Expression of type Lambda

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 Add, Integer, Interval, Mod, NaturalPos, one, subtract, zero

# build up the expression from sub-expressions
expr = Lambda(N, Conditional(Forall(instance_param_or_params = [a, b, c], instance_expr = InSet(Lambda(x, Mod(Add(x, c), 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())

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

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 48 body: 2
1	ExprTuple	48
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 28 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	48, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	40, 41, 45
11	Conditional	value: 12 condition: 13
12	Operation	operator: 28 operands: 14
13	Operation	operator: 15 operands: 16
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19, 20, 21
17	Lambda	parameter: 44 body: 23
18	Operation	operator: 24 operands: 25
19	Operation	operator: 28 operands: 26
20	Operation	operator: 28 operands: 27
21	Operation	operator: 28 operands: 29
22	ExprTuple	44
23	Operation	operator: 30 operands: 31
24	Literal
25	ExprTuple	32, 33
26	ExprTuple	40, 34
27	ExprTuple	41, 34
28	Literal
29	ExprTuple	45, 34
30	Literal
31	ExprTuple	35, 48
32	Operation	operator: 37 operands: 36
33	Operation	operator: 37 operands: 38
34	Literal
35	Operation	operator: 46 operands: 39
36	ExprTuple	40, 41
37	Literal
38	ExprTuple	42, 43
39	ExprTuple	44, 45
40	Variable
41	Variable
42	Literal
43	Operation	operator: 46 operands: 47
44	Variable
45	Variable
46	Literal
47	ExprTuple	48, 49
48	Variable
49	Operation	operator: 50 operand: 52
50	Literal
51	ExprTuple	52
52	Literal