def main():
# P = preference_generator(2, 3)
# print("Preferences P:")
# print(P)
f_rng = SCF("Random Choice", random_choice)
#f_con = SCF("Condorcet", condorcet)
f_plur = SCF("Plurality", plurality)
g = Generator(2, 2, mode="bruteforce", max_m=4, max_n=3)
a = Axiom("resolute", resolute)
a.check_axiom(g, f_plur)
def generate(self,
numAbstractions,
numNodesPerLayer,
abstractionsMax=False,
nodesPerLayerMax=False):
if abstractionsMax:
numAbstractions = randint(1, numAbstractions)
for i in range(numAbstractions):
aL = AbstractionLayer()
if nodesPerLayerMax:
numNodesPerLayer = randint(
len(self.outputNodes),
numNodesPerLayer) # ensures diminishing tree to the right
if i > 0:
self.abstractionLayers.append(
aL.create(self.abstractionLayers[i - 1].nodes))
else:
self.abstractionLayers.append(aL.create(self.inputNodes))
for j in range(numNodesPerLayer):
aL.addNewNode()
for outNode in self.outputNodes:
axOut = []
for prevN in self.abstractionLayers[-1].nodes:
ax = Axiom(prevN, outNode, sigmoid(uniform(-1, 1)))
axOut.append(ax)
self.axiomOutList.extend(axOut)
return self
def addNewNode(self):
axIn = []
node = NodeFactory.create()
for prevN in self.prevLayerNodes:
ax = Axiom(prevN, node, sigmoid(uniform(-1, 1)))
axIn.append(ax)
node.assignAxioms(axIn)
self.nodes.append(node)
self.axiomList.extend(axIn)
def dfs(narr,
axioms,
debug=False,
maxsteps=150,
animation_mode=False,
printTrim=False):
any_err = None
try:
next_steps = [(narr, None, Axiom(name='原式'), -1)]
return_steps = []
cnt = 0
while len(next_steps) > 0:
narr, ani_narr, axiom, axiom_idx = next_steps[0]
output_narr = deepcopy(narr)
#print('[output narr]', narr)
if animation_mode:
if printTrim:
rich.print('[blue]before trim[/]')
expression.narr_prettyprint(narr)
print('[tex]', expression.narr2tex(narr))
expression.trim_animations(narr)
if printTrim:
rich.print('[blue]after trim[/]')
expression.narr_prettyprint(narr)
print('[tex]', expression.narr2tex(narr))
return_steps.append((output_narr, ani_narr, axiom, axiom_idx))
next_steps = possible_next_steps(narr,
axioms,
state.value_v1,
animation_mode=animation_mode,
fast_return=True,
debug=debug)
if cnt > maxsteps:
any_err = "maximum steps reached."
break
cnt += 1
except KeyboardInterrupt:
pass
return return_steps, any_err
n_steps += len(steps)
print(f'steps: {len(steps)}')
print(f'test case: {i} / {len(testcases)}')
#input('Enter to continue...')
timer.show_stats(n_steps=n_steps)
if __name__ == '__main__':
if False:
all_axioms = [
Axiom(name='负号乘进因子',
recursive_apply=True,
strict_simplify=True,
disable=False).add_rule(
'-(x + *{1}) *{2} ',
'(-x - *{1}) *{2}',
animation='`-(x + *{1})`[replace]{-x - *{1}} *{2}').
add_test('-(3 - 2)x', '(-3 + 2) \\times x')
]
else:
from common_axioms import common_axioms
all_axioms = common_axioms()
args = sys.argv[1:]
if len(args) > 0:
tex = args[0]
narr = expression.tex2narr(tex)
steps, err = dfs(narr, all_axioms, debug=False, animation_mode=True)
b = get_atom_number(rewrite_rules['b'])
if a != None and b != None:
c = a + b
rewrite_rules['c'] = gen_atom_number(c)
new_narr = rewrite_by_alpha(output_tempalate, rewrite_rules)
return new_narr, True
return narr, False
calc_add = (
Axiom(name='加减法计算', root_sign_reduce=False)
.add_rule('#(a + b)', 'c', animation='`#1(a + b)`[replace]{c}', dynamic_procedure=_calc_add)
.add_test('-12 + 3', '-9')
.add_test('(-12 - 3 - 1)', ['-15 - 1', '-13 - 3', '-4 - 12'])
.add_test('-(12 - 1)', '-11')
.add_test('-(1 + 14 + 2)', ['-(15 + 2)', '-(3 + 14)', '-(16 + 1)'])
.add_test('0.25 - 3.25', '-3')
.add_test('-(1 + 3) x + 3 y', '-4 \\times x + 3 \\times y')
)
def _calc_mul(pattern_narr, signs, narr, rewrite_rules, output_tempalate):
a = get_atom_number(rewrite_rules['a'])
b = get_atom_number(rewrite_rules['b'])
if a != None and b != None:
c = a * b
rewrite_rules['c'] = gen_atom_number(c)
return rewrite_by_alpha(output_tempalate, rewrite_rules), True
print('\t', ani_tex)
print('\t', tex)
def nn_value(narr):
tex = expression.narr2tex(narr)
expr_val, _ = nn.predict_value(tex, nn_models)
return expr_val
if __name__ == '__main__':
if False:
all_axioms = [
Axiom(name='乘积写成乘方的形式', allow_complication=True).add_rule(
'#(#X)(#X)',
'#0 X^{2}',
animation='`#1(#2 X)(#3 X)`[replace]{#0 X^{2}}')
]
else:
from common_axioms import common_axioms
all_axioms = common_axioms(full=True)
testcase = (
"(-3 - \\frac{4}{17}) \\times (14 + \\frac{13}{15}) - (3 + \\frac{4}{17}) \\times (2 + \\frac{2}{15})"
)
debug_NN = True
if debug_NN:
global nn_models
def mcts(narr0,
all_axioms,
sample_depth=4,
n_sample_times=200,
n_maxsteps=100,
k=3,
debug=False,
nn_models=False,
training=False,
force_single_thread=False):
# q n narr father axiom axiomIdx children
root = [0, 1, narr0, None, None, -1, []]
moves = [root]
render_steps([(narr0, None, -1)])
global manager
if not force_single_thread:
# prepare proxy structure for parallel processes
root[6] = manager.list([])
root = manager.list(root)
moves = [root]
else:
manager = None
node = root
visited = set([expression.narr2tex(narr0)])
final_steps = []
while True:
q, n, narr, father, axiom, axiom_idx, children = node
# debug print
if True:
#if debug:
print('\033[94m', end='')
expr_val = state_value(narr)
print(f'[current] step={len(moves)}, val={expr_val:.1f}:',
expression.narr2tex(narr),
end='')
print('\033[0m', end='\n')
expression.narr_prettyprint(narr)
steps, step_probs = policy_steps(narr,
all_axioms,
k=k,
debug=debug,
nn_models=nn_models)
if debug:
rich.print(f'[magenta]Candidate steps: {len(steps)}[/]')
for i, (n, a, ai) in enumerate(steps):
val = state_value(n)
rich.print(f'[red]#{i+1}[/]', a.name(), ':', end=' ')
rich.print(f'val={val:.2f}', end=' ')
print(expression.narr2tex(n), end='\n\n')
if False:
from axiom import Axiom
render_steps([(narr, Axiom(), -1)] + steps, show_index=True)
choices = input('Limit choices: ')
choices = [
i for i in map(lambda x: int(x), choices.split(','))
]
rich.print(choices)
steps = [steps[i - 1] for i in choices]
if len(steps) == 0:
if debug: print('[no more candidate steps]')
if nn_models and training:
policy = 0
#policy_fine_tuning(nn_models, expr, policy, debug=debug, all_axioms=all_axioms)
break
if manager and not force_single_thread:
evaluate_parallel(node,
all_axioms,
steps,
n_sample_times,
sample_depth,
visited,
debug=debug,
nn_models=nn_models,
k=k,
step_probs=step_probs)
else:
evaluate(node,
all_axioms,
steps,
n_sample_times,
sample_depth,
visited,
debug=debug,
nn_models=nn_models,
k=k,
step_probs=step_probs)
# selection
move_choice, w, _ = best_child_of(node, c_param=.0, debug=debug)
move_to_expr = expression.narr2tex(move_choice[2])
if w == 0 or move_to_expr in visited:
print(
f'[abort] best w={w:.2f}, visited: {move_to_expr in visited}')
break
else:
if nn_models and training:
policy = move_choice[5] + 1
#policy_fine_tuning(nn_models, expr, policy, debug=debug, all_axioms=all_axioms)
moves.append(move_choice)
node = move_choice
# construct steps to be returned
final_steps = [(e, a, ai) for q, n, e, f, a, ai, c in moves]
render_steps(final_steps)
visited.add(move_to_expr)
#if debug: print('[visited]', visited)
if len(moves) >= n_maxsteps:
if debug: print('[exceed max steps]')
break
if len(final_steps) > 0:
final_steps = back_off_step(final_steps, debug=True)
if nn_models and training:
# fine-tune value network
for i, (e, _, _) in enumerate(final_steps):
value = -(len(final_steps) - i - 1)
#value_fine_tuning(nn_models, e, value, debug=debug)
return final_steps
def common_axioms(full=False):
"""
常用的规则集合（优先级已经按照数组元素的索引确定）
"""
axioms = []
axioms.append(dynamic_axioms.canonicalize)
axioms.append(
Axiom(name='一个数减去它本身是零', root_sign_reduce=False).add_rule(
'#(n - n)', '0', animation='`#1(n - n)`[replace]{0}').add_rule(
'#(a - \\frac{a}{1})',
'0',
animation='`#1(a - \\frac{a}{1})`[replace]{0}').add_rule(
'#(\\frac{a}{1} - a)',
'0',
animation='`#1(\\frac{a}{1} - a)`[replace]{0}').add_test(
'-(3 - 3 + 1)'))
axioms.append(
Axiom(name='任何数乘以零还是零',
recursive_apply=True).add_rule(
'# 0 \cdot *{1}',
'0',
animation='`#1 0 \cdot *{1}`[replace]{0}').add_rule(
'#\\frac{#0}{x}',
'0',
animation='`#1 \\frac{#2 0}{x}`[replace]{0}').add_test(
'0 a b'))
axioms.append(
Axiom(name='带分式的展开', allow_complication=True).add_rule(
'# & \\frac{a}{b}',
'#1 (v + \\frac{a}{b})',
animation="`#1 & \\frac{a}{b}`[replace]{#1 (v + \\frac{a}{b})}").
add_test('-3 \\frac{-2}{4}',
'-(3 + \\frac{-2}{4})').add_test('-3 \\frac{1}{2}',
'-(3 + \\frac{1}{2})'))
axioms.append(
Axiom(name='任何数加零还是它本身', recursive_apply=True,
root_sign_reduce=False).add_rule(
'#(0 + n)', 'n', animation='`0`[remove] + n').add_rule(
'#(n - 0)', 'n', animation='n - `0`[remove]').add_test(
'0+3+2').add_test('-(0+3+2)'))
axioms.append(
Axiom(name='一乘以任何数还是它本身', recursive_apply=True).add_rule(
'# 1 \\times *{1}',
'#1 *{1}', animation='#1 `1`[remove] *{1}').add_test(
'1 \cdot 4', '4').add_test('-4 \\times 1', '-4').add_test(
'- (a+b) \cdot 1',
'-a - b').add_test('- 1 \cdot 4 \cdot 1', '-4'))
axioms.append(
Axiom(name='根号的平方是其本身').add_rule(
'#(#\\sqrt{x})^{2}',
'#1 x',
animation='`#1 (#2\\sqrt{x})^{2}`[replace]{#1 x}'))
axioms.append(
Axiom(name='一的平方还是一').add_rule(
'#(# 1)^{2}', '#1 1',
animation='`#1 (#2 1)^{2}`[replace]{#1 1}').add_test(
'-1^{2}').add_test('(-1)^{2}'))
axioms.append(
Axiom(name='负数的平方是其相反数的平方', strict_simplify=(not full)).add_rule(
'#(-a)^{2}',
'#1 a^{2}',
animation='`#1 (-a)^{2}`[replace]{#1 a^{2}}').add_test(
'(-3)^{2} - (6 \div (-\\frac{2}{3})^{2}) - (-2)^{2}').
add_test('(6 \div (-\\frac{2}{3})^{2})').add_test(
'3^{2} - (1 + \\frac{1}{2}) \\times \\frac{2}{9} - (6 \div (-\\frac{2}{3})^{2}) - (-2)^{2}'
))
axioms.append(
Axiom(name='除以一个数等于乘上它的倒数', allow_complication=True).add_rule(
'# (#\\frac{w}{x}) \\div (# \\frac{y}{z})',
'#0 \\frac{wz}{xy}',
animation=
'`#1 (#2 \\frac{w}{x}) \\div (#3 \\frac{y}{z})`[replace]{#0 \\frac{wz}{xy}}'
).add_rule(
'# x \\div (# \\frac{y}{z})',
'#0 \\frac{xz}{y}',
animation='`#1 x \\div (#2 \\frac{y}{z})`[replace]{#0 \\frac{xz}{y}}'
).add_test('- 3 \\div (-\\frac{1}{2})',
'-\\frac{3 \\times 2}{1}').add_test(
'\\frac{2}{3} \div \\frac{4}{5}'))
axioms.append(
Axiom(name='以分数表示除法', allow_complication=True).add_rule(
'# (#x) \\div (#y)',
'#0 \\frac{x}{y}',
animation="`#1(#2 x) \\div (#3 y)`[replace]{#0 \\frac{x}{y}}").
add_test('1 \\div 2x + 3').add_test("6 \div 3"))
axioms.append(
Axiom(name='因子和分母消去').add_rule(
'# x \\times \\frac{a}{x}',
'#1 a',
animation="#1 `x`[remove] \\times \\frac{a}{`x`[removeDenom]}").
add_test('-3 \\times \\frac{-2}{3}',
'2').add_test('3 \\times \\frac{-2}{3}', '-2'))
axioms.append(
Axiom(name='绝对值分配到分子分母中', allow_complication=True).add_rule(
'# \\left| # \\frac{a}{b} \\right|',
'#1 \\frac{\\left| a \\right|}{\\left| b \\right|}',
animation=
'`#1 \\left| #2 \\frac{a}{b} \\right|`[replace]{#1 \\frac{\\left| a \\right|}{\\left| b \\right|}}'
).add_test('\\left| - \\frac{1}{2} \\right|'))
axioms.append(
Axiom(name='根号分配到分子分母中', allow_complication=True).add_rule(
'# \\sqrt{\\frac{a}{b}}',
'#1 \\frac{\\sqrt{a}}{\\sqrt{b}}',
animation=
'`#1 \\sqrt{\\frac{a}{b}}`[replace]{#1 \\frac{\\sqrt{a}}{\\sqrt{b}} }'
).add_test('- \sqrt{\\frac{1}{2}}'))
axioms.append(
Axiom(name='分子分母消除公因子', recursive_apply=True).add_rule(
'# \\frac{# x *{1} }{# x *{2} }',
'#1 \\frac{#2 *{1}}{#3 *{2}}',
animation='#1 \\frac{#2 `x`[remove] *{1}}{#3 `x`[remove] *{2}}'
).add_rule(
'# \\frac{# x }{# x *{2} }',
'#0 \\frac{1}{*{2}}',
animation='#0 \\frac{`x`[replace]{1} \\times 1}{`x`[remove] *{2}}'
).add_rule('# \\frac{# x *{1} }{# x }',
'#0 *{1}',
animation='#0 \\frac{`x`[remove] *{1} }{`x`[removeDenom]}').
add_rule('# \\frac{# x}{# x}',
'#0 1',
animation='`#1 \\frac{#2 x}{#3 x}`[replace]{#0 1}').add_test(
'- \\frac{bx}{ax}',
'-\\frac{b}{a}').add_test('- \\frac{-bx}{xa}',
'-\\frac{-b}{a}').add_test(
'- \\frac{-bxy}{-xay}',
'-\\frac{-b}{-a}').add_test(
'- \\frac{-x}{ax}',
'\\frac{1}{a}').
add_test('\\frac{-x}{xay}', '\\frac{-1}{a \\times y}').add_test(
'\\frac{3xy}{-xy}', '-3').add_test(
'-\\frac{-a}{-a}',
'-1').add_test('\\frac{-(a + b)}{a + b}', '-1').add_test(
'\\frac{-3 - \\frac{1}{3}}{-(3 + \\frac{1}{3}) \\times x}',
'\\frac{-1}{-x}'))
axioms.append(
Axiom(name='乘积写成乘方的形式', allow_complication=True).add_rule(
'#(#X)(#X)',
'#0 X^{2}',
animation='`#1(#2 X)(#3 X)`[replace]{#0 X^{2}}').add_test(
'xx').add_test('-\\frac{1}{2} \cdot \\frac{1}{2}',
'-(\\frac{1}{2})^{2}'))
axioms.append(dynamic_axioms.calc_add)
axioms.append(dynamic_axioms.calc_mul)
axioms.append(dynamic_axioms.calc_pow)
axioms.append(dynamic_axioms.calc_sqrt)
axioms.append(dynamic_axioms.calc_abs)
axioms.append(dynamic_axioms.collapse_fraction_add_float)
axioms.append(dynamic_axioms.simplify_fraction)
axioms.append(dynamic_axioms.collapse_fraction)
axioms.append(dynamic_axioms.calc_sqrt)
axioms.append(
Axiom(name='等式移项')
#.add_rule('#x # y = z', '#1 x #2 y -z=0', animation='#1 x #2 y - `z`[moveAfter,1] = `z`[moveBefore,1] + `0`[add]')
.add_rule('x = z',
'x - z = 0',
animation='x - `z`[add] = `z`[replace]{0}').add_test(
'1 - 2 = -3 + 5', '1 - 2 + 3 - 5 = 0').add_test(
'ax = bx', 'a \\times x - b \\times x = 0'))
axioms.append(
Axiom(name='等式两边同乘', allow_complication=True).add_rule(
'#\\frac{x}{y} + *{1} = z',
'#1 x + y(*{1}) = yz',
animation=
'#1 \\frac{x}{`y`[removeDenom]} + `y`[add](*{1}) = `y`[add] z').
add_rule(
'#\\frac{x}{y} = z',
'#1 x = yz',
animation='#1 \\frac{x}{`y`[removeDenom]} = `y`[add] z').add_test(
'-\\frac{-2}{-3} = -4', '2 = (-3) \\times (-4)').add_test(
'\\frac{x}{2} + a - b = z',
'x + 2 \\times (a - b) = 2 \\times z'))
if full:
tmp_axiom = (Axiom(
name='合并同类项',
recursive_apply=True,
allow_complication=True,
root_sign_reduce=False,
max_results=4
).add_rule(
'#(x + x)', '2x', animation='`#1(x + x)`[replace]{2x}'
).add_rule(
'#(-x - x)', '-2x', animation='`#1(-x - x)`[replace]{-2x}'
).add_rule(
'#(#x # kx)',
'(#2 1 #3 k) x',
animation='`#1(#2 x #3 kx)`[replace]{(#2 1 #3 k) x}'
).add_rule(
'#(#x # xk)',
'(#2 1 #3 k) x',
animation='`#1(#2 x #3 xk)`[replace]{(#2 1 #3 k) x}'
).add_rule(
'#(#x *{1} # x *{2})',
'(#2 *{1} #3 *{2}) x',
animation='`#1(#2 x *{1} #3 x *{2})`[replace]{(#2 *{1} #3 *{2}) x}'
))
else:
tmp_axiom = (Axiom(
name='合并同类项',
recursive_apply=True,
allow_complication=True,
root_sign_reduce=False,
max_results=4
).add_rule(
'#(X + X)', '2X', animation='`#1(X + X)`[replace]{2X}'
).add_rule(
'#(-X - X)', '-2X', animation='`#1(-X - X)`[replace]{-2X}'
).add_rule(
'#(#X # kX)',
'(#2 1 #3 k) X',
animation='`#1(#2 X #3 kX)`[replace]{(#2 1 #3 k) X}'
).add_rule(
'#(#X # Xk)',
'(#2 1 #3 k) X',
animation='`#1(#2 X #3 Xk)`[replace]{(#2 1 #3 k) X}'
).add_rule(
'#(#X *{1} # X *{2})',
'(#2 *{1} #3 *{2}) X',
animation='`#1(#2 X *{1} #3 X *{2})`[replace]{(#2 *{1} #3 *{2}) X}'
))
axioms.append(
tmp_axiom.add_test('a + a', '2 \\times a').add_test(
'-a - a', '2 \\times (-a)').add_test(
'x^{2} + x^{2}',
'2 \\times x^{2}').add_test('-(-x + 3x)').add_test(
'-x + 3x - 1x').add_test('(-1-1)x').add_test(
'-(-x + 3x - 1x)').add_test('x + 2x + 3x').add_test(
'x - 3 \cdot x', '(1 - 3) \\times x'))
axioms.append(
Axiom(name='负号提出括号', strict_simplify=True,
disable=(not full)).add_rule(
'+(x + *{1})',
'-(-x - *{1})',
animation='`+(x + *{1})`[replace]{-(-x - *{1})}').
add_test("(1 + 2 + 3)").add_test("(-a - b) x").add_test(
"(-a + b)x").add_test("(-a - b - c) x").add_test(
'-3-\\frac{4}{17}').add_test('(-3-\\frac{4}{17}) x').add_test(
'(-3-\\frac{4}{17}) x + y').add_test("(a - b - c) x"))
axioms.append(
Axiom(name='嵌套分式的化简').add_rule(
'#\\frac{#\\frac{x}{y}}{#\\frac{a}{b}}',
'#0 \\frac{bx}{ay}',
animation=
'`#1 \\frac{#2 \\frac{x}{y}}{#3 \\frac{a}{b}}`[replace]{#0 \\frac{bx}{ay}}'
).add_rule(
'#\\frac{#1}{#\\frac{x}{y}}',
'#0 \\frac{y}{x}',
animation=
'`#1 \\frac{#2 1}{#3 \\frac{x}{y}}`[replace]{#0 \\frac{y}{x}}'
).add_rule(
'#\\frac{#\\frac{x}{y}}{#1}',
'#0 \\frac{x}{y}',
animation=
'`#1 \\frac{#2 \\frac{x}{y}}{#3 1}`[replace]{#0 \\frac{x}{y}}').
add_rule('#\\frac{a}{#\\frac{x}{y}}',
'#0 \\frac{ay}{x}',
animation=
'`#1 \\frac{a}{#2 \\frac{x}{y}}`[replace]{#0 \\frac{ay}{x}}').
add_rule('#\\frac{#\\frac{x}{y}}{a}',
'#0 \\frac{x}{ay}',
animation=
'`#1 \\frac{#2 \\frac{x}{y}}{a}`[replace]{#0 \\frac{x}{ay}}').
add_test('-\\frac{-\\frac{4}{3}}{-\\frac{4}{3}}', '-1').add_test(
'-\\frac{-\\frac{4}{3}}{\\frac{1}{2}}',
'\\frac{2 \\times 4}{1 \\times 3}').add_test(
'-\\frac{-\\frac{4}{3}}{x}',
'\\frac{4}{x \\times 3}').add_test('-\\frac{4}{-\\frac{4}{3}}',
'\\frac{4 \\times 3}{4}').
add_test('\\frac{-8}{\\frac{1}{4}}').add_test(
'\\frac{-18}{\\frac{9 \\times (1 - \\frac{3}{4})}{4}}'))
axioms.append(dynamic_axioms.fraction_addition)
axioms.append(
Axiom(name='分母为一的分式化简').add_rule(
'#\\frac{k}{#1}',
'#0 k',
animation='`#1 \\frac{k}{#2 1}`[replace]{#0 k}').add_test(
'\\frac{3}{-1}', '-3'))
axioms.append(
Axiom(name='分式的乘法', allow_complication=True).add_rule(
'#\\frac{#1}{x} \\frac{#1}{y}',
'#0 \\frac{1}{xy}',
animation=
'`#1 \\frac{#2 1}{x} \\frac{#3 1}{y}`[replace]{#0 \\frac{1}{xy}}').
add_rule(
'#\\frac{#1}{x} \\frac{y}{#1}',
'#0 \\frac{y}{x}',
animation=
'`#1 \\frac{#2 1}{x} \\frac{y}{#3 1}`[replace]{#0 \\frac{y}{x}}').
add_rule(
'#\\frac{x}{#1} \\frac{y}{#1}',
'#0 xy',
animation='`#1 \\frac{x}{#2 1} \\frac{y}{#3 1}`[replace]{#0 xy}'
).add_rule('#\\frac{a}{c} \\frac{b}{d}',
'#1 \\frac{ab}{cd}',
animation=
'`#1 \\frac{a}{c} \\frac{b}{d}`[replace]{#1 \\frac{ab}{cd}}'
).add_test('-\\frac{1}{-2} \cdot \\frac{-2}{1}',
'-\\frac{-2}{-2}'))
axioms.append(
Axiom(name='乘数的指数是各个因子指数的乘数',
recursive_apply=True,
allow_complication=True).
add_rule(
'# (# a *{1})^{k}',
'#1 (#2 a)^{k} \\times (*{1})^{k}',
animation=
'`#1 (#2 a *{1})^{k}`[replace]{#1 (#2 a)^{k} \\times (*{1})^{k}}').
add_test('-(-3xy)^{2}', '-(-3)^{2} \\times x^{2} \\times y^{2}'))
axioms.append(
Axiom(name='将指数分配进分子分母', allow_complication=True).add_rule(
'#(-\\frac{#a}{#b})^{k}',
'#1 (-1)^{k} \\frac{(#2 a)^{k}}{(#3 b)^{k}}',
animation=
'`#1 (-\\frac{#2 a}{#3 b})^{k}`[replace]{#1 (-1)^{k} \\frac{(#2 a)^{k}}{(#3 b)^{k}}}'
).add_rule(
'#(+\\frac{#a}{#b})^{k}',
'#1 \\frac{(#2 a)^{k}}{(#3 b)^{k}}',
animation=
'`#1 (+\\frac{#2 a}{#3 b})^{k}`[replace]{#1 \\frac{(#2 a)^{k}}{(#3 b)^{k}}}'
).add_test('(-\\frac{-2}{3})^{2}',
'(-1)^{2} \\times \\frac{(-2)^{2}}{3^{2}}').add_test(
'-(\\frac{-2}{3})^{2}', '-\\frac{(-2)^{2}}{3^{2}}'))
axioms.append(
Axiom(name='系数乘进分式的分子', allow_complication=True).add_rule(
'# a \\times \\frac{# 1}{b}',
'#0 \\frac{a}{b}',
animation='`#1 a \\times \\frac{#2 1}{b}`[replace]{#0 \\frac{a}{b}}'
).add_rule(
'# a \\times \\frac{# c}{b}',
'#0 \\frac{ac}{b}',
animation='`#1 a \\times \\frac{#2 c}{b}`[replace]{#0 \\frac{ac}{b}}'
).add_test('-\\frac{-1}{3} \\times 12', '\\frac{4}{1}').add_test(
'12 \cdot \\frac{3}{2}', '\\frac{18}{1}').add_test(
'\\frac{9}{4} \\times (1 - \\frac{3}{4})',
'\\frac{9 \\times (1 - \\frac{3}{4})}{4}'))
axioms.append(
Axiom(name='乘法分配率',
allow_complication=True,
recursive_apply=True,
max_results=4).
add_rule(
'#a(x + *{1})',
'#1 ax #1 a *{1}',
animation=
'`#1 a`[moveBefore,1] `(#1 `a`[moveAfter,1] x #1 `a`[moveAfter,1] *{1})`[removeOnly]'
).add_test(
'3(a + b + c) + 1',
'1 + 3 \\times a + 3 \\times b + 3 \\times c').add_test(
'(a - b)(-2)',
'-2 \\times a - 2 \\times (-b)').add_test('2(a + 3(b + c))')
#.add_test('\\frac{60 \\times (1 - \\frac{2}{5}) + 3}{2}')
)
axioms.append(
Axiom(name='负号乘进因子',
recursive_apply=True,
strict_simplify=True,
disable=(not full)).add_rule(
'-(x + *{1}) *{2} ',
'(-x - *{1}) *{2}',
animation='`-(x + *{1})`[replace]{-x - *{1}} *{2}').add_test(
'-(3 - 2)x', '(-3 + 2) \\times x'))
axioms.append(dynamic_axioms.fraction_int_addition)
axioms.append(
Axiom(name='和平方公式', allow_complication=True).add_rule(
'#(#a + *{1})^{2}',
'#1 ( a^{2} #2 2 a *{1} + (*{1})^{2} )',
animation=
'`#1 (#2 a + *{1})^{2}`[replace]{#1 (a^{2} #2 2 a *{1} + (*{1})^{2})}'
).add_rule(
'# \\left| #a + *{1} \\right|^{2}',
'#1 ( a^{2} #2 2 a *{1} + (*{1})^{2} )',
animation=
'`#1 \\left| #2 a + *{1} \\right|^{2}`[replace]{#1 ( a^{2} #2 2 a *{1} + (*{1})^{2} )}'
).add_test('-(3 - 2)^{2}',
'-(3^{2} + 2 \\times 3 \\times (-2) + (-2)^{2})'))
axioms.append(
Axiom(name='平方差公式', allow_complication=True).add_rule(
'#(a + *{1})(a - *{1})',
'#1( a^{2} - (*{1})^{2} )',
animation=
'`#1 (a + *{1})(a - *{1})`[replace]{#1( a^{2} - (*{1})^{2} )}').
add_test('(3 + a + b)(-a - b + 3)',
['3^{2} - (a + b)^{2}', '3^{2} - (-a - b)^{2}']))
return axioms