Expression of type Conditional

from the theory of proveit.numbers.modular

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

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

expr:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\forall_{a, b, c \in \mathbb{Z}}~\left(\left[x \mapsto \left(x ~\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) \textrm{ if } N \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 26 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	44, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	38, 39, 32
9	Conditional	value: 10 condition: 11
10	Operation	operator: 26 operands: 12
11	Operation	operator: 13 operands: 14
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	17, 18, 19
15	Lambda	parameter: 34 body: 21
16	Operation	operator: 22 operands: 23
17	Operation	operator: 26 operands: 24
18	Operation	operator: 26 operands: 25
19	Operation	operator: 26 operands: 27
20	ExprTuple	34
21	Operation	operator: 28 operands: 29
22	Literal
23	ExprTuple	30, 31
24	ExprTuple	38, 33
25	ExprTuple	39, 33
26	Literal
27	ExprTuple	32, 33
28	Literal
29	ExprTuple	34, 44
30	Operation	operator: 36 operands: 35
31	Operation	operator: 36 operands: 37
32	Variable
33	Literal
34	Variable
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	40, 41
38	Variable
39	Variable
40	Literal
41	Operation	operator: 42 operands: 43
42	Literal
43	ExprTuple	44, 45
44	Variable
45	Operation	operator: 46 operand: 48
46	Literal
47	ExprTuple	48
48	Literal