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 N, a, b, x
from proveit.logic import Card, Equals, Forall, SetOfAll
from proveit.numbers import Integer, Interval, Mod, NaturalPos, greater_eq, one, subtract, zero

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
expr = Forall(instance_param_or_params = [N], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(SetOfAll(instance_param_or_params = [x], instance_element = Mod(x, N), domain = sub_expr1), Interval(zero, subtract(N, one))), domain = Integer, condition = greater_eq(Card(sub_expr1), N)), 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 \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(\left\{x ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}} = \{0~\ldotp \ldotp~N - 1\}\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: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 47 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	51, 11
10	Lambda	parameters: 56 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22, 23
19	Operation	operator: 24 operand: 31
20	Operation	operator: 55 operands: 26
21	Operation	operator: 47 operands: 27
22	Operation	operator: 47 operands: 28
23	Operation	operator: 29 operands: 30
24	Literal
25	ExprTuple	31
26	ExprTuple	32, 33
27	ExprTuple	57, 34
28	ExprTuple	58, 34
29	Literal
30	ExprTuple	51, 35
31	Lambda	parameter: 52 body: 37
32	Literal
33	Operation	operator: 38 operands: 39
34	Literal
35	Operation	operator: 40 operand: 53
36	ExprTuple	52
37	Conditional	value: 42 condition: 43
38	Literal
39	ExprTuple	51, 44
40	Literal
41	ExprTuple	53
42	Operation	operator: 45 operands: 46
43	Operation	operator: 47 operands: 48
44	Operation	operator: 49 operand: 54
45	Literal
46	ExprTuple	52, 51
47	Literal
48	ExprTuple	52, 53
49	Literal
50	ExprTuple	54
51	Variable
52	Variable
53	Operation	operator: 55 operands: 56
54	Literal
55	Literal
56	ExprTuple	57, 58
57	Variable
58	Variable