# 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 Lambda, N, a, b, c, x
from proveit.logic import Bijections, Forall, InSet
from proveit.numbers import Add, Integer, Interval, Mod, NaturalPos, one, subtract, zero

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [N], instance_expr = Forall(instance_param_or_params = [a, b, c], instance_expr = InSet(Lambda(x, Mod(Add(c, x), N)), Bijections(Interval(a, b), Interval(zero, subtract(N, one)))), domain = Integer), domain = NaturalPos)

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_{N \in \mathbb{N}^+}~\left[\forall_{a, b, c \in \mathbb{Z}}~\left(\left[x \mapsto \left(\left(c + x\right) ~\textup{mod}~ N\right)\right] \in \left[\{a~\ldotp \ldotp~b\} \xrightarrow[\text{onto}]{\text{1-to-1}} \{0~\ldotp \ldotp~N - 1\}\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 50
body: 4
3ExprTuple50
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 30
operands: 9
7Literal
8ExprTuple10
9ExprTuple50, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple42, 43, 46
13Conditionalvalue: 14
condition: 15
14Operationoperator: 30
operands: 16
15Operationoperator: 17
operands: 18
16ExprTuple19, 20
17Literal
18ExprTuple21, 22, 23
19Lambdaparameter: 47
body: 25
20Operationoperator: 26
operands: 27
21Operationoperator: 30
operands: 28
22Operationoperator: 30
operands: 29
23Operationoperator: 30
operands: 31
24ExprTuple47
25Operationoperator: 32
operands: 33
26Literal
27ExprTuple34, 35
28ExprTuple42, 36
29ExprTuple43, 36
30Literal
31ExprTuple46, 36
32Literal
33ExprTuple37, 50
34Operationoperator: 39
operands: 38
35Operationoperator: 39
operands: 40
36Literal
37Operationoperator: 48
operands: 41
38ExprTuple42, 43
39Literal
40ExprTuple44, 45
41ExprTuple46, 47
42Variable
43Variable
44Literal
45Operationoperator: 48
operands: 49
46Variable
47Variable
48Literal
49ExprTuple50, 51
50Variable
51Operationoperator: 52
operand: 54
52Literal
53ExprTuple54
54Literal