+(*********************************************************************************************************************************)
+(* NaturalDeduction: *)
+(* *)
+(* Structurally explicit natural deduction proofs. *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+Require Import Categories_ch1_3.
+Require Import Functors_ch1_4.
+Require Import Isomorphisms_ch1_5.
+Require Import ProductCategories_ch1_6_1.
+Require Import OppositeCategories_ch1_6_2.
+Require Import Enrichment_ch2_8.
+Require Import Subcategories_ch7_1.
+Require Import NaturalTransformations_ch7_4.
+Require Import NaturalIsomorphisms_ch7_5.
+Require Import MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+Require Import Enrichment_ch2_8.
+Require Import RepresentableStructure_ch7_2.
+Require Import NaturalDeduction.
+
+
+(*
+ * Everything in the rest of this section is just groundwork meant to
+ * build up to the definition of the AcceptableLanguage class, which
+ * appears at the end of the section. References to "the instance"
+ * mean instances of that class. Think of this section as being one
+ * big Class { ... } definition, except that we declare most of the
+ * stuff outside the curly brackets in order to take advantage of
+ * Coq's section mechanism.
+ *)
+Section Acceptable_Language.
+
+ (* Formalized Definition 4.1.1, production $\tau$ *)
+ Context {T : Type}. (* types of the language *)
+
+ Inductive Sequent := sequent : Tree ??T -> Tree ??T -> Sequent.
+ Notation "cs |= ss" := (sequent cs ss) : al_scope.
+ (* Because of term irrelevance we need only store the *erased* (def
+ * 4.4) trees; for this reason there is no Coq type directly
+ * corresponding to productions $e$ and $x$ of 4.1.1, and TreeOT can
+ * be used for productions $\Gamma$ and $\Sigma$ *)
+
+ (* to do: sequent calculus equals natural deduction over sequents, theorem equals sequent with null antecedent, *)
+
+ Context {Rule : Tree ??Sequent -> Tree ??Sequent -> Type}.
+
+ Notation "H /⋯⋯/ C" := (ND Rule H C) : al_scope.
+
+ Open Scope pf_scope.
+ Open Scope nd_scope.
+ Open Scope al_scope.
+
+ (* Formalized Definition 4.1
+ *
+ * Note that from this abstract interface, the terms (expressions)
+ * in the proof are not accessible at all; they don't need to be --
+ * so long as we have access to the equivalence relation upon
+ * proof-conclusions. Moreover, hiding the expressions actually
+ * makes the encoding in CiC work out easier for two reasons:
+ *
+ * 1. Because the denotation function is provided a proof rather
+ * than a term, it is a total function (the denotation function is
+ * often undefined for ill-typed terms).
+ *
+ * 2. We can define arr_composition of proofs without having to know how
+ * to compose expressions. The latter task is left up to the client
+ * function which extracts an expression from a completed proof.
+ *
+ * This also means that we don't need an explicit proof obligation for 4.1.2.
+ *)
+ Class AcceptableLanguage :=
+
+ (* Formalized Definition 4.1: denotational semantics equivalence relation on the conclusions of proofs *)
+ { al_eqv : @ND_Relation Sequent Rule
+ where "pf1 === pf2" := (@ndr_eqv _ _ al_eqv _ _ pf1 pf2)
+
+ (* Formalized Definition 4.1.3; note that t here is either $\top$ or a single type, not a Tree of types;
+ * we rely on "completeness of atomic initial segments" (http://en.wikipedia.org/wiki/Completeness_of_atomic_initial_sequents)
+ * to generate the rest *)
+ ; al_reflexive_seq : forall t, Rule [] [t|=t]
+
+ (* these can all be absorbed into a separate "sequent calculus" presentation *)
+ ; al_ant_assoc : forall {a b c d}, Rule [(a,,b),,c|=d] [(a,,(b,,c))|=d]
+ ; al_ant_cossa : forall {a b c d}, Rule [a,,(b,,c)|=d] [((a,,b),,c)|=d]
+ ; al_ant_cancell : forall {a b }, Rule [ [],,a |=b] [ a |=b]
+ ; al_ant_cancelr : forall {a b }, Rule [a,,[] |=b] [ a |=b]
+ ; al_ant_llecnac : forall {a b }, Rule [ a |=b] [ [],,a |=b]
+ ; al_ant_rlecnac : forall {a b }, Rule [ a |=b] [ a,,[] |=b]
+ ; al_suc_assoc : forall {a b c d}, Rule [d|=(a,,b),,c] [d|=(a,,(b,,c))]
+ ; al_suc_cossa : forall {a b c d}, Rule [d|=a,,(b,,c)] [d|=((a,,b),,c)]
+ ; al_suc_cancell : forall {a b }, Rule [a|=[],,b ] [a|= b ]
+ ; al_suc_cancelr : forall {a b }, Rule [a|=b,,[] ] [a|= b ]
+ ; al_suc_llecnac : forall {a b }, Rule [a|= b ] [a|=[],,b ]
+ ; al_suc_rlecnac : forall {a b }, Rule [a|= b ] [a|=b,,[] ]
+
+ ; al_horiz_expand_left : forall tau {Gamma Sigma}, Rule [ Gamma |= Sigma ] [tau,,Gamma|=tau,,Sigma]
+ ; al_horiz_expand_right : forall tau {Gamma Sigma}, Rule [ Gamma |= Sigma ] [Gamma,,tau|=Sigma,,tau]
+
+ (* these are essentially one way of formalizing
+ * "completeness of atomic initial segments" (http://en.wikipedia.org/wiki/Completeness_of_atomic_initial_sequents) *)
+ ; al_horiz_expand_left_reflexive : forall a b, [#al_reflexive_seq b#];;[#al_horiz_expand_left a#]===[#al_reflexive_seq (a,,b)#]
+ ; al_horiz_expand_right_reflexive : forall a b, [#al_reflexive_seq a#];;[#al_horiz_expand_right b#]===[#al_reflexive_seq (a,,b)#]
+ ; al_horiz_expand_right_then_cancel : forall a,
+ ((([#al_reflexive_seq (a,, [])#] ;; [#al_ant_cancelr#]);; [#al_suc_cancelr#]) === [#al_reflexive_seq a#])
+
+ ; al_vert_expand_ant_left : forall x `(pf:[a|=b]/⋯⋯/[c|=d]), [x,,a |= b ]/⋯⋯/[x,,c |= d ]
+ ; al_vert_expand_ant_right : forall x `(pf:[a|=b]/⋯⋯/[c|=d]), [ a,,x|= b ]/⋯⋯/[ c,,x|= d ]
+ ; al_vert_expand_suc_left : forall x `(pf:[a|=b]/⋯⋯/[c|=d]), [ a |=x,,b ]/⋯⋯/[ c |=x,,d ]
+ ; al_vert_expand_suc_right : forall x `(pf:[a|=b]/⋯⋯/[c|=d]), [ a |= b,,x]/⋯⋯/[ c |= d,,x]
+ ; al_vert_expand_ant_l_respects : forall x a b c d (f g:[a|=b]/⋯⋯/[c|=d]),
+ f===g -> al_vert_expand_ant_left x f === al_vert_expand_ant_left x g
+ ; al_vert_expand_ant_r_respects : forall x a b c d (f g:[a|=b]/⋯⋯/[c|=d]),
+ f===g -> al_vert_expand_ant_right x f === al_vert_expand_ant_right x g
+ ; al_vert_expand_suc_l_respects : forall x a b c d (f g:[a|=b]/⋯⋯/[c|=d]),
+ f===g -> al_vert_expand_suc_left x f === al_vert_expand_suc_left x g
+ ; al_vert_expand_suc_r_respects : forall x a b c d (f g:[a|=b]/⋯⋯/[c|=d]),
+ f===g -> al_vert_expand_suc_right x f === al_vert_expand_suc_right x g
+ ; al_vert_expand_ant_l_preserves_id : forall x a b, al_vert_expand_ant_left x (nd_id [a|=b]) === nd_id [x,,a|=b]
+ ; al_vert_expand_ant_r_preserves_id : forall x a b, al_vert_expand_ant_right x (nd_id [a|=b]) === nd_id [a,,x|=b]
+ ; al_vert_expand_suc_l_preserves_id : forall x a b, al_vert_expand_suc_left x (nd_id [a|=b]) === nd_id [a|=x,,b]
+ ; al_vert_expand_suc_r_preserves_id : forall x a b, al_vert_expand_suc_right x (nd_id [a|=b]) === nd_id [a|=b,,x]
+ ; al_vert_expand_ant_l_preserves_comp : forall x a b c d e f (h:[a|=b]/⋯⋯/[c|=d])(g:[c|=d]/⋯⋯/[e|=f]),
+ (al_vert_expand_ant_left x (h;;g)) === (al_vert_expand_ant_left x h);;(al_vert_expand_ant_left x g)
+ ; al_vert_expand_ant_r_preserves_comp : forall x a b c d e f (h:[a|=b]/⋯⋯/[c|=d])(g:[c|=d]/⋯⋯/[e|=f]),
+ (al_vert_expand_ant_right x (h;;g)) === (al_vert_expand_ant_right x h);;(al_vert_expand_ant_right x g)
+ ; al_vert_expand_suc_l_preserves_comp : forall x a b c d e f (h:[a|=b]/⋯⋯/[c|=d])(g:[c|=d]/⋯⋯/[e|=f]),
+ (al_vert_expand_suc_left x (h;;g)) === (al_vert_expand_suc_left x h);;(al_vert_expand_suc_left x g)
+ ; al_vert_expand_suc_r_preserves_comp : forall x a b c d e f (h:[a|=b]/⋯⋯/[c|=d])(g:[c|=d]/⋯⋯/[e|=f]),
+ (al_vert_expand_suc_right x (h;;g)) === (al_vert_expand_suc_right x h);;(al_vert_expand_suc_right x g)
+
+ ; al_subst : forall a b c, [ a |= b ] ,, [ b |= c ] /⋯⋯/ [ a |= c ]
+ ; al_subst_associativity : forall {a b c d},
+ ((al_subst a b c) ** (nd_id1 (c|=d))) ;;
+ (al_subst a c d)
+ ===
+ nd_assoc ;;
+ ((nd_id1 (a|=b)) ** (al_subst b c d) ;;
+ (al_subst a b d))
+ ; al_subst_associativity' : forall {a b c d},
+ nd_cossa ;;
+ ((al_subst a b c) ** (nd_id1 (c|=d))) ;;
+ (al_subst a c d)
+ ===
+ ((nd_id1 (a|=b)) ** (al_subst b c d) ;;
+ (al_subst a b d))
+
+ ; al_subst_left_identity : forall `(pf:h/⋯⋯/[t1|=t2]), nd_llecnac;;(( [#al_reflexive_seq t1#]**pf);; al_subst _ _ _) === pf
+ ; al_subst_right_identity : forall `(pf:h/⋯⋯/[t1|=t2]), nd_rlecnac;;((pf**[#al_reflexive_seq t2#] );; al_subst _ _ _) === pf
+ ; al_subst_commutes_with_horiz_expand_left : forall a b c d,
+ [#al_horiz_expand_left d#] ** [#al_horiz_expand_left d#];; al_subst (d,, a) (d,, b) (d,, c)
+ === al_subst a b c;; [#al_horiz_expand_left d#]
+ ; al_subst_commutes_with_horiz_expand_right : forall a b c d,
+ [#al_horiz_expand_right d#] ** [#al_horiz_expand_right d#] ;; al_subst (a,, d) (b,, d) (c,, d)
+ === al_subst a b c;; [#al_horiz_expand_right d#]
+ ; al_subst_commutes_with_vertical_expansion : forall t0 t1 t2, forall (f:[[]|=t1]/⋯⋯/[[]|=t0])(g:[[]|=t0]/⋯⋯/[[]|=t2]),
+ (((nd_rlecnac;;
+ ((([#al_reflexive_seq (t1,, [])#];; al_vert_expand_ant_left t1 (al_vert_expand_suc_right [] f));;
+ (nd_rule al_ant_cancelr));; (nd_rule al_suc_cancelr)) ** nd_id0);;
+ (nd_id [t1 |= t0]) **
+ ((([#al_reflexive_seq (t0,, [])#];; al_vert_expand_ant_left t0 (al_vert_expand_suc_right [] g));;
+ (nd_rule al_ant_cancelr));; (nd_rule al_suc_cancelr)));;
+ al_subst t1 t0 t2)
+ ===
+ ((([#al_reflexive_seq (t1,, [])#];;
+ (al_vert_expand_ant_left t1 (al_vert_expand_suc_right [] f);;
+ al_vert_expand_ant_left t1 (al_vert_expand_suc_right [] g)));;
+ (nd_rule al_ant_cancelr));; (nd_rule al_suc_cancelr))
+ }.
+
+ Notation "pf1 === pf2" := (@ndr_eqv _ _ al_eqv _ _ pf1 pf2) : temporary_scope3.
+ Open Scope temporary_scope3.
+
+ Lemma al_subst_respects :
+ forall {AL:AcceptableLanguage}{a b c},
+ forall
+ (f : [] /⋯⋯/ [a |= b])
+ (f' : [] /⋯⋯/ [a |= b])
+ (g : [] /⋯⋯/ [b |= c])
+ (g' : [] /⋯⋯/ [b |= c]),
+ (f === f') ->
+ (g === g') ->
+ (f ** g;; al_subst _ _ _) === (f' ** g';; al_subst _ _ _).
+ intros.
+ setoid_rewrite H.
+ setoid_rewrite H0.
+ reflexivity.
+ Defined.
+
+ (* a contextually closed language *)
+ (*
+ Class ContextuallyClosedAcceptableLanguage :=
+ { ccal_al : AcceptableLanguage
+ ; ccal_contextual_closure_operator : Tree ??T -> Tree ??T -> Tree ??T
+ where "a -~- b" := (ccal_contextual_closure_operator a b)
+ ; ccal_contextual_closure : forall {a b c d}(f:[a|=b]/⋯⋯/[c|=d]), [[]|=a-~-b]/⋯⋯/[[]|=c-~-d]
+ ; ccal_contextual_closure_respects : forall {a b c d}(f f':[a|=b]/⋯⋯/[c|=d]),
+ f===f' -> (ccal_contextual_closure f)===(ccal_contextual_closure f')
+ ; ccal_contextual_closure_preserves_comp : forall {a b c d e f}(f':[a|=b]/⋯⋯/[c|=d])(g':[c|=d]/⋯⋯/[e|=f]),
+ (ccal_contextual_closure f');;(ccal_contextual_closure g') === (ccal_contextual_closure (f';;g'))
+ ; ccal_contextual_closure_preserves_id : forall {a b}, ccal_contextual_closure (nd_id [a|=b]) === nd_id [[]|=a-~-b]
+ }.
+ Coercion ccal_al : ContextuallyClosedAcceptableLanguage >-> AcceptableLanguage.
+ *)
+
+ (* languages with unrestricted substructural rules (like that of Section 5) additionally implement this class *)
+ Class AcceptableLanguageWithUnrestrictedSubstructuralRules :=
+ { alwusr_al :> AcceptableLanguage
+ ; al_contr : forall a b, Rule [a,,a |= b ] [ a |= b]
+ ; al_exch : forall a b c, Rule [a,,b |= c ] [(b,,a)|= c]
+ ; al_weak : forall a b, Rule [[] |= b ] [ a |= b]
+ }.
+ Coercion alwusr_al : AcceptableLanguageWithUnrestrictedSubstructuralRules >-> AcceptableLanguage.
+
+ (* languages with a fixpoint operator *)
+ Class AcceptableLanguageWithFixpointOperator `(al:AcceptableLanguage) :=
+ { alwfpo_al := al
+ ; al_fix : forall a b x, Rule [a,,x |= b,,x] [a |= b]
+ }.
+ Coercion alwfpo_al : AcceptableLanguageWithFixpointOperator >-> AcceptableLanguage.
+
+ Close Scope temporary_scope3.
+ Close Scope al_scope.
+ Close Scope nd_scope.
+ Close Scope pf_scope.
+
+End Acceptable_Language.
+
+Implicit Arguments ND [ Judgment ].
+
+Open Scope nd_scope.
+ Add Parametric Morphism {T Rule AL a b c d e} : (@al_vert_expand_suc_right T Rule AL a b c d e)
+ with signature ((ndr_eqv(ND_Relation:=al_eqv)) ==> (ndr_eqv(ND_Relation:=al_eqv)))
+ as parametric_morphism_al_vert_expand_suc_right.
+ intros; apply al_vert_expand_suc_r_respects; auto.
+ Defined.
+ Add Parametric Morphism {T Rule AL a b c d e} : (@al_vert_expand_suc_left T Rule AL a b c d e)
+ with signature ((ndr_eqv(ND_Relation:=al_eqv)) ==> (ndr_eqv(ND_Relation:=al_eqv)))
+ as parametric_morphism_al_vert_expand_suc_left.
+ intros; apply al_vert_expand_suc_l_respects; auto.
+ Defined.
+ Add Parametric Morphism {T Rule AL a b c d e} : (@al_vert_expand_ant_right T Rule AL a b c d e)
+ with signature ((ndr_eqv(ND_Relation:=al_eqv)) ==> (ndr_eqv(ND_Relation:=al_eqv)))
+ as parametric_morphism_al_vert_expand_ant_right.
+ intros; apply al_vert_expand_ant_r_respects; auto.
+ Defined.
+ Add Parametric Morphism {T Rule AL a b c d e} : (@al_vert_expand_ant_left T Rule AL a b c d e)
+ with signature ((ndr_eqv(ND_Relation:=al_eqv)) ==> (ndr_eqv(ND_Relation:=al_eqv)))
+ as parametric_morphism_al_vert_expand_ant_left.
+ intros; apply al_vert_expand_ant_l_respects; auto.
+ Defined.
+Close Scope nd_scope.
+
+Notation "cs |= ss" := (@sequent _ cs ss) : al_scope.
+(*
+Definition mapSequent {T R:Type}(f:Tree ??T -> Tree ??R)(seq:@Sequent T) : @Sequent R :=
+ match seq with sequentpair a b => pair (f a) (f b) end.
+Implicit Arguments Sequent [ ].
+*)
+
+
+(* proofs which are generic and apply to any acceptable langauge (most of section 4) *)
+Section Acceptable_Language_Facts.
+
+ (* the ambient language about which we are proving facts *)
+ Context `(Lang : @AcceptableLanguage T Rule).
+
+ (* just for this section *)
+ Open Scope nd_scope.
+ Open Scope al_scope.
+ Open Scope pf_scope.
+ Notation "H /⋯⋯/ C" := (@ND Sequent Rule H C) : temporary_scope4.
+ Notation "a === b" := (@ndr_eqv _ _ al_eqv _ _ a b) : temporary_scope4.
+ Open Scope temporary_scope4.
+
+ Definition lang_al_eqv := al_eqv(AcceptableLanguage:=Lang).
+ Existing Instance lang_al_eqv.
+
+ Ltac distribute :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ context ct [(?A ** ?B) ;; (?C ** ?D)] =>
+ setoid_rewrite <- (ndr_prod_preserves_comp A B C D)
+ end
+ end.
+
+ Ltac sequentialize_product A B :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ | context ct [(A ** B)] =>
+ setoid_replace (A ** B)
+ with ((A ** (nd_id _)) ;; ((nd_id _) ** B))
+ (*with ((A ** (nd_id _)) ;; ((nd_id _) ** B))*)
+ end end.
+ Ltac sequentialize_product' A B :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ | context ct [(A ** B)] =>
+ setoid_replace (A ** B)
+ with (((nd_id _) ** B) ;; (A ** (nd_id _)))
+ (*with ((A ** (nd_id _)) ;; ((nd_id _) ** B))*)
+ end end.
+ Ltac distribute' :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ context ct [(?A ;; ?B) ** (?C ;; ?D)] =>
+ setoid_rewrite (ndr_prod_preserves_comp A B C D)
+ end
+ end.
+ Ltac distribute_left_product_with_id :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ context ct [(nd_id ?A) ** (?C ;; ?D)] =>
+ setoid_replace ((nd_id A) ** (C ;; D)) with ((nd_id A ;; nd_id A) ** (C ;; D));
+ [ setoid_rewrite (ndr_prod_preserves_comp (nd_id A) C (nd_id A) D) | idtac ]
+ end
+ end.
+ Ltac distribute_right_product_with_id :=
+ match goal with
+ [ |- ?G ] =>
+ match G with
+ context ct [(?C ;; ?D) ** (nd_id ?A)] =>
+ setoid_replace ((C ;; D) ** (nd_id A)) with ((C ;; D) ** (nd_id A ;; nd_id A));
+ [ setoid_rewrite (ndr_prod_preserves_comp C (nd_id A) D (nd_id A)) | idtac ]
+ end
+ end.
+
+ (* another phrasing of al_subst_associativity; obligations tend to show up in this form *)
+ Lemma al_subst_associativity'' :
+ forall (a b : T) (f : [] /⋯⋯/ [[a] |= [b]]) (c : T) (g : [] /⋯⋯/ [[b] |= [c]])
+ (d : T) (h : [] /⋯⋯/ [[c] |= [d]]),
+ nd_llecnac;; ((nd_llecnac;; (f ** g;; al_subst [a] [b] [c])) ** h;; al_subst [a] [c] [d]) ===
+ nd_llecnac;; (f ** (nd_llecnac;; (g ** h;; al_subst [b] [c] [d]));; al_subst [a] [b] [d]).
+ intros.
+ sequentialize_product' (nd_llecnac;; (f ** g;; al_subst [a] [b] [c])) h.
+ repeat setoid_rewrite <- ndr_comp_associativity.
+ distribute_right_product_with_id.
+ repeat setoid_rewrite ndr_comp_associativity.
+ set (@al_subst_associativity) as q. setoid_rewrite q. clear q.
+ apply ndr_comp_respects; try reflexivity.
+ repeat setoid_rewrite <- ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ sequentialize_product f ((nd_llecnac;; g ** h);; al_subst [b] [c] [d]).
+ distribute_left_product_with_id.
+ repeat setoid_rewrite <- ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ setoid_rewrite <- ndr_prod_preserves_comp.
+ repeat setoid_rewrite ndr_comp_left_identity.
+ repeat setoid_rewrite ndr_comp_right_identity.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ Qed.
+
+ (* Formalized Definition 4.6 *)
+ Section Types1.
+ Instance Types1 : Category T (fun t1 t2 => [ ] /⋯⋯/ [ [t1] |= [t2] ]) :=
+ { eqv := fun ta tb pf1 pf2 => pf1 === pf2
+ ; id := fun t => [#al_reflexive_seq [t]#]
+ ; comp := fun {ta tb tc:T}(pf1:[]/⋯⋯/[[ta]|=[tb]])(pf2:[]/⋯⋯/[[tb]|=[tc]]) => nd_llecnac ;; ((pf1 ** pf2) ;; (al_subst _ _ _))
+ }.
+ intros; apply Build_Equivalence;
+ [ unfold Reflexive; intros; reflexivity
+ | unfold Symmetric; intros; symmetry; auto
+ | unfold Transitive; intros; transitivity y; auto ].
+ unfold Proper; unfold respectful; intros; simpl.
+ apply ndr_comp_respects. reflexivity.
+ apply al_subst_respects; auto.
+ intros; simpl. apply al_subst_left_identity.
+ intros; simpl.
+ assert (@nd_llecnac _ Rule [] === @nd_rlecnac _ _ []).
+ apply ndr_structural_indistinguishable; auto.
+ setoid_rewrite H.
+ apply al_subst_right_identity.
+ intros; apply al_subst_associativity''.
+ Defined.
+ End Types1.
+
+ (* Formalized Definition 4.10 *)
+ Instance Judgments : Category (Tree ??Sequent) (fun h c => h /⋯⋯/ c) :=
+ { id := fun h => nd_id _
+ ; comp := fun a b c f g => f ;; g
+ ; eqv := fun a b f g => f===g
+ }.
+ intros; apply Build_Equivalence;
+ [ unfold Reflexive; intros; reflexivity
+ | unfold Symmetric; intros; symmetry; auto
+ | unfold Transitive; intros; transitivity y; auto ].
+ unfold Proper; unfold respectful; intros; simpl; apply ndr_comp_respects; auto.
+ intros; apply ndr_comp_left_identity.
+ intros; apply ndr_comp_right_identity.
+ intros; apply ndr_comp_associativity.
+ Defined.
+
+ (* a "primitive" proof has exactly one hypothesis and one conclusion *)
+ Inductive IsPrimitive : forall (h_:Tree ??(@Sequent T)), Type :=
+ isPrimitive : forall h, IsPrimitive [h].
+ Hint Constructors IsPrimitive.
+ Instance IsPrimitiveSubCategory : SubCategory Judgments IsPrimitive (fun _ _ _ _ _ => True).
+ apply Build_SubCategory; intros; auto.
+ Defined.
+
+ (* The primitive judgments form a subcategory; nearly all of the
+ * functors we build that go into Judgments will factor through the
+ * inclusion functor for this subcategory. Explicitly constructing
+ * it makes the formalization easier, but distracts from what's
+ * actually going on (from an expository perspective) *)
+ Definition PrimitiveJudgments := SubCategoriesAreCategories Judgments IsPrimitiveSubCategory.
+ Definition PrimitiveInclusion := InclusionFunctor Judgments IsPrimitiveSubCategory.
+
+ Section Types0.
+ Inductive IsNil : Tree ??(@Sequent T) -> Prop := isnil : IsNil [].
+ Inductive IsClosed : Tree ??(@Sequent T) -> Prop := isclosed:forall t, IsClosed [[]|=[t]].
+ Inductive IsIdentity : forall h c, (h /⋯⋯/ c) -> Prop :=
+ | isidentity0 : forall t, IsIdentity t t (nd_id t)
+ | isidentity1 : forall t pf1 pf2, IsIdentity t t pf1 -> IsIdentity t t pf2 -> IsIdentity t t (pf1 ;; pf2).
+ Inductive IsInTypes0 (h c:Tree ??Sequent)(pf:h /⋯⋯/ c) : Prop :=
+ | iit0_id0 : IsNil h -> IsNil c -> IsIdentity _ _ pf -> IsInTypes0 _ _ pf
+ | iit0_id1 : @IsClosed h -> @IsClosed c -> IsIdentity _ _ pf -> IsInTypes0 _ _ pf
+ | iit0_term : IsNil h -> @IsClosed c -> IsInTypes0 _ _ pf.
+ Instance Types0P : SubCategory Judgments
+ (fun x:Judgments => IsInTypes0 _ _ (id(Category:=Judgments) x))
+ (fun h c _ _ f => IsInTypes0 h c f).
+ intros.
+ apply Build_SubCategory; intros; simpl.
+ auto.
+ inversion H0.
+ inversion H1; subst.
+ inversion H2; subst.
+ inversion H; subst. inversion H4; subst.
+ apply iit0_id0; auto. apply isidentity1; auto.
+ inversion H5.
+ inversion H5.
+ inversion H1; subst.
+ inversion H2; subst.
+ inversion H3; subst. clear H8. clear H7.
+ inversion H; subst. inversion H5.
+ inversion H4; subst.
+ inversion H6; subst.
+ apply iit0_id1; auto. apply isidentity1; auto.
+ clear H10. clear H8.
+ apply iit0_id1; auto. apply isidentity1; auto.
+ inversion H4; subst. inversion H; subst.
+ inversion H8.
+ inversion H6.
+ apply iit0_term; auto.
+ clear H7; subst.
+ inversion H; subst.
+ inversion H4; subst.
+ apply iit0_term; auto.
+ inversion H4; subst.
+ inversion H7; subst. clear H14.
+ apply iit0_id1; auto. apply isidentity1; auto.
+ clear H13.
+ apply iit0_id1; auto. apply isidentity1; auto.
+ inversion H4; subst.
+ inversion H; subst.
+ inversion H10.
+ inversion H7.
+ apply iit0_term; auto.
+ inversion H1; subst.
+ inversion H; subst.
+ inversion H3; subst. apply iit0_term; auto.
+ inversion H4.
+ inversion H4.
+ Qed.
+
+ (* Formalized Definition 4.8 *)
+ Definition Types0 := SubCategoriesAreCategories Judgments Types0P.
+ End Types0.
+
+ (* Formalized Definition 4.11 *)
+ Instance Judgments_binoidal : BinoidalCat Judgments (fun a b:Tree ??Sequent => a,,b) :=
+ { bin_first := fun x => @Build_Functor _ _ Judgments _ _ Judgments (fun a => a,,x) (fun a b (f:a/⋯⋯/b) => f**(nd_id x)) _ _ _
+ ; bin_second := fun x => @Build_Functor _ _ Judgments _ _ Judgments (fun a => x,,a) (fun a b (f:a/⋯⋯/b) => (nd_id x)**f) _ _ _
+ }.
+ intros. simpl. simpl in H. setoid_rewrite H. reflexivity.
+ intros. simpl. reflexivity.
+ intros. simpl. setoid_rewrite <- ndr_prod_preserves_comp. setoid_rewrite ndr_comp_left_identity. reflexivity.
+ intros. simpl. simpl in H. setoid_rewrite H. reflexivity.
+ intros. simpl. reflexivity.
+ intros. simpl. setoid_rewrite <- ndr_prod_preserves_comp. setoid_rewrite ndr_comp_left_identity. reflexivity.
+ Defined.
+
+ Definition jud_assoc_iso (a b c:Judgments) : @Isomorphic _ _ Judgments ((a,,b),,c) (a,,(b,,c)).
+ apply (@Build_Isomorphic _ _ Judgments _ _ nd_assoc nd_cossa); simpl; auto.
+ Defined.
+ Definition jud_cancelr_iso (a:Judgments) : @Isomorphic _ _ Judgments (a,,[]) a.
+ apply (@Build_Isomorphic _ _ Judgments _ _ nd_cancelr nd_rlecnac); simpl; auto.
+ Defined.
+ Definition jud_cancell_iso (a:Judgments) : @Isomorphic _ _ Judgments ([],,a) a.
+ apply (@Build_Isomorphic _ _ Judgments _ _ nd_cancell nd_llecnac); simpl; auto.
+ Defined.
+
+ (* just for this section *)
+ Notation "a ⊗ b" := (@bin_obj _ _ Judgments _ Judgments_binoidal a b).
+ Notation "c ⋊ -" := (@bin_second _ _ Judgments _ Judgments_binoidal c).
+ Notation "- ⋉ c" := (@bin_first _ _ Judgments _ Judgments_binoidal c).
+ Notation "c ⋊ f" := ((c ⋊ -) \ f).
+ Notation "g ⋉ c" := ((- ⋉ c) \ g).
+
+ Hint Extern 1 => apply (@nd_structural_id0 _ Rule).
+ Hint Extern 1 => apply (@nd_structural_id1 _ Rule).
+ Hint Extern 1 => apply (@nd_structural_weak _ Rule).
+ Hint Extern 1 => apply (@nd_structural_copy _ Rule).
+ Hint Extern 1 => apply (@nd_structural_prod _ Rule).
+ Hint Extern 1 => apply (@nd_structural_comp _ Rule).
+ Hint Extern 1 => apply (@nd_structural_cancell _ Rule).
+ Hint Extern 1 => apply (@nd_structural_cancelr _ Rule).
+ Hint Extern 1 => apply (@nd_structural_llecnac _ Rule).
+ Hint Extern 1 => apply (@nd_structural_rlecnac _ Rule).
+ Hint Extern 1 => apply (@nd_structural_assoc _ Rule).
+ Hint Extern 1 => apply (@nd_structural_cossa _ Rule).
+ Hint Extern 2 => apply (@ndr_structural_indistinguishable _ Rule).
+
+ Program Instance Judgments_premonoidal : PreMonoidalCat Judgments_binoidal [ ] :=
+ { pmon_assoc := fun a b => @Build_NaturalIsomorphism _ _ _ _ _ _ _ _ _ _ (fun x => (jud_assoc_iso a x b)) _
+ ; pmon_cancell := @Build_NaturalIsomorphism _ _ _ _ _ _ _ _ _ _ (fun x => (jud_cancell_iso x)) _
+ ; pmon_cancelr := @Build_NaturalIsomorphism _ _ _ _ _ _ _ _ _ _ (fun x => (jud_cancelr_iso x)) _
+ ; pmon_assoc_rr := fun a b => @Build_NaturalIsomorphism _ _ _ _ _ _ _ _ _ _ (fun x => (jud_assoc_iso x a b)⁻¹) _
+ ; pmon_assoc_ll := fun a b => @Build_NaturalIsomorphism _ _ _ _ _ _ _ _ _ _ (fun x => jud_assoc_iso a b x) _
+ }.
+ Next Obligation.
+ setoid_rewrite (ndr_prod_associativity (nd_id a) f (nd_id b)).
+ repeat setoid_rewrite ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ symmetry.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; try reflexivity; simpl; auto.
+ Defined.
+ Next Obligation.
+ setoid_rewrite (ndr_prod_right_identity f).
+ repeat setoid_rewrite ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ symmetry.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; try reflexivity; simpl; auto.
+ Defined.
+ Next Obligation.
+ setoid_rewrite (ndr_prod_left_identity f).
+ repeat setoid_rewrite ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ symmetry.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; try reflexivity; simpl; auto.
+ Defined.
+ Next Obligation.
+ apply Build_Pentagon; intros.
+ simpl; apply ndr_structural_indistinguishable; auto.
+ Defined.
+ Next Obligation.
+ apply Build_Triangle; intros;
+ simpl; apply ndr_structural_indistinguishable; auto.
+ Defined.
+ Next Obligation.
+ setoid_rewrite (ndr_prod_associativity f (nd_id a) (nd_id b)).
+ repeat setoid_rewrite <- ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ eapply transitivity; [ idtac | apply ndr_comp_left_identity ].
+ apply ndr_comp_respects; try reflexivity; simpl; auto.
+ Defined.
+ Next Obligation.
+ setoid_rewrite (ndr_prod_associativity (nd_id a) (nd_id b) f).
+ repeat setoid_rewrite ndr_comp_associativity.
+ apply ndr_comp_respects; try reflexivity.
+ symmetry.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; try reflexivity; simpl; auto.
+ Defined.
+ Check (@Judgments_premonoidal). (* to force Coq to verify that we've finished all the obligations *)
+
+ Definition Judgments_monoidal_endofunctor_fobj : Judgments ×× Judgments -> Judgments :=
+ (fun xy =>
+ match xy with
+ | pair_obj x y => T_Branch x y
+ end).
+ Definition Judgments_monoidal_endofunctor_fmor :
+ forall a b, (a~~{Judgments ×× Judgments}~~>b) ->
+ ((Judgments_monoidal_endofunctor_fobj a)~~{Judgments}~~>(Judgments_monoidal_endofunctor_fobj b)).
+ intros.
+ destruct a.
+ destruct b.
+ destruct X.
+ exact (h**h0).
+ Defined.
+ Definition Judgments_monoidal_endofunctor : Functor (Judgments ×× Judgments) Judgments Judgments_monoidal_endofunctor_fobj.
+ refine {| fmor := Judgments_monoidal_endofunctor_fmor |}; intros; simpl.
+ abstract (destruct a; destruct b; destruct f; destruct f'; auto; destruct H; apply ndr_prod_respects; auto).
+ abstract (destruct a; simpl; reflexivity).
+ abstract (destruct a; destruct b; destruct c; destruct f; destruct g; symmetry; apply ndr_prod_preserves_comp).
+ Defined.
+
+ Instance Judgments_monoidal : MonoidalCat _ _ Judgments_monoidal_endofunctor [ ].
+ admit.
+ Defined.
+
+ (* all morphisms in the category of Judgments are central; there's probably a very short route from here to CartesianCat *)
+ Lemma all_central : forall a b:Judgments, forall (f:a~>b), CentralMorphism f.
+ intros; apply Build_CentralMorphism; intros.
+ simpl.
+ setoid_rewrite <- (ndr_prod_preserves_comp f (nd_id _) (nd_id _) g).
+ setoid_rewrite <- (ndr_prod_preserves_comp (nd_id _) g f (nd_id _)).
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ simpl.
+ setoid_rewrite <- (ndr_prod_preserves_comp g (nd_id _) (nd_id _) f).
+ setoid_rewrite <- (ndr_prod_preserves_comp (nd_id _) f g (nd_id _)).
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ Defined.
+
+ (*
+ Instance NoHigherOrderFunctionTypes : SubCategory Judgments
+ Instance NoFunctionTypes : SubCategory Judgments
+ Lemma first_order_functions_eliminable : IsomorphicCategories NoHigherOrderFunctionTypes NoFunctionTypes
+ *)
+
+ (* Formalized Theorem 4.19 *)
+ Instance Types_omega_e : ECategory Judgments_monoidal (Tree ??T) (fun tt1 tt2 => [ tt1 |= tt2 ]) :=
+ { eid := fun tt => [#al_reflexive_seq tt#]
+ ; ecomp := fun a b c => al_subst a b c
+ }.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ Definition Types_omega_monoidal_functor
+ : Functor (Types_omega_e ×× Types_omega_e) Types_omega_e (fun a => match a with pair_obj a1 a2 => a1,,a2 end).
+ admit.
+ Defined.
+
+ Instance Types_omega_monoidal : MonoidalCat Types_omega_e _ Types_omega_monoidal_functor [].
+ admit.
+ Defined.
+
+ Definition AL_Enrichment : Enrichment.
+ refine {| enr_c := Types_omega_e |}.
+ Defined.
+
+ Definition AL_SurjectiveEnrichment : SurjectiveEnrichment.
+ refine {| se_enr := AL_Enrichment |}.
+ unfold treeDecomposition.
+ intros; induction d; simpl.
+ destruct a.
+ destruct s.
+ exists [pair t t0]; auto.
+ exists []; auto.
+ destruct IHd1.
+ destruct IHd2.
+ exists (x,,x0); subst; auto.
+ Defined.
+
+ Definition AL_MonoidalEnrichment : MonoidalEnrichment.
+ refine {| me_enr := AL_SurjectiveEnrichment ; me_mon := Types_omega_monoidal |}.
+ admit.
+ Defined.
+
+ Definition AL_MonicMonoidalEnrichment : MonicMonoidalEnrichment.
+ refine {| ffme_enr := AL_MonoidalEnrichment |}.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ (*
+ Instance Types_omega_be : BinoidalECategory Types_omega_e :=
+ { bec_obj := fun tt1 tt2 => tt1,,tt2
+ ; bec_efirst := fun a b c => nd_rule (@al_horiz_expand_right _ _ Lang _ _ _)
+ ; bec_esecond := fun a b c => nd_rule (@al_horiz_expand_left _ _ Lang _ _ _)
+ }.
+ intros; apply all_central.
+ intros; apply all_central.
+ intros. unfold eid. simpl.
+ setoid_rewrite <- al_horiz_expand_right_reflexive.
+ reflexivity.
+ intros. unfold eid. simpl.
+ setoid_rewrite <- al_horiz_expand_left_reflexive.
+ reflexivity.
+ intros. simpl.
+ set (@al_subst_commutes_with_horiz_expand_right _ _ _ a b c d) as q.
+ setoid_rewrite <- q. clear q.
+ apply ndr_comp_respects; try reflexivity.
+ distribute.
+ apply ndr_prod_respects.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; reflexivity.
+ eapply transitivity; [ idtac | apply ndr_comp_left_identity ].
+ apply ndr_comp_respects; reflexivity.
+ intros. simpl.
+ set (@al_subst_commutes_with_horiz_expand_left _ _ _ a b c d) as q.
+ setoid_rewrite <- q. clear q.
+ apply ndr_comp_respects; try reflexivity.
+ distribute.
+ apply ndr_prod_respects.
+ eapply transitivity; [ idtac | apply ndr_comp_right_identity ].
+ apply ndr_comp_respects; reflexivity.
+ eapply transitivity; [ idtac | apply ndr_comp_left_identity ].
+ apply ndr_comp_respects; reflexivity.
+ Defined.
+ *)
+
+ Definition Types_omega : Category _ (fun tt1 tt2 => [ ]/⋯⋯/[ tt1 |= tt2 ]) := Underlying Types_omega_e.
+ Existing Instance Types_omega.
+
+ (*
+ Definition Types_omega_binoidal : BinoidalCat Types_omega (fun tt1 tt2 => tt1,,tt2) := Underlying_binoidal Types_omega_be.
+ Existing Instance Types_omega_binoidal.
+ *)
+
+ (* takes an "operation in the context" (proof from [b|=Top]/⋯⋯/[a|=Top]) and turns it into a function a-->b; note the variance *)
+ Definition context_operation_as_function
+ : forall {a}{b} (f:[b|=[]]~~{Judgments}~~>[a|=[]]), []~~{Judgments}~~>[a|=b].
+ intros.
+ apply (@al_vert_expand_suc_right _ _ _ b _ _) in f.
+ simpl in f.
+ apply (@al_vert_expand_ant_left _ _ _ [] _ _) in f.
+ simpl in f.
+ set ([#al_reflexive_seq _#] ;; f ;; [#al_ant_cancell#] ;; [#al_suc_cancell#]) as f'.
+ exact f'.
+ Defined.
+
+ (* takes an "operation in the context" (proof from [Top|=a]/⋯⋯/[Top|=b]) and turns it into a function a-->b; note the variance *)
+ Definition cocontext_operation_as_function
+ : forall {a}{b} (f:[[]|=a]~~{Judgments}~~>[[]|=b]), []~~{Judgments}~~>[a|=b].
+ intros. unfold hom. unfold hom in f.
+ apply al_vert_expand_ant_right with (x:=a) in f.
+ simpl in f.
+ apply al_vert_expand_suc_left with (x:=[]) in f.
+ simpl in f.
+ set ([#al_reflexive_seq _#] ;; f ;; [#al_ant_cancell#] ;; [#al_suc_cancell#]) as f'.
+ exact f'.
+ Defined.
+
+
+ Definition function_as_context_operation
+ : forall {a}{b}{c} (f:[]~~{Judgments}~~>[a|=b]), [b|=c]~~{Judgments}~~>[a|=c]
+ := fun a b c f => RepresentableFunctorºᑭ Types_omega_e c \ f.
+ Definition function_as_cocontext_operation
+ : forall {a}{b}{c} (f:[]/⋯⋯/[a|=b]), [c|=a]~~{Judgments}~~>[c|=b]
+ := fun a b c f => RepresentableFunctor Types_omega_e c \ f.
+
+ Close Scope temporary_scope4.
+ Close Scope al_scope.
+ Close Scope nd_scope.
+ Close Scope pf_scope.
+ Close Scope isomorphism_scope.
+End Acceptable_Language_Facts.
+
+Coercion AL_SurjectiveEnrichment : AcceptableLanguage >-> SurjectiveEnrichment.
+Coercion AL_MonicMonoidalEnrichment : AcceptableLanguage >-> MonicMonoidalEnrichment.
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