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.
232 Hint Unfold StructuralND.
234 Lemma nd_id_structural : forall sl, StructuralND (nd_id sl).
236 induction sl; simpl; auto.
240 (* An equivalence relation on proofs which is sensitive only to the logical content of the proof -- insensitive to
241 * structural variations *)
243 { ndr_eqv : forall {h c }, h /⋯⋯/ c -> h /⋯⋯/ c -> Prop where "pf1 === pf2" := (@ndr_eqv _ _ pf1 pf2)
244 ; ndr_eqv_equivalence : forall h c, Equivalence (@ndr_eqv h c)
246 (* the relation must respect composition, be associative wrt composition, and be left and right neutral wrt the identity proof *)
247 ; ndr_comp_respects : forall {a b c}(f f':a/⋯⋯/b)(g g':b/⋯⋯/c), f === f' -> g === g' -> f;;g === f';;g'
248 ; ndr_comp_associativity : forall `(f:a/⋯⋯/b)`(g:b/⋯⋯/c)`(h:c/⋯⋯/d), (f;;g);;h === f;;(g;;h)
250 (* the relation must respect products, be associative wrt products, and be left and right neutral wrt the identity proof *)
251 ; ndr_prod_respects : forall {a b c d}(f f':a/⋯⋯/b)(g g':c/⋯⋯/d), f===f' -> g===g' -> f**g === f'**g'
252 ; ndr_prod_associativity : forall `(f:a/⋯⋯/a')`(g:b/⋯⋯/b')`(h:c/⋯⋯/c'), (f**g)**h === nd_assoc ;; f**(g**h) ;; nd_cossa
254 (* products and composition must distribute over each other *)
255 ; 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')
257 (* Given a proof f, any two proofs built from it using only structural rules are indistinguishable. Keep in mind that
258 * nd_weak and nd_copy aren't considered structural, so the hypotheses and conclusions of such proofs will be an identical
259 * list, differing only in the "parenthesization" and addition or removal of empty leaves. *)
260 ; ndr_builtfrom_structural : forall `(f:a/⋯⋯/b){a' b'}(g1 g2:a'/⋯⋯/b'),
261 BuiltFrom f (@StructuralND) g1 ->
262 BuiltFrom f (@StructuralND) g2 ->
265 (* proofs of nothing are not distinguished from each other *)
266 ; ndr_void_proofs_irrelevant : forall `(f:a/⋯⋯/[])(g:a/⋯⋯/[]), f === g
268 (* products and duplication must distribute over each other *)
269 ; ndr_prod_preserves_copy : forall `(f:a/⋯⋯/b), nd_copy a;; f**f === f ;; nd_copy b
271 (* duplicating a hypothesis and discarding it is irrelevant *)
272 ; ndr_copy_then_weak_left : forall a, nd_copy a;; (nd_weak _ ** nd_id _) ;; nd_cancell === nd_id _
273 ; ndr_copy_then_weak_right : forall a, nd_copy a;; (nd_id _ ** nd_weak _) ;; nd_cancelr === nd_id _
277 * Natural Deduction proofs which are Structurally Implicit on the
278 * level of judgments. These proofs have poor compositionality
279 * properties (vertically, they look more like lists than trees) but
280 * are easier to do induction over.
282 Inductive SIND : Tree ??Judgment -> Tree ??Judgment -> Type :=
283 | scnd_weak : forall c , SIND c []
284 | scnd_comp : forall ht ct c , SIND ht ct -> Rule ct [c] -> SIND ht [c]
285 | scnd_branch : forall ht c1 c2, SIND ht c1 -> SIND ht c2 -> SIND ht (c1,,c2)
287 Hint Constructors SIND.
289 (* Any ND whose primitive Rules have at most one conclusion (note that nd_prod is allowed!) can be turned into an SIND. *)
290 Definition mkSIND (all_rules_one_conclusion : forall h c1 c2, Rule h (c1,,c2) -> False)
291 : forall h x c, ND x c -> SIND h x -> SIND h c.
293 induction nd; intro k.
297 eapply scnd_branch; apply k.
299 apply (scnd_branch _ _ _ (IHnd1 X) (IHnd2 X0)).
303 inversion k; subst; auto.
304 inversion k; subst; auto.
305 apply scnd_branch; auto.
306 apply scnd_branch; auto.
307 inversion k; subst; inversion X; subst; auto.
308 inversion k; subst; inversion X0; subst; auto.
311 eapply scnd_comp. apply k. apply r.
313 set (all_rules_one_conclusion _ _ _ r) as bogus.
317 (* Natural Deduction systems whose judgments happen to be pairs of the same type *)
319 Context {S:Type}. (* type of sequent components *)
320 Context {sequent:S->S->Judgment}. (* pairing operation which forms a sequent from its halves *)
321 Notation "a |= b" := (sequent a b).
323 (* a SequentND is a natural deduction whose judgments are sequents, has initial sequents, and has a cut rule *)
325 { snd_initial : forall a, [ ] /⋯⋯/ [ a |= a ]
326 ; snd_cut : forall a b c, [ a |= b ] ,, [ b |= c ] /⋯⋯/ [ a |= c ]
329 Context (sequentND:SequentND).
330 Context (ndr:ND_Relation).
333 * A predicate singling out structural rules, initial sequents,
336 * Proofs using only structural rules cannot add or remove
337 * judgments - their hypothesis and conclusion judgment-trees will
338 * differ only in "parenthesization" and the presence/absence of
339 * empty leaves. This means that a proof involving only
340 * structural rules, cut, and initial sequents can ADD new
341 * non-empty judgment-leaves only via snd_initial, and can only
342 * REMOVE non-empty judgment-leaves only via snd_cut. Since the
343 * initial sequent is a left and right identity for cut, and cut
344 * is associative, any two proofs (with the same hypotheses and
345 * conclusions) using only structural rules, cut, and initial
346 * sequents are logically indistinguishable - their differences
347 * are logically insignificant.
349 * Note that it is important that nd_weak and nd_copy aren't
350 * considered to be "structural".
352 Inductive SequentND_Inert : forall h c, h/⋯⋯/c -> Prop :=
353 | snd_inert_initial : forall a, SequentND_Inert _ _ (snd_initial a)
354 | snd_inert_cut : forall a b c, SequentND_Inert _ _ (snd_cut a b c)
355 | snd_inert_structural: forall a b f, Structural f -> SequentND_Inert a b f
358 (* An ND_Relation for a sequent deduction should not distinguish between two proofs having the same hypotheses and conclusions
359 * if those proofs use only initial sequents, cut, and structural rules (see comment above) *)
360 Class SequentND_Relation :=
362 ; sndr_inert : forall a b (f g:a/⋯⋯/b),
363 NDPredicateClosure SequentND_Inert f ->
364 NDPredicateClosure SequentND_Inert g ->
369 (* Deductions on sequents whose antecedent is a tree of propositions (i.e. a context) *)
371 Context {P:Type}{sequent:Tree ??P -> Tree ??P -> Judgment}.
372 Context {snd:SequentND(sequent:=sequent)}.
373 Notation "a |= b" := (sequent a b).
375 (* Note that these rules mirror nd_{cancell,cancelr,rlecnac,llecnac,assoc,cossa} but are completely separate from them *)
377 { cnd_ant_assoc : forall x a b c, ND [((a,,b),,c) |= x] [(a,,(b,,c)) |= x ]
378 ; cnd_ant_cossa : forall x a b c, ND [(a,,(b,,c)) |= x] [((a,,b),,c) |= x ]
379 ; cnd_ant_cancell : forall x a , ND [ [],,a |= x] [ a |= x ]
380 ; cnd_ant_cancelr : forall x a , ND [a,,[] |= x] [ a |= x ]
381 ; cnd_ant_llecnac : forall x a , ND [ a |= x] [ [],,a |= x ]
382 ; cnd_ant_rlecnac : forall x a , ND [ a |= x] [ a,,[] |= x ]
383 ; cnd_expand_left : forall a b c , ND [ a |= b] [ c,,a |= c,,b]
384 ; cnd_expand_right : forall a b c , ND [ a |= b] [ a,,c |= b,,c]
388 Context `(ContextND).
391 * A predicate singling out initial sequents, cuts, context expansion,
392 * and structural rules.
394 * Any two proofs (with the same hypotheses and conclusions) whose
395 * non-structural rules do nothing other than expand contexts,
396 * re-arrange contexts, or introduce additional initial-sequent
397 * conclusions are indistinguishable. One important consequence
398 * is that asking for a small initial sequent and then expanding
399 * it using cnd_expand_{right,left} is no different from simply
400 * asking for the larger initial sequent in the first place.
403 Inductive ContextND_Inert : forall h c, h/⋯⋯/c -> Prop :=
404 | cnd_inert_initial : forall a, ContextND_Inert _ _ (snd_initial a)
405 | cnd_inert_cut : forall a b c, ContextND_Inert _ _ (snd_cut a b c)
406 | cnd_inert_structural : forall a b f, Structural f -> ContextND_Inert a b f
407 | cnd_inert_cnd_ant_assoc : forall x a b c, ContextND_Inert _ _ (cnd_ant_assoc x a b c)
408 | cnd_inert_cnd_ant_cossa : forall x a b c, ContextND_Inert _ _ (cnd_ant_cossa x a b c)
409 | cnd_inert_cnd_ant_cancell : forall x a , ContextND_Inert _ _ (cnd_ant_cancell x a)
410 | cnd_inert_cnd_ant_cancelr : forall x a , ContextND_Inert _ _ (cnd_ant_cancelr x a)
411 | cnd_inert_cnd_ant_llecnac : forall x a , ContextND_Inert _ _ (cnd_ant_llecnac x a)
412 | cnd_inert_cnd_ant_rlecnac : forall x a , ContextND_Inert _ _ (cnd_ant_rlecnac x a)
413 | cnd_inert_se_expand_left : forall t g s , ContextND_Inert _ _ (@cnd_expand_left _ t g s)
414 | cnd_inert_se_expand_right : forall t g s , ContextND_Inert _ _ (@cnd_expand_right _ t g s).
416 Class ContextND_Relation {ndr}{sndr:SequentND_Relation _ ndr} :=
417 { cndr_inert : forall {a}{b}(f g:a/⋯⋯/b),
418 NDPredicateClosure ContextND_Inert f ->
419 NDPredicateClosure ContextND_Inert g ->
424 (* a proof is Analytic if it does not use cut *)
426 Definition Analytic_Rule : NDPredicate :=
427 fun h c f => forall c, not (snd_cut _ _ c = f).
428 Definition AnalyticND := NDPredicateClosure Analytic_Rule.
430 (* a proof system has cut elimination if, for every proof, there is an analytic proof with the same conclusion *)
431 Class CutElimination :=
432 { ce_eliminate : forall h c, h/⋯⋯/c -> h/⋯⋯/c
433 ; ce_analytic : forall h c f, AnalyticND (ce_eliminate h c f)
436 (* cut elimination is strong if the analytic proof is furthermore equivalent to the original proof *)
437 Class StrongCutElimination :=
438 { sce_ce <: CutElimination
439 ; ce_strong : forall h c f, f === ce_eliminate h c f
445 Close Scope nd_scope.
448 End Natural_Deduction.
450 Coercion snd_cut : SequentND >-> Funclass.
451 Coercion cnd_snd : ContextND >-> SequentND.
452 Coercion sndr_ndr : SequentND_Relation >-> ND_Relation.
453 Coercion cndr_sndr : ContextND_Relation >-> SequentND_Relation.
455 Implicit Arguments ND [ Judgment ].
457 (* This first notation gets its own scope because it can be confusing when we're working with multiple different kinds
458 * of proofs. When only one kind of proof is in use, it's quite helpful though. *)
459 Notation "H /⋯⋯/ C" := (@ND _ _ H C) : pf_scope.
460 Notation "a ;; b" := (nd_comp a b) : nd_scope.
461 Notation "a ** b" := (nd_prod a b) : nd_scope.
462 Notation "[# a #]" := (nd_rule a) : nd_scope.
463 Notation "a === b" := (@ndr_eqv _ _ _ _ _ a b) : nd_scope.
465 Hint Constructors Structural.
466 Hint Constructors ND_Relation.
467 Hint Constructors BuiltFrom.
468 Hint Constructors NDPredicateClosure.
469 Hint Constructors ContextND_Inert.
470 Hint Constructors SequentND_Inert.
471 Hint Unfold StructuralND.
473 (* enable setoid rewriting *)
477 Hint Extern 2 (StructuralND (nd_id _)) => apply nd_id_structural.
478 Hint Extern 2 (NDPredicateClosure _ ( _ ;; _ ) ) => apply ndpc_comp.
479 Hint Extern 2 (NDPredicateClosure _ ( _ ** _ ) ) => apply ndpc_prod.
480 Hint Extern 2 (NDPredicateClosure (@Structural _ _) (nd_id _)) => apply nd_id_structural.
481 Hint Extern 2 (BuiltFrom _ _ ( _ ;; _ ) ) => apply builtfrom_comp1.
482 Hint Extern 2 (BuiltFrom _ _ ( _ ;; _ ) ) => apply builtfrom_comp2.
483 Hint Extern 2 (BuiltFrom _ _ ( _ ** _ ) ) => apply builtfrom_prod1.
484 Hint Extern 2 (BuiltFrom _ _ ( _ ** _ ) ) => apply builtfrom_prod2.
486 (* Hint Constructors has failed me! *)
487 Hint Extern 2 (@Structural _ _ _ _ (@nd_id0 _ _)) => apply nd_structural_id0.
488 Hint Extern 2 (@Structural _ _ _ _ (@nd_id1 _ _ _)) => apply nd_structural_id1.
489 Hint Extern 2 (@Structural _ _ _ _ (@nd_cancell _ _ _)) => apply nd_structural_cancell.
490 Hint Extern 2 (@Structural _ _ _ _ (@nd_cancelr _ _ _)) => apply nd_structural_cancelr.
491 Hint Extern 2 (@Structural _ _ _ _ (@nd_llecnac _ _ _)) => apply nd_structural_llecnac.
492 Hint Extern 2 (@Structural _ _ _ _ (@nd_rlecnac _ _ _)) => apply nd_structural_rlecnac.
493 Hint Extern 2 (@Structural _ _ _ _ (@nd_assoc _ _ _ _ _)) => apply nd_structural_assoc.
494 Hint Extern 2 (@Structural _ _ _ _ (@nd_cossa _ _ _ _ _)) => apply nd_structural_cossa.
496 Hint Extern 4 (NDPredicateClosure _ _) => apply ndpc_p.
498 Add Parametric Relation {jt rt ndr h c} : (h/⋯⋯/c) (@ndr_eqv jt rt ndr h c)
499 reflexivity proved by (@Equivalence_Reflexive _ _ (ndr_eqv_equivalence h c))
500 symmetry proved by (@Equivalence_Symmetric _ _ (ndr_eqv_equivalence h c))
501 transitivity proved by (@Equivalence_Transitive _ _ (ndr_eqv_equivalence h c))
502 as parametric_relation_ndr_eqv.
503 Add Parametric Morphism {jt rt ndr h x c} : (@nd_comp jt rt h x c)
504 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
505 as parametric_morphism_nd_comp.
506 intros; apply ndr_comp_respects; auto.
508 Add Parametric Morphism {jt rt ndr a b c d} : (@nd_prod jt rt a b c d)
509 with signature ((ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)) ==> (ndr_eqv(ND_Relation:=ndr)))
510 as parametric_morphism_nd_prod.
511 intros; apply ndr_prod_respects; auto.
514 Section ND_Relation_Facts.
515 Context `{ND_Relation}.
518 Lemma ndr_comp_right_identity : forall h c (f:h/⋯⋯/c), ndr_eqv (f ;; nd_id c) f.
519 intros; apply (ndr_builtfrom_structural f). auto.
524 Lemma ndr_comp_left_identity : forall h c (f:h/⋯⋯/c), ndr_eqv (nd_id h ;; f) f.
525 intros; apply (ndr_builtfrom_structural f); auto.
528 Ltac nd_prod_preserves_comp_ltac P EQV :=
530 [ |- context [ (?A ** ?B) ;; (?C ** ?D) ] ] =>
531 set (@ndr_prod_preserves_comp _ _ EQV _ _ A _ _ B _ C _ D) as P
534 Lemma nd_swap A B C D (f:ND _ A B) (g:ND _ C D) :
535 (f ** nd_id C) ;; (nd_id B ** g) ===
536 (nd_id A ** g) ;; (f ** nd_id D).
537 setoid_rewrite <- ndr_prod_preserves_comp.
538 setoid_rewrite ndr_comp_left_identity.
539 setoid_rewrite ndr_comp_right_identity.
543 (* this tactical searches the environment; setoid_rewrite doesn't seem to be able to do that properly sometimes *)
544 Ltac nd_swap_ltac P EQV :=
546 [ |- context [ (?F ** nd_id _) ;; (nd_id _ ** ?G) ] ] =>
547 set (@nd_swap _ _ EQV _ _ _ _ F G) as P
550 Lemma nd_prod_split_left A B C D (f:ND _ A B) (g:ND _ B C) :
551 nd_id D ** (f ;; g) ===
552 (nd_id D ** f) ;; (nd_id D ** g).
553 setoid_rewrite <- ndr_prod_preserves_comp.
554 setoid_rewrite ndr_comp_left_identity.
558 Lemma nd_prod_split_right A B C D (f:ND _ A B) (g:ND _ B C) :
559 (f ;; g) ** nd_id D ===
560 (f ** nd_id D) ;; (g ** nd_id D).
561 setoid_rewrite <- ndr_prod_preserves_comp.
562 setoid_rewrite ndr_comp_left_identity.
566 End ND_Relation_Facts.
568 (* a generalization of the procedure used to build (nd_id n) from nd_id0 and nd_id1 *)
569 Definition nd_replicate
575 (forall (o:Ob), @ND Judgment Rule [h o] [c o]) ->
576 @ND Judgment Rule (mapOptionTree h j) (mapOptionTree c j).
585 (* "map" over natural deduction proofs, where the result proof has the same judgments (but different rules) *)
588 {Judgment}{Rule0}{Rule1}
589 (r:forall h c, Rule0 h c -> @ND Judgment Rule1 h c)
591 (pf:@ND Judgment Rule0 h c)
593 @ND Judgment Rule1 h c.
594 intros Judgment Rule0 Rule1 r.
596 refine ((fix nd_map h c pf {struct pf} :=
600 @ND Judgment Rule1 H C
602 | nd_id0 => let case_nd_id0 := tt in _
603 | nd_id1 h => let case_nd_id1 := tt in _
604 | nd_weak1 h => let case_nd_weak := tt in _
605 | nd_copy h => let case_nd_copy := tt in _
606 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
607 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
608 | nd_rule _ _ rule => let case_nd_rule := tt in _
609 | nd_cancell _ => let case_nd_cancell := tt in _
610 | nd_cancelr _ => let case_nd_cancelr := tt in _
611 | nd_llecnac _ => let case_nd_llecnac := tt in _
612 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
613 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
614 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
615 end))) ); simpl in *.
617 destruct case_nd_id0. apply nd_id0.
618 destruct case_nd_id1. apply nd_id1.
619 destruct case_nd_weak. apply nd_weak.
620 destruct case_nd_copy. apply nd_copy.
621 destruct case_nd_prod. apply (nd_prod (nd_map _ _ lpf) (nd_map _ _ rpf)).
622 destruct case_nd_comp. apply (nd_comp (nd_map _ _ top) (nd_map _ _ bot)).
623 destruct case_nd_cancell. apply nd_cancell.
624 destruct case_nd_cancelr. apply nd_cancelr.
625 destruct case_nd_llecnac. apply nd_llecnac.
626 destruct case_nd_rlecnac. apply nd_rlecnac.
627 destruct case_nd_assoc. apply nd_assoc.
628 destruct case_nd_cossa. apply nd_cossa.
632 (* "map" over natural deduction proofs, where the result proof has different judgments *)
635 {Judgment0}{Rule0}{Judgment1}{Rule1}
636 (f:Judgment0->Judgment1)
637 (r:forall h c, Rule0 h c -> @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c))
639 (pf:@ND Judgment0 Rule0 h c)
641 @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c).
642 intros Judgment0 Rule0 Judgment1 Rule1 f r.
644 refine ((fix nd_map' h c pf {struct pf} :=
648 @ND Judgment1 Rule1 (mapOptionTree f H) (mapOptionTree f C)
650 | nd_id0 => let case_nd_id0 := tt in _
651 | nd_id1 h => let case_nd_id1 := tt in _
652 | nd_weak1 h => let case_nd_weak := tt in _
653 | nd_copy h => let case_nd_copy := tt in _
654 | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
655 | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
656 | nd_rule _ _ rule => let case_nd_rule := tt in _
657 | nd_cancell _ => let case_nd_cancell := tt in _
658 | nd_cancelr _ => let case_nd_cancelr := tt in _
659 | nd_llecnac _ => let case_nd_llecnac := tt in _
660 | nd_rlecnac _ => let case_nd_rlecnac := tt in _
661 | nd_assoc _ _ _ => let case_nd_assoc := tt in _
662 | nd_cossa _ _ _ => let case_nd_cossa := tt in _
663 end))) ); simpl in *.
665 destruct case_nd_id0. apply nd_id0.
666 destruct case_nd_id1. apply nd_id1.
667 destruct case_nd_weak. apply nd_weak.
668 destruct case_nd_copy. apply nd_copy.
669 destruct case_nd_prod. apply (nd_prod (nd_map' _ _ lpf) (nd_map' _ _ rpf)).
670 destruct case_nd_comp. apply (nd_comp (nd_map' _ _ top) (nd_map' _ _ bot)).
671 destruct case_nd_cancell. apply nd_cancell.
672 destruct case_nd_cancelr. apply nd_cancelr.
673 destruct case_nd_llecnac. apply nd_llecnac.
674 destruct case_nd_rlecnac. apply nd_rlecnac.
675 destruct case_nd_assoc. apply nd_assoc.
676 destruct case_nd_cossa. apply nd_cossa.
680 (* witnesses the fact that every Rule in a particular proof satisfies the given predicate *)
681 Inductive nd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h}{c}, @ND Judgment Rule h c -> Prop :=
682 | nd_property_structural : forall h c pf, Structural pf -> @nd_property _ _ P h c pf
683 | nd_property_prod : forall h0 c0 pf0 h1 c1 pf1,
684 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P h1 c1 pf1 -> @nd_property _ _ P _ _ (nd_prod pf0 pf1)
685 | nd_property_comp : forall h0 c0 pf0 c1 pf1,
686 @nd_property _ _ P h0 c0 pf0 -> @nd_property _ _ P c0 c1 pf1 -> @nd_property _ _ P _ _ (nd_comp pf0 pf1)
687 | nd_property_rule : forall h c r, P h c r -> @nd_property _ _ P h c (nd_rule r).
688 Hint Constructors nd_property.
690 (* witnesses the fact that every Rule in a particular proof satisfies the given predicate (for SIND) *)
691 Inductive scnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h c}, @SIND Judgment Rule h c -> Prop :=
692 | scnd_property_weak : forall c, @scnd_property _ _ P _ _ (scnd_weak c)
693 | scnd_property_comp : forall h x c r cnd',
695 @scnd_property _ _ P h x cnd' ->
696 @scnd_property _ _ P h _ (scnd_comp _ _ _ cnd' r)
697 | scnd_property_branch :
698 forall x c1 c2 cnd1 cnd2,
699 @scnd_property _ _ P x c1 cnd1 ->
700 @scnd_property _ _ P x c2 cnd2 ->
701 @scnd_property _ _ P x _ (scnd_branch _ _ _ cnd1 cnd2).
703 (* renders a proof as LaTeX code *)
706 Context {Judgment : Type}.
707 Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}.
708 Context {JudgmentToLatexMath : ToLatexMath Judgment}.
709 Context {RuleToLatexMath : forall h c, ToLatexMath (Rule h c)}.
711 Open Scope string_scope.
713 Definition judgments2latex (j:Tree ??Judgment) := treeToLatexMath (mapOptionTree toLatexMath j).
715 Definition eolL : LatexMath := rawLatexMath eol.
717 (* invariant: each proof shall emit its hypotheses visibly, except nd_id0 *)
718 Section SIND_toLatex.
720 (* indicates which rules should be hidden (omitted) from the rendered proof; useful for structural operations *)
721 Context (hideRule : forall h c (r:Rule h c), bool).
723 Fixpoint SIND_toLatexMath {h}{c}(pns:SIND(Rule:=Rule) h c) : LatexMath :=
725 | scnd_branch ht c1 c2 pns1 pns2 => SIND_toLatexMath pns1 +++ rawLatexMath " \hspace{1cm} " +++ SIND_toLatexMath pns2
726 | scnd_weak c => rawLatexMath ""
727 | scnd_comp ht ct c pns rule => if hideRule _ _ rule
728 then SIND_toLatexMath pns
729 else rawLatexMath "\trfrac["+++ toLatexMath rule +++ rawLatexMath "]{" +++ eolL +++
730 SIND_toLatexMath pns +++ rawLatexMath "}{" +++ eolL +++
731 toLatexMath c +++ rawLatexMath "}" +++ eolL
735 (* this is a work-in-progress; please use SIND_toLatexMath for now *)
736 Fixpoint nd_toLatexMath {h}{c}(nd:@ND _ Rule h c)(indent:string) :=
738 | nd_id0 => rawLatexMath indent +++
739 rawLatexMath "% nd_id0 " +++ eolL
740 | nd_id1 h' => rawLatexMath indent +++
741 rawLatexMath "% nd_id1 "+++ judgments2latex h +++ eolL
742 | nd_weak1 h' => rawLatexMath indent +++
743 rawLatexMath "\inferrule*[Left=ndWeak]{" +++ toLatexMath h' +++ rawLatexMath "}{ }" +++ eolL
744 | nd_copy h' => rawLatexMath indent +++
745 rawLatexMath "\inferrule*[Left=ndCopy]{"+++judgments2latex h+++
746 rawLatexMath "}{"+++judgments2latex c+++rawLatexMath "}" +++ eolL
747 | nd_prod h1 h2 c1 c2 pf1 pf2 => rawLatexMath indent +++
748 rawLatexMath "% prod " +++ eolL +++
749 rawLatexMath indent +++
750 rawLatexMath "\begin{array}{c c}" +++ eolL +++
751 (nd_toLatexMath pf1 (" "+++indent)) +++
752 rawLatexMath indent +++
753 rawLatexMath " & " +++ eolL +++
754 (nd_toLatexMath pf2 (" "+++indent)) +++
755 rawLatexMath indent +++
756 rawLatexMath "\end{array}"
757 | nd_comp h m c pf1 pf2 => rawLatexMath indent +++
758 rawLatexMath "% comp " +++ eolL +++
759 rawLatexMath indent +++
760 rawLatexMath "\begin{array}{c}" +++ eolL +++
761 (nd_toLatexMath pf1 (" "+++indent)) +++
762 rawLatexMath indent +++
763 rawLatexMath " \\ " +++ eolL +++
764 (nd_toLatexMath pf2 (" "+++indent)) +++
765 rawLatexMath indent +++
766 rawLatexMath "\end{array}"
767 | nd_cancell a => rawLatexMath indent +++
768 rawLatexMath "% nd-cancell " +++ (judgments2latex a) +++ eolL
769 | nd_cancelr a => rawLatexMath indent +++
770 rawLatexMath "% nd-cancelr " +++ (judgments2latex a) +++ eolL
771 | nd_llecnac a => rawLatexMath indent +++
772 rawLatexMath "% nd-llecnac " +++ (judgments2latex a) +++ eolL
773 | nd_rlecnac a => rawLatexMath indent +++
774 rawLatexMath "% nd-rlecnac " +++ (judgments2latex a) +++ eolL
775 | nd_assoc a b c => rawLatexMath ""
776 | nd_cossa a b c => rawLatexMath ""
777 | nd_rule h c r => toLatexMath r
782 Close Scope pf_scope.
783 Close Scope nd_scope.