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Expression of type Implies

from the theory of proveit.numbers.summation

In [1]:
import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Qx, fx, x
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Integer, Sum
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(fx, Integer), condition = Qx), InSet(Sum(index_or_indices = sub_expr1, summand = fx, condition = Qx), Integer))
expr:
In [3]:
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")
Passed sanity check: expr matches stored_expr
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\left[\forall_{x~|~Q\left(x\right)}~\left(f\left(x\right) \in \mathbb{Z}\right)\right] \Rightarrow \left(\left[\sum_{x~|~Q\left(x\right)}~f\left(x\right)\right] \in \mathbb{Z}\right)
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()()('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 5
operand: 8
4Operationoperator: 15
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 18
8Lambdaparameter: 24
body: 10
9Operationoperator: 11
operand: 14
10Conditionalvalue: 13
condition: 20
11Literal
12ExprTuple14
13Operationoperator: 15
operands: 16
14Lambdaparameter: 24
body: 17
15Literal
16ExprTuple19, 18
17Conditionalvalue: 19
condition: 20
18Literal
19Operationoperator: 21
operand: 24
20Operationoperator: 22
operand: 24
21Variable
22Variable
23ExprTuple24
24Variable