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The ring and field tactic families¶

Author: Bruno Barras, Benjamin Grégoire, Assia Mahboubi, Laurent Théry 1

This chapter presents the tactics dedicated to dealing with ring and field equations.

What does this tactic do?¶

ring does associative-commutative rewriting in ring and semiring structures. Assume you have two binary functions $\oplus$ and $\otimes$ that are associative and commutative, with $\oplus$ distributive on $\otimes$, and two constants 0 and 1 that are unities for $\oplus$ and $\otimes$. A polynomial is an expression built on variables $V_0$, $V_1$, $\dots$ and constants by application of $\oplus$ and $\otimes$.

Let an ordered product be a product of variables $V_{i_1} \otimes \dots \otimes V_{i_n}$ verifying $i_1 ≤ i_2 ≤ \dots ≤ i_n$ . Let a monomial be the product of a constant and an ordered product. We can order the monomials by the lexicographic order on products of variables. Let a canonical sum be an ordered sum of monomials that are all different, i.e. each monomial in the sum is strictly less than the following monomial according to the lexicographic order. It is an easy theorem to show that every polynomial is equivalent (modulo the ring properties) to exactly one canonical sum. This canonical sum is called the normal form of the polynomial. In fact, the actual representation shares monomials with same prefixes. So what does the ring tactic do? It normalizes polynomials over any ring or semiring structure. The basic use of ring is to simplify ring expressions, so that the user does not have to deal manually with the theorems of associativity and commutativity.

Example

In the ring of integers, the normal form of: $x (3 + yx + 25(1 − z)) + zx$
is: $28x + (−24)xz + xxy$.

ring is also able to compute a normal form modulo monomial equalities. For example, under the hypothesis that $2x^2 = yz+1$, the normal form of $2(x + 1)x − x − zy$ is $x+1$.

The variables map¶

It is frequent to have an expression built with $+$ and $\times$, but rarely on variables only. Let us associate a number to each subterm of a ring expression in the Gallina language. For example, consider this expression in the semiring nat:

(plus (mult (plus (f (5)) x) x)
      (mult (if b then (4) else (f (3))) (2)))

As a ring expression, it has 3 subterms. Give each subterm a number in an arbitrary order:

0	$\mapsto$	if b then (4) else (f (3))
1	$\mapsto$	(f (5))
2	$\mapsto$	x

Then normalize the “abstract” polynomial $((V_1 \oplus V_2 ) \otimes V_2) \oplus (V_0 \otimes 2)$ In our example the normal form is: $(2 \otimes V_0 ) \oplus (V_1 \otimes V_2) \oplus (V_2 \otimes V_2 )$. Then substitute the variables by their values in the variables map to get the concrete normal polynomial:

(plus (mult (2) (if b then (4) else (f (3))))
      (plus (mult (f (5)) x) (mult x x)))

Is it automatic?¶

Yes, building the variables map and doing the substitution after normalizing is automatically done by the tactic. So you can just forget this paragraph and use the tactic according to your intuition.

Concrete usage in Coq¶

ring: This tactic solves equations upon polynomial expressions of a ring (or semiring) structure. It proceeds by normalizing both sides of the equation (w.r.t. associativity, commutativity and distributivity, constant propagation, rewriting of monomials) and comparing syntactically the results.

ring_simplify

This tactic applies the normalization procedure described above to the given terms. The tactic then replaces all occurrences of the terms given in the conclusion of the goal by their normal forms. If no term is given, then the conclusion should be an equation and both sides are normalized. The tactic can also be applied in a hypothesis.

The tactic must be loaded by Require Import Ring. The ring structures must be declared with the Add Ring command (see below). The ring of booleans is predefined; if one wants to use the tactic on nat one must first require the module ArithRing exported by Arith); for Z, do Require Import ZArithRing or simply Require Import ZArith; for N, do Require Import NArithRing or Require Import NArith.

Example

Require Import ZArith.
Open Scope Z_scope.
Goal forall a b c:Z,
    (a + b + c) ^ 2 =
     a * a + b ^ 2 + c * c + 2 * a * b + 2 * a * c + 2 * b * c.
forall a b c : Z,
(a + b + c) ^ 2 =
a * a + b ^ 2 + c * c + 2 * a * b + 2 * a * c +
2 * b * c
intros; ring.
Abort.
Goal forall a b:Z,
     2 * a * b = 30 -> (a + b) ^ 2 = a ^ 2 + b ^ 2 + 30.
forall a b : Z,
2 * a * b = 30 -> (a + b) ^ 2 = a ^ 2 + b ^ 2 + 30
intros a b H; ring [H].
Abort.

Variant ring [term*]: This tactic decides the equality of two terms modulo ring operations and the equalities defined by the terms. Each term has to be a proof of some equality m = p, where m is a monomial (after “abstraction”), p a polynomial and = the corresponding equality of the ring structure.

Variant ring_simplify [term*] term* in ident: This tactic performs the simplification in the hypothesis named ident.

Note

ring_simplify term₁; ring_simplify term₂ is not equivalent to ring_simplify term₁ term₂.

In the latter case the variables map is shared between the two terms, and common subterm t of term₁ and term₂ will have the same associated variable number. So the first alternative should be avoided for terms belonging to the same ring theory.

Error messages:

Error Not a valid ring equation.¶: The conclusion of the goal is not provable in the corresponding ring theory.

Error Arguments of ring_simplify do not have all the same type.¶: ring_simplify cannot simplify terms of several rings at the same time. Invoke the tactic once per ring structure.

Error Cannot find a declared ring structure over term.¶: No ring has been declared for the type of the terms to be simplified. Use Add Ring first.

Error Cannot find a declared ring structure for equality term.¶: Same as above in the case of the ring tactic.

Adding a ring structure¶

Declaring a new ring consists in proving that a ring signature (a carrier set, an equality, and ring operations: Ring_theory.ring_theory and Ring_theory.semi_ring_theory) satisfies the ring axioms. Semi- rings (rings without + inverse) are also supported. The equality can be either Leibniz equality, or any relation declared as a setoid (see Tactics enabled on user provided relations). The definitions of ring and semiring (see module Ring_theory) are:

Record ring_theory : Prop := mk_rt {
  Radd_0_l    : forall x, 0 + x == x;
  Radd_sym    : forall x y, x + y == y + x;
  Radd_assoc  : forall x y z, x + (y + z) == (x + y) + z;
  Rmul_1_l    : forall x, 1 * x == x;
  Rmul_sym    : forall x y, x * y == y * x;
  Rmul_assoc  : forall x y z, x * (y * z) == (x * y) * z;
  Rdistr_l    : forall x y z, (x + y) * z == (x * z) + (y * z);
  Rsub_def    : forall x y, x - y == x + -y;
  Ropp_def    : forall x, x + (- x) == 0
}.

Record semi_ring_theory : Prop := mk_srt {
  SRadd_0_l   : forall n, 0 + n == n;
  SRadd_sym   : forall n m, n + m == m + n ;
  SRadd_assoc : forall n m p, n + (m + p) == (n + m) + p;
  SRmul_1_l   : forall n, 1*n == n;
  SRmul_0_l   : forall n, 0*n == 0;
  SRmul_sym   : forall n m, n*m == m*n;
  SRmul_assoc : forall n m p, n*(m*p) == (n*m)*p;
  SRdistr_l   : forall n m p, (n + m)*p == n*p + m*p
}.

This implementation of ring also features a notion of constant that can be parameterized. This can be used to improve the handling of closed expressions when operations are effective. It consists in introducing a type of coefficients and an implementation of the ring operations, and a morphism from the coefficient type to the ring carrier type. The morphism needs not be injective, nor surjective.

As an example, one can consider the real numbers. The set of coefficients could be the rational numbers, upon which the ring operations can be implemented. The fact that there exists a morphism is defined by the following properties:

Record ring_morph : Prop := mkmorph {
  morph0    : [cO] == 0;
  morph1    : [cI] == 1;
  morph_add : forall x y, [x +! y] == [x]+[y];
  morph_sub : forall x y, [x -! y] == [x]-[y];
  morph_mul : forall x y, [x *! y] == [x]*[y];
  morph_opp : forall x, [-!x] == -[x];
  morph_eq  : forall x y, x?=!y = true -> [x] == [y]
}.

Record semi_morph : Prop := mkRmorph {
  Smorph0 : [cO] == 0;
  Smorph1 : [cI] == 1;
  Smorph_add : forall x y, [x +! y] == [x]+[y];
  Smorph_mul : forall x y, [x *! y] == [x]*[y];
  Smorph_eq  : forall x y, x?=!y = true -> [x] == [y]
}.

where c0 and cI denote the 0 and 1 of the coefficient set, +!, *!, -! are the implementations of the ring operations, == is the equality of the coefficients, ?+! is an implementation of this equality, and [x] is a notation for the image of x by the ring morphism.

Since Z is an initial ring (and N is an initial semiring), it can always be considered as a set of coefficients. There are basically three kinds of (semi-)rings:

abstract rings: to be used when operations are not effective. The set of coefficients is Z (or N for semirings).
computational rings: to be used when operations are effective. The set of coefficients is the ring itself. The user only has to provide an implementation for the equality.
customized ring: for other cases. The user has to provide the coefficient set and the morphism.

This implementation of ring can also recognize simple power expressions as ring expressions. A power function is specified by the following property:

Require Import Reals.
Section POWER.
  Variable Cpow : Set.
  Variable Cp_phi : N -> Cpow.
  Variable rpow : R -> Cpow -> R.

  Record power_theory : Prop := mkpow_th {
    rpow_pow_N : forall r n, rpow r (Cp_phi n) = pow_N 1%R Rmult r n
  }.

End POWER.

The syntax for adding a new ring is

Command Add Ring ident : term ( ring_mod , ring_mod* )?¶

The ident is not relevant. It is used just for error messages. The term is a proof that the ring signature satisfies the (semi-)ring axioms. The optional list of modifiers is used to tailor the behavior of the tactic. The following list describes their syntax and effects:

ring_mod ::=  abstract | decidable term | morphism term
              setoid term term
              constants [ tactic ]
              preprocess [ tactic ]
              postprocess [ tactic ]
              power_tac term [ tactic ]
              sign term
              div term

abstract: declares the ring as abstract. This is the default.
decidable term: declares the ring as computational. The expression term is the correctness proof of an equality test ?=! (which should be evaluable). Its type should be of the form forall x y, x ?=! y = true → x == y.
morphism term: declares the ring as a customized one. The expression term is a proof that there exists a morphism between a set of coefficient and the ring carrier (see Ring_theory.ring_morph and Ring_theory.semi_morph).
setoid term term: forces the use of given setoid. The first term is a proof that the equality is indeed a setoid (see Setoid.Setoid_Theory), and the second term a proof that the ring operations are morphisms (see Ring_theory.ring_eq_ext and Ring_theory.sring_eq_ext). This modifier needs not be used if the setoid and morphisms have been declared.
constants [ tactic ]: specifies a tactic expression tactic that, given a term, returns either an object of the coefficient set that is mapped to the expression via the morphism, or returns InitialRing.NotConstant. The default behavior is to map only 0 and 1 to their counterpart in the coefficient set. This is generally not desirable for non trivial computational rings.
preprocess [ tactic ]: specifies a tactic tactic that is applied as a preliminary step for ring and ring_simplify. It can be used to transform a goal so that it is better recognized. For instance, S n can be changed to plus 1 n.
postprocess [ tactic ]: specifies a tactic tactic that is applied as a final step for ring_simplify. For instance, it can be used to undo modifications of the preprocessor.
power_tac term [ tactic ]: allows ring and ring_simplify to recognize power expressions with a constant positive integer exponent (example: $x^2$ ). The term term is a proof that a given power function satisfies the specification of a power function (term has to be a proof of Ring_theory.power_theory) and tactic specifies a tactic expression that, given a term, “abstracts” it into an object of type N whose interpretation via Cp_phi (the evaluation function of power coefficient) is the original term, or returns InitialRing.NotConstant if not a constant coefficient (i.e. L_tac is the inverse function of Cp_phi). See files plugins/setoid_ring/ZArithRing.v and plugins/setoid_ring/RealField.v for examples. By default the tactic does not recognize power expressions as ring expressions.
sign term: allows ring_simplify to use a minus operation when outputting its normal form, i.e writing x − y instead of x + (− y). The term term is a proof that a given sign function indicates expressions that are signed (term has to be a proof of Ring_theory.get_sign). See plugins/setoid_ring/InitialRing.v for examples of sign function.
div term: allows ring and ring_simplify to use monomials with coefficients other than 1 in the rewriting. The term term is a proof that a given division function satisfies the specification of an euclidean division function (term has to be a proof of Ring_theory.div_theory). For example, this function is called when trying to rewrite $7x$ by $2x = z$ to tell that $7 = 3 \times 2 + 1$. See plugins/setoid_ring/InitialRing.v for examples of div function.

Error messages:

Error Bad ring structure.¶: The proof of the ring structure provided is not of the expected type.

Error Bad lemma for decidability of equality.¶: The equality function provided in the case of a computational ring has not the expected type.

Error Ring operation should be declared as a morphism.¶: A setoid associated to the carrier of the ring structure has been found, but the ring operation should be declared as morphism. See Tactics enabled on user provided relations.

How does it work?¶

The code of ring is a good example of a tactic written using reflection. What is reflection? Basically, using it means that a part of a tactic is written in Gallina, Coq's language of terms, rather than L_tac or OCaml. From the philosophical point of view, reflection is using the ability of the Calculus of Constructions to speak and reason about itself. For the ring tactic we used Coq as a programming language and also as a proof environment to build a tactic and to prove its correctness.

The interested reader is strongly advised to have a look at the file Ring_polynom.v. Here a type for polynomials is defined:

Inductive PExpr : Type :=
  | PEc : C -> PExpr
  | PEX : positive -> PExpr
  | PEadd : PExpr -> PExpr -> PExpr
  | PEsub : PExpr -> PExpr -> PExpr
  | PEmul : PExpr -> PExpr -> PExpr
  | PEopp : PExpr -> PExpr
  | PEpow : PExpr -> N -> PExpr.

Polynomials in normal form are defined as:

Inductive Pol : Type :=
  | Pc : C -> Pol
  | Pinj : positive -> Pol -> Pol
  | PX : Pol -> positive -> Pol -> Pol.

where Pinj n P denotes P in which $V_i$ is replaced by $V_{i+n}$ , and PX P n Q denotes $P \otimes V_1^n \oplus Q'$, Q' being Q where $V_i$ is replaced by $V_{i+1}$.

Variable maps are represented by lists of ring elements, and two interpretation functions, one that maps a variables map and a polynomial to an element of the concrete ring, and the second one that does the same for normal forms:

Definition PEeval : list R -> PExpr -> R := [...].
Definition Pphi_dev : list R -> Pol -> R := [...].

A function to normalize polynomials is defined, and the big theorem is its correctness w.r.t interpretation, that is:

Definition norm : PExpr -> Pol := [...].
Lemma Pphi_dev_ok :
   forall l pe npe, norm pe = npe -> PEeval l pe == Pphi_dev l npe.

So now, what is the scheme for a normalization proof? Let p be the polynomial expression that the user wants to normalize. First a little piece of ML code guesses the type of p, the ring theory T to use, an abstract polynomial ap and a variables map v such that p is $\beta\delta\iota$- equivalent to (PEeval v ap). Then we replace it by (Pphi_dev v (norm ap)), using the main correctness theorem and we reduce it to a concrete expression p’, which is the concrete normal form of p. This is summarized in this diagram:

`p`	$\rightarrow_{\beta\delta\iota}$	`(PEeval` `v` `ap)`
	`=`_{(by the main correctness theorem)}
`p’`	$\leftarrow_{\beta\delta\iota}$	`(Pphi_dev` `v` `(norm` `ap))`

The user does not see the right part of the diagram. From outside, the tactic behaves like a $\beta\delta\iota$ simplification extended with rewriting rules for associativity and commutativity. Basically, the proof is only the application of the main correctness theorem to well-chosen arguments.

Dealing with fields¶

field

This tactic is an extension of the ring tactic that deals with rational expressions. Given a rational expression $F = 0$. It first reduces the expression F to a common denominator $N/D = 0$ where N and D are two ring expressions. For example, if we take $F = (1 − 1/x) x − x + 1$, this gives $N = (x − 1) x − x^2 + x$ and $D = x$. It then calls ring to solve $N = 0$.

Note that field also generates nonzero conditions for all the denominators it encounters in the reduction. In our example, it generates the condition $x \neq 0$. These conditions appear as one subgoal which is a conjunction if there are several denominators. Nonzero conditions are always polynomial expressions. For example when reducing the expression $1/(1 + 1/x)$, two side conditions are generated: $x \neq 0$ and $x + 1 \neq 0$. Factorized expressions are broken since a field is an integral domain, and when the equality test on coefficients is complete w.r.t. the equality of the target field, constants can be proven different from zero automatically.

The tactic must be loaded by Require Import Field. New field structures can be declared to the system with the Add Field command (see below). The field of real numbers is defined in module RealField (in plugins/setoid_ring). It is exported by module Rbase, so that requiring Rbase or Reals is enough to use the field tactics on real numbers. Rational numbers in canonical form are also declared as a field in the module Qcanon.

Example

Require Import Reals.
Open Scope R_scope.
Goal forall x,
       x <> 0 -> (1 - 1 / x) * x - x + 1 = 0.
forall x : R, x <> 0 -> (1 - 1 / x) * x - x + 1 = 0
intros; field; auto.
Abort.
Goal forall x y,
       y <> 0 -> y = x -> x / y = 1.
forall x y : R, y <> 0 -> y = x -> x / y = 1
intros x y H H1; field [H1]; auto.
Abort.

Variant field [term*]: This tactic decides the equality of two terms modulo field operations and the equalities defined by the terms. Each term has to be a proof of some equality m = p, where m is a monomial (after “abstraction”), p a polynomial and = the corresponding equality of the field structure.

Note

Rewriting works with the equality m = p only if p is a polynomial since rewriting is handled by the underlying ring tactic.

Variant field_simplify: performs the simplification in the conclusion of the goal, $F_1 = F_2$ becomes $N_1 / D_1 = N_2 / D_2$. A normalization step (the same as the one for rings) is then applied to $N_1$, $D_1$, $N_2$ and $D_2$. This way, polynomials remain in factorized form during fraction simplification. This yields smaller expressions when reducing to the same denominator since common factors can be canceled.

Variant field_simplify [term*]: This variant performs the simplification in the conclusion of the goal using the equalities defined by the terms.

Variant field_simplify [term*] term*: This variant performs the simplification in the terms terms of the conclusion of the goal using the equalities defined by terms inside the brackets.

Variant field_simplify in ident: This variant performs the simplification in the assumption ident.

Variant field_simplify [term*] in ident: This variant performs the simplification in the assumption ident using the equalities defined by the terms.

Variant field_simplify [term*] term* in ident: This variant performs the simplification in the terms of the assumption ident using the equalities defined by the terms inside the brackets.

Variant field_simplify_eq: performs the simplification in the conclusion of the goal removing the denominator. $F_1 = F_2$ becomes $N_1 D_2 = N_2 D_1$.

Variant field_simplify_eq [ term*]: This variant performs the simplification in the conclusion of the goal using the equalities defined by terms.

Variant field_simplify_eq in ident: This variant performs the simplification in the assumption ident.

Variant field_simplify_eq [term*] in ident: This variant performs the simplification in the assumption ident using the equalities defined by terms and removing the denominator.

Adding a new field structure¶

Declaring a new field consists in proving that a field signature (a carrier set, an equality, and field operations: Field_theory.field_theory and Field_theory.semi_field_theory) satisfies the field axioms. Semi-fields (fields without + inverse) are also supported. The equality can be either Leibniz equality, or any relation declared as a setoid (see Tactics enabled on user provided relations). The definition of fields and semifields is:

Record field_theory : Prop := mk_field {
  F_R : ring_theory rO rI radd rmul rsub ropp req;
  F_1_neq_0 : ~ 1 == 0;
  Fdiv_def : forall p q, p / q == p * / q;
  Finv_l : forall p, ~ p == 0 ->  / p * p == 1
}.

Record semi_field_theory : Prop := mk_sfield {
  SF_SR : semi_ring_theory rO rI radd rmul req;
  SF_1_neq_0 : ~ 1 == 0;
  SFdiv_def : forall p q, p / q == p * / q;
  SFinv_l : forall p, ~ p == 0 ->  / p * p == 1
}.

The result of the normalization process is a fraction represented by the following type:

Record linear : Type := mk_linear {
  num : PExpr C;
  denum : PExpr C;
  condition : list (PExpr C)
}.

where num and denum are the numerator and denominator; condition is a list of expressions that have appeared as a denominator during the normalization process. These expressions must be proven different from zero for the correctness of the algorithm.

The syntax for adding a new field is

Command Add Field ident : term ( field_mod , field_mod* )?¶

The ident is not relevant. It is used just for error messages. term is a proof that the field signature satisfies the (semi-)field axioms. The optional list of modifiers is used to tailor the behavior of the tactic.

field_mod ::=  ring_mod | completeness term

Since field tactics are built upon ring tactics, all modifiers of the Add Ring apply. There is only one specific modifier:

completeness term: allows the field tactic to prove automatically that the image of nonzero coefficients are mapped to nonzero elements of the field. term is a proof of forall x y, [x] == [y] -> x ?=! y = true, which is the completeness of equality on coefficients w.r.t. the field equality.

History of ring¶

First Samuel Boutin designed the tactic ACDSimpl. This tactic did lot of rewriting. But the proofs terms generated by rewriting were too big for Coq’s type checker. Let us see why:

Require Import ZArith.
Open Scope Z_scope.
Goal forall x y z : Z,
       x + 3 + y + y * z = x + 3 + y + z * y.
forall x y z : Z,
x + 3 + y + y * z = x + 3 + y + z * y
intros; rewrite (Zmult_comm y z); reflexivity.
Save foo.
Print foo.foo = 
fun x y z : Z =>
eq_ind_r
  (fun z0 : Z => x + 3 + y + z0 = x + 3 + y + z * y)
  eq_refl (Z.mul_comm y z)
     : forall x y z : Z,
       x + 3 + y + y * z = x + 3 + y + z * y

Argument scopes are [Z_scope Z_scope Z_scope]

At each step of rewriting, the whole context is duplicated in the proof term. Then, a tactic that does hundreds of rewriting generates huge proof terms. Since ACDSimpl was too slow, Samuel Boutin rewrote it using reflection (see [Bou97]). Later, it was rewritten by Patrick Loiseleur: the new tactic does not any more require ACDSimpl to compile and it makes use of $\beta\delta\iota$-reduction not only to replace the rewriting steps, but also to achieve the interleaving of computation and reasoning (see Discussion). He also wrote some ML code for the Add Ring command that allows registering new rings dynamically.

Proofs terms generated by ring are quite small, they are linear in the number of $\oplus$ and $\otimes$ operations in the normalized terms. Type checking those terms requires some time because it makes a large use of the conversion rule, but memory requirements are much smaller.

Discussion¶

Efficiency is not the only motivation to use reflection here. ring also deals with constants, it rewrites for example the expression 34 + 2 * x − x + 12 to the expected result x + 46. For the tactic ACDSimpl, the only constants were 0 and 1. So the expression 34 + 2 * (x − 1) + 12 is interpreted as $V_0 \oplus V_1 \otimes (V_2 \ominus 1) \oplus V_3$, with the variables mapping $\{V_0 \mapsto 34; V_1 \mapsto 2; V_2 \mapsto x; V_3 \mapsto 12\}$. Then it is rewritten to 34 − x + 2 * x + 12, very far from the expected result. Here rewriting is not sufficient: you have to do some kind of reduction (some kind of computation) to achieve the normalization.

The tactic ring is not only faster than the old one: by using reflection, we get for free the integration of computation and reasoning that would be very difficult to implement without it.

Is it the ultimate way to write tactics? The answer is: yes and no. The ring tactic intensively uses the conversion rules of the Calculus of Inductive Constructions, i.e. it replaces proofs by computations as much as possible. It can be useful in all situations where a classical tactic generates huge proof terms, like symbolic processing and tautologies. But there are also tactics like auto or linear that do many complex computations, using side-effects and backtracking, and generate a small proof term. Clearly, it would be significantly less efficient to replace them by tactics using reflection.

Another idea suggested by Benjamin Werner: reflection could be used to couple an external tool (a rewriting program or a model checker) with Coq. We define (in Coq) a type of terms, a type of traces, and prove a correctness theorem that states that replaying traces is safe with respect to some interpretation. Then we let the external tool do every computation (using side-effects, backtracking, exception, or others features that are not available in pure lambda calculus) to produce the trace. Now we can check in Coq that the trace has the expected semantics by applying the correctness theorem.

Footnotes

1: based on previous work from Patrick Loiseleur and Samuel Boutin

0	\(\mapsto\)	if b then (4) else (f (3))
1	\(\mapsto\)	(f (5))
2	\(\mapsto\)	x