Expression of type Forall

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 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 = Forall(instance_param_or_params = [N], instance_expr = 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), domain = 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())

\forall_{N \in \mathbb{N}^+}~\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)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 50 body: 4
3	ExprTuple	50
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 30 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	50, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	42, 43, 47
13	Conditional	value: 14 condition: 15
14	Operation	operator: 30 operands: 16
15	Operation	operator: 17 operands: 18
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22, 23
19	Lambda	parameter: 46 body: 25
20	Operation	operator: 26 operands: 27
21	Operation	operator: 30 operands: 28
22	Operation	operator: 30 operands: 29
23	Operation	operator: 30 operands: 31
24	ExprTuple	46
25	Operation	operator: 32 operands: 33
26	Literal
27	ExprTuple	34, 35
28	ExprTuple	42, 36
29	ExprTuple	43, 36
30	Literal
31	ExprTuple	47, 36
32	Literal
33	ExprTuple	37, 50
34	Operation	operator: 39 operands: 38
35	Operation	operator: 39 operands: 40
36	Literal
37	Operation	operator: 48 operands: 41
38	ExprTuple	42, 43
39	Literal
40	ExprTuple	44, 45
41	ExprTuple	46, 47
42	Variable
43	Variable
44	Literal
45	Operation	operator: 48 operands: 49
46	Variable
47	Variable
48	Literal
49	ExprTuple	50, 51
50	Variable
51	Operation	operator: 52 operand: 54
52	Literal
53	ExprTuple	54
54	Literal