X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FProgrammingLanguage.v;fp=src%2FProgrammingLanguage.v;h=073dfef0d36e64c0769d6311bd60958a6511faa8;hp=0000000000000000000000000000000000000000;hb=85e4f0fd6b0673c1cc763eeb2585b7dc3d388455;hpb=70939a4eb9560ceeea3e9cf176ac5a36f9201ac4 diff --git a/src/ProgrammingLanguage.v b/src/ProgrammingLanguage.v new file mode 100644 index 0000000..073dfef --- /dev/null +++ b/src/ProgrammingLanguage.v @@ -0,0 +1,766 @@ +(*********************************************************************************************************************************) +(* 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.