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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 a, b, j
from proveit.logic import Forall, Implies, InSet
from proveit.numbers import Interval, Natural, Sum
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [j]
sub_expr2 = Interval(a, b)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(j, Natural), domain = sub_expr2), InSet(Sum(index_or_indices = sub_expr1, summand = j, domain = sub_expr2), Natural))
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_{j \in \{a~\ldotp \ldotp~b\}}~\left(j \in \mathbb{N}\right)\right] \Rightarrow \left(\left(\sum_{j = a}^{b} j\right) \in \mathbb{N}\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: 20
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 18
8Lambdaparameter: 22
body: 10
9Operationoperator: 11
operand: 14
10Conditionalvalue: 13
condition: 19
11Literal
12ExprTuple14
13Operationoperator: 20
operands: 15
14Lambdaparameter: 22
body: 17
15ExprTuple22, 18
16ExprTuple22
17Conditionalvalue: 22
condition: 19
18Literal
19Operationoperator: 20
operands: 21
20Literal
21ExprTuple22, 23
22Variable
23Operationoperator: 24
operands: 25
24Literal
25ExprTuple26, 27
26Variable
27Variable