-(*DOCBEGIN
+(*D
Matita Tutorial: inductively generated formal topologies
========================================================
-This is a not so short introduction to Matita, based on
+This is a not so short introduction to [Matita][2], based on
the formalization of the paper
> 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.
-
-Orientering
------------
-
-TODO
-
-buttons, PG-interaction-model, sequent window, script window
-
-The library, inclusion of `sets/sets.ma`, notation defined: Ω^A.
-Symbols (see menu: View ▹ TeX/UTF-8 Table):
-
-- ∀ `\Forall`
-- λ `\lambda`
-- ≝ `\def`
-- → `->`
+by Stefano Berardi and Silvio 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 is a important part of every formalization, not only from the aesthetic
+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.
+
+The formalization uses the ng (new generation) version of Matita
+(that will be named 1.x when finally released).
+Last stable release of the "old" system is named 0.5.7; the ng system
+is coexisting with the old one in every development release
+(named "nightly builds" in the download page of Matita)
+with a version strictly greater than 0.5.7.
+
+To read this tutorial in HTML format, you need a decent browser
+equipped with a unicode capable font. Use the PDF format if some
+symbols are not displayed correctly.
+
+Orienteering
+------------
+
+The graphical interface of Matita is composed of three windows:
+the script window, on the left, is where you type; the sequent
+window on the top right is where the system shows you the ongoing proof;
+the error window, on the bottom right, is where the system complains.
+On the top of the script window five buttons drive the processing of
+the proof script. From left to right they request the system to:
+
+- go back to the beginning of the script
+- go back one step
+- go to the current cursor position
+- advance one step
+- advance to the end of the script
+
+When the system processes a command, it locks the part of the script
+corresponding to the command, such that you cannot edit it anymore
+(without going back). Locked parts are coloured in blue.
+
+The sequent window is hyper textual, i.e. you can click on symbols
+to jump to their definition, or switch between different notations
+for the same expression (for example, equality has two notations,
+one of them makes the type of the arguments explicit).
+
+Everywhere in the script you can use the `ncheck (term).` command to
+ask for the type a given term. If you do that in the middle of a proof,
+the term is assumed to live in the current proof context (i.e. can use
+variables introduced so far).
+
+To ease the typing of mathematical symbols, the script window
+implements two unusual input facilities:
+
+- some TeX symbols can be typed using their TeX names, and are
+ automatically converted to UTF-8 characters. For a list of
+ the supported TeX names, see the menu: View ▹ TeX/UTF-8 Table.
+ Moreover some ASCII-art is understood as well, like `=>` and `->`
+ to mean double or single arrows.
+ Here we recall some of these "shortcuts":
+
+ - ∀ can be typed with `\forall`
+ - λ can be typed with `\lambda`
+ - ≝ can be typed with `\def` or `:=`
+ - → can be typed with `\to` or `->`
+
+- some symbols have variants, like the ≤ relation and ≼, ≰, ⋠.
+ The user can cycle between variants typing one of them and then
+ pressing ALT-L. Note that also letters do have variants, for
+ example W has Ω, 𝕎 and 𝐖, L has Λ, 𝕃, and 𝐋, F has Φ, …
+ Variants are listed in the aforementioned TeX/UTF-8 table.
+
+The syntax of terms (and types) is the one of the λ-calculus CIC
+on which Matita is based. The main syntactical difference w.r.t.
+the usual mathematical notation is the function application, written
+`(f x y)` in place of `f(x,y)`.
+
+Pressing `F1` opens the Matita manual.
+
+CIC (as [implemented in Matita][3]) in a nutshell
+-------------------------------------------------
+
+CIC is a full and functional Pure Type System (all products do exist,
+and their sort is is determined by the target) with an impredicative sort
+Prop and a predicative sort Type. It features both dependent types and
+polymorphism like the [Calculus of Constructions][4]. Proofs and terms share
+the same syntax, and they can occur in types.
+
+The environment used for in the typing judgement can be populated with
+well typed definitions or theorems, (co)inductive types validating positivity
+conditions and recursive functions provably total by simple syntactical
+analysis (recursive calls are allowed only on structurally smaller subterms).
+Co-recursive
+functions can be defined as well, and must satisfy the dual condition, i.e.
+performing the recursive call only after having generated a constructor (a piece
+of output).
+
+The CIC λ-calculus is equipped with a pattern matching construct (match) on inductive
+types defined in the environment. This construct, together with the possibility to
+definable total recursive functions, allows to define eliminators (or constructors)
+for (co)inductive types. The λ-calculus is also equipped with explicitly typed
+local definitions (let in) that in the degenerate case work as casts (i.e.
+the type annotation `(t : T)` is implemented as `let x : T ≝ t in x`).
+
+Types are compare up to conversion. Since types may depend on terms, conversion
+involves β-reduction, δ-reduction (definition unfolding), ζ-reduction (local
+definition unfolding), ι-reduction (pattern matching simplification),
+μ-reduction (recursive function computation) and ν-reduction (co-fixpoint
+computation).
+
+Since we are going to formalize constructive and predicative mathematics
+in an intensional type theory like CIC, we try to establish some terminology.
+Type is the sort of sets equipped with the `Id` equality (i.e. an intensional,
+not quotiented set). We will avoid using `Id` (Leibniz equality),
+thus we will explicitly equip a set with an equivalence relation when needed.
+We will call this structure a _setoid_. Note that we will
+attach the infix `=` symbol only to the equality of a setoid,
+not to Id.
+
+We write `Type[i]` to mention a Type in the predicative hierarchy
+of types. To ease the comprehension we will use `Type[0]` for sets,
+and `Type[1]` for classes. The index `i` is just a label: constraints among
+universes are declared by the user. The standard library defines
+
+> Type[0] < Type[1] < Type[2]
+
+For every `Type[i]` there is a corresponding level of predicative
+propositions `CProp[i]`. A predicative proposition cannot be eliminated toward
+`Type[j]` unless it holds no computational content (i.e. it is an inductive type
+with 0 or 1 constructors with propositional arguments, like `Id` and `And`
+but not like `Or`).
-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:
+These notions come with some standard notation attached to them:
-- A ∪ B `A \cup B`
-- A ∩ B `A \cap B`
-- x ∈ A `x \in A`
-- Ω^A, that is the type of the subsets of A, `\Omega ^ A`
+- A ∪ B can be typed with `A \cup B`
+- A ∩ B can be typed with `A \cap B`
+- A ≬ B can be typed with `A \between B`
+- x ∈ A can be typed with `x \in A`
+- Ω^A, that is the type of the subsets of A, can be typed with `\Omega ^ A`
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*)
+D*)
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
+(*D
Some basic results that we will use are also part of the sets library:
Defining Axiom set
------------------
-records, projections, types of projections..
+A set of axioms is made of a set(oid) `S`, a family of sets `I` and a
+family `C` of subsets of `S` indexed by elements `a` of `S`
+and elements of `I(a)`.
+
+It is desirable to state theorems like "for every set of axioms, …"
+without explicitly mentioning S, I and C. To do that, the three
+components have to be grouped into a record (essentially a dependently
+typed tuple). The system is able to generate the projections
+of the record automatically, and they are named as the fields of
+the record. So, given an axiom set `A` we can obtain the set
+with `S A`, the family of sets with `I A` and the family of subsets
+with `C A`.
-DOCEND*)
+D*)
nrecord Ax : Type[1] ≝ {
S :> setoid;
C : ∀a:S. I a → Ω ^ S
}.
-(*DOCBEGIN
+(*D
+
+Forget for a moment the `:>` that will be detailed later, and focus on
+the record definition. It is made of a list of pairs: a name, followed
+by `:` and the its type. It is a dependently typed tuple, thus
+already defined names (fields) can be used in the types that follow.
+
+Note that `S` is declared to be a `setoid` and not a Type. The original
+paper probably also considers I to generate setoids, and both I and C
+to be (dependent) morphisms. For the sake of simplicity, we will "cheat" and use
+setoids only when strictly needed (i.e. where we want to talk about
+equality). Setoids will play a role only when we will define
+the alternative version of the axiom set.
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.
+a "cast" function 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.
+system inserts the coercion around it, to make the whole term well typed.
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
+The quantification over `x` of type `G` is ill-typed, 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
+`carr` 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.
+obtaining `…∀x: carr G.…`.
-In this particular case, the coercion `S` allows to write
+Coercions are hidden by the system when it displays a term.
+In this particular case, the coercion `S` allows to write (and read):
∀A:Ax.∀a:A.…
Implicit arguments
------------------
-Something that is not still satisfactory, in that the dependent type
+Something that is not still satisfactory, is 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*)
+D*)
-(* ncheck I. *)
-(* ncheck C. *)
+(** ncheck I. *) (* shows: ∀A:Ax.A → Type[0] *)
+(** ncheck C. *) (* shows: ∀A:Ax.∀a:A.A → I A a → Ω^A *)
-(*DOCBEGIN
+(*D
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
+mechanism 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
+infer. Matita performs Hindley-Milner-style 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`.
+first one, the first (omitted) argument can be inferred just
+computing the type of `a` (that is `A`).
-DOCEND*)
+D*)
-(* ncheck (∀A:Ax.∀a:A.I ? a). *)
+(** ncheck (∀A:Ax.∀a:A.I ? a). *) (* shows: ∀A:Ax.∀a:A.I A a *)
-(*DOCBEGIN
+(*D
-This is still not completely satisfactory, since you have always type
+This is still not completely satisfactory, since you have always to type
`?`; to fix this minor issue we have to introduce the notational
support built in Matita.
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.
+mappings between the content 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*)
+D*)
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
+(*D
-The forst notation defines the writing `𝐈 a` where `a` is a generic
+The first 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
Content elements have to be interpreted, and possibly multiple,
incompatible, interpretations can be defined.
-DOCEND*)
+D*)
interpretation "I" 'I a = (I ? a).
interpretation "C" 'C a i = (C ? a i).
-(*DOCBEGIN
+(*D
The `interpretation` command allows to define the mapping between
the content level and the terms level. Here we associate the `I` and
`C _ a i`, the system looks for a presentation for the content element
`'C a i`.
-DOCEND*)
+D*)
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
+(*D
For output purposes we can define more complex notations, for example
-we can put bold parenteses around the arguments of `𝐈` and `𝐂`, decreasing
+we can put bold parentheses 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 understanding of the definition of the cover relation
+between subsets, we first define the inductive predicate unfolding the
+definition, and we later refine it with.
+
+D*)
+
+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.
+
+(*D
+
+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 for all 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 abstracted (in the arity
+of the inductive predicate) on the right of `:`.
+
+D*)
+
+(** ncheck xcreflexivity. *) (* shows: ∀A:Ax.∀U:Ω^A.∀a:A.a∈U → xcover A U a *)
+
+(*D
+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.
-DOCEND*)
+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 `◃`.
+
+D*)
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.
+(*D
+
+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 ignored 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.
+
+D*)
-(* 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).
+(*D
+
+The cover relation
+------------------
+
+We can now define the cover relation using the `◃` notation for
+the premise of infinity.
+
+D*)
+
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.
+(*D
+
+Note that the system accepts the definition
+but prompts the user for the relation the `cover_set` notion is
+abstracted on.
+
+
+
+The horizontal 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 first, then between two subsets (but this time
+we fix the relation `cover_set` is abstracted on).
+
+D*)
+
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).
+
+(*D
+
+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 build
+its body interactively. In the λ-calculus Matita is based on, CIC, proofs
+and terms share the same syntax, and it is thus possible to use the
+commands devoted to build proof term also to build regular definitions.
+A tentative semantics for the proof mode commands (called tactics)
+in terms of sequent calculus rules are given in the
+<a href="#appendix">appendix</a>.
+
+D*)
+
+ndefinition xcover_set :
+ ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0].
+ (** screenshot "xcover-set-1". *)
+#cover; #A; #C; #U; (** screenshot "xcover-set-2". *)
+napply (∀y:A.y ∈ C → ?); (** screenshot "xcover-set-3". *)
+napply cover; (** screenshot "xcover-set-4". *)
+##[ napply A;
+##| napply U;
+##| napply y;
+##]
+nqed.
+
+(*D[xcover-set-1]
+The system asks for a proof of the full statement, in an empty context.
+
+The `#` command is the ∀-introduction rule, it gives a name to an
+assumption putting it in the context, and generates a λ-abstraction
+in the proof term.
+
+D[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 for it later.
+Note that the type of `∀y:A.y ∈ C → ?` is a proposition
+whenever `?` is a proposition.
+
+D[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 `? ? ?`).
+
+D[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 subproof. The command `##|` switches to the next
+subproof and `##]` ends the branching.
+D*)
+
+(*D
+
+The fish relation
+-----------------
+
+The definition of fish works exactly the same way as for cover, except
+that it is defined as a coinductive proposition.
+D*)
ndefinition fish_set ≝ λf:∀A:Ax.Ω^A → A → CProp[0].
λA,U,V.
interpretation "fish set" 'fish A U = (fish_set fish ? U A).
interpretation "fish" 'fish a U = (fish ? U a).
+(*D
+
+Introduction rule for fish
+---------------------------
+
+Matita is able to generate elimination rules for inductive types,
+but not introduction rules for the coinductive case.
+
+D*)
+
+(** ncheck cover_rect_CProp0. *)
+
+(*D
+
+We thus have to define the introduction rule for fish by co-recursion.
+Here we again use the proof mode of Matita to exhibit the body of the
+corecursive function.
+
+D*)
+
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". *)
+#a; #p; napply cfish; (** screenshot "def-fish-rec-2". *)
+##[ nchange in H1 with (∀b.b∈P → b∈U); (** screenshot "def-fish-rec-2-1". *)
+ napply H1; (** screenshot "def-fish-rec-3". *)
+ nassumption;
+##| #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;
+ ##| napply (fish_rec ? U P); (** screenshot "def-fish-rec-9". *)
+ nassumption;
+ ##]
+ ##]
##]
nqed.
-notation "◃U" non associative with precedence 55
-for @{ 'coverage $U }.
+(*D
+D[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.
+
+D[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 system would be able to unfold the definition
+of inclusion by itself)
+
+D[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`.
+
+D[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.
+
+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`.
+
+D[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 conjunction.
+
+D[def-fish-rec-5-1]
+The goal is now the existence of a point in `𝐂 a i` fished by `U`.
+We thus need to use the introduction rule for the existential quantifier.
+In CIC it is a defined notion, that is an inductive type with just
+one constructor (one introduction rule) holding the witness and the proof
+that the witness satisfies a proposition.
+
+> ncheck Ex.
+
+Instead of trying to remember the name of the constructor, that should
+be used as the argument of `napply`, we can ask the system to find by
+itself the constructor name and apply it with the `@` tactic.
+Note that some inductive predicates, like the disjunction, have multiple
+introduction rules, and thus `@` can be followed by a number identifying
+the constructor.
+
+D[def-fish-rec-6]
+After choosing `x` as the witness, we have to prove a conjunction,
+and we again apply the introduction rule for the inductively defined
+predicate `∧`.
+
+D[def-fish-rec-7]
+The left hand side of the conjunction is trivial to prove, since it
+is already in the context. The right hand side needs to perform
+the co-recursive call.
+
+D[def-fish-rec-9]
+The co-recursive call needs some arguments, but all of them are
+in the context. Instead of explicitly mention them, we use the
+`nassumption` tactic, that simply tries to apply every context item.
+
+D*)
+
+(*D
+
+Subset of covered/fished points
+-------------------------------
+
+We now have to define the subset of `S` of points covered by `U`.
+We also define a prefix notation for it. Remember that the precedence
+of the prefix form of a symbol has to be higher than the precedence
+of its infix form.
+
+D*)
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).
+(*D
+
+Here we define the equation characterizing the cover relation.
+Even if it is not part of the paper, we proved that `◃(U)` is
+the minimum solution for
+such equation, the interested reader should be able to reply the proof
+with Matita.
+
+D*)
+
ndefinition cover_equation : ∀A:Ax.∀U,X:Ω^A.CProp[0] ≝ λA,U,X.
∀a.a ∈ X ↔ (a ∈ U ∨ ∃i:𝐈 a.∀y.y ∈ 𝐂 a i → y ∈ X).
ntheorem coverage_cover_equation : ∀A,U. cover_equation A U (◃U).
#A; #U; #a; @; #H;
-##[ nelim H; #b; (* manca clear *)
+##[ nelim H; #b;
##[ #bU; @1; nassumption;
##| #i; #CaiU; #IH; @2; @ i; #c; #cCbi; ncases (IH ? cCbi);
##[ #E; @; napply E;
##]
nqed.
+(*D
+
+We similarly define the subset of points "fished" by `F`, the
+equation characterizing `⋉(F)` and prove that fish is
+the biggest solution for such equation.
+
+D*)
+
notation "⋉F" non associative with precedence 55
for @{ 'fished $F }.
ndefinition fish_equation : ∀A:Ax.∀F,X:Ω^A.CProp[0] ≝ λA,F,X.
∀a. a ∈ X ↔ a ∈ F ∧ ∀i:𝐈 a.∃y.y ∈ 𝐂 a i ∧ y ∈ X.
-ntheorem fised_fish_equation : ∀A,F. fish_equation A F (⋉F).
-#A; #F; #a; @; (* bug, fare case sotto diverso da farlo sopra *) #H; ncases H;
+ntheorem fished_fish_equation : ∀A,F. fish_equation A F (⋉F).
+#A; #F; #a; @; (* *; non genera outtype che lega a *) #H; ncases H;
##[ #b; #bF; #H2; @ bF; #i; ncases (H2 i); #c; *; #cC; #cF; @c; @ cC;
napply cF;
##| #aF; #H1; @ aF; napply H1;
##]
nqed.
-ntheorem fised_max_fish_equation : ∀A,F,G. fish_equation A F G → G ⊆ ⋉F.
+ntheorem fished_max_fish_equation : ∀A,F,G. fish_equation A F G → G ⊆ ⋉F.
#A; #F; #G; #H; #a; #aG; napply (fish_rec … aG);
#b; ncases (H b); #H1; #_; #bG; ncases (H1 bG); #E1; #E2; nassumption;
nqed.
+(*D
+
+Part 2, the new set of axioms
+-----------------------------
+
+Since the name of defined objects (record included) has to be unique
+within the same file, we prefix every field name
+in the new definition of the axiom set with `n`.
+
+D*)
+
nrecord nAx : Type[2] ≝ {
- nS:> setoid; (*Type[0];*)
+ nS:> setoid;
nI: nS → Type[0];
nD: ∀a:nS. nI a → Type[0];
nd: ∀a:nS. ∀i:nI a. nD a i → nS
}.
-(*
-TYPE f A → B, g : B → A, f ∘ g = id, g ∘ g = id.
+(*D
+
+We again define a notation for the projections, making the
+projected record an implicit argument. Note that, since we already have
+a notation for `𝐈`, we just add another interpretation for it. The
+system, looking at the argument of `𝐈`, will be able to choose
+the correct interpretation.
-a = b → I a = I b
-*)
+D*)
notation "𝐃 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'D $a $i }.
notation "𝐝 \sub ( ❨a,\emsp i,\emsp j❩ )" non associative with precedence 70 for @{ 'd $a $i $j}.
interpretation "d" 'd a i j = (nd ? a i j).
interpretation "new I" 'I a = (nI ? a).
+(*D
+
+The first result the paper presents to motivate the new formulation
+of the axiom set is the possibility to define and old axiom set
+starting from a new one and vice versa. The key definition for
+such construction is the image of d(a,i).
+The paper defines the image as
+
+> Im[d(a,i)] = { d(a,i,j) | j : D(a,i) }
+
+but this not so formal notation poses some problems. The image is
+often used as the left hand side of the ⊆ predicate
+
+> Im[d(a,i)] ⊆ V
+
+Of course this writing is interpreted by the authors as follows
+
+> ∀j:D(a,i). d(a,i,j) ∈ V
+
+If we need to use the image to define `𝐂 ` (a subset of `S`) we are obliged to
+form a subset, i.e. to place a single variable `{ here | … }` of type `S`.
+
+> Im[d(a,i)] = { y | ∃j:D(a,i). y = d(a,i,j) }
+
+This poses no theoretical problems, since `S` is a setoid and thus equipped
+with an equality.
+
+Unless we define two different images, one for stating that the image is ⊆ of
+something and another one to define `𝐂`, we end up using always the latter.
+Thus the statement `Im[d(a,i)] ⊆ V` unfolds to
+
+> ∀x:S. ( ∃j.x = d(a,i,j) ) → x ∈ V
+
+That, up to rewriting with the equation defining `x`, is what we mean.
+The technical problem arises later, when `V` will be a complex
+construction that has to be proved extensional
+(i.e. ∀x,y. x = y → x ∈ V → y ∈ V).
+
+D*)
+
ndefinition image ≝ λA:nAx.λa:A.λi. { x | ∃j:𝐃 a i. x = 𝐝 a i j }.
notation > "𝐈𝐦 [𝐝 term 90 a term 90 i]" non associative with precedence 70 for @{ 'Im $a $i }.
-notation "𝐈𝐦 [𝐝 \sub ( ❨a,\emsp i❩ )]" non associative with precedence 70 for @{ 'Im $a $i }.
+notation < "𝐈𝐦 [𝐝 \sub ( ❨a,\emsp i❩ )]" non associative with precedence 70 for @{ 'Im $a $i }.
interpretation "image" 'Im a i = (image ? a i).
+(*D
+
+Thanks to our definition of image, we can define a function mapping a
+new axiom set to an old one and vice versa. Note that in the second
+definition, when we give the `𝐝` component, the projection of the
+Σ-type is inlined (constructed on the fly by `*;`)
+while in the paper it was named `fst`.
+
+D*)
+
ndefinition Ax_of_nAx : nAx → Ax.
#A; @ A (nI ?); #a; #i; napply (𝐈𝐦 [𝐝 a i]);
nqed.
-ninductive sigma (A : Type[0]) (P : A → CProp[0]) : Type[0] ≝
- sig_intro : ∀x:A.P x → sigma A P.
-
-interpretation "sigma" 'sigma \eta.p = (sigma ? p).
-
ndefinition nAx_of_Ax : Ax → nAx.
#A; @ A (I ?);
##[ #a; #i; napply (Σx:A.x ∈ 𝐂 a i);
##]
nqed.
+(*D
+
+We then define the inductive type of ordinals, parametrized over an axiom
+set. We also attach some notations to the constructors.
+
+D*)
+
ninductive Ord (A : nAx) : Type[0] ≝
| oO : Ord A
| oS : Ord A → Ord A
| oL : ∀a:A.∀i.∀f:𝐃 a i → Ord A. Ord A.
-notation "Λ term 90 f" non associative with precedence 50 for @{ 'oL $f }.
+notation "0" non associative with precedence 90 for @{ 'oO }.
notation "x+1" non associative with precedence 50 for @{'oS $x }.
+notation "Λ term 90 f" non associative with precedence 50 for @{ 'oL $f }.
-interpretation "ordinals Lambda" 'oL f = (oL ? ? ? f).
+interpretation "ordinals Zero" 'oO = (oO ?).
interpretation "ordinals Succ" 'oS x = (oS ? x).
+interpretation "ordinals Lambda" 'oL f = (oL ? ? ? f).
+
+(*D
+
+The definition of `U⎽x` is by recursion over the ordinal `x`.
+We thus define a recursive function using the `nlet rec` command.
+The `on x` directive tells
+the system on which argument the function is (structurally) recursive.
+
+In the `oS` case we use a local definition to name the recursive call
+since it is used twice.
+
+Note that Matita does not support notation in the left hand side
+of a pattern match, and thus the names of the constructors have to
+be spelled out verbatim.
+
+D*)
nlet rec famU (A : nAx) (U : Ω^A) (x : Ord A) on x : Ω^A ≝
match x with
[ oO ⇒ U
- | oS y ⇒ let Un ≝ famU A U y in Un ∪ { x | ∃i.𝐈𝐦[𝐝 x i] ⊆ Un}
+ | oS y ⇒ let U_n ≝ famU A U y in U_n ∪ { x | ∃i.𝐈𝐦[𝐝 x i] ⊆ U_n}
| oL a i f ⇒ { x | ∃j.x ∈ famU A U (f j) } ].
notation < "term 90 U \sub (term 90 x)" non associative with precedence 50 for @{ 'famU $U $x }.
notation > "U ⎽ term 90 x" non associative with precedence 50 for @{ 'famU $U $x }.
interpretation "famU" 'famU U x = (famU ? U x).
+
+(*D
+
+We attach as the input notation for U_x the similar `U⎽x` where underscore,
+that is a character valid for identifier names, has been replaced by `⎽` that is
+not. The symbol `⎽` can act as a separator, and can be typed as an alternative
+for `_` (i.e. pressing ALT-L after `_`).
+
+The notion ◃(U) has to be defined as the subset of elements `y`
+belonging to `U⎽x` for some `x`. Moreover, we have to define the notion
+of cover between sets again, since the one defined at the beginning
+of the tutorial works only for the old axiom set.
+
+D*)
ndefinition ord_coverage : ∀A:nAx.∀U:Ω^A.Ω^A ≝ λA,U.{ y | ∃x:Ord A. y ∈ famU ? U x }.
interpretation "new covers set" 'covers a U = (ord_cover_set ord_coverage ? a U).
interpretation "new covers" 'covers a U = (mem ? (ord_coverage ? U) a).
-ntheorem new_coverage_reflexive:
- ∀A:nAx.∀U:Ω^A.∀a. a ∈ U → a ◃ U.
-#A; #U; #a; #H; @ (oO A); napply H;
-nqed.
+(*D
+
+Before proving that this cover relation validates the reflexivity and infinity
+rules, we prove this little technical lemma that is used in the proof for the
+infinity rule.
+
+D*)
-nlemma ord_subset:
- ∀A:nAx.∀a:A.∀i,f,U.∀j:𝐃 a i.U⎽(f j) ⊆ U⎽(Λ f).
+nlemma ord_subset: ∀A:nAx.∀a:A.∀i,f,U.∀j:𝐃 a i. U⎽(f j) ⊆ U⎽(Λ f).
#A; #a; #i; #f; #U; #j; #b; #bUf; @ j; nassumption;
nqed.
-naxiom AC : ∀A,a,i,U.(∀j:𝐃 a i.∃x:Ord A.𝐝 a i j ∈ U⎽x) → (Σf.∀j:𝐃 a i.𝐝 a i j ∈ U⎽(f j)).
+(*D
-naxiom setoidification :
- ∀A:nAx.∀a,b:A.∀U.a=b → b ∈ U → a ∈ U.
+The proof of infinity uses the following form of the Axiom of Choice,
+that cannot be proved inside Matita, since the existential quantifier
+lives in the sort of predicative propositions while the sigma in the conclusion
+lives in the sort of data types, and thus the former cannot be eliminated
+to provide the witness for the second.
-(*DOCBEGIN
+D*)
-Bla Bla,
+naxiom AC : ∀A,a,i,U.
+ (∀j:𝐃 a i.∃x:Ord A.𝐝 a i j ∈ U⎽x) → (Σf.∀j:𝐃 a i.𝐝 a i j ∈ U⎽(f j)).
-<div id="figure1" class="img" ><img src="figure1.png"/>foo</div>
+(*D
+
+In the proof of infinity, we have to rewrite under the ∈ predicate.
+It is clearly possible to show that `U⎽x` is an extensional set:
+
+> a = b → a ∈ U⎽x → b ∈ U⎽x
+
+Anyway this proof is a non trivial induction over x, that requires `𝐈` and `𝐃` to be
+declared as morphisms. This poses no problem, but goes out of the scope of the
+tutorial, since dependent morphisms are hard to manipulate, and we thus assume it.
+
+D*)
+
+naxiom U_x_is_ext: ∀A:nAx.∀a,b:A.∀x.∀U. a = b → b ∈ U⎽x → a ∈ U⎽x.
+
+(*D
+
+The reflexivity proof is trivial, it is enough to provide the ordinal `0`
+as a witness, then `◃(U)` reduces to `U` by definition,
+hence the conclusion. Note that `0` is between `(` and `)` to
+make it clear that it is a term (an ordinal) and not the number
+of the constructor we want to apply (that is the first and only one
+of the existential inductive type).
+
+D*)
+ntheorem new_coverage_reflexive: ∀A:nAx.∀U:Ω^A.∀a. a ∈ U → a ◃ U.
+#A; #U; #a; #H; @ (0); napply H;
+nqed.
+
+(*D
+
+We now proceed with the proof of the infinity rule.
+
+D*)
-DOCEND*)
-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".
-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 "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';
-ncases (AC … H'); #f; #Hf;
+#A; #U; #a; (** screenshot "n-cov-inf-1". *)
+*; #i; #H; nnormalize in H; (** screenshot "n-cov-inf-2". *)
+ncut (∀y:𝐃 a i.∃x:Ord A.𝐝 a i y ∈ U⎽x); ##[ (** screenshot "n-cov-inf-3". *)
+ #z; napply H; @ z; napply #; ##] #H'; (** screenshot "n-cov-inf-4". *)
+ncases (AC … H'); #f; #Hf; (** screenshot "n-cov-inf-5". *)
ncut (∀j.𝐝 a i j ∈ U⎽(Λ f));
- ##[ #j; napply (ord_subset … f … (Hf j));##] #Hf';
-@ ((Λ f)+1); @2; nwhd; @i; #x; *; #d; #Hd;
-napply (setoidification … Hd); napply Hf';
+ ##[ #j; napply (ord_subset … f … (Hf j));##] #Hf';(** screenshot "n-cov-inf-6". *)
+@ (Λ f+1); (** screenshot "n-cov-inf-7". *)
+@2; (** screenshot "n-cov-inf-8". *)
+@i; #x; *; #d; #Hd; (** screenshot "n-cov-inf-9". *)
+napply (U_x_is_ext … Hd); napply Hf';
nqed.
+(*D
+D[n-cov-inf-1]
+We eliminate the existential, obtaining an `i` and a proof that the
+image of `𝐝 a i` is covered by U. The `nnormalize` tactic computes the normal
+form of `H`, thus expands the definition of cover between sets.
+
+D[n-cov-inf-2]
+When the paper proof considers `H`, it implicitly substitutes assumed
+equation defining `y` in its conclusion.
+In Matita this step is not completely trivial.
+We thus assert (`ncut`) the nicer form of `H` and prove it.
+
+D[n-cov-inf-3]
+After introducing `z`, `H` can be applied (choosing `𝐝 a i z` as `y`).
+What is the left to prove is that `∃j: 𝐃 a j. 𝐝 a i z = 𝐝 a i j`, that
+becomes trivial if `j` is chosen to be `z`. In the command `napply #`,
+the `#` is a standard notation for the reflexivity property of the equality.
+
+D[n-cov-inf-4]
+Under `H'` the axiom of choice `AC` can be eliminated, obtaining the `f` and
+its property. Note that the axiom `AC` was abstracted over `A,a,i,U` before
+assuming `(∀j:𝐃 a i.∃x:Ord A.𝐝 a i j ∈ U⎽x)`. Thus the term that can be eliminated
+is `AC ???? H'` where the system is able to infer every `?`. Matita provides
+a facility to specify a number of `?` in a compact way, i.e. `…`. The system
+expand `…` first to zero, then one, then two, three and finally four question
+marks, "guessing" how may of them are needed.
+
+D[n-cov-inf-5]
+The paper proof does now a forward reasoning step, deriving (by the ord_subset
+lemma we proved above) `Hf'` i.e. 𝐝 a i j ∈ U⎽(Λf).
+
+D[n-cov-inf-6]
+To prove that `a◃U` we have to exhibit the ordinal x such that `a ∈ U⎽x`.
+
+D[n-cov-inf-7]
+The definition of `U⎽(…+1)` expands to the union of two sets, and proving
+that `a ∈ X ∪ Y` is, by definition, equivalent to prove that `a` is in `X` or `Y`.
+Applying the second constructor `@2;` of the disjunction,
+we are left to prove that `a` belongs to the right hand side of the union.
+
+D[n-cov-inf-8]
+We thus provide `i` as the witness of the existential, introduce the
+element being in the image and we are
+left to prove that it belongs to `U⎽(Λf)`. In the meanwhile, since belonging
+to the image means that there exists an object in the domain …, we eliminate the
+existential, obtaining `d` (of type `𝐃 a i`) and the equation defining `x`.
+
+D[n-cov-inf-9]
+We just need to use the equational definition of `x` to obtain a conclusion
+that can be proved with `Hf'`. We assumed that `U⎽x` is extensional for
+every `x`, thus we are allowed to use `Hd` and close the proof.
+
+D*)
+
+(*D
+
+The next proof is that ◃(U) is minimal. The hardest part of the proof
+is to prepare the goal for the induction. The desiderata is to prove
+`U⎽o ⊆ V` by induction on `o`, but the conclusion of the lemma is,
+unfolding all definitions:
+
+> ∀x. x ∈ { y | ∃o:Ord A.y ∈ U⎽o } → x ∈ V
+
+D*)
+
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;
+ ∀A:nAx.∀U:Ω^A.∀V.U ⊆ V → (∀a:A.∀i.𝐈𝐦[𝐝 a i] ⊆ V → a ∈ V) → ◃U ⊆ V.
+#A; #U; #V; #HUV; #Im;#b; (** screenshot "n-cov-min-2". *)
+*; #o; (** screenshot "n-cov-min-3". *)
+ngeneralize in match b; nchange with (U⎽o ⊆ V); (** screenshot "n-cov-min-4". *)
+nelim o; (** screenshot "n-cov-min-5". *)
##[ #b; #bU0; napply HUV; napply bU0;
##| #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.
+(*D
+D[n-cov-min-2]
+After all the introductions, event the element hidden in the ⊆ definition,
+we have to eliminate the existential quantifier, obtaining the ordinal `o`
+
+D[n-cov-min-3]
+What is left is almost right, but the element `b` is already in the
+context. We thus generalize every occurrence of `b` in
+the current goal, obtaining `∀c.c ∈ U⎽o → c ∈ V` that is `U⎽o ⊆ V`.
+
+D[n-cov-min-4]
+We then proceed by induction on `o` obtaining the following goals
+
+D[n-cov-min-5]
+All of them can be proved using simple set theoretic arguments,
+the induction hypothesis and the assumption `Im`.
+
+D*)
+
+
+(*D
+
+The notion `F⎽x` is again defined by recursion over the ordinal `x`.
+
+D*)
+
nlet rec famF (A: nAx) (F : Ω^A) (x : Ord A) on x : Ω^A ≝
match x with
[ oO ⇒ F
- | oS o ⇒ let Fo ≝ famF A F o in Fo ∩ { x | ∀i:𝐈 x.∃j:𝐃 x i.𝐝 x i j ∈ Fo }
+ | oS o ⇒ let F_o ≝ famF A F o in F_o ∩ { x | ∀i:𝐈 x.∃j:𝐃 x i.𝐝 x i j ∈ F_o }
| oL a i f ⇒ { x | ∀j:𝐃 a i.x ∈ famF A F (f j) }
].
interpretation "fished new fish" 'fished U = (ord_fished ? U).
interpretation "new fish" 'fish a U = (mem ? (ord_fished ? U) a).
-ntheorem new_fish_antirefl:
- ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → a ∈ F.
-#A; #F; #a; #H; nlapply (H (oO ?)); #aFo; napply aFo;
-nqed.
+(*D
+
+The proof of compatibility uses this little result, that we
+proved outside the main proof.
-nlemma co_ord_subset:
- ∀A:nAx.∀F:Ω^A.∀a,i.∀f:𝐃 a i → Ord A.∀j. F⎽(Λ f) ⊆ F⎽(f j).
+D*)
+nlemma co_ord_subset: ∀A:nAx.∀F:Ω^A.∀a,i.∀f:𝐃 a i → Ord A.∀j. F⎽(Λ f) ⊆ F⎽(f j).
#A; #F; #a; #i; #f; #j; #x; #H; napply H;
nqed.
-naxiom AC_dual :
- ∀A:nAx.∀a:A.∀i,F. (∀f:𝐃 a i → Ord A.∃x:𝐃 a i.𝐝 a i x ∈ F⎽(f x)) → ∃j:𝐃 a i.∀x:Ord A.𝐝 a i j ∈ F⎽x.
+(*D
+We assume the dual of the axiom of choice, as in the paper proof.
+D*)
+naxiom AC_dual: ∀A:nAx.∀a:A.∀i,F.
+ (∀f:𝐃 a i → Ord A.∃x:𝐃 a i.𝐝 a i x ∈ F⎽(f x)) → ∃j:𝐃 a i.∀x:Ord A.𝐝 a i j ∈ F⎽x.
+
+(*D
+
+Proving the anti-reflexivity property is simple, since the
+assumption `a ⋉ F` states that for every ordinal `x` (and thus also 0)
+`a ∈ F⎽x`. If `x` is choose to be `0`, we obtain the thesis.
+
+D*)
+ntheorem new_fish_antirefl: ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → a ∈ F.
+#A; #F; #a; #H; nlapply (H 0); #aFo; napply aFo;
+nqed.
+
+(*D
+
+We now prove the compatibility property for the new fish relation.
+
+D*)
ntheorem new_fish_compatible:
∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ⋉ F.
-#A; #F; #a; #aF; #i; nnormalize;
-napply AC_dual; #f;
-nlapply (aF (Λf+1)); #aLf;
-nchange in aLf with (a ∈ F⎽(Λ f) ∧ ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ∈ F⎽(Λ f));
-ncut (∃j:𝐃 a i.𝐝 a i j ∈ F⎽(f j));##[
- ncases aLf; #_; #H; nlapply (H i); *; #j; #Hj; @j; napply Hj;##] #aLf';
-napply aLf';
+#A; #F; #a; #aF; #i; nnormalize; (** screenshot "n-f-compat-1". *)
+napply AC_dual; #f; (** screenshot "n-f-compat-2". *)
+nlapply (aF (Λf+1)); #aLf; (** screenshot "n-f-compat-3". *)
+nchange in aLf with
+ (a ∈ F⎽(Λ f) ∧ ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ∈ F⎽(Λ f)); (** screenshot "n-f-compat-4". *)
+ncases aLf; #_; #H; nlapply (H i); (** screenshot "n-f-compat-5". *)
+*; #j; #Hj; @j; (** screenshot "n-f-compat-6". *)
+napply (co_ord_subset … Hj);
nqed.
+(*D
+D[n-f-compat-1]
+After reducing to normal form the goal, we observe it is exactly the conclusion of
+the dual axiom of choice we just assumed. We thus apply it ad introduce the
+fcuntion `f`.
+
+D[n-f-compat-2]
+The hypothesis `aF` states that `a⋉F⎽x` for every `x`, and we choose `Λf+1`.
+
+D[n-f-compat-3]
+Since F_(Λf+1) is defined by recursion and we actually have a concrete input
+`Λf+1` for that recursive function, it can be computed.
+Anyway, using the `nnormalize`
+tactic would reduce too much (both the `+1` and the `Λf` steps would be performed);
+we thus explicitly give a convertible type for that hypothesis, corresponding
+the computation of the `+1` step, plus the unfolding the definition of
+the intersection.
+
+D[n-f-compat-4]
+We are interested in the right hand side of `aLf`, an in particular to
+its intance where the generic index in `𝐈 a` is `i`.
+
+D[n-f-compat-5]
+We then eliminate the existential, obtaining `j` and its property `Hj`. We provide
+the same witness
+
+D[n-f-compat-6]
+What is left to prove is exactly the `co_ord_subset` lemma we factored out
+of the main proof.
+
+D*)
+
+(*D
+
+The proof that `⋉(F)` is maximal is exactly the dual one of the
+minimality of `◃(U)`. Thus the main problem is to obtain `G ⊆ F⎽o`
+before doing the induction over `o`.
+
+Note that `G` is assumed to be of type `𝛀^A`, that means an extensional
+subset of `S`, while `Ω^A` means just a subset (note that the former is bold).
+
+D*)
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;
+ ∀A:nAx.∀G:𝛀^A.∀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;
nchange with (G ⊆ F⎽o);
nelim o;
##[ napply GF;
##| #p; #IH; napply (subseteq_intersection_r … IH);
#x; #Hx; #i; ncases (H … Hx i); #c; *; *; #d; #Ed; #cG;
- @d; napply IH; napply (setoidification … Ed^-1); napply cG;
+ @d; napply IH; (** screenshot "n-f-max-1". *)
+ alias symbol "prop2" = "prop21".
+ napply (. Ed^-1‡#); napply cG;
##| #a; #i; #f; #Hf; nchange with (G ⊆ { y | ∀x. y ∈ F⎽(f x) });
#b; #Hb; #d; napply (Hf d); napply Hb;
##]
nqed.
-(*DOCBEGIN
+(*D
+D[n-f-max-1]
+Here the situation looks really similar to the one of the dual proof where
+we had to apply the assumption `U_x_is_ext`, but here the set is just `G`
+not `F_x`. Since we assumed `G` to be extensional we can
+exploit the facilities
+Matita provides to perform rewriting in the general setting of setoids.
+
+The `.` notation simply triggers the mechanism, while the given argument has to
+mimic the context under which the rewriting has to happen. In that case
+we want to rewrite the left hand side of the binary morphism `∈`.
+The infix notation
+to represent the context of a binary morphism is `‡`. The right hand side
+has to be left untouched, and the identity rewriting step is represented with
+`#` (actually a reflexivity proof for the subterm identified by the context).
+
+We want to rewrite the left hand side using `Ed` right-to-left (the default
+is left-to-right). We thus write `Ed^-1`, that is a proof that `𝐝 x i d = c`.
+
+The complete command is `napply (. Ed^-1‡#)` that has to be read like:
+
+> perform some rewritings under a binary morphism,
+> on the right do nothing,
+> on the left rewrite with Ed right-to-left.
+
+After the rewriting step the goal is exactly the `cG` assumption.
+
+D*)
+
+
+(*D
+<div id="appendix" class="anchor"></div>
+Appendix: tactics explanation
+-----------------------------
+
+In this appendix we try to give a description of tactics
+in terms of sequent calculus rules annotated with proofs.
+The `:` separator has to be read as _is a proof of_, in the spirit
+of the Curry-Howard isomorphism.
+
+ Γ ⊢ f : A1 → … → An → B Γ ⊢ ?1 : A1 … ?n : An
+ napply f; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ (f ?1 … ?n ) : B
+
+ Γ ⊢ ? : F → B Γ ⊢ f : F
+ nlapply f; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ (? f) : B
+
+
+ Γ; x : T ⊢ ? : P(x)
+ #x; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ λx:T.? : ∀x:T.P(x)
+
+
+ Γ ⊢ ?_i : args_i → P(k_i args_i)
+ ncases x; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ match x with [ k1 args1 ⇒ ?_1 | … | kn argsn ⇒ ?_n ] : P(x)
+
+
+ Γ ⊢ ?i : ∀t. P(t) → P(k_i … t …)
+ nelim x; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ (T_rect_CProp0 ?_1 … ?_n) : P(x)
+
+Where `T_rect_CProp0` is the induction principle for the
+inductive type `T`.
+
+
+ Γ ⊢ ? : Q Q ≡ P
+ nchange with Q; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ ? : P
+
+Where the equivalence relation between types `≡` keeps into account
+β-reduction, δ-reduction (definition unfolding), ζ-reduction (local
+definition unfolding), ι-reduction (pattern matching simplification),
+μ-reduction (recursive function computation) and ν-reduction (co-fixpoint
+computation).
+
+
+ Γ; H : Q; Δ ⊢ ? : G Q ≡ P
+ nchange in H with Q; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ; H : P; Δ ⊢ ? : G
+
+
+ Γ; H : Q; Δ ⊢ ? : G P →* Q
+ nnormalize in H; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ; H : P; Δ ⊢ ? : G
+
+Where `Q` is the normal form of `P` considering βδζιμν-reduction steps.
+
+
+ Γ ⊢ ? : Q P →* Q
+ nnormalize; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ ? : P
+
+
+ Γ ⊢ ?_2 : T → G Γ ⊢ ?_1 : T
+ ncut T; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ (?_2 ?_1) : G
+
+
+ Γ ⊢ ? : ∀x.P(x)
+ ngeneralize in match t; ⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼⎼
+ Γ ⊢ (? t) : P(t)
+
+D*)
+
+
+(*D
+
+<date>
+Last updated: $Date$
+</date>
[1]: http://upsilon.cc/~zack/research/publications/notation.pdf
+[2]: http://matita.cs.unibo.it
+[3]: http://www.cs.unibo.it/~tassi/smallcc.pdf
+[4]: http://www.inria.fr/rrrt/rr-0530.html
-*)
\ No newline at end of file
+D*)
projects / helm.git / blobdiff