancell a => rawLatexMath indent +++
785 rawLatexMath "% nd-cancell " +++ (judgments2latex a) +++ eolL
786 | nd_cancelr a => rawLatexMath indent +++
787 rawLatexMath "% nd-cancelr " +++ (judgments2latex a) +++ eolL
788 | nd_llecnac a => rawLatexMath indent +++
789 rawLatexMath "% nd-llecnac " +++ (judgments2latex a) +++ eolL
790 | nd_rlecnac a => rawLatexMath indent +++
791 rawLatexMath "% nd-rlecnac " +++ (judgments2latex a) +++ eolL
792 | nd_assoc a b c => rawLatexMath ""
793 | nd_cossa a b c => rawLatexMath ""
794 | nd_rule h c r => toLatexMath r
799 Close Scope pf_scope.
800 Close Scope nd_scope.