# from the theory of proveit.numbers.modular¶

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.
# import Expression classes needed to build the expression
from proveit import a, b, q
from proveit.logic import Equals, Exists, Forall
from proveit.numbers import Add, Integer, Mod, Mult, Real, RealPos

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, b], instance_expr = Exists(instance_param_or_params = [q], instance_expr = Equals(a, Add(Mult(q, b), Mod(a, b))), domain = Integer), domains = [Real, RealPos])

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 \in \mathbb{R}, b \in \mathbb{R}^+}~\left[\exists_{q \in \mathbb{Z}}~\left(a = \left(\left(q \cdot b\right) + \left(a ~\textup{mod}~ b\right)\right)\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: 35
body: 4
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple12, 13
11Lambdaparameter: 36
body: 15
12Operationoperator: 24
operands: 16
13Operationoperator: 24
operands: 17
14ExprTuple36
15Conditionalvalue: 18
condition: 19
16ExprTuple37, 20
17ExprTuple38, 21
18Operationoperator: 22
operands: 23
19Operationoperator: 24
operands: 25
20Literal
21Literal
22Literal
23ExprTuple37, 26
24Literal
25ExprTuple36, 27
26Operationoperator: 28
operands: 29
27Literal
28Literal
29ExprTuple30, 31
30Operationoperator: 32
operands: 33
31Operationoperator: 34
operands: 35
32Literal
33ExprTuple36, 38
34Literal
35ExprTuple37, 38
36Variable
37Variable
38Variable