(* 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
* 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
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)
(* 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.
| 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 ClosedND) *)
-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).
-(* witnesses the fact that every Rule in a particular proof satisfies the given predicate (for SCND) *)
-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 ->
Definition eolL : LatexMath := rawLatexMath eol.
(* invariant: each proof shall emit its hypotheses visibly, except nd_id0 *)
- Section SCND_toLatex.
+ 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 SCND_toLatexMath {h}{c}(pns:SCND(Rule:=Rule) h c) : LatexMath :=
+ Fixpoint SIND_toLatexMath {h}{c}(pns:SIND(Rule:=Rule) h c) : LatexMath :=
match pns with
- | scnd_branch ht c1 c2 pns1 pns2 => SCND_toLatexMath pns1 +++ rawLatexMath " \hspace{1cm} " +++ SCND_toLatexMath pns2
+ | 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 SCND_toLatexMath pns
+ then SIND_toLatexMath pns
else rawLatexMath "\trfrac["+++ toLatexMath rule +++ rawLatexMath "]{" +++ eolL +++
- SCND_toLatexMath pns +++ rawLatexMath "}{" +++ eolL +++
+ SIND_toLatexMath pns +++ rawLatexMath "}{" +++ eolL +++
toLatexMath c +++ rawLatexMath "}" +++ eolL
end.
- End SCND_toLatex.
+ End SIND_toLatex.
- (* this is a work-in-progress; please use SCND_toLatexMath for now *)
+ (* 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 +++
projects / coq-hetmet.git / blobdiff