X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FNaturalDeduction.v;h=ced66c4b30d8efe99749bf21dc11f0ceabfd098d;hp=2bc361f1bb0daa5c034322fa48c735d04146ae4f;hb=6ef9f270b138fc7aab48013d55a8192ff022c0f1;hpb=358d686692c40722885eaeffc7203e32142af7c8 diff --git a/src/NaturalDeduction.v b/src/NaturalDeduction.v index 2bc361f..ced66c4 100644 --- a/src/NaturalDeduction.v +++ b/src/NaturalDeduction.v @@ -1,5 +1,8 @@ (*********************************************************************************************************************************) -(* NaturalDeduction: structurally explicit proofs in Coq *) +(* NaturalDeduction: *) +(* *) +(* Structurally explicit natural deduction proofs. *) +(* *) (*********************************************************************************************************************************) Generalizable All Variables. @@ -9,9 +12,7 @@ Require Import Coq.Strings.Ascii. Require Import Coq.Strings.String. (* - * IMPORTANT!!! - * - * Unlike most formalizations, this library offers TWO different ways + * Unlike most formalizations, this library offers two different ways * to represent a natural deduction proof. To demonstrate this, * consider the signature of the propositional calculus: * @@ -79,6 +80,22 @@ Require Import Coq.Strings.String. * (NaturalDeduction.v) and are designed specifically in order to * circumvent the problem in the previous paragraph. * + * These proofs are actually structurally explicit on (potentially) + * two different levels. The beginning of this file formalizes + * natural deduction proofs with explicit structural operations for + * manipulating lists of judgments – for example, the open + * hypotheses of an incomplete proof. The class + * TreeStructuralRules further down in the file instantiates ND + * such that Judgments is actually a pair of trees of propositions, + * and there will be a whole *other* set of rules for manipulating + * the structure of a tree of propositions *within* a single + * judgment. + * + * The flattening functor ends up mapping the first kind of + * structural operation (moving around judgments) onto the second + * kind (moving around propositions/types). That's why everything + * is so laboriously explicit - there's important information in + * those structural operations. *) (* @@ -113,8 +130,10 @@ Section Natural_Deduction. forall conclusions:Tree ??Judgment, Type := - (* natural deduction: you may infer anything from itself -- "identity proof" *) + (* natural deduction: you may infer nothing from nothing *) | nd_id0 : [ ] /⋯⋯/ [ ] + + (* natural deduction: you may infer anything from itself -- "identity proof" *) | nd_id1 : forall h, [ h ] /⋯⋯/ [ h ] (* natural deduction: you may discard conclusions *) @@ -136,7 +155,9 @@ Section Natural_Deduction. `(pf2: x /⋯⋯/ c), ( h /⋯⋯/ c) - (* structural rules on lists of judgments *) + (* Structural rules on lists of judgments - note that this is completely separate from the structural + * rules for *contexts* within a sequent. The rules below manipulate lists of *judgments* rather than + * lists of *propositions*. *) | nd_cancell : forall {a}, [] ,, a /⋯⋯/ a | nd_cancelr : forall {a}, a ,, [] /⋯⋯/ a | nd_llecnac : forall {a}, a /⋯⋯/ [] ,, a @@ -179,6 +200,13 @@ Section Natural_Deduction. | T_Branch a b => nd_prod (nd_id a) (nd_id b) end. + Fixpoint nd_weak' (sl:Tree ??Judgment) : sl /⋯⋯/ [] := + match sl as SL return SL /⋯⋯/ [] with + | T_Leaf None => nd_id0 + | T_Leaf (Some x) => nd_weak x + | T_Branch a b => nd_prod (nd_weak' a) (nd_weak' b) ;; nd_cancelr + end. + Hint Constructors Structural. Lemma nd_id_structural : forall sl, Structural (nd_id sl). intros. @@ -186,6 +214,16 @@ Section Natural_Deduction. destruct a; auto. Defined. + Lemma weak'_structural : forall a, Structural (nd_weak' a). + intros. + induction a. + destruct a; auto. + simpl. + auto. + simpl. + auto. + Qed. + (* An equivalence relation on proofs which is sensitive only to the logical content of the proof -- insensitive to * structural variations *) Class ND_Relation := @@ -207,27 +245,32 @@ Section Natural_Deduction. (* products and composition must distribute over each other *) ; 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') + (* products and duplication must distribute over each other *) + ; ndr_prod_preserves_copy : forall `(f:a/⋯⋯/b), nd_copy a;; f**f === f ;; nd_copy b + (* any two _structural_ proofs with the same hypotheses/conclusions must be considered equal *) ; ndr_structural_indistinguishable : forall `(f:a/⋯⋯/b)(g:a/⋯⋯/b), Structural f -> Structural g -> f===g + + (* any two proofs of nothing are "equally good" *) + ; ndr_void_proofs_irrelevant : forall `(f:a/⋯⋯/[])(g:a/⋯⋯/[]), f === g }. (* - * Single-conclusion proofs; this is an alternate representation - * where each inference has only a single conclusion. These have - * worse compositionality properties than ND's, but are easier to - * emit as LaTeX code. + * Natural Deduction proofs which are Structurally Implicit on the + * level of judgments. These proofs have poor compositionality + * properties (vertically, they look more like lists than trees) but + * are easier to do induction over. *) - Inductive SCND : Tree ??Judgment -> Tree ??Judgment -> Type := - | scnd_comp : forall ht ct c , SCND ht ct -> Rule ct [c] -> SCND ht [c] - | scnd_weak : forall c , SCND c [] - | scnd_leaf : forall ht c , SCND ht [c] -> SCND ht [c] - | scnd_branch : forall ht c1 c2, SCND ht c1 -> SCND ht c2 -> SCND ht (c1,,c2) + Inductive SIND : Tree ??Judgment -> Tree ??Judgment -> Type := + | scnd_weak : forall c , SIND c [] + | scnd_comp : forall ht ct c , SIND ht ct -> Rule ct [c] -> SIND ht [c] + | scnd_branch : forall ht c1 c2, SIND ht c1 -> SIND ht c2 -> SIND ht (c1,,c2) . - Hint Constructors SCND. + Hint Constructors SIND. - (* Any ND whose primitive Rules have at most one conclusion (note that nd_prod is allowed!) can be turned into an SCND. *) - Definition mkSCND (all_rules_one_conclusion : forall h c1 c2, Rule h (c1,,c2) -> False) - : forall h x c, ND x c -> SCND h x -> SCND h c. + (* Any ND whose primitive Rules have at most one conclusion (note that nd_prod is allowed!) can be turned into an SIND. *) + Definition mkSIND (all_rules_one_conclusion : forall h c1 c2, Rule h (c1,,c2) -> False) + : forall h x c, ND x c -> SIND h x -> SIND h c. intros h x c nd. induction nd; intro k. apply k. @@ -247,50 +290,43 @@ Section Natural_Deduction. inversion k; subst; inversion X0; subst; auto. destruct c. destruct o. - apply scnd_leaf. eapply scnd_comp. apply k. apply r. + eapply scnd_comp. apply k. apply r. apply scnd_weak. set (all_rules_one_conclusion _ _ _ r) as bogus. inversion bogus. Defined. - (* a "ClosedND" is a proof with no open hypotheses and no multi-conclusion rules *) - Inductive ClosedND : Tree ??Judgment -> Type := - | cnd_weak : ClosedND [] - | cnd_rule : forall h c , ClosedND h -> Rule h c -> ClosedND c - | cnd_branch : forall c1 c2, ClosedND c1 -> ClosedND c2 -> ClosedND (c1,,c2) + (* a "ClosedSIND" is a proof with no open hypotheses and no multi-conclusion rules *) + Inductive ClosedSIND : Tree ??Judgment -> Type := + | cnd_weak : ClosedSIND [] + | cnd_rule : forall h c , ClosedSIND h -> Rule h c -> ClosedSIND c + | cnd_branch : forall c1 c2, ClosedSIND c1 -> ClosedSIND c2 -> ClosedSIND (c1,,c2) . - (* we can turn an SCND without hypotheses into a ClosedND *) - Definition closedFromSCND h c (pn2:SCND h c)(cnd:ClosedND h) : ClosedND c. - refine ((fix closedFromPnodes h c (pn2:SCND h c)(cnd:ClosedND h) {struct pn2} := - (match pn2 in SCND H C return H=h -> C=c -> _ with + (* we can turn an SIND without hypotheses into a ClosedSIND *) + Definition closedFromSIND h c (pn2:SIND h c)(cnd:ClosedSIND h) : ClosedSIND c. + refine ((fix closedFromPnodes h c (pn2:SIND h c)(cnd:ClosedSIND h) {struct pn2} := + (match pn2 in SIND H C return H=h -> C=c -> _ with | scnd_weak c => let case_weak := tt in _ - | scnd_leaf ht z pn' => let case_leaf := tt in let qq := closedFromPnodes _ _ pn' in _ | scnd_comp ht ct c pn' rule => let case_comp := tt in let qq := closedFromPnodes _ _ pn' in _ | scnd_branch ht c1 c2 pn' pn'' => let case_branch := tt in - let q1 := closedFromPnodes _ _ pn' in - let q2 := closedFromPnodes _ _ pn'' in _ + let q1 := closedFromPnodes _ _ pn' in + let q2 := closedFromPnodes _ _ pn'' in _ end (refl_equal _) (refl_equal _))) h c pn2 cnd). - destruct case_comp. - intros. - clear pn2. - apply (cnd_rule ct). - apply qq. - subst. - apply cnd0. - apply rule. - destruct case_weak. intros; subst. apply cnd_weak. - destruct case_leaf. + destruct case_comp. intros. + clear pn2. + apply (cnd_rule ct). apply qq. subst. apply cnd0. + apply rule. destruct case_branch. intros. @@ -300,18 +336,78 @@ Section Natural_Deduction. Defined. (* undo the above *) - Fixpoint closedNDtoNormalND {c}(cnd:ClosedND c) : ND [] c := - match cnd in ClosedND C return ND [] C with + Fixpoint closedNDtoNormalND {c}(cnd:ClosedSIND c) : ND [] c := + match cnd in ClosedSIND C return ND [] C with | cnd_weak => nd_id0 | cnd_rule h c cndh rhc => closedNDtoNormalND cndh ;; nd_rule rhc | cnd_branch c1 c2 cnd1 cnd2 => nd_llecnac ;; nd_prod (closedNDtoNormalND cnd1) (closedNDtoNormalND cnd2) end. + (* Natural Deduction systems whose judgments happen to be pairs of the same type *) + Section Sequents. + Context {S:Type}. (* type of sequent components *) + Context {sequent:S->S->Judgment}. + Context {ndr:ND_Relation}. + Notation "a |= b" := (sequent a b). + Notation "a === b" := (@ndr_eqv ndr _ _ a b) : nd_scope. + + (* Sequent systems with initial sequents *) + Class SequentCalculus := + { nd_seq_reflexive : forall a, ND [ ] [ a |= a ] + }. + + (* Sequent systems with a cut rule *) + Class CutRule (nd_cutrule_seq:SequentCalculus) := + { nd_cut : forall a b c, [ a |= b ] ,, [ b |= c ] /⋯⋯/ [ a |= c ] + ; nd_cut_left_identity : forall a b, (( (nd_seq_reflexive a)**(nd_id _));; nd_cut _ _ b) === nd_cancell + ; nd_cut_right_identity : forall a b, (((nd_id _)**(nd_seq_reflexive a) );; nd_cut b _ _) === nd_cancelr + ; nd_cut_associativity : forall {a b c d}, + (nd_id1 (a|=b) ** nd_cut b c d) ;; (nd_cut a b d) === nd_cossa ;; (nd_cut a b c ** nd_id1 (c|=d)) ;; nd_cut a c d + }. + + End Sequents. + + (* Sequent systems in which each side of the sequent is a tree of something *) + Section SequentsOfTrees. + Context {T:Type}{sequent:Tree ??T -> Tree ??T -> Judgment}. + Context (ndr:ND_Relation). + Notation "a |= b" := (sequent a b). + Notation "a === b" := (@ndr_eqv ndr _ _ a b) : nd_scope. + + (* Sequent systems in which we can re-arrange the tree to the left of the turnstile - note that these rules + * mirror nd_{cancell,cancelr,rlecnac,llecnac,assoc,cossa} but are completely separate from them *) + Class TreeStructuralRules := + { tsr_ant_assoc : forall {x a b c}, ND [((a,,b),,c) |= x] [(a,,(b,,c)) |= x] + ; tsr_ant_cossa : forall {x a b c}, ND [(a,,(b,,c)) |= x] [((a,,b),,c) |= x] + ; tsr_ant_cancell : forall {x a }, ND [ [],,a |= x] [ a |= x] + ; tsr_ant_cancelr : forall {x a }, ND [a,,[] |= x] [ a |= x] + ; tsr_ant_llecnac : forall {x a }, ND [ a |= x] [ [],,a |= x] + ; tsr_ant_rlecnac : forall {x a }, ND [ a |= x] [ a,,[] |= x] + }. + + Notation "[# a #]" := (nd_rule a) : nd_scope. + + (* Sequent systems in which we can add any proposition to both sides of the sequent (sort of a "horizontal weakening") *) + Context `{se_cut : @CutRule _ sequent ndr sc}. + Class SequentExpansion := + { se_expand_left : forall tau {Gamma Sigma}, ND [ Gamma |= Sigma ] [tau,,Gamma|=tau,,Sigma] + ; se_expand_right : forall tau {Gamma Sigma}, ND [ Gamma |= Sigma ] [Gamma,,tau|=Sigma,,tau] + + (* left and right expansion must commute with cut *) + ; se_reflexive_left : ∀ a c, nd_seq_reflexive a;; se_expand_left c === nd_seq_reflexive (c,, a) + ; se_reflexive_right : ∀ a c, nd_seq_reflexive a;; se_expand_right c === nd_seq_reflexive (a,, c) + ; se_cut_left : ∀ a b c d, (se_expand_left _)**(se_expand_left _);;nd_cut _ _ _===nd_cut a b d;;(se_expand_left c) + ; se_cut_right : ∀ a b c d, (se_expand_right _)**(se_expand_right _);;nd_cut _ _ _===nd_cut a b d;;(se_expand_right c) + }. + End SequentsOfTrees. + Close Scope nd_scope. Open Scope pf_scope. End Natural_Deduction. +Coercion nd_cut : CutRule >-> Funclass. + Implicit Arguments ND [ Judgment ]. Hint Constructors Structural. Hint Extern 1 => apply nd_id_structural. @@ -467,5 +563,110 @@ Inductive nd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall | nd_property_rule : forall h c r, P h c r -> @nd_property _ _ P h c (nd_rule r). Hint Constructors nd_property. +(* witnesses the fact that every Rule in a particular proof satisfies the given predicate (for ClosedSIND) *) +Inductive cnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {c}, @ClosedSIND Judgment Rule c -> Prop := +| cnd_property_weak : @cnd_property _ _ P _ cnd_weak +| cnd_property_rule : forall h c r cnd', + P h c r -> + @cnd_property _ _ P h cnd' -> + @cnd_property _ _ P c (cnd_rule _ _ cnd' r) +| cnd_property_branch : + forall c1 c2 cnd1 cnd2, + @cnd_property _ _ P c1 cnd1 -> + @cnd_property _ _ P c2 cnd2 -> + @cnd_property _ _ P _ (cnd_branch _ _ cnd1 cnd2). + +(* witnesses the fact that every Rule in a particular proof satisfies the given predicate (for SIND) *) +Inductive scnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h c}, @SIND Judgment Rule h c -> Prop := +| scnd_property_weak : forall c, @scnd_property _ _ P _ _ (scnd_weak c) +| scnd_property_comp : forall h x c r cnd', + P x [c] r -> + @scnd_property _ _ P h x cnd' -> + @scnd_property _ _ P h _ (scnd_comp _ _ _ cnd' r) +| scnd_property_branch : + forall x c1 c2 cnd1 cnd2, + @scnd_property _ _ P x c1 cnd1 -> + @scnd_property _ _ P x c2 cnd2 -> + @scnd_property _ _ P x _ (scnd_branch _ _ _ cnd1 cnd2). + +(* renders a proof as LaTeX code *) +Section ToLatex. + + Context {Judgment : Type}. + Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}. + Context {JudgmentToLatexMath : ToLatexMath Judgment}. + Context {RuleToLatexMath : forall h c, ToLatexMath (Rule h c)}. + + Open Scope string_scope. + + Definition judgments2latex (j:Tree ??Judgment) := treeToLatexMath (mapOptionTree toLatexMath j). + + Definition eolL : LatexMath := rawLatexMath eol. + + (* invariant: each proof shall emit its hypotheses visibly, except nd_id0 *) + Section SIND_toLatex. + + (* indicates which rules should be hidden (omitted) from the rendered proof; useful for structural operations *) + Context (hideRule : forall h c (r:Rule h c), bool). + + Fixpoint SIND_toLatexMath {h}{c}(pns:SIND(Rule:=Rule) h c) : LatexMath := + match pns with + | scnd_branch ht c1 c2 pns1 pns2 => SIND_toLatexMath pns1 +++ rawLatexMath " \hspace{1cm} " +++ SIND_toLatexMath pns2 + | scnd_weak c => rawLatexMath "" + | scnd_comp ht ct c pns rule => if hideRule _ _ rule + then SIND_toLatexMath pns + else rawLatexMath "\trfrac["+++ toLatexMath rule +++ rawLatexMath "]{" +++ eolL +++ + SIND_toLatexMath pns +++ rawLatexMath "}{" +++ eolL +++ + toLatexMath c +++ rawLatexMath "}" +++ eolL + end. + End SIND_toLatex. + + (* this is a work-in-progress; please use SIND_toLatexMath for now *) + Fixpoint nd_toLatexMath {h}{c}(nd:@ND _ Rule h c)(indent:string) := + match nd with + | nd_id0 => rawLatexMath indent +++ + rawLatexMath "% nd_id0 " +++ eolL + | nd_id1 h' => rawLatexMath indent +++ + rawLatexMath "% nd_id1 "+++ judgments2latex h +++ eolL + | nd_weak h' => rawLatexMath indent +++ + rawLatexMath "\inferrule*[Left=ndWeak]{" +++ toLatexMath h' +++ rawLatexMath "}{ }" +++ eolL + | nd_copy h' => rawLatexMath indent +++ + rawLatexMath "\inferrule*[Left=ndCopy]{"+++judgments2latex h+++ + rawLatexMath "}{"+++judgments2latex c+++rawLatexMath "}" +++ eolL + | nd_prod h1 h2 c1 c2 pf1 pf2 => rawLatexMath indent +++ + rawLatexMath "% prod " +++ eolL +++ + rawLatexMath indent +++ + rawLatexMath "\begin{array}{c c}" +++ eolL +++ + (nd_toLatexMath pf1 (" "+++indent)) +++ + rawLatexMath indent +++ + rawLatexMath " & " +++ eolL +++ + (nd_toLatexMath pf2 (" "+++indent)) +++ + rawLatexMath indent +++ + rawLatexMath "\end{array}" + | nd_comp h m c pf1 pf2 => rawLatexMath indent +++ + rawLatexMath "% comp " +++ eolL +++ + rawLatexMath indent +++ + rawLatexMath "\begin{array}{c}" +++ eolL +++ + (nd_toLatexMath pf1 (" "+++indent)) +++ + rawLatexMath indent +++ + rawLatexMath " \\ " +++ eolL +++ + (nd_toLatexMath pf2 (" "+++indent)) +++ + rawLatexMath indent +++ + rawLatexMath "\end{array}" + | nd_cancell a => rawLatexMath indent +++ + rawLatexMath "% nd-cancell " +++ (judgments2latex a) +++ eolL + | nd_cancelr a => rawLatexMath indent +++ + rawLatexMath "% nd-cancelr " +++ (judgments2latex a) +++ eolL + | nd_llecnac a => rawLatexMath indent +++ + rawLatexMath "% nd-llecnac " +++ (judgments2latex a) +++ eolL + | nd_rlecnac a => rawLatexMath indent +++ + rawLatexMath "% nd-rlecnac " +++ (judgments2latex a) +++ eolL + | nd_assoc a b c => rawLatexMath "" + | nd_cossa a b c => rawLatexMath "" + | nd_rule h c r => toLatexMath r + end. + +End ToLatex. + Close Scope pf_scope. Close Scope nd_scope.