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=50a62bff4ec049e749c5429ef2b16707bb44ddfd;hpb=4286d368b6d902a4b54c4cf8590c387f2cdb90ea;p=helm.git

diff --git a/helm/software/matita/nlibrary/topology/igft.ma b/helm/software/matita/nlibrary/topology/igft.ma
index 50a62bff4..66c7dd3a1 100644
--- a/helm/software/matita/nlibrary/topology/igft.ma
+++ b/helm/software/matita/nlibrary/topology/igft.ma
@@ -3,88 +3,412 @@
 Matita Tutorial: inductively generated formal topologies
 ======================================================== 
 
-Small intro...
+This is a not so short introduction to Matita, based on
+the formalization of the paper
 
-Initial setup
--------------
+> Between formal topology and game theory: an
+> explicit solution for the conditions for an
+> inductive generation of formal topologies
+
+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, ncheck
 
 The library, inclusion of `sets/sets.ma`, notation defined: Ω^A.
 Symbols (see menu: View ▹ TeX/UTF-8 Table):
 
-- `Ω` can be typed \Omega 
-- `∀` \Forall
-- `λ` \lambda
-- `≝` \def
-- `→` ->
+- ∀ `\Forall`
+- λ `\lambda`
+- ≝ `\def`
+- → `->`
+
+Virtuals, ALT-L, for example `I` changes into `𝕀`, finally `𝐈`.
+
+The standard library and the `include` command
+----------------------------------------------
+
+Some basic notions, like subset, membership, intersection and union
+are part of the standard library of Matita.
+
+These notions come with
+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` 
 
-Virtuals, ALT-L, for example I changes into 𝕀, finally 𝐈.
+The `include` command tells Matita to load a part of the library, 
+in particular the part that we will use can be loaded as follows: 
 
 DOCEND*)
 
 include "sets/sets.ma".
 
+(*HIDE*)
+(* move away *)
+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
+nqed.
+
+nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
+nqed. 
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
+nqed. 
+(*UNHIDE*)
+
 (*DOCBEGIN
 
-Axiom set
----------
+Some basic results that we will use are also part of the sets library:
+
+- subseteq\_union\_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W
+- subseteq\_intersection\_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V
 
-records, ...
+Defining Axiom set
+------------------
+
+records, projections, types of projections..
 
 DOCEND*)
 
 nrecord Ax : Type[1] ≝ { 
-  S:> setoid;
-  I: S → Type[0];
-  C: ∀a:S. I a → Ω ^ S
+  S :> setoid;
+  I :  S → Type[0];
+  C :  ∀a:S. I a → Ω ^ S
 }.
 
-(*HIDE*)
-notation "𝐈  \sub( ❨a❩ )" non associative with precedence 70 for @{ 'I $a }.
-notation "𝐂 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'C $a $i }.
-(*UNHIDE*)
+(*DOCBEGIN
+
+Note that the field `S` was declared with `:>` instead of a simple `:`.
+This declares the `S` projection to be a coercion. A coercion is 
+a function case the system automatically inserts when it is needed.
+In that case, the projection `S` has type `Ax → setoid`, and whenever
+the expected type of a term is `setoid` while its type is `Ax`, the 
+system inserts the coercion around it, to make the whole term well types.
+
+When formalizing an algebraic structure, declaring the carrier as a 
+coercion is a common practice, since it allows to write statements like
+
+    ∀G:Group.∀x:G.x * x^-1 = 1 
+
+The quantification over `x` of type `G` is illtyped, since `G` is a term
+(of type `Group`) and thus not a type. Since the carrier projection 
+`carr` of `G` is a coercion, that maps a `Group` into the type of 
+its elements, the system automatically inserts `carr` around `G`, 
+obtaining `…∀x: carr G.…`. Coercions are also hidden by the system
+when it displays a term.
+
+In this particular case, the coercion `S` allows to write
+
+    ∀A:Ax.∀a:A.…
+
+Since `A` is not a type, but it can be turned into a `setoid` by `S`
+and a `setoid` can be turned into a type by its `carr` projection, the 
+composed coercion `carr ∘ S` is silently inserted.
+
+Implicit arguments
+------------------
+
+Something that is not still satisfactory, in that the dependent type
+of `I` and `C` are abstracted over the Axiom set. To obtain the
+precise type of a term, you can use the `ncheck` command as follows.
+
+DOCEND*) 
+
+(* ncheck I. *)
+(* ncheck C. *)
 
 (*DOCBEGIN
 
-Notation for the axiom set
---------------------------
+One would like to write `I a` and not `I A a` under a context where
+`A` is an axiom set and `a` has type `S A` (or thanks to the coercion
+mecanism simply `A`). In Matita, a question mark represents an implicit
+argument, i.e. a missing piece of information the system is asked to
+infer. Matita performs some sort of type inference, thus writing
+`I ? a` is enough: since the second argument of `I` is typed by the 
+first one, the first one can be inferred just computing the type of `a`.
 
-bla bla
+DOCEND*) 
+
+(* ncheck (∀A:Ax.∀a:A.I ? a). *)
+
+(*DOCBEGIN
+
+This is still not completely satisfactory, since you have always type 
+`?`; to fix this minor issue we have to introduce the notational
+support built in Matita.
+
+Notation for I and C
+--------------------
+
+Matita is quipped with a quite complex notational support,
+allowing the user to define and use mathematical notations 
+([From Notation to Semantics: There and Back Again][1]). 
+
+Since notations are usually ambiguous (e.g. the frequent overloading of 
+symbols) Matita distinguishes between the term level, the 
+content level, and the presentation level, allowing multiple 
+mappings between the contenet and the term level. 
+
+The mapping between the presentation level (i.e. what is typed on the 
+keyboard and what is displayed in the sequent window) and the content
+level is defined with the `notation` command. When followed by
+`>`, it defines an input (only) notation.   
 
 DOCEND*)
 
 notation > "𝐈 term 90 a" non associative with precedence 70 for @{ 'I $a }.
 notation > "𝐂 term 90 a term 90 i" non associative with precedence 70 for @{ 'C $a $i }.
 
+(*DOCBEGIN
+
+The forst notation defines the writing `𝐈 a` where `a` is a generic
+term of precedence 90, the maximum one. This high precedence forces
+parentheses around any term of a lower precedence. For example `𝐈 x`
+would be accepted, since identifiers have precedence 90, but
+`𝐈 f x` would be interpreted as `(𝐈 f) x`. In the latter case, parentheses
+have to be put around `f x`, thus the accepted writing would be `𝐈 (f x)`.
+
+To obtain the `𝐈` is enough to type `I` and then cycle between its
+similar symbols with ALT-L. The same for `𝐂`. Notations cannot use
+regular letters or the round parentheses, thus their variants (like the 
+bold ones) have to be used.
+
+The first notation associates `𝐈 a` with `'I $a` where `'I` is a 
+new content element to which a term `$a` is passed.
+
+Content elements have to be interpreted, and possibly multiple, 
+incompatible, interpretations can be defined.
+
+DOCEND*)
+
 interpretation "I" 'I a = (I ? a).
 interpretation "C" 'C a i = (C ? a i).
 
 (*DOCBEGIN
 
-The first definition
---------------------
+The `interpretation` command allows to define the mapping between
+the content level and the terms level. Here we associate the `I` and
+`C` projections of the Axiom set record, where the Axiom set is an implicit 
+argument `?` to be inferred by the system.
+
+Interpretation are bi-directional, thus when displaying a term like 
+`C _ a i`, the system looks for a presentation for the content element
+`'C a i`. 
+
+DOCEND*)
+
+notation < "𝐈  \sub( ❨a❩ )" non associative with precedence 70 for @{ 'I $a }.
+notation < "𝐂 \sub( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'C $a $i }.
+
+(*DOCBEGIN
+
+For output purposes we can define more complex notations, for example
+we can put bold parenteses around the arguments of `𝐈` and `𝐂`, decreasing
+the size of the arguments and lowering their baseline (i.e. putting them
+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:
 
-![bla bla][def-fish-rec]
+>  (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*)
 
-ndefinition cover_set ≝ λc:∀A:Ax.Ω^A → A → CProp[0].λA,C,U.
-  ∀y.y ∈ C → c A U y.
+(* 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 : 
+  ∀cover: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0] 
+≝ 
+  λcover.                           λA,    C,U.     ∀y.y ∈ C → cover A U y.
+
+(*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).
 
+(*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.
   ∃a.a ∈ V ∧ f A U a.
@@ -103,22 +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. 
@@ -263,7 +661,6 @@ naxiom setoidification :
 
 Bla Bla, 
 
-<div id="figure1" class="img" ><img src="figure1.png"/>foo</div>
 
 DOCEND*)
 
@@ -274,9 +671,11 @@ alias symbol "covers" = "new covers set".
 alias symbol "covers" = "new covers".
 alias symbol "covers" = "new covers set".
 alias symbol "covers" = "new covers".
+alias symbol "covers" = "new covers set".
+alias symbol "covers" = "new covers".
 ntheorem new_coverage_infinity:
   ∀A:nAx.∀U:Ω^A.∀a:A. (∃i:𝐈 a. 𝐈𝐦[𝐝 a i] ◃ U) → a ◃ U.
-#A; #U; #a;(** screenshot "topology/figure1". *)  
+#A; #U; #a;(** screenshot "figure1". *)  
 *; #i; #H; nnormalize in H;
 ncut (∀y:𝐃 a i.∃x:Ord A.𝐝 a i y ∈ U⎽x); ##[
   #y; napply H; @ y; napply #; ##] #H'; 
@@ -287,17 +686,12 @@ ncut (∀j.𝐝 a i j ∈ U⎽(Λ f));
 napply (setoidification … Hd); napply Hf';
 nqed.
 
-(* move away *)
-nlemma subseteq_union: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
-#A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
-nqed. 
-
 nlemma new_coverage_min :  
   ∀A:nAx.∀U:qpowerclass A.∀V.U ⊆ V → (∀a:A.∀i.𝐈𝐦[𝐝 a i] ⊆ V → a ∈ V) → ◃(pc ? U) ⊆ V.
 #A; #U; #V; #HUV; #Im;  #b; *; #o; ngeneralize in match b; nchange with ((pc ? U)⎽o ⊆ V);
 nelim o;
 ##[ #b; #bU0; napply HUV; napply bU0;
-##| #p; #IH; napply subseteq_union; ##[ nassumption; ##]
+##| #p; #IH; napply subseteq_union_l; ##[ nassumption; ##]
     #x; *; #i; #H; napply (Im ? i); napply (subseteq_trans … IH); napply H;
 ##| #a; #i; #f; #IH; #x; *; #d; napply IH; ##]
 nqed.
@@ -341,15 +735,6 @@ ncut (∃j:𝐃 a i.𝐝 a i j ∈ F⎽(f j));##[
 napply aLf';
 nqed.
 
-(* move away *)
-nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
-#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
-nqed.
-
-nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
-#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
-nqed. 
- 
 ntheorem max_new_fished: 
   ∀A:nAx.∀G,F:Ω^A.G ⊆ F → (∀a.a ∈ G → ∀i.𝐈𝐦[𝐝 a i] ≬ G) → G ⊆ ⋉F.
 #A; #G; #F; #GF; #H; #b; #HbG; #o; ngeneralize in match HbG; ngeneralize in match b;
@@ -364,3 +749,8 @@ nelim o;
 ##]
 nqed.
 
+(*DOCBEGIN
+
+[1]: http://upsilon.cc/~zack/research/publications/notation.pdf 
+
+*)
\ No newline at end of file