X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Ftopology%2Figft.ma;h=66c7dd3a14e9dd9811d43847bb3aef37d7c0d829;hb=8d321a03cf328b85fe7c084bb22685673633d2ee;hp=3c97b9c25000735971c09018c1a1f4c42a6e86a3;hpb=a8aaa095aad443c8eca8f64e3f22f54615e8dd9b;p=helm.git

diff --git a/helm/software/matita/nlibrary/topology/igft.ma b/helm/software/matita/nlibrary/topology/igft.ma
index 3c97b9c25..66c7dd3a1 100644
--- a/helm/software/matita/nlibrary/topology/igft.ma
+++ b/helm/software/matita/nlibrary/topology/igft.ma
@@ -12,12 +12,20 @@ the formalization of the paper
 
 by S.Berardi and S. Valentini. The tutorial is by Enrico Tassi. 
 
+The tutorial spends a considerable amount of effort in defining 
+notations that resemble the ones used in the original paper. We believe
+this a important part of every formalization, not only for the estetic 
+point of view, but also from the practical point of view. Being 
+consistent allows to follow the paper in a pedantic way, and hopefully
+to make the formalization (at least the definitions and proved
+statements) readable to the author of the paper. 
+
 Orientering
 -----------
 
 TODO 
 
-buttons, PG-interaction-model, sequent window, script window
+buttons, PG-interaction-model, sequent window, script window, ncheck
 
 The library, inclusion of `sets/sets.ma`, notation defined: Ω^A.
 Symbols (see menu: View ▹ TeX/UTF-8 Table):
@@ -40,6 +48,7 @@ some notation attached to them:
 
 - A ∪ B `A \cup B`
 - A ∩ B `A \cap B` 
+- A ≬ B `A \between B`
 - x ∈ A `x \in A` 
 - Ω^A, that is the type of the subsets of A, `\Omega ^ A` 
 
@@ -219,38 +228,186 @@ as subscript), separating them with a comma followed by a little space.
 The first (technical) definition
 --------------------------------
 
+Before defining the cover relation as an inductive predicate, one
+has to notice that the infinity rule uses, in its hypotheses, the 
+cover relation between two subsets, while the inductive predicate 
+we are going to define relates an element and a subset.
+
+An option would be to unfold the definition of cover between subsets,
+but we prefer to define the abstract notion of cover between subsets
+(so that we can attach a (ambiguous) notation to it).
+
+Anyway, to ease the understaing of the definition of the cover relation 
+between subsets, we first define the inductive predicate unfolding the 
+definition, and we later refine it with.
+
+DOCEND*)
+
+ninductive xcover (A : Ax) (U : Ω^A) : A → CProp[0] ≝ 
+| xcreflexivity : ∀a:A. a ∈ U → xcover A U a
+| xcinfinity    : ∀a:A.∀i:𝐈 a. (∀y.y ∈ 𝐂 a i → xcover A U y) → xcover A U a.
+
+(*DOCBEGIN
+
+We defined the xcover (x will be removed in the final version of the 
+definition) as an inductive predicate. The arity of the inductive
+predicate has to be carefully analyzed:
+
+>  (A : Ax) (U : Ω^A) : A → CProp[0]
+
+The syntax separates with `:` abstractions that are fixed for every
+constructor (introduction rule) and abstractions that can change. In that 
+case the parameter `U` is abstracted once and forall in front of every 
+constructor, and every occurrence of the inductive predicate is applied to
+`U` in a consistent way. Arguments abstracted on the right of `:` are not
+constant, for example the xcinfinity constructor introduces `a ◃ U`,
+but under the assumption that (for every y) `y ◃ U`. In that rule, the left
+had side of the predicate changes, thus it has to be abstrated (in the arity
+of the inductive predicate) on the right of `:`.
 
+DOCEND*)
+
+(* ncheck xcreflexivity. *)
+
+(*DOCBEGIN
+
+We want now to abstract out `(∀y.y ∈ 𝐂 a i → xcover A U y)` and define
+a notion `cover_set` to which a notation `𝐂 a i ◃ U` can be attached.
+
+This notion has to be abstracted over the cover relation (whose
+type is the arity of the inductive `xcover` predicate just defined).
+
+Then it has to be abstracted over the arguments of that cover relation,
+i.e. the axiom set and the set U, and the subset (in that case `𝐂 a i`)
+sitting on the left hand side of `◃`. 
 
 DOCEND*)
 
 ndefinition cover_set : 
-  ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0] 
+  ∀cover: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0] 
 ≝ 
-  λc: ∀A:Ax.Ω^A → A → CProp[0]. λA,C,U.∀y.y ∈ C → c A U y.
+  λcover.                           λA,    C,U.     ∀y.y ∈ C → cover A U y.
 
-ndefinition cover_set_interactive : 
-  ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0]. 
-#cover; #A; #C; #U; napply (∀y:A.y ∈ C → ?); napply cover;
-##[ napply A;
-##| napply U;
-##| napply y;
-##]
-nqed.
+(*DOCBEGIN
+
+The `ndefinition` command takes a name, a type and body (of that type).
+The type can be omitted, and in that case it is inferred by the system.
+If the type is given, the system uses it to infer implicit arguments
+of the body. In that case all types are left implicit in the body.
+
+We now define the notation `a ◃ b`. Here the keywork `hvbox`
+and `break` tell the system how to wrap text when it does not
+fit the screen (they can be safely ignore for the scope of
+this tutorial). we also add an interpretation for that notation, 
+where the (abstracted) cover relation is implicit. The system
+will not be able to infer it from the other arguments `C` and `U`
+and will thus prompt the user for it. This is also why we named this 
+interpretation `covers set temp`: we will later define another 
+interpretation in which the cover relation is the one we are going to 
+define.
+
+DOCEND*)
 
-(* a \ltri b *)
 notation "hvbox(a break ◃ b)" non associative with precedence 45
 for @{ 'covers $a $b }.
 
 interpretation "covers set temp" 'covers C U = (cover_set ?? C U).
 
+(*DOCBEGIN
+
+We can now define the cover relation using the `◃` notation for 
+the premise of infinity. 
+
+DOCEND*)
+
 ninductive cover (A : Ax) (U : Ω^A) : A → CProp[0] ≝ 
-| creflexivity : ∀a:A. a ∈ U → cover ? U a
-| cinfinity    : ∀a:A.∀i:𝐈 a. 𝐂 a i ◃ U → cover ? U a.
+| creflexivity : ∀a. a ∈ U → cover ? U a
+| cinfinity    : ∀a. ∀i. 𝐂 a i ◃ U → cover ? U a.
+(** screenshot "cover". *) 
 napply cover;
 nqed.
 
+(*DOCBEGIN
+
+Note that the system accepts the definition
+but prompts the user for the relation the `cover_set` notion is
+abstracted on.
+
+![The system asks for a cover relation][cover]
+
+The orizontal line separates the hypotheses from the conclusion.
+The `napply cover` command tells the system that the relation
+it is looking for is exactly our first context entry (i.e. the inductive
+predicate we are defining, up to α-conversion); while the `nqed` command
+ends a definition or proof.
+
+We can now define the interpretation for the cover relation between an
+element and a subset fist, then between two subsets (but this time
+we fixed the relation `cover_set` is abstracted on).
+
+DOCEND*)
+
 interpretation "covers" 'covers a U = (cover ? U a).
-(* interpretation "covers set" 'covers a U = (cover_set cover ? a U). *)
+interpretation "covers set" 'covers a U = (cover_set cover ? a U).
+
+(*DOCBEGIN
+
+We will proceed similarly for the fish relation, but before going
+on it is better to give a short introduction to the proof mode of Matita.
+We define again the `cover_set` term, but this time we will build
+its body interactively. In λ-calculus Matita is based on, CIC, proofs
+and terms share the same syntax, and it thus possible to use the
+commands devoted to build proof term to build regular definitions.
+
+DOCEND*)
+
+
+ndefinition xcover_set : 
+  ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0]. 
+(** screenshot "xcover-set-1". *)
+(*DOCBEGIN
+The system asks for a proof of the full statement, in an empty context.
+![xcover_set proof step ][xcover-set-1]
+The `#` command in the ∀-introduction rule, it gives a name to an 
+assumption putting it in the context, and generates a λ-abstraction
+in the proof term.
+DOCEND*)
+#cover; #A; #C; #U; (** screenshot "xcover-set-2". *) 
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-2]
+We have now to provide a proposition, and we exhibit it. We left
+a part of it implicit; since the system cannot infer it it will
+ask it later. Note that the type of `∀y:A.y ∈ C → ?` is a proposition
+whenever `?` is.
+DOCEND*)
+napply (∀y:A.y ∈ C → ?); (** screenshot "xcover-set-3". *)
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-3]
+The proposition we want to provide is an application of the
+cover relation we have abstracted in the context. The command
+`napply`, if the given term has not the expected type (in that
+case it is a product versus a proposition) it applies it to as many 
+implicit arguments as necessary (in that case `? ? ?`).
+DOCEND*)
+napply cover; (** screenshot "xcover-set-4". *)
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-4]
+The system will now ask in turn the three implicit arguments 
+passed to cover. The syntax `##[` allows to start a branching
+to tackle every sub proof individually, otherwise every command
+is applied to every subrpoof. The command `##|` switches to the next
+subproof and `##]` ends the branching.  
+DOCEND*)
+##[ napply A;
+##| napply U;
+##| napply y;
+##]
+nqed.
+
+(*DOCBEGIN
+The definition of fish works exactly the same way as for cover, except 
+that it is defined as a coinductive proposition.
+DOCEND*)
 
 ndefinition fish_set ≝ λf:∀A:Ax.Ω^A → A → CProp[0].
  λA,U,V.
@@ -270,22 +427,96 @@ nqed.
 interpretation "fish set" 'fish A U = (fish_set fish ? U A).
 interpretation "fish" 'fish a U = (fish ? U a).
 
+(*DOCBEGIN
+
+Matita is able to generate elimination rules for inductive types,
+but not introduction rules for the coinductive case. 
+
+DOCEND*)
+
+(* ncheck cover_rect_CProp0. *) 
+
+(*DOCBEGIN
+
+We thus have to define the introduction rule for fish by corecursion.
+Here we again use the proof mode of Matita to exhibit the body of the
+corecursive function.
+
+DOCEND*)
+
 nlet corec fish_rec (A:Ax) (U: Ω^A)
  (P: Ω^A) (H1: P ⊆ U)
-  (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P):
-   ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
-#a; #p; napply cfish; (** screenshot "def-fish-rec". *)
-##[ napply H1; napply p;
-##| #i; ncases (H2 a p i); #x; *; #xC; #xP; @; ##[napply x]
-    @; ##[ napply xC ] napply (fish_rec ? U P); nassumption;
+  (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P): ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
+(** screenshot "def-fish-rec-1". *)
+(*DOCBEGIN
+![fish proof step][def-fish-rec-1]
+Note the first item of the context, it is the corecursive function we are 
+defining. This item allows to perform the recursive call, but we will be
+allowed to do such call only after having generated a constructor of
+the fish coinductive type.
+
+We introduce `a` and `p`, and then return the fish constructor `cfish`.
+Since the constructor accepts two arguments, the system asks for them.
+DOCEND*)
+#a; #p; napply cfish; (** screenshot "def-fish-rec-2". *)
+(*DOCBEGIN
+![fish proof step][def-fish-rec-2]
+The first one is a proof that `a ∈ U`. This can be proved using `H1` and `p`.
+With the `nchange` tactic we change `H1` into an equivalent form (this step
+can be skipped, since the systeem would be able to unfold the definition
+of inclusion by itself)
+DOCEND*) 
+##[ nchange in H1 with (∀b.b∈P → b∈U);
+
+    (** screenshot "def-fish-rec-2-1". *) napply H1;
+    (** screenshot "def-fish-rec-3". *) nassumption;
+(*DOCBEGIN
+![fish proof step][def-fish-rec-2-1]
+It is now clear that `H1` can be applied. Again `napply` adds two 
+implicit arguments to `H1 ? ?`, obtaining a proof of `? ∈ U` given a proof
+that `? ∈ P`. Thanks to unification, the system understands that `?` is actually
+`a`, and it asks a proof that `a ∈ P`.
+![fish proof step][def-fish-rec-3]
+The `nassumption` tactic looks for the required proof in the context, and in
+that cases finds it in the last context position. 
+
+We move now to the second branch of the proof, corresponding to the second
+argument of the `cfish` constructor.
+![fish proof step][def-fish-rec-4]
+DOCEND*)
+##| (** screenshot "def-fish-rec-4". *) #i; ncases (H2 a p i); 
+    (** screenshot "def-fish-rec-5". *) #x; *; #xC; #xP;
+    (** screenshot "def-fish-rec-5-1". *) @; 
+    ##[ (** screenshot "def-fish-rec-6". *) napply x
+    ##| (** screenshot "def-fish-rec-7". *)
+        @; ##[ napply xC; 
+           ##| (** screenshot "def-fish-rec-8". *) 
+               napply (fish_rec ? U P); 
+               (** screenshot "def-fish-rec-9". *)
+               nassumption;
+           ##]
+    ##]
 ##]
 nqed.
-
-notation "◃U" non associative with precedence 55
-for @{ 'coverage $U }.
+(*DOCBEGIN
+We introduce `i` and then we destruct `H2 a p i`, that being a proof
+of an overlap predicate, give as an element and a proof that it is 
+both in `𝐂 a i` and `P`.
+![fish proof step][def-fish-rec-5]
+We then introduce `x`, break the conjunction (the `*;` command is the
+equivalent of `ncases` but operates on the first hypothesis that can
+be introduced. We then introduce the two sides of the conjuction.
+![fish proof step][def-fish-rec-5-1]
+![fish proof step][def-fish-rec-6]
+![fish proof step][def-fish-rec-7]
+![fish proof step][def-fish-rec-8]
+![fish proof step][def-fish-rec-9]
+DOCEND*)
 
 ndefinition coverage : ∀A:Ax.∀U:Ω^A.Ω^A ≝ λA,U.{ a | a ◃ U }.
 
+notation "◃U" non associative with precedence 55 for @{ 'coverage $U }.
+
 interpretation "coverage cover" 'coverage U = (coverage ? U).
 
 ndefinition cover_equation : ∀A:Ax.∀U,X:Ω^A.CProp[0] ≝  λA,U,X. 
@@ -430,7 +661,6 @@ naxiom setoidification :
 
 Bla Bla, 
 
-<div id="figure1" class="img" ><img src="figure1.png"/>foo</div>
 
 DOCEND*)