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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 S, k, l
from proveit.logic import Forall, Implies
from proveit.numbers import Less, Sum
from proveit.numbers.summation import ak, al, bk, bl
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [l]
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Less(ak, bk), domain = S), Less(Sum(index_or_indices = sub_expr1, summand = al, domain = S), Sum(index_or_indices = sub_expr1, summand = bl, domain = S)))
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_{k \in S}~\left(a\left(k\right) < b\left(k\right)\right)\right] \Rightarrow \left(\left[\sum_{l \in S}~a\left(l\right)\right] < \left[\sum_{l \in S}~b\left(l\right)\right]\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: 19
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Lambdaparameter: 35
body: 11
9Operationoperator: 13
operand: 17
10Operationoperator: 13
operand: 18
11Conditionalvalue: 15
condition: 16
12ExprTuple17
13Literal
14ExprTuple18
15Operationoperator: 19
operands: 20
16Operationoperator: 33
operands: 21
17Lambdaparameter: 36
body: 22
18Lambdaparameter: 36
body: 23
19Literal
20ExprTuple24, 25
21ExprTuple35, 37
22Conditionalvalue: 26
condition: 28
23Conditionalvalue: 27
condition: 28
24Operationoperator: 30
operand: 35
25Operationoperator: 31
operand: 35
26Operationoperator: 30
operand: 36
27Operationoperator: 31
operand: 36
28Operationoperator: 33
operands: 34
29ExprTuple35
30Variable
31Variable
32ExprTuple36
33Literal
34ExprTuple36, 37
35Variable
36Variable
37Variable