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IntrospectiveCalculus.v
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IntrospectiveCalculus.v
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Set Implicit Arguments.
Set Asymmetric Patterns.
Require Import Unicode.Utf8.
Require Import List.
Require Import Coq.Program.Equality.
Require Import Ascii String.
Open Scope string_scope.
(* Hack - remove later *)
Parameter cheat : ∀ {A}, A.
(****************************************************
Typeclass practicalities
****************************************************)
(* We often want to view IC formulas as "extensible". That is, we know that the formula type has certain constructors, but don't guarantee that those are the ONLY constructors.
Theoretically, this is "simpler" than Coq's `Inductive` types (constructors come _before_ inductive instances), but in Coq, it's not elementary. So to represent a "set of constructors", we use a typeclass, where the typeclass methods are the constructors. Thus, anytime we have a type F with an instance of FormulaConstructors, we can view F as a legitimate "extension" of the concept of formulas.
Here we must do some bureaucracy about how we'll use typeclasses. *)
(* Sometimes we need to pass a typeclass as a function argument. This is fine because a typeclass is an ordinary value, but Coq's features for using typeclasses _implicitly_ don't have a built-in way to treat a local variable as a typeclass. To work around this, we make a wrapper class "Class", which is technically the identity function but is always treated as a typeclass. So when we want a local variable `C` to be treated as a typeclass, we say `Class C` instead. This is the same thing, but counts as a typeclass for implicits. *)
Definition Class [P] (p:P) := p.
Existing Class Class.
Hint Unfold Class : typeclass_instances.
Definition use_class {T P t} (c:@Class (T -> Type) P t) : P t := c.
(* Definition use_class {P T} (c:Class P T) : P T := c. *)
(* We also often want to extend a typeclass with additional constructors, while still leaving it open to still-further constructors. This gives us a _second typeclass_ that is a subclass of the first. It's useful to represent the subclass relation explicitly: *)
Definition SubclassOf {T} (Superclass Subclass : T -> Type) := ∀ T (_:Class Subclass T), Class Superclass T.
Existing Class SubclassOf.
Notation "R ⊆ S" := (SubclassOf S R) (at level 70).
Instance subclass_refl {T} {A : T -> Type}
: (A ⊆ A)
:= λ x xA, xA.
(* Once we make an instance for subclass transitivity (A ⊆ B -> B ⊆ C -> A ⊆ C), we risk making instance-search diverge: The subgoal looking for (A ⊆ B) tries looking for some additional transitivity-instance (A ⊆ X) and (X ⊆ B), ad infinitum. So we have to set a limit on typeclass search: *)
Set Typeclasses Iterative Deepening.
Set Typeclasses Depth 3.
Instance subclass_trans {T} {A B C : T -> Type}
: (A ⊆ B) ->
(B ⊆ C) ->
(A ⊆ C)
:= λ ab bc, λ x xA, bc _ (ab _ xA).
Definition test_subclass_trans {T} {A B C : T -> Type}
: (A ⊆ B) -> (B ⊆ C) -> (A ⊆ C) := _.
(* Set Typeclasses Debug Verbosity 2. *)
Definition test_subclass_trans_2 {T} {A B C D : T -> Type}
: (A ⊆ B) -> (B ⊆ C) -> (C ⊆ D) -> (A ⊆ D) := λ _ _ _, _.
(* Lemma subclass_trans_refl1 [T] [A B : T -> Type] (s : A ⊆ B) :
subclass_trans subclass_refl s = s.
reflexivity.
Qed.
Lemma subclass_trans_refl2 [T] [A B : T -> Type] (s : A ⊆ B) :
subclass_trans s subclass_refl = s.
reflexivity.
Qed. *)
Instance subclass_apply {T} {A B : T -> Type} {t}
: (A ⊆ B) -> Class A t -> Class B t
:= λ ab At, ab _ At.
(* The definitions above treat a class as an arbitrary constraint, but *constructor* classes also have a positivity requirement, which we sometimes need to require explicitly: *)
(* Class ConstructorClass (C : Type -> Type) := {
cc_embed : ∀ {a b} {_:C a}, (a -> b) -> C b
}. *)
(****************************************************
Relations between internal and external meaning
****************************************************)
Class ApplyConstructor F := {
apply : F -> F -> F
}.
Class FunctionConstructors F := {
const : F
; fuse : F
; fc_ac :: ApplyConstructor F
}.
Class PropositionConstructors F := {
pc_fc :: FunctionConstructors F
; f_implies : F
; f_and : F
; f_forall_quoted_formulas : F
}.
(* When we want to use PropositionConstructors methods, but only have our wrapper Class PropositionConstructors, we need to be able to unwrap it: *)
Instance pc_class_unwrap :
∀ {F}, Class PropositionConstructors F ->
PropositionConstructors F := λ _ b, b.
Instance aplc_class_unwrap :
∀ {F}, Class ApplyConstructor F ->
ApplyConstructor F := λ _ b, b.
Instance func_aplc_subclass : FunctionConstructors ⊆ ApplyConstructor := λ _, _.
(* And because we use a low depth limit, we need to provide some "shortcuts": *)
Instance shortcut_cpc_fnc F : Class PropositionConstructors F -> FunctionConstructors F := _.
Instance shortcut_cpc_aplc F : Class PropositionConstructors F -> ApplyConstructor F := λ _, _.
Notation "[ x y .. z ]" := (apply .. (apply x y) .. z)
(at level 0, x at next level, y at next level).
Notation "[ x -> y ]" := [f_implies x y] (at level 0, x at next level, y at next level).
Notation "[ x & y ]" := [f_and x y]
(at level 0, x at next level, y at next level).
Notation "[ ⋀ x ]" := [f_forall_quoted_formulas x]
(at level 0, x at next level).
Inductive UnfoldStep {F} {_f:FunctionConstructors F} : F -> F -> Prop :=
| unfold_const a b : UnfoldStep [const a b] a
| unfold_fuse a b c : UnfoldStep [fuse a b c] [[a c] [b c]]
| unfold_in_lhs a b c : UnfoldStep a b -> UnfoldStep [a c] [b c].
(* Notation "R ∧1 S" := (λ x, R x ∧ S x) (at level 70). *)
(* Notation "R ×1 S" := (λ x, prod (R x) (S x)) (at level 70). *)
(* Notation "R ∧3 S" := (λ x y z, R x y z ∧ S x y z) (at level 70). *)
(* Notation "R ∧4 S" := (λ x y z w, R x y z w ∧ S x y z w) (at level 70). *)
(* Notation "R ⊆ S" := (∀ x, R x -> S x) (at level 70). *)
(* Definition Construction Constructors := ∀ {T} `{Constructors T}, T. *)
Class OneMoreConstructor Constrs F := {
omc_embed : Class Constrs F
; omc_new : F
}.
Instance onemore_class_unwrap :
∀ {Constrs F}, Class (OneMoreConstructor Constrs) F ->
OneMoreConstructor Constrs F := λ _ _ c, c.
Instance OneMoreConstructor_subclass {C}
: OneMoreConstructor C ⊆ C
:= λ _ _, omc_embed.
Instance shortcut_omc_trans A B
: (A ⊆ B) -> (OneMoreConstructor A ⊆ B)
:= _.
Instance shortcut_omc_trans_apply A B T
: (A ⊆ B) -> (Class (OneMoreConstructor A) T) -> Class B T
:= λ _ _, _.
Instance shortcut_omc_func_aplc A T
: (A ⊆ FunctionConstructors) -> (Class (OneMoreConstructor A) T) -> Class ApplyConstructor T
:= λ _ _, _.
(* Propositions are formulas that say things about other formulas, but there's no intrinsic reason those formulas have to use the _same grammar_. So we will often be passing around _two_ formula-constructor-classes, which I call the "viewing type" (conventionally named V, with constructors named VC) and the "target type" (conventionally named T, with constructors named TC).
The gotcha: We don't want one of them to be accidentally passed where the other is expected (especially implicitly!)... but they are the same type!
So, we make the types different _in name_, which means Coq will implicitly pass them only in the correct role. *)
Definition VConsClass := Type -> Type.
Definition TConsClass := Type -> Type.
Implicit Type VC : VConsClass.
Implicit Type TC : TConsClass.
Existing Class VConsClass.
Existing Class TConsClass.
(* ...but when a Type -> Type is needed in a context that can disambiguate which one, we do want Coq to know that a VConsClass or TConsClass will suffice: *)
Hint Unfold VConsClass : typeclass_instances.
Hint Unfold TConsClass : typeclass_instances.
(* Set Typeclasses Debug Verbosity 2. *)
Definition test_subclass_apply_with_VConsClass
{VC VC2:VConsClass} {V} (v:Class VC2 V) `{VC2 ⊆ VC}
: Class VC V := _.
Definition test_subclass_apply_with_VConsClass_2
{C : Type -> Type} {VC2 : VConsClass} {V:Type} (v:@Class VConsClass VC2 V) `{VC2 ⊆ C}
: @Class (Type -> Type) C V := _.
Definition test_OneMoreConstructor_apply_and_unwrap_with_VConsClass
{VC oVC2:VConsClass} `{oVC2 ⊆ OneMoreConstructor VC}
{V} (_v:Class oVC2 V)
: OneMoreConstructor VC V := _.
(* Given any constructor-class, we may also be interested in the inductive type with those constructors. We will be using this a lot, so it gets a short name.
Unfortunately, we can't express the statement `C (Ind C)`, because it results in universe inconsistency. This would work fine in the Prop universe; I think I could theoretically rewrite most of the code here to use Prop instead of Type, and have it work. But Coq generally expects you to be working in the predicative hierarchy instead, and can't abstract over Prop vs Type in the way we'd need. But, I think it turns out to be unnecessary to express `C (Ind C)` anyway (we can just convert to other forms). *)
Definition Ind C := ∀ T (_:Class C T), T.
Definition ind_on_stricter_constructors C C2
: (C2 ⊆ C) -> Ind C -> Ind C2
:= λ _c x T eT, x T _.
(* We also deal with "quoted formulas". To say that a V-formula is a quoted version of a T-formula, we need to define the MeansQuoted relation (MQ for short). This relation is only well-defined once you first specify the V and T constructors: *)
Definition MQT V TC := V -> Ind TC -> Prop.
(* Definition mqt_on_looser_constructors [V TC TC2] [eT : (TC ⊆ TC2)]
: MQT V TC -> MQT V TC2
:= λ MQ v t, MQ v (ind_on_stricter_constructors eT t). *)
(* ...and usually, we need to be talking about _constructors_ (MQC) for that relation, because it must be just as extensible as the formula types: *)
Definition MQCT VC TC :=
∀ V (_v:Class VC V) (TC2 : TConsClass) (eT : TC2 ⊆ TC), MQT V TC2 -> Prop.
Existing Class MQCT.
(* We also define how an MQCT relates _inductive instances_ of VC and TC - which is the same thing as saying an inductive instance of those MQ constructors: *)
(* We also need to be able to say an inductive instance of those MQ constructors ("this concrete qx is forced by the constructors to mean a quoted version of this concrete x, no matter what embedding we're in..."); "any MQ that obeys MQC (bearing in mind that MQC can judge any MQ on any extension of TC) must hold (MQ qx x)" *)
Definition MQInd {VC TC V} {_v:Class VC V} (qx : V) (x : Ind TC) {MQC : MQCT VC TC}
:=
∀ TC2 eT MQ (_mq : MQC V _v TC2 eT MQ), MQ qx (ind_on_stricter_constructors eT x).
Print MQInd.
(* ...and to say that a particular Ind VC always construct something that quotes an Ind TC: *)
Definition VInd_MQInd {VC TC V} {_v:Class VC V} (qx : Ind VC) (x : Ind TC) {MQC : MQCT VC TC}
:= ∀ V _v, MQInd (qx V _v) x.
(* Definition MQInd {VC TC} (qx : Ind VC) (x : Ind TC) {MQC : MQCT}
:=
∀ V (_v:Class VC V) T (_t:Class TC T) (MQ : MQT),
MQC _ _ _ _ MQ ->
MQ _ _ _ _ (qx _ _) (x _ _). *)
(* ...the main case of which is to add one additional variable to each of V and T, and say that the new V-variable is a quoted version of the new T-variable. *)
Definition OneMoreMQConstructor [VC TC] (qx : Ind VC) (x: Ind TC) (MQC : MQCT VC TC) : MQCT VC TC
:= λ V _v TC2 eT MQ,
(MQC _ _ _ _ MQ) ∧
(∀ T (_t:Class (OneMoreConstructor TC) T), (MQ (qx _ _) (λ _ _, x _ _))).
Definition MQCOnStricterFormulaTypes
{VC TC VC2 TC2} {eV: VC2 ⊆ VC} {eT: TC2 ⊆ TC}
(MQC : MQCT VC TC) : MQCT VC2 TC2
:= λ _ _ _ _ MQ, (MQC _ _ _ _ MQ).
Print MQCOnStricterFormulaTypes.
Definition MQCSubclassOf [VC TC] (MQC1 MQC2 : MQCT VC TC)
:= ∀ V _v TC2 eT MQ, MQC1 V _v TC2 eT MQ -> MQC2 V _v TC2 eT MQ.
Lemma MQCSubclassOf_refl [VC TC] (MQC : MQCT VC TC) : MQCSubclassOf MQC MQC.
unfold MQCSubclassOf. trivial.
Defined.
Lemma OneMoreMQConstructor_embed
{VC TC} (qx : Ind VC) (x: Ind TC) (MQC : MQCT VC TC)
: MQCSubclassOf (OneMoreMQConstructor qx x MQC) MQC.
unfold MQCSubclassOf, OneMoreMQConstructor.
intros.
destruct H; assumption.
Qed.
Lemma MQC_embed_under_stricter
{VC TC VC2 TC2} {eV: VC2 ⊆ VC} {eT: TC2 ⊆ TC}
(MQC1 MQC2 : MQCT VC TC)
: MQCSubclassOf MQC1 MQC2 ->
MQCSubclassOf (MQCOnStricterFormulaTypes MQC1) (MQCOnStricterFormulaTypes MQC2).
unfold MQCSubclassOf, MQCOnStricterFormulaTypes.
intros.
exact (H _ _ _ _ _ H0).
Qed.
(* Lemma MQC_unembed_under_stricter
{VC TC VC2 TC2} {eV: VC2 ⊆ VC} {eT: TC2 ⊆ TC}
V (_v : Class VC2 V) (MQ : MQT V TC2)
(MQC1 MQC2 : MQCT VC TC)
:
MQCSubclassOf (MQCOnStricterFormulaTypes MQC1) (MQCOnStricterFormulaTypes MQC2) ->
MQCSubclassOf MQC1 MQC2.
unfold MQCSubclassOf, MQCOnStricterFormulaTypes.
intros.
unfold subclass_apply, mqt_on_looser_constructors, ind_on_stricter_constructors, subclass_apply in H.
specialize (H V _ MQ).
exact (H _ _ _ H0).
Qed. *)
(* Lemma MQC_stricter_trans
{VC TC VC2 TC2 VC3 TC3}
{eV12: VC2 ⊆ VC} {eT12: TC2 ⊆ TC}
{eV23: VC3 ⊆ VC2} {eT23: TC3 ⊆ TC2}
(MQC : MQCT VC TC)
: MQCSubclassOf
(@MQCOnStricterFormulaTypes VC2 TC2 VC3 TC3 _ _ (MQCOnStricterFormulaTypes MQC))
(@MQCOnStricterFormulaTypes VC TC VC3 TC3 _ _ MQC).
unfold MQCSubclassOf, MQCOnStricterFormulaTypes.
intros.
assumption.
Qed. *)
(* Lemma MQC_stricter_unwrap
{VC TC} (MQC : MQCT VC TC)
V _v TC2 eT MQ
: @MQCOnStricterFormulaTypes VC TC VC TC _ _ MQC V _v TC2 eT MQ
-> MQC V _v TC2 eT MQ.
unfold MQCOnStricterFormulaTypes.
intros.
assumption.
Qed. *)
(* Lemma MQInd_erjekr {VC TC V} {_v:Class VC V} (qx : V) (x : Ind TC) {MQC : MQCT VC TC} TC2 (eT : TC2 ⊆ TC) :
MQInd qx x -> @MQInd VC TC2 _ _ qx (ind_on_stricter_constructors eT x) (MQCOnStricterFormulaTypes _).
intro _mqi. unfold MQInd in *. intros.
apply _mqi.
unfold MQCOnStricterFormulaTypes in H.
exact H. *)
Definition OneMoreQuotvar {VC TC} (MQC : MQCT VC TC)
: MQCT (OneMoreConstructor VC) (OneMoreConstructor TC)
:=
OneMoreMQConstructor
(@omc_new _) (@omc_new _)
(MQCOnStricterFormulaTypes MQC).
(* λ oVC2 oTC2 _v _t MQ,
(MQC _ _ _ _ MQ) ∧
(∀ V (_v:Class oVC2 V) T (_t:Class oTC2 T),
MQ _ _ _ _ omc_new omc_new). *)
Print OneMoreQuotvar.
Eval compute in OneMoreQuotvar.
(* Propositions represent rules of inference. A Rule is a constraint on what inferences may be valid: for example, the rule (A & B) |- (B & A) says that for all values of B and A, the inference (A & B) |- (B & A) must be valid.
In practice, we don't use the full generality of arbitrary constraints. Our only recursive rule is transitivity ((A |- B) and (B |- C) imply (A |- C)), and there isn't a proposition that represents it, it's just always true. All propositions represent simple positive constraints, which just say that certain inferences must be valid.
Nevertheless, the simplest way to define Rule is as a predicate on InfSets, which are predicates on inferences. (An InfSet takes two formulas P,C and says whether it holds inference P |- C as valid.)
We must ask what the actual type of Rule is. A Rule must be agnostic to grammar-extensions, but you may express a rule that assumes particular constructors exist (otherwise, our example rule wouldn't be able to express &). Therefore: *)
Definition InfSet T := T -> T -> Prop.
Existing Class InfSet.
(* Definition _proves {T} {infs:InfSet T} p c := infs p c.
Notation "p ⊢ c" := (_proves p c) (at level 70). *)
Definition Rule {TC} := ∀ T (_:Class TC T), InfSet T -> Prop.
(* We can now make the big definition of what propositions mean. *)
(* definition specific to T, although we will often be dealing in values of type (∀ T (_t:Class TC T), Ruleset) *)
Inductive Ruleset TC :=
| ruleset_implies : Ind TC -> Ind TC -> Ruleset TC
| ruleset_and : Ruleset TC -> Ruleset TC -> Ruleset TC
| ruleset_forall : Ruleset (OneMoreConstructor TC) -> Ruleset TC
.
(* Definition specialize_ind {TC} (f : Ind (OneMoreConstructor TC)) (x : Ind TC) : Ind TC :=
λ T _t, f T {| omc_new := (x _ _) |}. *)
(* wrong variable order:
Fixpoint specialize_ruleset {TC} (R : @Ruleset (OneMoreConstructor TC)) (x : Ind TC) : @Ruleset TC :=
match R return Ruleset with
| ruleset_implies p c => ruleset_implies (specialize_ind p x) (specialize_ind c x)
| ruleset_and A B => ruleset_and (specialize_ruleset A x) (specialize_ruleset B x)
| ruleset_forall F => ruleset_forall (specialize_ruleset F (ind_on_stricter_constructors _ x))
end. *)
Fixpoint Ruleset_to_Coq [TC] (R : Ruleset TC) {T} {_t: Class TC T} (infs : InfSet T) : Prop :=
match R with
| ruleset_implies p c => infs (p _ _) (c _ _)
| ruleset_and A B => Ruleset_to_Coq A _ ∧ Ruleset_to_Coq B _
| ruleset_forall F => ∀ x, @Ruleset_to_Coq _ F _ {| omc_new := x |} _
end.
Fixpoint Ruleset_on_stricter_constructors [TC TC2] [eT : TC2 ⊆ TC] (R : Ruleset TC) : Ruleset TC2 :=
match R with
| ruleset_implies p c => ruleset_implies (ind_on_stricter_constructors _ p) (ind_on_stricter_constructors _ c)
| ruleset_and A B => ruleset_and (Ruleset_on_stricter_constructors A) (Ruleset_on_stricter_constructors B)
| ruleset_forall F => ruleset_forall (@Ruleset_on_stricter_constructors _ (OneMoreConstructor TC2) (λ _ _, {| omc_new := omc_new |}) F)
end.
(* Set Printing Implicit. *)
(* Set Typeclasses Debug Verbosity 2. *)
(* Set Typeclasses Depth 4. *)
Inductive FormulaMeansProp {VC TC} {MQC : MQCT VC TC}
{_vp:VC ⊆ PropositionConstructors}
{V} {_v: Class VC V}
(* {T} {_t: Class TC T}
{MQ : ∀ {V} {_v:Class VC V} {T} {_t:Class TC T}, V -> T -> Prop}
{_mq : MQC _ _ _ _ (@MQ)} *)
: V -> Ruleset TC -> Prop :=
| mi_implies [qp qc p c]
:
MQInd qp p ->
MQInd qc c ->
FormulaMeansProp [qp -> qc] (ruleset_implies p c)
| mi_unfold [a b B] :
UnfoldStep a b ->
FormulaMeansProp b B ->
FormulaMeansProp a B
| mi_and [a b A B] :
FormulaMeansProp a A ->
FormulaMeansProp b B ->
FormulaMeansProp [a & b] (ruleset_and A B)
| mi_forall_quoted_formulas
(f : Ind VC)
(F : Ruleset (OneMoreConstructor TC))
:
(∀ V _v,
@FormulaMeansProp _ _ (OneMoreQuotvar MQC) _ V _v
[(f _ _) omc_new]
F)
->
FormulaMeansProp
[⋀ (f _ _)]
(ruleset_forall (@F))
.
Definition VIndMeansProp {VC TC} {MQC : MQCT VC TC}
{_vp:VC ⊆ PropositionConstructors}
: Ind VC -> Ruleset TC -> Prop
:= λ p P, ∀ V _v, FormulaMeansProp (p V _v) P.
(****************************************************
Definitions of concrete formula types
****************************************************)
(* The above has been very abstract. We now proceed to the concrete formulas of IC. These belong to the Set universe, for convenience of programming.
All formulas of IC are apply-trees where the leaves are standard atoms... *)
Inductive Atom :=
| atom_const
| atom_fuse
| atom_implies
| atom_and
| atom_forall_quoted_formulas
| atom_quote
.
(* ...but our metatheorems may also extend the formula type with additional constructors, which represent metavariables. Each such variable is essentially an additional constructor, so our formula type is parameterized with the extension type. *)
Inductive Formula Ext :=
| f_atm : Atom -> Formula Ext
| f_ext : Ext -> Formula Ext
| fo_apl : Formula Ext -> Formula Ext -> Formula Ext.
Arguments f_atm [Ext] _.
Inductive OneMoreExt OldExt :=
| ome_embed : OldExt -> OneMoreExt OldExt
| ome_new : OneMoreExt OldExt.
Arguments ome_embed {OldExt} _.
Arguments ome_new {OldExt}.
(* We'll also need to "be able to talk to" the abstract definitions above, so we make an equivalent definition out of typeclasses. *)
Class FormulaConstructors F := {
fc_atm : Atom -> F
; fc_apl :: ApplyConstructor F
}.
(* Set Printing Implicit.
Set Typeclasses Debug Verbosity 2.
Definition test_subclass_apply_with_VConsClass_FormulaConstructors
{VC2 : (Type -> Type)} {TC2:(Type -> Type)} {V:Type} (v:@Class (Type -> Type) VC2 V) {_:VC2 ⊆ FormulaConstructors} {_:TC2 ⊆ FormulaConstructors}
: @Class (Type -> Type) FormulaConstructors V.
(* clear X0. *)
typeclasses eauto.
eauto with typeclass_instances. *)
(* Definition test_subclass_apply_with_VConsClass_FormulaConstructors
{VC2 : VConsClass} {TC2:TConsClass} {V:Type} (v:@Class VConsClass VC2 V) {_:VC2 ⊆ FormulaConstructors} {_:TC2 ⊆ FormulaConstructors}
: @Class (Type -> Type) FormulaConstructors V := _.
Definition test_subclass_apply_with_VConsClass_FormulaConstructors_2 :∀ (VC2:VConsClass) (TC2:TConsClass),
(VC2 ⊆ FormulaConstructors) -> (TC2 ⊆ FormulaConstructors) ->
Prop :=
λ VC2 TC2 eV eT,
∀ V (_v:Class VC2 V)
(* T (_t:Class TC2 T) *)
,
let _ : Class FormulaConstructors V := _ in
(* let _ : Class FormulaConstructors T := _ in
let MQ := MQ V _ T _ in *)
False. *)
Instance f_unwrap :
∀ {F}, Class FormulaConstructors F ->
FormulaConstructors F := λ _ b, b.
(* Instance fc_cc : ConstructorClass (Class FormulaConstructors) := {
cc_embed := λ a b FCa ab, {|
fc_atm := λ a, ab (fc_atm a)
; fc_apl := λ a b, ab (fc_apl a b)
|}
}. *)
Instance f_apli {Ext} : Class ApplyConstructor (Formula Ext) := {|
apply := @fo_apl Ext
|}.
Instance f_fc {Ext} : Class FormulaConstructors (Formula Ext) := {|
fc_atm := @f_atm Ext
|}.
Instance fc_func F : FormulaConstructors F ->
FunctionConstructors F := {
const := fc_atm atom_const
; fuse := fc_atm atom_fuse
}.
Instance fc_prop F : FormulaConstructors F ->
PropositionConstructors F := {
f_implies := (fc_atm atom_implies)
; f_and := (fc_atm atom_and)
; f_forall_quoted_formulas := (fc_atm atom_forall_quoted_formulas)
}.
Instance fc_subset_prop
: FormulaConstructors ⊆ PropositionConstructors
:= fc_prop.
(* Definition OneMoreConstructor_fc_trans_apply {C} {_vp : C ⊆ FormulaConstructors} {V} {_v : Class (OneMoreConstructor C) V}
: Class FormulaConstructors V := let _ : Class C V := _ in _.
Hint Immediate OneMoreConstructor_fc_trans_apply : typeclass_instances. *)
Instance shortcut_cfc_pc F : Class FormulaConstructors F -> PropositionConstructors F := _.
Instance shortcut_cfc_func F : Class FormulaConstructors F -> FunctionConstructors F := _.
Instance shortcut_cfc_aplc F : Class FormulaConstructors F -> ApplyConstructor F := _.
(* Definition ind_to_sf (i : Ind FormulaConstructors)
: StandardFormula := i _ _. *)
(* Definition ome_to_ind {Ext} {Constrs}
(embed : Ext -> Ind Constrs)
(e : @OneMoreExt Ext)
: Ind (OneMoreConstructor Constrs)
:= λ _ _, match e with
| ome_embed e => (embed e _ _)
| ome_new => omc_new
end. *)
Fixpoint NMoreConstructors n Constrs :=
match n with
| 0 => Constrs
| S pred => OneMoreConstructor (NMoreConstructors pred Constrs)
end.
(* Fixpoint NMoreConstructorsL n Constrs :=
match n with
| 0 => Constrs
| S pred => (NMoreConstructorsL pred (OneMoreConstructor Constrs))
end. *)
Fixpoint NMoreExt n Ext :=
match n with
| 0 => Ext
| S pred => @OneMoreExt (NMoreExt pred Ext)
end.
Fixpoint NMoreQuotvars n {VC} {TC} (MQC : MQCT VC TC) : MQCT (NMoreConstructors n VC) (NMoreConstructors n TC) :=
match n with
| 0 => MQC
| S p => OneMoreQuotvar (NMoreQuotvars p MQC)
end.
(* Fixpoint NMoreQuotvars_without_new_MQCs n {VC} {TC} (MQC : MQCT VC TC) : MQCT (NMoreConstructors n VC) (NMoreConstructors n TC) :=
match n with
| 0 => MQC
| S p => MQCOnStricterFormulaTypes (NMoreQuotvars_without_new_MQCs p MQC)
end. *)
Definition FormulaWithNVars n : Type := Formula (NMoreExt n False).
Definition FCWithNVars n : Type -> Type := NMoreConstructors n FormulaConstructors.
(* we could write (Formula False) instead, but this makes inference easier *)
Definition StandardFormula := FormulaWithNVars 0.
Definition StandardRuleset := Ruleset (FCWithNVars 0).
(* Fixpoint sf_to_ind (f : StandardFormula)
: Ind FormulaConstructors
:= λ _ _, match f with
| f_atm a => fc_atm a
| f_ext a => match a with end
| fo_apl a b => [(sf_to_ind a _) (sf_to_ind b _)]
end. *)
(* Instance fplus_fcplus {Ext} : Class (OneMoreConstructor FormulaConstructors) (Formula (@OneMoreExt Ext)) :=
{|
omc_embed := _
; omc_new := f_ext ome_new
|}.
Instance fplus2_fcplus2 {Ext} : Class (OneMoreConstructor (OneMoreConstructor FormulaConstructors)) (Formula (@OneMoreExt (@OneMoreExt Ext))) :=
{|
omc_embed := _
; omc_new := f_ext (ome_embed ome_new)
|}.
Print fplus2_fcplus2.
Definition test1 : Formula (@OneMoreExt (@OneMoreExt False))
:= @omc_new (OneMoreConstructor FormulaConstructors) _ _.
Definition test2 : Formula (@OneMoreExt (@OneMoreExt False))
:= @omc_new (FormulaConstructors) _ _.
Eval compute in (test1, test2). *)
(* Set Printing Implicit. *)
(* Fixpoint nth_new_ext {Ext} n : (@NMoreExt (S n) Ext) :=
match n with
| 0 => ome_new
| S pred => ome_embed (nth_new_ext pred)
end. *)
(* Instance fplus__eq_add_S {n m Ext} :
Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt (n + S m) Ext))
->
Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt (S n + m) Ext)).
intro.
change (S n + m) with (S (n + m)).
rewrite (plus_n_Sm n m).
assumption.
Defined. *)
(* Fixpoint fplusnm_fcplusn n m :
Class (FCWithNVars n)
(Formula (@NMoreExt n (@NMoreExt m False))) := match n return Class (FCWithNVars n)
(Formula (@NMoreExt n (@NMoreExt m False))) with
| 0 => f_fc
| S pred =>
{|
omc_embed :=
(* we can technically get away with writing just `_` for this, but that's too magical *)
fplusnm_fcplusn ()
; omc_new := f_ext ome_new
|}
end. *)
(* Instance fplusnm_fcplusn n m :
Class (FCWithNVars n)
(Formula (@NMoreExt n (@NMoreExt m False))).
induction n. exact _.
(* Set Printing Implicit. *)
cbn.
refine
{|
omc_embed :=
_
; omc_new := f_ext ome_new
|}.
:=
match n return Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt (n + m) Ext)) with
| 0 => f_fc
| S pred =>
{|
omc_embed :=
(* we can technically get away with writing just `_` for this, but that's too magical *)
fplus__eq_add_S (fplusnm_fcplusn pred (S m))
; omc_new := f_ext (nth_new_ext (pred + m))
|}
end.
Existing Instance fplusnm_fcplusn.
Print fplusnm_fcplusn.
Print fplus__eq_add_S. *)
(* Instance fplus__plus_n_0 n {Ext} :
Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt (n + 0) Ext))
->
Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt n Ext)).
intro.
refine (eq_rect_r (λ m, Class (NMoreConstructors n FormulaConstructors)
(Formula (@NMoreExt m Ext))) X _).
apply plus_n_O.
Defined.
Print fplus__plus_n_0. *)
Fixpoint NMoreConstructors_subclass n C
: NMoreConstructors n C ⊆ C
:= match n with
| 0 => subclass_refl
| S p => subclass_trans OneMoreConstructor_subclass (NMoreConstructors_subclass p C)
end.
Existing Instance NMoreConstructors_subclass.
(* Lemma NMoreConstructors_flop n C : NMoreConstructors (S n) C = NMoreConstructors n (OneMoreConstructor C).
induction n; [trivial|].
cbn in *.
rewrite <- IHn.
reflexivity.
Qed. *)
(* Set Printing Implicit. *)
Instance nmore_fc n Constrs T : (Constrs ⊆ FormulaConstructors) -> Class (NMoreConstructors n Constrs) T -> Class FormulaConstructors T := λ _ _, _.
(* Definition fwn_to_ind n
: FormulaWithNVars n -> Ind (FCWithNVars n)
:=
nmf_to_ind n (False_rect _). *)
Instance fcn_sub_fc n
: FCWithNVars n ⊆ FormulaConstructors
:= NMoreConstructors_subclass n _.
Instance fcn_pc n : FCWithNVars n ⊆ PropositionConstructors
:= _.
Instance fn_cpc n : Class PropositionConstructors (FormulaWithNVars n)
:= _.
Instance fn_pc n : PropositionConstructors (FormulaWithNVars n)
:= _.
Fixpoint embed_formula
Ext1 Ext2 (embed : Ext1 -> Ext2)
(f : (Formula Ext1)) : (Formula Ext2)
:= match f with
| f_atm a => f_atm a
| f_ext a => f_ext (embed a)
| fo_apl a b => [(embed_formula embed a) (embed_formula embed b)]
end.
(* Fixpoint FCn_embed
n Ext1 Ext2 (embed : Ext1 -> Ext2)
: Class (FCWithNVars n) (Formula Ext1) ->
Class (FCWithNVars n) (Formula Ext2)
:= λ _f, match n return Class (FCWithNVars n) (Formula Ext1) -> Class (FCWithNVars n) (Formula Ext2) with
| 0 => λ _, f_fc
| S n => λ _, {|
omc_embed := FCn_embed ( OneMoreConstructor_subclass _) ;
omc_new := embed_formula embed omc_new
|}
end _f. *)
(* Definition FCn_embed n Ext1 Ext2 (embed : Ext1 -> Ext2) :
Class (FCWithNVars n) (Formula Ext1) ->
Class (FCWithNVars n) (Formula Ext2).
intro.
induction n.
exact f_fc.
exact {|
omc_embed := IHn (OneMoreConstructor_subclass X) ;
omc_new := embed_formula embed omc_new
|}.
Defined. *)
Definition FCn_embed [n m] :
Class (FCWithNVars n) (FormulaWithNVars m) ->
Class (FCWithNVars n) (FormulaWithNVars (S m)).
intro.
induction n.
exact f_fc.
exact {|
omc_embed := IHn (OneMoreConstructor_subclass X) ;
omc_new := embed_formula ome_embed omc_new
|}.
Defined.
(* Inductive ReifiedFCWithNVarsInM : nat -> nat -> Type :=
| rfcn_0 m : ReifiedFCWithNVarsInM 0 m
| rfcn_S n m : ReifiedFCWithNVarsInM n m -> ReifiedFCWithNVarsInM (S n) (S m)
.
Locate "≤".
Print le.
Print le_ind.
Print le_rect.
Fixpoint FCn_embed22 n m (_c : Class (FCWithNVars n) (FormulaWithNVars m)) :
Class (FCWithNVars n) (FormulaWithNVars (S m)) :=
match n with
| 0 => f_fc
| (S pred) => FCn_embed22 pred m
end.
intro.
induction n.
exact f_fc.
exact {|
omc_embed := IHn (OneMoreConstructor_subclass X) ;
omc_new := embed_formula ome_embed omc_new
|}.
Defined. *)
(* Fixpoint ome_new_embed3 {Ext} n R :
(@NMoreExt (S n) Ext -> R) -> R :=
match n with
| 0 => λ embed, embed ome_new
| S pred => λ embed, (@ome_new_embed3 _ pred R (λ e, embed (ome_embed e)))
end.
Print ome_new_embed3. *)
(* Definition next_new_var_in {Ext} m
(prev_var : (NMoreExt m Ext))
: (NMoreExt (S m) Ext)
:= ome_embed prev_var. *)
(* Fixpoint nth_new_var_in_wrong {Ext} n m
: (NMoreExt n (NMoreExt (S m) Ext))
:= match n with
| 0 => ome_new
| S pred => next_new_var_in pred (nth_new_var_in_wrong pred m)
end. *)
(* Fixpoint nth_new_var_in {Ext} n m
: (NMoreExt (S m) (NMoreExt n Ext))
:= match n with
| 0 => ome_new
| S pred => next_new_var_in pred (nth_new_var_in pred m)
end. *)
(* Definition FCn_embed3_impl n m :
Class (FCWithNVars (S n)) (FormulaWithNVars m) ->
Class (FCWithNVars (S n)) (FormulaWithNVars (S m)).
intro.
(* induction n.
exact f_fc. *)
notypeclasses refine {|
omc_embed := _ ;
omc_new := f_ext (nth_new_var_in n m)
|}.
change (Class (FCWithNVars n) (FormulaWithNVars (S m))).
IHn (OneMoreConstructor_subclass X)
Defined. *)
(* Require Import Vector.
Fixpoint nth_var_in n m (vars : Vector.t (NMoreExt (S m) False) n) : (NMoreExt (S m) False).
destruct vars.
exact ome_new.
exact (nth_var_in n m vars).
Defined.
Definition embed_vars n m (vars : Vector.t (NMoreExt m False) n) : Vector.t (NMoreExt (S m) False) n
:= map ome_embed vars. *)
(*
Definition vars_to_ n (vars : Vector.t (NMoreExt n False) n) :
(FCWithNVars n)
(FormulaWithNVars n).
induction n.
exact f_fc.
destruct vars.
exact {|
omc_embed := IHvars ;
omc_new := f_ext ome_new
|}. *)
(* Fixpoint vars_to_ n m (vars : Vector.t (NMoreExt m False) n) :
(FCWithNVars n)
(FormulaWithNVars m) :=
match vars with
| nil => f_fc
| cons h pred t => {|
omc_embed := vars_to_ m t ;
omc_new := f_ext h
|}
end.
Fixpoint right_vars n : Vector.t (NMoreExt n False) n :=
match n with
| 0 => Vector.nil _
| S pred => Vector.cons _ ome_new _ (map ome_embed (right_vars pred))
end. *)
(* Instance fn_fcn n : Class
(FCWithNVars n)
(FormulaWithNVars n)
:= vars_to_ n (right_vars n). *)
Inductive ler (n : nat) : nat -> Type :=
| ler_nn : ler n n
| ler_Sm m : ler n m → ler n (S m).
Definition ltr n m := ler (S n) m.
Inductive upto : nat -> nat -> Type :=
| upto_0m m : upto 0 m
| upto_Sn n m : ltr n m -> upto n m -> upto (S n) m.
(* Inductive leroute : nat -> nat -> Type :=
| leroute_00 : leroute 0 0
| leroute_Sm n m : leroute n m -> leroute n (S m)
| leroute_Sn n m : ltr n m -> leroute n m -> leroute (S n) m. *)
(* Fixpoint ler_0m m : ler 0 m :=
match m with
| 0 => ler_nn 0
| S pred => ler_Sm (ler_0m pred)
end. *)
(* Search le.
Lemma ler_SnSm n m (l : ler n m) : ler (S n) (S m).
induction l; constructor.
assumption.
Defined. *)
Fixpoint ler_Pn [n m] (l : ler (S n) m) : ler n m :=
match l with
| ler_nn => ler_Sm (ler_nn _)
| ler_Sm m l => ler_Sm (ler_Pn l)
end.
(* Set Printing Implicit.
Print ler_Pn. *)
(* Fixpoint ler_pboth n m (l : ler (S n) (S m)) : ler n m :=
match l in ler _ m0 return (m0 = (S m)) -> ler n m with
| ler_nn => λ eq, match eq with eq_refl => ler_nn n end
| ler_Sm m l => λ eq, _
end eq_refl.
dependent induction l.
constructor.
assumption.
Defined. *)
Lemma ler_pboth n m (l : ler (S n) (S m)) : ler n m.
dependent induction l.
constructor.
induction m.
dependent destruction l.
apply ler_Sm.
apply IHl.
reflexivity.
Qed.
Lemma S_not_ler n : ler (S n) n -> False.
intro l. induction n.
dependent destruction l.
exact (IHn (ler_pboth l) ).
Qed.
Lemma ler_unique n m (l0 l1 : ler n m) : l0 = l1.
induction l0.
dependent destruction l1.
reflexivity.
contradict l1; apply S_not_ler.
dependent destruction l1.
clear IHl0; contradict l0; apply S_not_ler.
rewrite (IHl0 l1); reflexivity.
Qed.
Fixpoint upto_Sm n m (u : upto n m) : upto n (S m) :=
match u with
| upto_0m _ => upto_0m _
| upto_Sn _ _ l u => upto_Sn (ler_Sm l) (upto_Sm u)
end.
Fixpoint upto_nn n : upto n n :=
match n with
| 0 => upto_0m _
| S pred => upto_Sn (ler_nn _) (upto_Sm (upto_nn pred))
end.
Fixpoint ler_upto n m (l : ler n m) : upto n m :=
match l with
| ler_nn => upto_nn n
| ler_Sm m l => upto_Sm (ler_upto l)
end.
(* Definition upto_ler n m (u : upto n m) : ler n m :=
match u with
| upto_0m m => ler_0m m
| upto_Sn _ _ l u => l
end. *)
(* Fixpoint leroute_ler n m (r : leroute n m) : ler n m :=
match r with
| leroute_00 => ler_nn _
| leroute_Sm _ _ r => ler_Sm (leroute_ler r)
| leroute_Sn _ _ l r => l
end. *)
Lemma upto_unique n m (u0 u1 : upto n m) : u0 = u1.
induction u0.
dependent destruction u1.
reflexivity.
dependent destruction u1.
rewrite (ler_unique l0 l).
rewrite (IHu0 u1); reflexivity.
Qed.
(* Lemma upto_SnSm n m (u : upto n m) : upto (S n) (S m).
induction u; constructor.
apply ler_SnSm; apply ler_0m.
assumption.
Defined. *)
(* Fixpoint leroute_0m m : leroute 0 m :=
match m with
| 0 => leroute_00