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+---
+jupyter:
+  jupytext:
+    formats: ipynb,md
+    text_representation:
+      extension: .md
+      format_name: markdown
+      format_version: '1.2'
+      jupytext_version: 1.6.0
+  kernelspec:
+    display_name: Python 3
+    language: python
+    name: python3
+---
+
+
+#  $\lambda$-calcul
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Analyseur lexical
+<!-- #endregion -->
+
+```python
+from sly import Lexer
+```
+
+```python
+class LambdaSyntaxError(Exception):
+    def __init__(self, msg):
+        self.message = msg
+```
+
+```python
+class Lambda_lexer(Lexer):
+    tokens = {VAR, LPAR, RPAR, LAMBDA, POINT}
+    VAR = r'[a-zA-Z][a-zA-Z0-9]*'
+    LPAR = r'\('
+    RPAR = r'\)'
+    LAMBDA = r'\!|λ'
+    POINT = r'\.'
+    ignore = ' \t'
+    
+    # Define a rule so we can track line numbers
+    @_(r'\n+')
+    def ignore_newline(self, t):
+        self.lineno += len(t.value)
+        
+    def error(self, t):
+        raise LambdaSyntaxError('Caractère illégal : {:s} at position {:d}'.format(t.value[0], t.index))
+        self.index += 1
+```
+
+```python
+lexer = Lambda_lexer()
+```
+
+```python
+l = list(lexer.tokenize('''((!x.(x x) y)
+λx.y)'''))
+l
+```
+
+```python
+list(lexer.tokenize('!x.[y z]'))
+```
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Analyseur syntaxique
+<!-- #endregion -->
+
+Voici la grammaire du langage décrivant les $\lambda$-termes
+
+    term ::= VAR | LAMBDA VAR POINT term | LPAR term term RPAR
+
+```python
+from sly import Parser
+```
+
+```python
+class Lambda_parser(Parser):
+    tokens = Lambda_lexer.tokens
+    
+    @_('VAR')
+    def term(self, p):
+        return Lambda_terme(0, p[0])
+    
+    @_('LAMBDA VAR POINT term')
+    def term(self, p):
+        return Lambda_terme(1, p[1], p.term)
+    
+    @_('LPAR term term RPAR')
+    def term(self, p):
+        return Lambda_terme(2, p.term0, p.term1)
+```
+
+```python
+parser = Lambda_parser()
+```
+
+## La classe `Lambda_terme`
+
+```python
+def autre_variable(var, variables):
+    n = 0
+    while var + '{:d}'.format(n) in variables:
+        n += 1
+    return var + '{:d}'.format(n)
+
+class Lambda_terme():
+    def __init__(self, categorie, *args):
+        if categorie not in (0, 1, 2):
+            raise Exception('categorie non valide')
+        if categorie == 0: 
+            if len(args) != 1 or not isinstance(args[0], str):
+                raise Exception('mauvaise construction pour une variable')
+        elif categorie == 1:
+            if len(args) != 2 or not isinstance(args[0], str) or not isinstance(args[1], Lambda_terme):
+                raise Exception('mauvaise construction pour une abstraction')
+        else:
+            if len(args) != 2 or not isinstance(args[0], Lambda_terme) or not isinstance(args[1], Lambda_terme):
+                raise Exception('mauvaise construction pour une application')
+        self._content = (categorie,) + tuple(args)
+        
+    @staticmethod
+    def cree(descr):
+        return parser.parse(lexer.tokenize(descr))
+    
+    def est_variable(self):
+        return self._content[0] == 0
+
+    def est_abstraction(self):
+        return self._content[0] == 1
+
+    def est_application(self):
+        return self._content[0] == 2
+
+    def applique(self, terme):
+        if not isinstance(terme, Lambda_terme):
+            raise Exception('Application impossible')
+        return Lambda_terme(2, self, terme)
+    
+    def abstrait(self, var):
+        if not isinstance(var, str):
+            raise Exception("Variable d'Abstraction invalide")
+        return Lambda_terme(1, var, self)
+    
+    def est_redex(self):
+        return self.est_application() and self._content[1].est_abstraction()
+    
+    def est_forme_normale(self):
+        if self.est_variable():
+            return True
+        elif self.est_abstraction():
+            return self._content[2].est_forme_normale()
+        else:
+            return not self._content[1].est_abstraction() and all(self._content[k].est_forme_normale() for k in (1, 2))
+        
+    def variables_libres(self):
+        if self.est_variable():
+            return {self._content[1]}
+        elif self.est_application():
+            var_libres = self._content[2].variables_libres()
+            return var_libres.union(self._content[1].variables_libres())
+        else:
+            var_libres = self._content[2].variables_libres()
+            var_libres.discard(self._content[1])
+            return var_libres
+    
+    def subs(self, var, terme):
+        if not isinstance(var, str):
+            raise Exception('subst possible uniquement pour les variables')
+        if not isinstance(terme, Lambda_terme):
+            raise Exception('subst possible uniquement sur un lambda-terme')
+        if self.est_variable():
+            if var == self._content[1]:
+                return terme
+            else:
+                return self
+        elif self.est_application():
+            return Lambda_terme(2, self._content[1].subs(var, terme), self._content[2].subs(var, terme))
+        else:
+            var_abstr = self._content[1]
+            corps_abstr = self._content[2]
+            var_libres_corps = corps_abstr.variables_libres()
+            if var == var_abstr or var not in var_libres_corps:
+                return self
+            elif var_abstr not in terme.variables_libres():
+                return Lambda_terme(1, var_abstr, corps_abstr.subs(var, terme))
+            else:
+                nouvelle_var = autre_variable(var_abstr, corps_abstr.variables_libres())
+                return Lambda_terme(1, 
+                                    nouvelle_var, 
+                                    corps_abstr.subs(var_abstr, Lambda_terme(0, nouvelle_var)).subs(var, terme))
+                    
+                
+    def _reduit(self):
+        if self.est_variable():
+            return self, False
+        elif self.est_abstraction():
+            corps_reduit, est_reduit = self._content[2]._reduit()
+            return (Lambda_terme(1, self._content[1], corps_reduit) if est_reduit else self, est_reduit)
+        else:
+            termegauche = self._content[1]
+            termedroit = self._content[2]
+            if termegauche.est_abstraction():
+                var_abstr = termegauche._content[1]
+                corps = termegauche._content[2]
+                return corps.subs(var_abstr, termedroit), True
+            else:
+                termegauche_reduit, est_reduit = termegauche._reduit()
+                if est_reduit:
+                    return Lambda_terme(2, termegauche_reduit, termedroit), est_reduit
+                else:
+                    termedroit_reduit, est_reduit = termedroit._reduit()
+                    return (Lambda_terme(2, termegauche, termedroit_reduit) if est_reduit else self, est_reduit)
+                
+    def reduit(self):
+        return self._reduit()
+            
+    def __str__(self):
+        if self.est_variable():
+            return self._content[1]
+        elif self.est_abstraction():
+            return 'λ{:s}.{:s}'.format(self._content[1], str(self._content[2]))
+        else:
+            return '({:s} {:s})'.format(str(self._content[1]), str(self._content[2]))
+```
+
+```python
+T1 = Lambda_terme(0, "x")
+T2 = Lambda_terme(1, "x", T1)
+T3 = Lambda_terme.cree('(!x.x x)')
+```
+
+```python
+print(T1)
+print(T2)
+print(T3)
+```
+
+```python
+tuple(t.est_variable() for t in (T1, T2, T3))
+```
+
+```python
+tuple(t.est_abstraction() for t in (T1, T2, T3))
+```
+
+```python
+tuple(t.est_application() for t in (T1, T2, T3))
+```
+
+```python
+tuple(t.est_redex() for t in (T1, T2, T3))
+```
+
+```python
+tuple(t.est_forme_normale() for t in (T1, T2, T3))
+```
+
+```python
+tuple(t.variables_libres() for t in (T1, T2, T3))
+```
+
+```python
+print(T1, '-->', T1.subs('y', Lambda_terme.cree('(y x)')))
+print(T1, '-->', T1.subs('x', Lambda_terme.cree('(y x)')))
+```
+
+```python
+T4 = Lambda_terme.cree('!x.y')
+print(T4, '-->', T4.subs('x', Lambda_terme.cree('(y z)')))
+print(T4, '-->', T4.subs('y', Lambda_terme.cree('(t z)')))
+print(T4, '-->', T4.subs('y', Lambda_terme.cree('(x z)')))
+```
+
+```python
+print(T3, '-->', T3.subs('y', Lambda_terme.cree('(y x)')))
+print(T3, '-->', T3.subs('x', Lambda_terme.cree('(y x)')))
+```
+
+```python
+
+```
+
+```python
+OMEGA = Lambda_terme.cree('(!x.(x x) !x.(x x))')
+```
+
+```python
+print(OMEGA)
+```
+
+```python
+OMEGA.est_redex()
+```
+
+```python
+OMEGA.est_forme_normale()
+```
+
+```python
+OMEGA.variables_libres()
+```
+
+```python
+
+```
+
+```python
+res, est_red = OMEGA.reduit()
+print(res)
+print(est_red)
+```
+
+```python
+res, est_red = Lambda_terme.cree('(!x.(eric x) vero)').reduit()
+print(res, est_red)
+```
+
+```python
+def calcul(lambda_terme, nb_etapes_max=100, verbose=False):
+    etape = 0
+    forme_normale_atteinte = False
+    if verbose: print(lambda_terme)
+    while not forme_normale_atteinte and etape < nb_etapes_max:
+        etape += 1
+        terme_reduit, est_reduit = lambda_terme.reduit()
+        if verbose: print('{:3d}: ---> {:s}'.format(etape, str(lambda_terme), str(terme_reduit)))
+        forme_normale_atteinte = not est_reduit
+        lambda_terme = terme_reduit
+    if forme_normale_atteinte:
+        if verbose: print('Forme normale calculée : {:s}'.format(str(terme_reduit)))
+        return terme_reduit
+    else:
+        if verbose: print('Pas de forme normale atteinte après {:d} étapes de réduction'.format(etape))
+        return None
+```
+
+```python
+calcul(OMEGA, nb_etapes_max=10, verbose=True)
+```
+
+## Entiers, successeurs, addition, multiplication et exponentiation
+
+```python
+ZERO = Lambda_terme.cree('!f.!x.x')
+```
+
+```python
+UN = Lambda_terme.cree('!f.!x.(f x)')
+```
+
+```python
+DEUX = Lambda_terme.cree('!f.!x.(f (f x))')
+```
+
+```python
+SUC = Lambda_terme.cree('!n.!f.!x.(f ((n f) x))')
+```
+
+```python
+TROIS = calcul(SUC.applique(DEUX), verbose=True)
+```
+
+```python
+calcul(TROIS.applique(SUC).applique(ZERO), verbose=True)
+```
+
+```python
+ADD = Lambda_terme.cree('!n.!m.!f.!x.((n f) ((m f) x))')
+```
+
+```python
+QUATRE = calcul(ADD.applique(UN).applique(TROIS), verbose=True)
+```
+
+```python
+CINQ = calcul(ADD.applique(TROIS).applique(DEUX), verbose=True)
+```
+
+```python
+SEPT = calcul(ADD.applique(QUATRE).applique(TROIS), verbose=True)
+```
+
+```python
+MUL = Lambda_terme.cree('!n.!m.!f.(n (m f))')
+```
+
+```python
+SIX = calcul(MUL.applique(DEUX).applique(TROIS), verbose=True)
+```
+
+```python
+EXP = Lambda_terme.cree('!n.!m.(m n)')
+```
+
+```python
+HUIT = calcul(EXP.applique(DEUX).applique(TROIS), verbose=True)
+```
+
+```python
+NEUF = calcul(EXP.applique(TROIS).applique(DEUX), verbose=True)
+```
+
+## Booléens, opérateurs logiques et conditionnelles
+
+```python
+VRAI = Lambda_terme.cree('!x.!y.x')
+```
+
+```python
+FAUX = Lambda_terme.cree('!x.!y.y')
+```
+
+```python
+COND = Lambda_terme.cree('!c.!a.!s.((c a) s)') 
+```
+
+```python
+calcul(COND.applique(VRAI).applique(UN).applique(DEUX), verbose=True)
+```
+
+```python
+calcul(COND.applique(FAUX).applique(UN).applique(DEUX), verbose=True)
+```
+
+```python
+ET = COND.applique(Lambda_terme.cree('a')).applique(Lambda_terme.cree('b')).applique(FAUX).abstrait('b').abstrait('a')
+```
+
+```python
+print(ET)
+```
+
+```python
+calcul(ET.applique(VRAI).applique(VRAI), verbose=True)
+```
+
+```python
+calcul(ET.applique(VRAI).applique(FAUX), verbose=True)
+```
+
+```python
+calcul(ET.applique(FAUX).applique(VRAI), verbose=True)
+```
+
+```python
+calcul(ET.applique(FAUX).applique(FAUX), verbose=True)
+```
+
+```python
+OU = COND.applique(Lambda_terme.cree('a')).applique(VRAI).applique(Lambda_terme.cree('b')).abstrait('b').abstrait('a')
+```
+
+```python
+calcul(OU.applique(VRAI).applique(VRAI), verbose=True)
+```
+
+```python
+calcul(OU.applique(VRAI).applique(FAUX), verbose=True)
+```
+
+```python
+calcul(OU.applique(FAUX).applique(VRAI), verbose=True)
+```
+
+```python
+calcul(OU.applique(FAUX).applique(FAUX), verbose=True)
+```
+
+```python
+NON = COND.applique(Lambda_terme.cree('a')).applique(FAUX).applique(VRAI).abstrait('a')
+```
+
+```python
+calcul(NON.applique(VRAI), verbose=True)
+```
+
+```python
+calcul(NON.applique(FAUX), verbose=True)
+```
+
+```python
+NUL = Lambda_terme.cree('!n.((n !x.{:s}) {:s})'.format(str(FAUX), str(VRAI)))
+```
+
+```python
+print(NUL)
+```
+
+```python
+calcul(NUL.applique(ZERO), verbose=True)
+```
+
+```python
+calcul(NUL.applique(TROIS), verbose=True)
+```
+
+## Couples
+
+```python
+CONS = COND.applique(Lambda_terme.cree('s')).applique(Lambda_terme.cree('x')).applique(Lambda_terme.cree('y')).abstrait('s').abstrait('y').abstrait('x')
+```
+
+```python
+print(CONS)
+```
+
+```python
+UN_DEUX = calcul(CONS.applique(UN).applique(DEUX), verbose=True)
+```
+
+```python
+CAR = Lambda_terme.cree('c').applique(VRAI).abstrait('c')
+```
+
+```python
+print(CAR)
+```
+
+```python
+calcul(CAR.applique(UN_DEUX), verbose=True)
+```
+
+```python
+CDR = Lambda_terme.cree('c').applique(FAUX).abstrait('c')
+```
+
+```python
+calcul(CDR.applique(UN_DEUX), verbose=True)
+```
+
+```python
+M_PRED = Lambda_terme.cree('!n.(CAR ((n !c.((CONS (CDR c)) (SUC (CDR c)))) ((CONS ZERO) ZERO)))')
+print(M_PRED)
+PRED = M_PRED.subs('CAR', CAR).subs('CONS', CONS).subs('CDR', CDR).subs('SUC', SUC).subs('ZERO', ZERO)
+print(PRED)
+```
+
+```python
+calcul(PRED.applique(DEUX), verbose=True)
+```
+
+```python
+calcul(PRED.applique(ZERO), verbose=True)
+```
+
+```python
+M_SUB = Lambda_terme.cree('!n.!m.((m PRED) n)')
+print(M_SUB)
+SUB = M_SUB.subs('PRED', PRED)
+print(SUB)
+```
+
+```python
+calcul(SUB.applique(TROIS).applique(UN), verbose=True)
+```
+
+```python
+M_INF = Lambda_terme.cree('!n.!m.(NUL ((SUB m) n))')
+print(M_INF)
+INF = M_INF.subs('NUL', NUL).subs('SUB', SUB)
+#lambda n: lambda m: est_nul(sub(m)(n))
+```
+
+```python
+calcul(INF.applique(TROIS).applique(UN), verbose=True)
+```
+
+```python
+calcul(INF.applique(UN).applique(TROIS), verbose=True)
+```
+
+```python
+calcul(INF.applique(UN).applique(UN), verbose=True)
+```
+
+```python
+M_EGAL = Lambda_terme.cree('!n.!m.((ET ((INF n) m)) ((INF m) n))')
+print(M_EGAL)
+EGAL = M_EGAL.subs('ET', ET).subs('INF', INF)
+print(EGAL)
+```
+
+```python
+print(calcul(EGAL.applique(UN).applique(UN)))
+```
+
+```python
+print(calcul(EGAL.applique(UN).applique(DEUX)))
+```
+
+## Itération
+
+```python
+M_FACTv1 = Lambda_terme.cree('!n.(CDR ((n !c.((CONS (SUC (CAR c))) ((MUL (SUC (CAR c))) (CDR c)))) ((CONS ZERO) UN)))')
+print(M_FACTv1)
+FACTv1 = M_FACTv1.subs('CONS', CONS).subs('CAR', CAR).subs('CDR', CDR).subs('SUC', SUC).subs('MUL', MUL).subs('UN', UN).subs('ZERO', ZERO)
+print(FACTv1)
+```
+
+```python
+print(calcul(FACTv1.applique(ZERO)))
+```
+
+```python
+print(calcul(FACTv1.applique(UN)))
+```
+
+```python
+print(calcul(FACTv1.applique(DEUX)))
+```
+
+```python
+print(calcul(FACTv1.applique(DEUX), nb_etapes_max=200))
+```
+
+```python
+print(calcul(FACTv1.applique(TROIS), nb_etapes_max=500))
+```
+
+```python
+print(calcul(FACTv1.applique(QUATRE), nb_etapes_max=1700))
+```
+
+## Et la récursivité ? 
+
+```python
+M_PHI_FACT = Lambda_terme.cree('!f.!n.(((COND ((EGAL n) ZERO)) UN) ((MUL n) (f (PRED n))))')
+print(M_PHI_FACT)
+PHI_FACT = M_PHI_FACT.subs('COND', COND).subs('EGAL', EGAL).subs('ZERO', ZERO).subs('UN', UN).subs('MUL', MUL).subs('PRED', PRED)
+```
+
+```python
+BOTTOM = Lambda_terme.cree('!y.OMEGA').subs('OMEGA', OMEGA)
+print(BOTTOM)
+```
+
+```python
+FACT0 = PHI_FACT.applique(BOTTOM)
+```
+
+```python
+print(calcul(FACT0.applique(ZERO)))
+```
+
+```python
+calcul(FACT0.applique(UN), verbose=True)
+```
+
+```python
+FACT1 = PHI_FACT.applique(FACT0)
+```
+
+```python
+print(calcul(FACT1.applique(ZERO)))
+```
+
+```python
+print(calcul(FACT1.applique(UN), nb_etapes_max=200))
+```
+
+```python
+FIX_CURRY = Lambda_terme.cree('!f.(!x.(f (x x)) !x.(f (x x)))')
+print(FIX_CURRY)
+```
+
+```python
+FACTv2 = FIX_CURRY.applique(PHI_FACT)
+```
+
+```python
+print(calcul(FACTv2.applique(ZERO)))
+```
+
+```python
+print(calcul(FACTv2.applique(UN), nb_etapes_max=200))
+```
+
+```python
+print(calcul(FACTv2.applique(DEUX), nb_etapes_max=700))
+```
+
+```python
+print(calcul(FACTv2.applique(TROIS), nb_etapes_max=4000))
+```
+
+```python
+print(calcul(FACTv2.applique(QUATRE), nb_etapes_max=25000))
+```
+
+```python
+PF = FIX_CURRY.applique(Lambda_terme.cree('M'))
+```
+
+```python
+calcul(PF, verbose=True, nb_etapes_max=10)
+```
+
+```python
+
+```
+
+
+# $\lambda$-calcul avec les lambda-expressions de Python
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Les entiers de Church
+<!-- #endregion -->
+
+```python
+zero = lambda f: lambda x: x
+```
+
+```python
+un = lambda f: lambda x: f(x)
+```
+
+```python
+deux = lambda f: lambda x: f(f(x))
+```
+
+```python
+trois = lambda f: lambda x: f(f(f(x)))
+```
+
+```python
+def entier_church_en_int(ec):
+    return ec(lambda n: n+1)(0)
+```
+
+```python
+tuple(entier_church_en_int(n) for n in (zero, un, deux, trois))
+```
+
+```python
+suc = lambda n: lambda f: lambda x: f(n(f)(x))
+```
+
+```python
+tuple(entier_church_en_int(suc(n)) for n in (zero, un, deux, trois)) 
+```
+
+```python
+def int_en_entier_church(n):
+    if n == 0:
+        return zero
+    else:
+        return suc(int_en_entier_church(n - 1))
+```
+
+```python
+tuple(entier_church_en_int(int_en_entier_church(n)) for n in range(10))
+```
+
+```python
+add = lambda n: lambda m: lambda f: lambda x: n(f)(m(f)(x))
+```
+
+```python
+cinq = add(deux)(trois)
+entier_church_en_int(cinq)
+```
+
+```python
+mul = lambda n: lambda m: lambda f: n(m(f))
+```
+
+```python
+six = mul(deux)(trois)
+entier_church_en_int(six)
+```
+
+```python
+exp = lambda n: lambda m: m(n)
+```
+
+```python
+huit = exp(deux)(trois)
+entier_church_en_int(huit)
+```
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Les booléens
+<!-- #endregion -->
+
+```python
+vrai = lambda x: lambda y: x
+faux = lambda x: lambda y: y
+```
+
+```python
+def booleen_en_bool(b):
+    return b(True)(False)
+    
+```
+
+```python
+tuple(booleen_en_bool(b) for b in (vrai, faux))
+```
+
+```python
+cond = lambda c: lambda a: lambda s: c(a)(s) 
+```
+
+```python
+cond(vrai)(1)(2)
+```
+
+```python
+cond(faux)(1)(2)
+```
+
+```python
+cond(vrai)(1)(1/0)
+```
+
+```python
+non = lambda b: cond(b)(faux)(vrai)
+```
+
+```python
+tuple(booleen_en_bool(non(b)) for b in (vrai, faux))
+```
+
+```python
+et = lambda b1: lambda b2: cond(b1)(b2)(faux)
+```
+
+```python
+tuple(booleen_en_bool(et(b1)(b2)) for b1 in (vrai, faux) for b2 in (vrai, faux))
+```
+
+```python
+ou = lambda b1: lambda b2: cond(b1)(vrai)(b2)
+```
+
+```python
+tuple(booleen_en_bool(ou(b1)(b2)) for b1 in (vrai, faux) for b2 in (vrai, faux))
+```
+
+```python
+est_nul = lambda n : n(lambda x: faux)(vrai)
+```
+
+```python
+tuple(booleen_en_bool(est_nul(n)) for n in (zero, un, deux, trois, cinq, six, huit))
+```
+
+## Les couples
+
+```python
+cons = lambda x: lambda y: lambda z: z(x)(y)
+```
+
+```python
+un_deux = cons(un)(deux)
+```
+
+```python
+car = lambda c: c(vrai)
+cdr = lambda c: c(faux)
+```
+
+```python
+entier_church_en_int(car(un_deux)), entier_church_en_int(cdr(un_deux))
+```
+
+```python
+pred = lambda n: car(n(lambda c: cons(cdr(c))(suc(cdr(c))))(cons(zero)(zero)))
+```
+
+```python
+tuple(entier_church_en_int(pred(int_en_entier_church(n))) for n in range(10))
+```
+
+```python
+sub = lambda n: lambda m: m(pred)(n)
+```
+
+```python
+entier_church_en_int(sub(huit)(trois))
+```
+
+```python
+est_inf_ou_egal = lambda n: lambda m: est_nul(sub(m)(n))
+```
+
+```python
+tuple(booleen_en_bool(est_inf_ou_egal(cinq)(int_en_entier_church(n))) for n in range(10))
+```
+
+```python
+est_egal = lambda n: lambda m: et(est_inf_ou_egal(n)(m))(est_inf_ou_egal(m)(n))
+```
+
+```python
+tuple(booleen_en_bool(est_egal(cinq)(int_en_entier_church(n))) for n in range(10))
+```
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Itération
+<!-- #endregion -->
+
+```python
+fact = lambda n: cdr(n(lambda c: (cons(suc(car(c)))(mul(suc(car(c)))(cdr(c)))))(cons(zero)(un)))
+```
+
+```python
+tuple(entier_church_en_int(fact(int_en_entier_church(n))) for n in range(7))
+```
+
+## Combinateur de point fixe
+
+```python
+phi_fact = lambda f: lambda n: 1 if n == 0 else n*f(n-1)
+```
+
+```python
+bottom = lambda x: (lambda y: y(y))(lambda y:y(y))
+```
+
+```python
+f0 = phi_fact(bottom)
+f1 = phi_fact(f0)
+f2 = phi_fact(f1)
+f3 = phi_fact(f2)
+f4 = phi_fact(f3)
+```
+
+```python
+tuple(f4(n) for n in range(4))
+```
+
+```python
+def fact_rec(n):
+    if n == 0:
+        return 1
+    else:
+        return n * fact_rec(n - 1)
+```
+
+```python
+fact2 = phi_fact(fact_rec)
+```
+
+```python
+tuple(fact2(n) for n in range(7))
+```
+
+```python
+fix_curry = lambda f: (lambda x: lambda y: f(x(x))(y))(lambda x: lambda y: f(x(x))(y))
+```
+
+```python
+fact3 = fix_curry(phi_fact)
+```
+
+```python
+tuple(fact3(n) for n in range(7))
+```
+
+<!-- #region toc-hr-collapsed=true toc-nb-collapsed=true -->
+## Un programme obscur
+<!-- #endregion -->
+
+```python
+print((lambda x: (lambda y: lambda z: x(y(y))(z))(lambda y: lambda z: x(y(y))(z))) 
+      (lambda x: lambda y: '' if y == [] else chr(y[0])+x(y[1:]))
+      (((lambda x: (lambda y: lambda z: x(y(y))(z)) (lambda y: lambda z: x(y(y))(z)))
+        (lambda x: lambda y: lambda z: [] if z == [] else [y(z[0])]+x(y)(z[1:])))      
+       (lambda x: (lambda x: (lambda y: lambda z: x(y(y))(z))(lambda y: lambda z: x(y(y))(z)))
+        (lambda x: lambda y: lambda z: lambda t: 1 if t == 0 else (lambda x: ((lambda u: 1 if u == 0 else z)(t % 2)) * x * x % y)
+         (x(y)(z)(t // 2)))(989)(x)(761))
+       ([377, 900, 27, 27, 182, 647, 163, 182, 390, 27, 726, 937])))
+```
+
+```python
+phiListEnChaine = lambda x: lambda y: '' if y == [] else chr(y[0]) + x(y[1:])
+```
+
+```python
+fix_curry(phiListEnChaine)([65+k for k in range(26)])
+```
+
+```python
+phiMap = lambda x: lambda y: lambda z: [] if z == [] else [y(z[0])] + x(y)(z[1:])
+```
+
+```python
+fix_curry(phiMap)(lambda x: x*x)([1, 2, 3, 4])
+```
+
+```python
+phiExpoMod = lambda x: lambda y: lambda z: lambda t: 1 if z == 0 else (lambda u: 1 if u == 0 else y)(z % 2) * x(y)(z//2)(t) ** 2 % t
+```
+
+```python
+fix_curry(phiExpoMod)(2)(10)(1000)
+```