3100 lines
125 KiB
Plaintext
3100 lines
125 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"# $\\lambda$-calcul"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"from lambda_calcul import Lambda_terme"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"lines_to_next_cell": 0
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},
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"source": [
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"Voici la grammaire du langage décrivant les $\\lambda$-termes\n",
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"\n",
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" term ::= VAR | LAMBDA VAR POINT term | LPAR term term RPAR\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## La classe `Lambda_terme`"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"T1 = Lambda_terme(\"x\")\n",
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"T2 = Lambda_terme(\"(x x)\")\n",
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"T3 = Lambda_terme(\"!x.x\")\n",
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"T4 = Lambda_terme('!x.(x x)')\n",
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"T5 = Lambda_terme('(!x.(x y) z)')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"x\n",
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"(x x)\n",
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"λx.x\n",
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"λx.(x x)\n",
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"(λx.(x y) z)\n"
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]
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}
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],
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"source": [
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"print(T1)\n",
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"print(T2)\n",
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"print(T3)\n",
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"print(T4)\n",
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"print(T5)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"termes = (T1, T2, T3, T4, T5)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(True, False, False, False, False)"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.est_variable() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(False, False, True, True, False)"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.est_abstraction() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(False, True, False, False, True)"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.est_application() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(False, False, False, False, True)"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.est_redex() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(True, True, True, True, False)"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.est_forme_normale() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"({'x'}, {'x'}, set(), set(), {'y', 'z'})"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tuple(t.variables_libres() for t in termes)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"x --> x\n",
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"x --> (y x)\n"
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]
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}
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],
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"source": [
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"print(T1, '-->', T1.subs('y', Lambda_terme('(y x)')))\n",
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"print(T1, '-->', T1.subs('x', Lambda_terme('(y x)')))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"λx.y --> λx.y\n",
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"λx.y --> λx.(t z)\n",
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"λx.y --> λx0.(x z)\n"
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]
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}
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],
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"source": [
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"T5 = Lambda_terme('!x.y')\n",
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"print(T5, '-->', T5.subs('x', Lambda_terme('(y z)')))\n",
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"print(T5, '-->', T5.subs('y', Lambda_terme('(t z)')))\n",
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"print(T5, '-->', T5.subs('y', Lambda_terme('(x z)')))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"λx.x --> λx.x\n",
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"λx.x --> λx.x\n"
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]
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}
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],
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"source": [
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"print(T3, '-->', T3.subs('y', Lambda_terme('(y x)')))\n",
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"print(T3, '-->', T3.subs('x', Lambda_terme('(y x)')))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"λx.x\n",
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"λy.x\n",
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"λx.(x x)\n",
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"λy.(x x)\n",
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"λx.λx.x\n",
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"λy.λx.x\n",
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"λx.λx.(x x)\n",
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"λy.λx.(x x)\n",
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"λx.(λx.(x y) z)\n",
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"λy.(λx.(x y) z)\n"
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]
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}
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],
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"source": [
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"for t in termes:\n",
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" for v in ('x', 'y'):\n",
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" print(t.abstrait(v))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(x x)\n",
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"(x (x x))\n",
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"(x λx.x)\n",
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"(x λx.(x x))\n",
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"(x (λx.(x y) z))\n",
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"((x x) x)\n",
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"((x x) (x x))\n",
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"((x x) λx.x)\n",
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"((x x) λx.(x x))\n",
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"((x x) (λx.(x y) z))\n",
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"(λx.x x)\n",
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"(λx.x (x x))\n",
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"(λx.x λx.x)\n",
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"(λx.x λx.(x x))\n",
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"(λx.x (λx.(x y) z))\n",
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"(λx.(x x) x)\n",
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"(λx.(x x) (x x))\n",
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"(λx.(x x) λx.x)\n",
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"(λx.(x x) λx.(x x))\n",
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"(λx.(x x) (λx.(x y) z))\n",
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"((λx.(x y) z) x)\n",
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"((λx.(x y) z) (x x))\n",
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"((λx.(x y) z) λx.x)\n",
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"((λx.(x y) z) λx.(x x))\n",
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"((λx.(x y) z) (λx.(x y) z))\n"
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]
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}
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],
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"source": [
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"for t1 in termes:\n",
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" for t2 in termes:\n",
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" print(t1.applique(t2))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
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"OMEGA = Lambda_terme('(!x.(x x) !x.(x x))')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(λx.(x x) λx.(x x))\n"
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]
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}
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],
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"source": [
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"print(OMEGA)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 18,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"OMEGA.est_redex()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"False"
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]
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},
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"execution_count": 19,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"OMEGA.est_forme_normale()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"set()"
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]
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},
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"execution_count": 20,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"OMEGA.variables_libres()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": 21,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(λx.(x x) λx.(x x))\n",
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"True\n"
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]
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}
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],
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"source": [
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"res, est_red = OMEGA.reduit()\n",
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"print(res)\n",
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"print(est_red)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(eric vero) True\n"
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]
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}
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],
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"source": [
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"res, est_red = Lambda_terme('(!x.(eric x) vero)').reduit()\n",
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"print(res, est_red)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
|
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"(λx.(x x) λx.(x x))\n",
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" 1: ---> (λx.(x x) λx.(x x))\n",
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" 2: ---> (λx.(x x) λx.(x x))\n",
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" 3: ---> (λx.(x x) λx.(x x))\n",
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" 4: ---> (λx.(x x) λx.(x x))\n",
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" 5: ---> (λx.(x x) λx.(x x))\n",
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" 6: ---> (λx.(x x) λx.(x x))\n",
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" 7: ---> (λx.(x x) λx.(x x))\n",
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" 8: ---> (λx.(x x) λx.(x x))\n",
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" 9: ---> (λx.(x x) λx.(x x))\n",
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" 10: ---> (λx.(x x) λx.(x x))\n",
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"Pas de forme normale atteinte après 10 étapes de réduction\n"
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]
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}
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],
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"source": [
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"OMEGA.forme_normale(nb_etapes_max=10, verbose=True)"
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]
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},
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{
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|
"cell_type": "markdown",
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"metadata": {},
|
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"source": [
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"## Entiers, successeurs, addition, multiplication et exponentiation"
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]
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},
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{
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"cell_type": "code",
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|
"execution_count": 25,
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"metadata": {},
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"outputs": [],
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"source": [
|
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"ZERO = Lambda_terme('!f.!x.x')"
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]
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},
|
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{
|
|
"cell_type": "code",
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|
"execution_count": 26,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
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"UN = Lambda_terme('!f.!x.(f x)')"
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|
]
|
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},
|
|
{
|
|
"cell_type": "code",
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|
"execution_count": 27,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
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"DEUX = Lambda_terme('!f.!x.(f (f x))')"
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]
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},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 28,
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|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
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"SUC = Lambda_terme('!n.!f.!x.(f ((n f) x))')"
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]
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},
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{
|
|
"cell_type": "code",
|
|
"execution_count": 29,
|
|
"metadata": {},
|
|
"outputs": [
|
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
|
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"(λn.λf.λx.(f ((n f) x)) λf.λx.(f (f x)))\n",
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" 1: ---> (λn.λf.λx.(f ((n f) x)) λf.λx.(f (f x)))\n",
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" 2: ---> λf.λx.(f ((λf.λx.(f (f x)) f) x))\n",
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" 3: ---> λf.λx.(f (λx.(f (f x)) x))\n",
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" 4: ---> λf.λx.(f (f (f x)))\n",
|
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"Forme normale calculée : λf.λx.(f (f (f x)))\n"
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]
|
|
}
|
|
],
|
|
"source": [
|
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"TROIS = SUC.applique(DEUX).forme_normale(verbose=True)"
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]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 30,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
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"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λf.λx.(f (f (f x))) λn.λf.λx.(f ((n f) x))) λf.λx.x)\n",
|
|
" 1: ---> ((λf.λx.(f (f (f x))) λn.λf.λx.(f ((n f) x))) λf.λx.x)\n",
|
|
" 2: ---> (λx.(λn.λf.λx.(f ((n f) x)) (λn.λf.λx.(f ((n f) x)) (λn.λf.λx.(f ((n f) x)) x))) λf.λx.x)\n",
|
|
" 3: ---> (λn.λf.λx.(f ((n f) x)) (λn.λf.λx.(f ((n f) x)) (λn.λf.λx.(f ((n f) x)) λf.λx.x)))\n",
|
|
" 4: ---> λf.λx.(f (((λn.λf.λx.(f ((n f) x)) (λn.λf.λx.(f ((n f) x)) λf.λx.x)) f) x))\n",
|
|
" 5: ---> λf.λx.(f ((λf.λx.(f (((λn.λf.λx.(f ((n f) x)) λf.λx.x) f) x)) f) x))\n",
|
|
" 6: ---> λf.λx.(f (λx.(f (((λn.λf.λx.(f ((n f) x)) λf.λx.x) f) x)) x))\n",
|
|
" 7: ---> λf.λx.(f (f (((λn.λf.λx.(f ((n f) x)) λf.λx.x) f) x)))\n",
|
|
" 8: ---> λf.λx.(f (f ((λf.λx.(f ((λf.λx.x f) x)) f) x)))\n",
|
|
" 9: ---> λf.λx.(f (f (λx.(f ((λf.λx.x f) x)) x)))\n",
|
|
" 10: ---> λf.λx.(f (f (f ((λf.λx.x f) x))))\n",
|
|
" 11: ---> λf.λx.(f (f (f (λx.x x))))\n",
|
|
" 12: ---> λf.λx.(f (f (f x)))\n",
|
|
"Forme normale calculée : λf.λx.(f (f (f x)))\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed71d9b7f0>"
|
|
]
|
|
},
|
|
"execution_count": 30,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"TROIS.applique(SUC).applique(ZERO).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 31,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"ADD = Lambda_terme('!n.!m.!f.!x.((n f) ((m f) x))')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 32,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f x)) λf.λx.(f (f (f x))))\n",
|
|
" 1: ---> ((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f x)) λf.λx.(f (f (f x))))\n",
|
|
" 2: ---> (λm.λf.λx.((λf.λx.(f x) f) ((m f) x)) λf.λx.(f (f (f x))))\n",
|
|
" 3: ---> λf.λx.((λf.λx.(f x) f) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 4: ---> λf.λx.(λx.(f x) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 5: ---> λf.λx.(f ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 6: ---> λf.λx.(f (λx.(f (f (f x))) x))\n",
|
|
" 7: ---> λf.λx.(f (f (f (f x))))\n",
|
|
"Forme normale calculée : λf.λx.(f (f (f (f x))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"QUATRE = ADD.applique(UN).applique(TROIS).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 33,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f (f (f x)))) λf.λx.(f (f x)))\n",
|
|
" 1: ---> ((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f (f (f x)))) λf.λx.(f (f x)))\n",
|
|
" 2: ---> (λm.λf.λx.((λf.λx.(f (f (f x))) f) ((m f) x)) λf.λx.(f (f x)))\n",
|
|
" 3: ---> λf.λx.((λf.λx.(f (f (f x))) f) ((λf.λx.(f (f x)) f) x))\n",
|
|
" 4: ---> λf.λx.(λx.(f (f (f x))) ((λf.λx.(f (f x)) f) x))\n",
|
|
" 5: ---> λf.λx.(f (f (f ((λf.λx.(f (f x)) f) x))))\n",
|
|
" 6: ---> λf.λx.(f (f (f (λx.(f (f x)) x))))\n",
|
|
" 7: ---> λf.λx.(f (f (f (f (f x)))))\n",
|
|
"Forme normale calculée : λf.λx.(f (f (f (f (f x)))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"CINQ = ADD.applique(TROIS).applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 34,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f (f (f (f x))))) λf.λx.(f (f (f x))))\n",
|
|
" 1: ---> ((λn.λm.λf.λx.((n f) ((m f) x)) λf.λx.(f (f (f (f x))))) λf.λx.(f (f (f x))))\n",
|
|
" 2: ---> (λm.λf.λx.((λf.λx.(f (f (f (f x)))) f) ((m f) x)) λf.λx.(f (f (f x))))\n",
|
|
" 3: ---> λf.λx.((λf.λx.(f (f (f (f x)))) f) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 4: ---> λf.λx.(λx.(f (f (f (f x)))) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 5: ---> λf.λx.(f (f (f (f ((λf.λx.(f (f (f x))) f) x)))))\n",
|
|
" 6: ---> λf.λx.(f (f (f (f (λx.(f (f (f x))) x)))))\n",
|
|
" 7: ---> λf.λx.(f (f (f (f (f (f (f x)))))))\n",
|
|
"Forme normale calculée : λf.λx.(f (f (f (f (f (f (f x)))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"SEPT = ADD.applique(QUATRE).applique(TROIS).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 35,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"MUL = Lambda_terme('!n.!m.!f.(n (m f))')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 36,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.λf.(n (m f)) λf.λx.(f (f x))) λf.λx.(f (f (f x))))\n",
|
|
" 1: ---> ((λn.λm.λf.(n (m f)) λf.λx.(f (f x))) λf.λx.(f (f (f x))))\n",
|
|
" 2: ---> (λm.λf.(λf.λx.(f (f x)) (m f)) λf.λx.(f (f (f x))))\n",
|
|
" 3: ---> λf.(λf.λx.(f (f x)) (λf.λx.(f (f (f x))) f))\n",
|
|
" 4: ---> λf.λx.((λf.λx.(f (f (f x))) f) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 5: ---> λf.λx.(λx.(f (f (f x))) ((λf.λx.(f (f (f x))) f) x))\n",
|
|
" 6: ---> λf.λx.(f (f (f ((λf.λx.(f (f (f x))) f) x))))\n",
|
|
" 7: ---> λf.λx.(f (f (f (λx.(f (f (f x))) x))))\n",
|
|
" 8: ---> λf.λx.(f (f (f (f (f (f x))))))\n",
|
|
"Forme normale calculée : λf.λx.(f (f (f (f (f (f x))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"SIX = MUL.applique(DEUX).applique(TROIS).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 37,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"EXP = Lambda_terme('!n.!m.(m n)')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 38,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.(m n) λf.λx.(f (f x))) λf.λx.(f (f (f x))))\n",
|
|
" 1: ---> ((λn.λm.(m n) λf.λx.(f (f x))) λf.λx.(f (f (f x))))\n",
|
|
" 2: ---> (λm.(m λf.λx.(f (f x))) λf.λx.(f (f (f x))))\n",
|
|
" 3: ---> (λf.λx.(f (f (f x))) λf.λx.(f (f x)))\n",
|
|
" 4: ---> λx.(λf.λx.(f (f x)) (λf.λx.(f (f x)) (λf.λx.(f (f x)) x)))\n",
|
|
" 5: ---> λx.λx0.((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0))\n",
|
|
" 6: ---> λx.λx0.(λx0.((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) x) x0)) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0))\n",
|
|
" 7: ---> λx.λx0.((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0)))\n",
|
|
" 8: ---> λx.λx0.(λx0.(x (x x0)) ((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0)))\n",
|
|
" 9: ---> λx.λx0.(x (x ((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0))))\n",
|
|
" 10: ---> λx.λx0.(x (x (λx0.(x (x x0)) ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0))))\n",
|
|
" 11: ---> λx.λx0.(x (x (x (x ((λf.λx.(f (f x)) (λf.λx.(f (f x)) x)) x0)))))\n",
|
|
" 12: ---> λx.λx0.(x (x (x (x (λx0.((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) x) x0)) x0)))))\n",
|
|
" 13: ---> λx.λx0.(x (x (x (x ((λf.λx.(f (f x)) x) ((λf.λx.(f (f x)) x) x0))))))\n",
|
|
" 14: ---> λx.λx0.(x (x (x (x (λx0.(x (x x0)) ((λf.λx.(f (f x)) x) x0))))))\n",
|
|
" 15: ---> λx.λx0.(x (x (x (x (x (x ((λf.λx.(f (f x)) x) x0)))))))\n",
|
|
" 16: ---> λx.λx0.(x (x (x (x (x (x (λx0.(x (x x0)) x0)))))))\n",
|
|
" 17: ---> λx.λx0.(x (x (x (x (x (x (x (x x0))))))))\n",
|
|
"Forme normale calculée : λx.λx0.(x (x (x (x (x (x (x (x x0))))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"HUIT = EXP.applique(DEUX).applique(TROIS).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 39,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.(m n) λf.λx.(f (f (f x)))) λf.λx.(f (f x)))\n",
|
|
" 1: ---> ((λn.λm.(m n) λf.λx.(f (f (f x)))) λf.λx.(f (f x)))\n",
|
|
" 2: ---> (λm.(m λf.λx.(f (f (f x)))) λf.λx.(f (f x)))\n",
|
|
" 3: ---> (λf.λx.(f (f x)) λf.λx.(f (f (f x))))\n",
|
|
" 4: ---> λx.(λf.λx.(f (f (f x))) (λf.λx.(f (f (f x))) x))\n",
|
|
" 5: ---> λx.λx0.((λf.λx.(f (f (f x))) x) ((λf.λx.(f (f (f x))) x) ((λf.λx.(f (f (f x))) x) x0)))\n",
|
|
" 6: ---> λx.λx0.(λx0.(x (x (x x0))) ((λf.λx.(f (f (f x))) x) ((λf.λx.(f (f (f x))) x) x0)))\n",
|
|
" 7: ---> λx.λx0.(x (x (x ((λf.λx.(f (f (f x))) x) ((λf.λx.(f (f (f x))) x) x0)))))\n",
|
|
" 8: ---> λx.λx0.(x (x (x (λx0.(x (x (x x0))) ((λf.λx.(f (f (f x))) x) x0)))))\n",
|
|
" 9: ---> λx.λx0.(x (x (x (x (x (x ((λf.λx.(f (f (f x))) x) x0)))))))\n",
|
|
" 10: ---> λx.λx0.(x (x (x (x (x (x (λx0.(x (x (x x0))) x0)))))))\n",
|
|
" 11: ---> λx.λx0.(x (x (x (x (x (x (x (x (x x0)))))))))\n",
|
|
"Forme normale calculée : λx.λx0.(x (x (x (x (x (x (x (x (x x0)))))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"NEUF = EXP.applique(TROIS).applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Booléens, opérateurs logiques et conditionnelles"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 40,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"VRAI = Lambda_terme('!x.!y.x')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 41,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"FAUX = Lambda_terme('!x.!y.y')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 42,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"COND = Lambda_terme('!c.!a.!s.((c a) s)') "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 43,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(((λc.λa.λs.((c a) s) λx.λy.x) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 1: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 2: ---> ((λa.λs.((λx.λy.x a) s) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 3: ---> (λs.((λx.λy.x λf.λx.(f x)) s) λf.λx.(f (f x)))\n",
|
|
" 4: ---> ((λx.λy.x λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 5: ---> (λy.λf.λx.(f x) λf.λx.(f (f x)))\n",
|
|
" 6: ---> λf.λx.(f x)\n",
|
|
"Forme normale calculée : λf.λx.(f x)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed720bf438>"
|
|
]
|
|
},
|
|
"execution_count": 43,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"COND.applique(VRAI).applique(UN).applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 44,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(((λc.λa.λs.((c a) s) λx.λy.y) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 1: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 2: ---> ((λa.λs.((λx.λy.y a) s) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 3: ---> (λs.((λx.λy.y λf.λx.(f x)) s) λf.λx.(f (f x)))\n",
|
|
" 4: ---> ((λx.λy.y λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 5: ---> (λy.y λf.λx.(f (f x)))\n",
|
|
" 6: ---> λf.λx.(f (f x))\n",
|
|
"Forme normale calculée : λf.λx.(f (f x))\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed720bf748>"
|
|
]
|
|
},
|
|
"execution_count": 44,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"COND.applique(FAUX).applique(UN).applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 45,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"ET = COND.applique(Lambda_terme('a')).applique(Lambda_terme('b')).applique(FAUX).abstrait('b').abstrait('a')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 46,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(ET)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 47,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.x) λx.λy.x)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.x) λx.λy.x)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.x) b) λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.x) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.x a) s) λx.λy.x) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.x λx.λy.x) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.x λx.λy.x) λx.λy.y)\n",
|
|
" 7: ---> (λy.λx.λy.x λx.λy.y)\n",
|
|
" 8: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72488c88>"
|
|
]
|
|
},
|
|
"execution_count": 47,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"ET.applique(VRAI).applique(VRAI).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 48,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.x) b) λx.λy.y) λx.λy.y)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.y) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.x a) s) λx.λy.y) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.x λx.λy.y) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.x λx.λy.y) λx.λy.y)\n",
|
|
" 7: ---> (λy.λx.λy.y λx.λy.y)\n",
|
|
" 8: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72397b38>"
|
|
]
|
|
},
|
|
"execution_count": 48,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"ET.applique(VRAI).applique(FAUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 49,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.y) λx.λy.x)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.y) λx.λy.x)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.y) b) λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.y a) s) λx.λy.x) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.y λx.λy.x) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.y λx.λy.x) λx.λy.y)\n",
|
|
" 7: ---> (λy.y λx.λy.y)\n",
|
|
" 8: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72397b38>"
|
|
]
|
|
},
|
|
"execution_count": 49,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"ET.applique(FAUX).applique(VRAI).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 50,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.y) λx.λy.y)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) λx.λy.y) λx.λy.y)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.y) b) λx.λy.y) λx.λy.y)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.y) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.y a) s) λx.λy.y) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.y λx.λy.y) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.y λx.λy.y) λx.λy.y)\n",
|
|
" 7: ---> (λy.y λx.λy.y)\n",
|
|
" 8: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72397b38>"
|
|
]
|
|
},
|
|
"execution_count": 50,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"ET.applique(FAUX).applique(FAUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 51,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"OU = COND.applique(Lambda_terme('a')).applique(VRAI).applique(Lambda_terme('b')).abstrait('b').abstrait('a')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 52,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(OU)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 53,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.x) λx.λy.x)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.x) λx.λy.x)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.x) b) λx.λy.x)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.x) λx.λy.x)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.x a) s) λx.λy.x) λx.λy.x)\n",
|
|
" 5: ---> (λs.((λx.λy.x λx.λy.x) s) λx.λy.x)\n",
|
|
" 6: ---> ((λx.λy.x λx.λy.x) λx.λy.x)\n",
|
|
" 7: ---> (λy.λx.λy.x λx.λy.x)\n",
|
|
" 8: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72488c88>"
|
|
]
|
|
},
|
|
"execution_count": 53,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"OU.applique(VRAI).applique(VRAI).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 54,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.x) λx.λy.y)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.x) λx.λy.y)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.x) b) λx.λy.y)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.x) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.x a) s) λx.λy.x) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.x λx.λy.x) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.x λx.λy.x) λx.λy.y)\n",
|
|
" 7: ---> (λy.λx.λy.x λx.λy.y)\n",
|
|
" 8: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72488c88>"
|
|
]
|
|
},
|
|
"execution_count": 54,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"OU.applique(VRAI).applique(FAUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 55,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.y) λx.λy.x)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.y) λx.λy.x)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.x) b) λx.λy.x)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.x) λx.λy.x)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.y a) s) λx.λy.x) λx.λy.x)\n",
|
|
" 5: ---> (λs.((λx.λy.y λx.λy.x) s) λx.λy.x)\n",
|
|
" 6: ---> ((λx.λy.y λx.λy.x) λx.λy.x)\n",
|
|
" 7: ---> (λy.y λx.λy.x)\n",
|
|
" 8: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72488c88>"
|
|
]
|
|
},
|
|
"execution_count": 55,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"OU.applique(FAUX).applique(VRAI).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 56,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.y) λx.λy.y)\n",
|
|
" 1: ---> ((λa.λb.(((λc.λa.λs.((c a) s) a) λx.λy.x) b) λx.λy.y) λx.λy.y)\n",
|
|
" 2: ---> (λb.(((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.x) b) λx.λy.y)\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 4: ---> ((λa.λs.((λx.λy.y a) s) λx.λy.x) λx.λy.y)\n",
|
|
" 5: ---> (λs.((λx.λy.y λx.λy.x) s) λx.λy.y)\n",
|
|
" 6: ---> ((λx.λy.y λx.λy.x) λx.λy.y)\n",
|
|
" 7: ---> (λy.y λx.λy.y)\n",
|
|
" 8: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72397b38>"
|
|
]
|
|
},
|
|
"execution_count": 56,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"OU.applique(FAUX).applique(FAUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 57,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"NON = COND.applique(Lambda_terme('a')).applique(FAUX).applique(VRAI).abstrait('a')"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 58,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λa.(((λc.λa.λs.((c a) s) a) λx.λy.y) λx.λy.x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(NON)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 60,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λa.(((λc.λa.λs.((c a) s) a) λx.λy.y) λx.λy.x) λx.λy.x)\n",
|
|
" 1: ---> (λa.(((λc.λa.λs.((c a) s) a) λx.λy.y) λx.λy.x) λx.λy.x)\n",
|
|
" 2: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> ((λa.λs.((λx.λy.x a) s) λx.λy.y) λx.λy.x)\n",
|
|
" 4: ---> (λs.((λx.λy.x λx.λy.y) s) λx.λy.x)\n",
|
|
" 5: ---> ((λx.λy.x λx.λy.y) λx.λy.x)\n",
|
|
" 6: ---> (λy.λx.λy.y λx.λy.x)\n",
|
|
" 7: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72397b38>"
|
|
]
|
|
},
|
|
"execution_count": 60,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"NON.applique(VRAI).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 61,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λa.(((λc.λa.λs.((c a) s) a) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 1: ---> (λa.(((λc.λa.λs.((c a) s) a) λx.λy.y) λx.λy.x) λx.λy.y)\n",
|
|
" 2: ---> (((λc.λa.λs.((c a) s) λx.λy.y) λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> ((λa.λs.((λx.λy.y a) s) λx.λy.y) λx.λy.x)\n",
|
|
" 4: ---> (λs.((λx.λy.y λx.λy.y) s) λx.λy.x)\n",
|
|
" 5: ---> ((λx.λy.y λx.λy.y) λx.λy.x)\n",
|
|
" 6: ---> (λy.y λx.λy.x)\n",
|
|
" 7: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72488c88>"
|
|
]
|
|
},
|
|
"execution_count": 61,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"NON.applique(FAUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 62,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"NUL = Lambda_terme('!n.((n !x.{:s}) {:s})'.format(str(FAUX), str(VRAI)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 63,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.((n λx.λx.λy.y) λx.λy.x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(NUL)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 64,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λn.((n λx.λx.λy.y) λx.λy.x) λf.λx.x)\n",
|
|
" 1: ---> (λn.((n λx.λx.λy.y) λx.λy.x) λf.λx.x)\n",
|
|
" 2: ---> ((λf.λx.x λx.λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> (λx.x λx.λy.x)\n",
|
|
" 4: ---> λx.λy.x\n",
|
|
"Forme normale calculée : λx.λy.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed72233cf8>"
|
|
]
|
|
},
|
|
"execution_count": 64,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"NUL.applique(ZERO).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 65,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λn.((n λx.λx.λy.y) λx.λy.x) λf.λx.(f (f (f x))))\n",
|
|
" 1: ---> (λn.((n λx.λx.λy.y) λx.λy.x) λf.λx.(f (f (f x))))\n",
|
|
" 2: ---> ((λf.λx.(f (f (f x))) λx.λx.λy.y) λx.λy.x)\n",
|
|
" 3: ---> (λx.(λx.λx.λy.y (λx.λx.λy.y (λx.λx.λy.y x))) λx.λy.x)\n",
|
|
" 4: ---> (λx.λx.λy.y (λx.λx.λy.y (λx.λx.λy.y λx.λy.x)))\n",
|
|
" 5: ---> λx.λy.y\n",
|
|
"Forme normale calculée : λx.λy.y\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed722334e0>"
|
|
]
|
|
},
|
|
"execution_count": 65,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"NUL.applique(TROIS).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Couples"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 66,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"CONS = Lambda_terme('!x.!y.!s.((({:s} s) x) y)'.format(str(COND)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 67,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(CONS)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 68,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 1: ---> ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 2: ---> (λy.λs.(((λc.λa.λs.((c a) s) s) λf.λx.(f x)) y) λf.λx.(f (f x)))\n",
|
|
" 3: ---> λs.(((λc.λa.λs.((c a) s) s) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 4: ---> λs.((λa.λs0.((s a) s0) λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 5: ---> λs.(λs0.((s λf.λx.(f x)) s0) λf.λx.(f (f x)))\n",
|
|
" 6: ---> λs.((s λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
"Forme normale calculée : λs.((s λf.λx.(f x)) λf.λx.(f (f x)))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"UN_DEUX = CONS.applique(UN).applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 70,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"CAR = Lambda_terme('!c.(c {:s})'.format(str(VRAI)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 71,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λc.(c λx.λy.x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(CAR)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 72,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λc.(c λx.λy.x) λs.((s λf.λx.(f x)) λf.λx.(f (f x))))\n",
|
|
" 1: ---> (λc.(c λx.λy.x) λs.((s λf.λx.(f x)) λf.λx.(f (f x))))\n",
|
|
" 2: ---> (λs.((s λf.λx.(f x)) λf.λx.(f (f x))) λx.λy.x)\n",
|
|
" 3: ---> ((λx.λy.x λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 4: ---> (λy.λf.λx.(f x) λf.λx.(f (f x)))\n",
|
|
" 5: ---> λf.λx.(f x)\n",
|
|
"Forme normale calculée : λf.λx.(f x)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed720bf438>"
|
|
]
|
|
},
|
|
"execution_count": 72,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"CAR.applique(UN_DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 73,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"CDR = Lambda_terme('!c.(c {:s})'.format(str(FAUX)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 74,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λc.(c λx.λy.y)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(CDR)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 76,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λc.(c λx.λy.y) λs.((s λf.λx.(f x)) λf.λx.(f (f x))))\n",
|
|
" 1: ---> (λc.(c λx.λy.y) λs.((s λf.λx.(f x)) λf.λx.(f (f x))))\n",
|
|
" 2: ---> (λs.((s λf.λx.(f x)) λf.λx.(f (f x))) λx.λy.y)\n",
|
|
" 3: ---> ((λx.λy.y λf.λx.(f x)) λf.λx.(f (f x)))\n",
|
|
" 4: ---> (λy.y λf.λx.(f (f x)))\n",
|
|
" 5: ---> λf.λx.(f (f x))\n",
|
|
"Forme normale calculée : λf.λx.(f (f x))\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed720bf748>"
|
|
]
|
|
},
|
|
"execution_count": 76,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"CDR.applique(UN_DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 77,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.(CAR ((n λc.((CONS (CDR c)) (SUC (CDR c)))) ((CONS ZERO) ZERO)))\n",
|
|
"λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_PRED = Lambda_terme('!n.(CAR ((n !c.((CONS (CDR c)) (SUC (CDR c)))) ((CONS ZERO) ZERO)))')\n",
|
|
"print(M_PRED)\n",
|
|
"PRED = M_PRED.subs('CAR', CAR).subs('CONS', CONS).subs('CDR', CDR).subs('SUC', SUC).subs('ZERO', ZERO)\n",
|
|
"print(PRED)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 78,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f (f x)))\n",
|
|
" 1: ---> (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f (f x)))\n",
|
|
" 2: ---> (λc.(c λx.λy.x) ((λf.λx.(f (f x)) λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n",
|
|
" 3: ---> (((λf.λx.(f (f x)) λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
|
|
" 4: ---> ((λx.(λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) x)) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
|
|
" 5: ---> ((λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λx.λy.x)\n",
|
|
" 6: ---> (((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.x)\n",
|
|
" 7: ---> ((λy.λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.x)\n",
|
|
" 8: ---> (λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.x)\n",
|
|
" 9: ---> (((λc.λa.λs.((c a) s) λx.λy.x) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
|
|
" 10: ---> ((λa.λs.((λx.λy.x a) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
|
|
" 11: ---> (λs.((λx.λy.x (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) s) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
|
|
" 12: ---> ((λx.λy.x (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
|
|
" 13: ---> (λy.(λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
|
|
" 14: ---> (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n",
|
|
" 15: ---> ((λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.y)\n",
|
|
" 16: ---> (((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y)\n",
|
|
" 17: ---> ((λy.λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y)\n",
|
|
" 18: ---> (λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y)\n",
|
|
" 19: ---> (((λc.λa.λs.((c a) s) λx.λy.y) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
|
|
" 20: ---> ((λa.λs.((λx.λy.y a) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
|
|
" 21: ---> (λs.((λx.λy.y (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) s) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
|
|
" 22: ---> ((λx.λy.y (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
|
|
" 23: ---> (λy.y (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
|
|
" 24: ---> (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n",
|
|
" 25: ---> λf.λx.(f (((λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) f) x))\n",
|
|
" 26: ---> λf.λx.(f (((((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x) λx.λy.y) f) x))\n",
|
|
" 27: ---> λf.λx.(f ((((λy.λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) y) λf.λx.x) λx.λy.y) f) x))\n",
|
|
" 28: ---> λf.λx.(f (((λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) λf.λx.x) λx.λy.y) f) x))\n",
|
|
" 29: ---> λf.λx.(f (((((λc.λa.λs.((c a) s) λx.λy.y) λf.λx.x) λf.λx.x) f) x))\n",
|
|
" 30: ---> λf.λx.(f ((((λa.λs.((λx.λy.y a) s) λf.λx.x) λf.λx.x) f) x))\n",
|
|
" 31: ---> λf.λx.(f (((λs.((λx.λy.y λf.λx.x) s) λf.λx.x) f) x))\n",
|
|
" 32: ---> λf.λx.(f ((((λx.λy.y λf.λx.x) λf.λx.x) f) x))\n",
|
|
" 33: ---> λf.λx.(f (((λy.y λf.λx.x) f) x))\n",
|
|
" 34: ---> λf.λx.(f ((λf.λx.x f) x))\n",
|
|
" 35: ---> λf.λx.(f (λx.x x))\n",
|
|
" 36: ---> λf.λx.(f x)\n",
|
|
"Forme normale calculée : λf.λx.(f x)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
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"<lambda_calcul.Lambda_terme at 0x7fed726b20f0>"
|
|
]
|
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},
|
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"execution_count": 78,
|
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"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"PRED.applique(DEUX).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 79,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.x)\n",
|
|
" 1: ---> (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.x)\n",
|
|
" 2: ---> (λc.(c λx.λy.x) ((λf.λx.x λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n",
|
|
" 3: ---> (((λf.λx.x λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
|
|
" 4: ---> ((λx.x ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
|
|
" 5: ---> (((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x) λx.λy.x)\n",
|
|
" 6: ---> ((λy.λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) y) λf.λx.x) λx.λy.x)\n",
|
|
" 7: ---> (λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) λf.λx.x) λx.λy.x)\n",
|
|
" 8: ---> (((λc.λa.λs.((c a) s) λx.λy.x) λf.λx.x) λf.λx.x)\n",
|
|
" 9: ---> ((λa.λs.((λx.λy.x a) s) λf.λx.x) λf.λx.x)\n",
|
|
" 10: ---> (λs.((λx.λy.x λf.λx.x) s) λf.λx.x)\n",
|
|
" 11: ---> ((λx.λy.x λf.λx.x) λf.λx.x)\n",
|
|
" 12: ---> (λy.λf.λx.x λf.λx.x)\n",
|
|
" 13: ---> λf.λx.x\n",
|
|
"Forme normale calculée : λf.λx.x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed720bf0b8>"
|
|
]
|
|
},
|
|
"execution_count": 79,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"PRED.applique(ZERO).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 80,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.λm.((m PRED) n)\n",
|
|
"λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_SUB = Lambda_terme('!n.!m.((m PRED) n)')\n",
|
|
"print(M_SUB)\n",
|
|
"SUB = M_SUB.subs('PRED', PRED)\n",
|
|
"print(SUB)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 81,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) λf.λx.(f (f (f x)))) λf.λx.(f x))\n",
|
|
" 1: ---> ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) λf.λx.(f (f (f x)))) λf.λx.(f x))\n",
|
|
" 2: ---> (λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λf.λx.(f (f (f x)))) λf.λx.(f x))\n",
|
|
" 3: ---> ((λf.λx.(f x) λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λf.λx.(f (f (f x))))\n",
|
|
" 4: ---> (λx.(λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) x) λf.λx.(f (f (f x))))\n",
|
|
" 5: ---> (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f (f (f x))))\n",
|
|
" 6: ---> (λc.(c λx.λy.x) ((λf.λx.(f (f (f x))) λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))\n",
|
|
" 7: ---> (((λf.λx.(f (f (f x))) λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
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" 8: ---> ((λx.(λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) x))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.x)\n",
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" 9: ---> ((λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.x)\n",
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" 10: ---> (((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))) λx.λy.x)\n",
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" 11: ---> ((λy.λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))) λx.λy.x)\n",
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" 12: ---> (λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))) λx.λy.x)\n",
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" 13: ---> (((λc.λa.λs.((c a) s) λx.λy.x) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))))\n",
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" 14: ---> ((λa.λs.((λx.λy.x a) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))))\n",
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" 15: ---> (λs.((λx.λy.x (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) s) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))))\n",
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" 16: ---> ((λx.λy.x (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))))\n",
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" 17: ---> (λy.(λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))))\n",
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" 18: ---> (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
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" 19: ---> ((λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λx.λy.y)\n",
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" 20: ---> (((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.y)\n",
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" 21: ---> ((λy.λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.y)\n",
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" 22: ---> (λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))) λx.λy.y)\n",
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" 23: ---> (((λc.λa.λs.((c a) s) λx.λy.y) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
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" 24: ---> ((λa.λs.((λx.λy.y a) s) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
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" 25: ---> (λs.((λx.λy.y (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) s) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
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" 26: ---> ((λx.λy.y (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
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" 27: ---> (λy.y (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))))\n",
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" 28: ---> (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))))\n",
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" 29: ---> λf.λx.(f (((λc.(c λx.λy.y) (λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) f) x))\n",
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" 30: ---> λf.λx.(f ((((λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) λx.λy.y) f) x))\n",
|
|
" 31: ---> λf.λx.(f (((((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y) f) x))\n",
|
|
" 32: ---> λf.λx.(f ((((λy.λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y) f) x))\n",
|
|
" 33: ---> λf.λx.(f (((λs.(((λc.λa.λs.((c a) s) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λx.λy.y) f) x))\n",
|
|
" 34: ---> λf.λx.(f (((((λc.λa.λs.((c a) s) λx.λy.y) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) f) x))\n",
|
|
" 35: ---> λf.λx.(f ((((λa.λs.((λx.λy.y a) s) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) f) x))\n",
|
|
" 36: ---> λf.λx.(f (((λs.((λx.λy.y (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) s) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) f) x))\n",
|
|
" 37: ---> λf.λx.(f ((((λx.λy.y (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) f) x))\n",
|
|
" 38: ---> λf.λx.(f (((λy.y (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) f) x))\n",
|
|
" 39: ---> λf.λx.(f (((λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) f) x))\n",
|
|
" 40: ---> λf.λx.(f ((λf.λx.(f (((λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) f) x)) f) x))\n",
|
|
" 41: ---> λf.λx.(f (λx.(f (((λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) f) x)) x))\n",
|
|
" 42: ---> λf.λx.(f (f (((λc.(c λx.λy.y) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)) f) x)))\n",
|
|
" 43: ---> λf.λx.(f (f (((((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x) λx.λy.y) f) x)))\n",
|
|
" 44: ---> λf.λx.(f (f ((((λy.λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) y) λf.λx.x) λx.λy.y) f) x)))\n",
|
|
" 45: ---> λf.λx.(f (f (((λs.(((λc.λa.λs.((c a) s) s) λf.λx.x) λf.λx.x) λx.λy.y) f) x)))\n",
|
|
" 46: ---> λf.λx.(f (f (((((λc.λa.λs.((c a) s) λx.λy.y) λf.λx.x) λf.λx.x) f) x)))\n",
|
|
" 47: ---> λf.λx.(f (f ((((λa.λs.((λx.λy.y a) s) λf.λx.x) λf.λx.x) f) x)))\n",
|
|
" 48: ---> λf.λx.(f (f (((λs.((λx.λy.y λf.λx.x) s) λf.λx.x) f) x)))\n",
|
|
" 49: ---> λf.λx.(f (f ((((λx.λy.y λf.λx.x) λf.λx.x) f) x)))\n",
|
|
" 50: ---> λf.λx.(f (f (((λy.y λf.λx.x) f) x)))\n",
|
|
" 51: ---> λf.λx.(f (f ((λf.λx.x f) x)))\n",
|
|
" 52: ---> λf.λx.(f (f (λx.x x)))\n",
|
|
" 53: ---> λf.λx.(f (f x))\n",
|
|
"Forme normale calculée : λf.λx.(f (f x))\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<lambda_calcul.Lambda_terme at 0x7fed71d9b860>"
|
|
]
|
|
},
|
|
"execution_count": 81,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"SUB.applique(TROIS).applique(UN).forme_normale(verbose=True)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 86,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.λm.(NUL ((SUB n) m))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_INF = Lambda_terme('!n.!m.(NUL ((SUB n) m))')\n",
|
|
"print(M_INF)\n",
|
|
"INF = M_INF.subs('NUL', NUL).subs('SUB', SUB)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 87,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.y\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(INF.applique(TROIS).applique(UN).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 88,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.x\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(INF.applique(UN).applique(TROIS).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 89,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.x\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(INF.applique(UN).applique(UN).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 91,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.λm.((ET ((INF n) m)) ((INF m) n))\n",
|
|
"λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_EGAL = Lambda_terme('!n.!m.((ET ((INF n) m)) ((INF m) n))')\n",
|
|
"print(M_EGAL)\n",
|
|
"EGAL = M_EGAL.subs('ET', ET).subs('INF', INF)\n",
|
|
"print(EGAL)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 93,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.x\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(EGAL.applique(UN).applique(UN).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 94,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λx.λy.y\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(EGAL.applique(UN).applique(DEUX).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Itération"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 95,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λn.(CDR ((n λc.((CONS (SUC (CAR c))) ((MUL (SUC (CAR c))) (CDR c)))) ((CONS ZERO) UN)))\n",
|
|
"λn.(λc.(c λx.λy.y) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.x) c))) ((λn.λm.λf.(n (m f)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.x) c))) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.(f x))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_FACTv1 = Lambda_terme('!n.(CDR ((n !c.((CONS (SUC (CAR c))) ((MUL (SUC (CAR c))) (CDR c)))) ((CONS ZERO) UN)))')\n",
|
|
"print(M_FACTv1)\n",
|
|
"FACTv1 = M_FACTv1.subs('CONS', CONS).subs('CAR', CAR).subs('CDR', CDR).subs('SUC', SUC).subs('MUL', MUL).subs('UN', UN).subs('ZERO', ZERO)\n",
|
|
"print(FACTv1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 96,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(ZERO).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 97,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(UN).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 98,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"None\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(DEUX).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 99,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f x))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(DEUX).forme_normale(nb_etapes_max=200))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 100,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f (f (f (f (f x))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(TROIS).forme_normale(nb_etapes_max=500))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 101,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f x))))))))))))))))))))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv1.applique(QUATRE).forme_normale(nb_etapes_max=1700))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Et la récursivité ? "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 102,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λn.(((COND ((EGAL n) ZERO)) UN) ((MUL n) (f (PRED n))))\n",
|
|
"λf.λn.(((λc.λa.λs.((c a) s) ((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) n) λf.λx.x)) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) n) (f (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) n))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"M_PHI_FACT = Lambda_terme('!f.!n.(((COND ((EGAL n) ZERO)) UN) ((MUL n) (f (PRED n))))')\n",
|
|
"print(M_PHI_FACT)\n",
|
|
"PHI_FACT = M_PHI_FACT.subs('COND', COND).subs('EGAL', EGAL).subs('ZERO', ZERO).subs('UN', UN).subs('MUL', MUL).subs('PRED', PRED)\n",
|
|
"print(PHI_FACT)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 103,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λy.(λx.(x x) λx.(x x))\n"
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|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"BOTTOM = Lambda_terme('!y.OMEGA').subs('OMEGA', OMEGA)\n",
|
|
"print(BOTTOM)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 104,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"FACT0 = PHI_FACT.applique(BOTTOM)"
|
|
]
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|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 105,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACT0.applique(ZERO).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 107,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"((λf.λn.(((λc.λa.λs.((c a) s) ((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) n) λf.λx.x)) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) n) (f (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) n)))) λy.(λx.(x x) λx.(x x))) λf.λx.(f x))\n",
|
|
" 1: ---> ((λf.λn.(((λc.λa.λs.((c a) s) ((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) n) λf.λx.x)) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) n) (f (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) n)))) λy.(λx.(x x) λx.(x x))) λf.λx.(f x))\n",
|
|
" 2: ---> (λn.(((λc.λa.λs.((c a) s) ((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) n) λf.λx.x)) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) n) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) n)))) λf.λx.(f x))\n",
|
|
" 3: ---> (((λc.λa.λs.((c a) s) ((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) λf.λx.(f x)) λf.λx.x)) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 4: ---> ((λa.λs.((((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) λf.λx.(f x)) λf.λx.x) a) s) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 5: ---> (λs.((((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) λf.λx.(f x)) λf.λx.x) λf.λx.(f x)) s) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 6: ---> ((((λn.λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) n) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) n)) λf.λx.(f x)) λf.λx.x) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 7: ---> (((λm.((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) m)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) m) λf.λx.(f x))) λf.λx.x) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 8: ---> ((((λa.λb.(((λc.λa.λs.((c a) s) a) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 9: ---> (((λb.(((λc.λa.λs.((c a) s) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x)) b) λx.λy.y) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 10: ---> (((((λc.λa.λs.((c a) s) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 11: ---> ((((λa.λs.((((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x) a) s) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 12: ---> (((λs.((((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) s) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 13: ---> ((((((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.(f x)) λf.λx.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 14: ---> (((((λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) λf.λx.(f x)) m)) λf.λx.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 15: ---> (((((λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) λf.λx.(f x)) λf.λx.x)) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 16: ---> ((((((((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) λf.λx.(f x)) λf.λx.x) λx.λx.λy.y) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 17: ---> (((((((λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λf.λx.(f x)) λf.λx.x) λx.λx.λy.y) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 18: ---> ((((((((λf.λx.x λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) λf.λx.(f x)) λx.λx.λy.y) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 19: ---> (((((((λx.x λf.λx.(f x)) λx.λx.λy.y) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 20: ---> ((((((λf.λx.(f x) λx.λx.λy.y) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 21: ---> (((((λx.(λx.λx.λy.y x) λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 22: ---> (((((λx.λx.λy.y λx.λy.x) ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 23: ---> ((((λx.λy.y ((λn.λm.(λn.((n λx.λx.λy.y) λx.λy.x) ((λn.λm.((m λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x)))) n) n) m)) λf.λx.x) λf.λx.(f x))) λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
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" 24: ---> (((λy.y λx.λy.y) λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 25: ---> ((λx.λy.y λf.λx.(f x)) ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 26: ---> (λy.y ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x)))))\n",
|
|
" 27: ---> ((λn.λm.λf.(n (m f)) λf.λx.(f x)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x))))\n",
|
|
" 28: ---> (λm.λf.(λf.λx.(f x) (m f)) (λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x))))\n",
|
|
" 29: ---> λf.(λf.λx.(f x) ((λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x))) f))\n",
|
|
" 30: ---> λf.λx.(((λy.(λx.(x x) λx.(x x)) (λn.(λc.(c λx.λy.x) ((n λc.((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) (λc.(c λx.λy.y) c)) (λn.λf.λx.(f ((n f) x)) (λc.(c λx.λy.y) c)))) ((λx.λy.λs.(((λc.λa.λs.((c a) s) s) x) y) λf.λx.x) λf.λx.x))) λf.λx.(f x))) f) x)\n",
|
|
" 31: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 32: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 33: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 34: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 35: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 36: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 37: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 38: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 39: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
" 40: ---> λf.λx.(((λx.(x x) λx.(x x)) f) x)\n",
|
|
"Pas de forme normale atteinte après 40 étapes de réduction\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"FACT0.applique(UN).forme_normale(verbose=True, nb_etapes_max=40)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 108,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"FACT1 = PHI_FACT.applique(FACT0)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 109,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACT1.applique(ZERO).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 110,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACT1.applique(UN).forme_normale(nb_etapes_max=200))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 111,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.(λx.(f (x x)) λx.(f (x x)))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"FIX_CURRY = Lambda_terme('!f.(!x.(f (x x)) !x.(f (x x)))')\n",
|
|
"print(FIX_CURRY)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 112,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"FACTv2 = FIX_CURRY.applique(PHI_FACT)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 113,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv2.applique(ZERO).forme_normale())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 115,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f x)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv2.applique(UN).forme_normale(nb_etapes_max=200))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 116,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f x))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv2.applique(DEUX).forme_normale(nb_etapes_max=700))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 117,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f (f (f (f (f x))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv2.applique(TROIS).forme_normale(nb_etapes_max=4000))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 118,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"λf.λx.(f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f (f x))))))))))))))))))))))))\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"print(FACTv2.applique(QUATRE).forme_normale(nb_etapes_max=25000))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 120,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"PF = FIX_CURRY.applique(Lambda_terme('M'))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 121,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(λf.(λx.(f (x x)) λx.(f (x x))) M)\n",
|
|
" 1: ---> (λf.(λx.(f (x x)) λx.(f (x x))) M)\n",
|
|
" 2: ---> (λx.(M (x x)) λx.(M (x x)))\n",
|
|
" 3: ---> (M (λx.(M (x x)) λx.(M (x x))))\n",
|
|
" 4: ---> (M (M (λx.(M (x x)) λx.(M (x x)))))\n",
|
|
" 5: ---> (M (M (M (λx.(M (x x)) λx.(M (x x))))))\n",
|
|
" 6: ---> (M (M (M (M (λx.(M (x x)) λx.(M (x x)))))))\n",
|
|
" 7: ---> (M (M (M (M (M (λx.(M (x x)) λx.(M (x x))))))))\n",
|
|
" 8: ---> (M (M (M (M (M (M (λx.(M (x x)) λx.(M (x x)))))))))\n",
|
|
" 9: ---> (M (M (M (M (M (M (M (λx.(M (x x)) λx.(M (x x))))))))))\n",
|
|
" 10: ---> (M (M (M (M (M (M (M (M (λx.(M (x x)) λx.(M (x x)))))))))))\n",
|
|
"Pas de forme normale atteinte après 10 étapes de réduction\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"PF.forme_normale(verbose=True, nb_etapes_max=10)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"lines_to_next_cell": 2
|
|
},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# $\\lambda$-calcul avec les lambda-expressions de Python"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"incorrectly_encoded_metadata": "toc-hr-collapsed=true toc-nb-collapsed=true"
|
|
},
|
|
"source": [
|
|
"## Les entiers de Church"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"zero = lambda f: lambda x: x"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"un = lambda f: lambda x: f(x)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"deux = lambda f: lambda x: f(f(x))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"trois = lambda f: lambda x: f(f(f(x)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def entier_church_en_int(ec):\n",
|
|
" return ec(lambda n: n+1)(0)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(entier_church_en_int(n) for n in (zero, un, deux, trois))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"suc = lambda n: lambda f: lambda x: f(n(f)(x))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(entier_church_en_int(suc(n)) for n in (zero, un, deux, trois)) "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def int_en_entier_church(n):\n",
|
|
" if n == 0:\n",
|
|
" return zero\n",
|
|
" else:\n",
|
|
" return suc(int_en_entier_church(n - 1))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(entier_church_en_int(int_en_entier_church(n)) for n in range(10))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"add = lambda n: lambda m: lambda f: lambda x: n(f)(m(f)(x))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cinq = add(deux)(trois)\n",
|
|
"entier_church_en_int(cinq)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"mul = lambda n: lambda m: lambda f: n(m(f))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"six = mul(deux)(trois)\n",
|
|
"entier_church_en_int(six)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"exp = lambda n: lambda m: m(n)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"huit = exp(deux)(trois)\n",
|
|
"entier_church_en_int(huit)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"incorrectly_encoded_metadata": "toc-hr-collapsed=true toc-nb-collapsed=true"
|
|
},
|
|
"source": [
|
|
"## Les booléens"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"vrai = lambda x: lambda y: x\n",
|
|
"faux = lambda x: lambda y: y"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def booleen_en_bool(b):\n",
|
|
" return b(True)(False)\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(b) for b in (vrai, faux))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cond = lambda c: lambda a: lambda s: c(a)(s) "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cond(vrai)(1)(2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cond(faux)(1)(2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cond(vrai)(1)(1/0)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"non = lambda b: cond(b)(faux)(vrai)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(non(b)) for b in (vrai, faux))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"et = lambda b1: lambda b2: cond(b1)(b2)(faux)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(et(b1)(b2)) for b1 in (vrai, faux) for b2 in (vrai, faux))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"ou = lambda b1: lambda b2: cond(b1)(vrai)(b2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(ou(b1)(b2)) for b1 in (vrai, faux) for b2 in (vrai, faux))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"est_nul = lambda n : n(lambda x: faux)(vrai)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(est_nul(n)) for n in (zero, un, deux, trois, cinq, six, huit))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Les couples"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"cons = lambda x: lambda y: lambda z: z(x)(y)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"un_deux = cons(un)(deux)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"car = lambda c: c(vrai)\n",
|
|
"cdr = lambda c: c(faux)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"entier_church_en_int(car(un_deux)), entier_church_en_int(cdr(un_deux))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"pred = lambda n: car(n(lambda c: cons(cdr(c))(suc(cdr(c))))(cons(zero)(zero)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(entier_church_en_int(pred(int_en_entier_church(n))) for n in range(10))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"sub = lambda n: lambda m: m(pred)(n)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"entier_church_en_int(sub(huit)(trois))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"est_inf_ou_egal = lambda n: lambda m: est_nul(sub(m)(n))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(est_inf_ou_egal(cinq)(int_en_entier_church(n))) for n in range(10))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"est_egal = lambda n: lambda m: et(est_inf_ou_egal(n)(m))(est_inf_ou_egal(m)(n))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(booleen_en_bool(est_egal(cinq)(int_en_entier_church(n))) for n in range(10))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"incorrectly_encoded_metadata": "toc-hr-collapsed=true toc-nb-collapsed=true"
|
|
},
|
|
"source": [
|
|
"## Itération"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fact = lambda n: cdr(n(lambda c: (cons(suc(car(c)))(mul(suc(car(c)))(cdr(c)))))(cons(zero)(un)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(entier_church_en_int(fact(int_en_entier_church(n))) for n in range(7))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Combinateur de point fixe"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"phi_fact = lambda f: lambda n: 1 if n == 0 else n*f(n-1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"bottom = lambda x: (lambda y: y(y))(lambda y:y(y))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"f0 = phi_fact(bottom)\n",
|
|
"f1 = phi_fact(f0)\n",
|
|
"f2 = phi_fact(f1)\n",
|
|
"f3 = phi_fact(f2)\n",
|
|
"f4 = phi_fact(f3)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(f4(n) for n in range(4))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def fact_rec(n):\n",
|
|
" if n == 0:\n",
|
|
" return 1\n",
|
|
" else:\n",
|
|
" return n * fact_rec(n - 1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fact2 = phi_fact(fact_rec)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(fact2(n) for n in range(7))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fix_curry = lambda f: (lambda x: lambda y: f(x(x))(y))(lambda x: lambda y: f(x(x))(y))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fact3 = fix_curry(phi_fact)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"tuple(fact3(n) for n in range(7))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"incorrectly_encoded_metadata": "toc-hr-collapsed=true toc-nb-collapsed=true"
|
|
},
|
|
"source": [
|
|
"## Un programme obscur"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"print((lambda x: (lambda y: lambda z: x(y(y))(z))(lambda y: lambda z: x(y(y))(z))) \n",
|
|
" (lambda x: lambda y: '' if y == [] else chr(y[0])+x(y[1:]))\n",
|
|
" (((lambda x: (lambda y: lambda z: x(y(y))(z)) (lambda y: lambda z: x(y(y))(z)))\n",
|
|
" (lambda x: lambda y: lambda z: [] if z == [] else [y(z[0])]+x(y)(z[1:]))) \n",
|
|
" (lambda x: (lambda x: (lambda y: lambda z: x(y(y))(z))(lambda y: lambda z: x(y(y))(z)))\n",
|
|
" (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)\n",
|
|
" (x(y)(z)(t // 2)))(989)(x)(761))\n",
|
|
" ([377, 900, 27, 27, 182, 647, 163, 182, 390, 27, 726, 937])))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"phiListEnChaine = lambda x: lambda y: '' if y == [] else chr(y[0]) + x(y[1:])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fix_curry(phiListEnChaine)([65+k for k in range(26)])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"phiMap = lambda x: lambda y: lambda z: [] if z == [] else [y(z[0])] + x(y)(z[1:])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fix_curry(phiMap)(lambda x: x*x)([1, 2, 3, 4])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"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"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"fix_curry(phiExpoMod)(2)(10)(1000)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"jupytext": {
|
|
"formats": "ipynb,md"
|
|
},
|
|
"kernelspec": {
|
|
"display_name": "Python 3",
|
|
"language": "python",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.7.3"
|
|
},
|
|
"toc-autonumbering": true,
|
|
"widgets": {
|
|
"application/vnd.jupyter.widget-state+json": {
|
|
"state": {},
|
|
"version_major": 2,
|
|
"version_minor": 0
|
|
}
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|