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:
*
* (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.
*)
(*
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 *)
`(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
(* 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
+ ; ndr_comp_preserves_cancell : forall `(f:a/⋯⋯/b), nd_cancell;; f === nd_id _ ** f ;; nd_cancell
+ ; ndr_comp_preserves_cancelr : forall `(f:a/⋯⋯/b), nd_cancelr;; f === f ** nd_id _ ;; nd_cancelr
+ ; ndr_comp_preserves_assoc : forall `(f:a/⋯⋯/b)`(g:a1/⋯⋯/b1)`(h:a2/⋯⋯/b2),
+ nd_assoc;; (f ** (g ** h)) === ((f ** g) ** h) ;; nd_assoc
+
(* 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
}.
(*
- * Single-conclusion proofs; this is an alternate representation
- * where each inference has only a single conclusion. These have
- * worse compositionality properties than ND's (they don't form a
- * category), 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_weak : forall c , SCND c []
- | scnd_comp : forall ht ct c , SCND ht ct -> Rule ct [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.
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_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).
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}.
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
}.
End Sequents.
-(*Implicit Arguments SequentCalculus [ S ]*)
-(*Implicit Arguments CutRule [ S ]*)
+
+ (* 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]
+ { 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]
| 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 *)
-Inductive cnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {c}, @ClosedND Judgment Rule c -> Prop :=
+(* 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 c2 cnd2 ->
@cnd_property _ _ P _ (cnd_branch _ _ cnd1 cnd2).
-Inductive scnd_property {Judgment}{Rule}(P:forall h c, @Rule h c -> Prop) : forall {h c}, @SCND Judgment Rule h c -> Prop :=
+(* 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 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.
projects / coq-hetmet.git / blobdiff