# from the theory of proveit.numbers.multiplication¶

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, x, y
from proveit.logic import Forall
from proveit.numbers import LessEq, Mult, Real, greater_eq, zero

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, x, y], instance_expr = LessEq(Mult(x, a), Mult(y, a)), domain = Real, conditions = [LessEq(x, y), greater_eq(a, zero)])

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())

\forall_{a, x, y \in \mathbb{R}~|~x \leq y, a \geq 0}~\left(\left(x \cdot a\right) \leq \left(y \cdot a\right)\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 4
body: 5
4ExprTuple32, 29, 30
5Conditionalvalue: 6
condition: 7
6Operationoperator: 26
operands: 8
7Operationoperator: 9
operands: 10
8ExprTuple11, 12
9Literal
10ExprTuple13, 14, 15, 16, 17
11Operationoperator: 19
operands: 18
12Operationoperator: 19
operands: 20
13Operationoperator: 23
operands: 21
14Operationoperator: 23
operands: 22
15Operationoperator: 23
operands: 24
16Operationoperator: 26
operands: 25
17Operationoperator: 26
operands: 27
18ExprTuple29, 32
19Literal
20ExprTuple30, 32
21ExprTuple32, 28
22ExprTuple29, 28
23Literal
24ExprTuple30, 28
25ExprTuple29, 30
26Literal
27ExprTuple31, 32
28Literal
29Variable
30Variable
31Literal
32Variable