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 L, S, c, x
from proveit.logic import Card, Equals, Forall, SetOfAll
from proveit.numbers import Add, Mod, Real

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [c], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(Add(c, x), L), domain = S)), Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(x, L), domain = S))), domain = Real)

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_{c \in \mathbb{R}}~\left(|\left\{\left(c + x\right) ~\textup{mod}~ L\right\}_{x \in S}| = |\left\{x ~\textup{mod}~ L\right\}_{x \in S}|\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 40 body: 5
4	ExprTuple	40
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 33 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	40, 13
11	Operation	operator: 15 operand: 17
12	Operation	operator: 15 operand: 18
13	Literal
14	ExprTuple	17
15	Literal
16	ExprTuple	18
17	Operation	operator: 20 operand: 22
18	Operation	operator: 20 operand: 23
19	ExprTuple	22
20	Literal
21	ExprTuple	23
22	Lambda	parameter: 41 body: 24
23	Lambda	parameter: 41 body: 26
24	Conditional	value: 27 condition: 29
25	ExprTuple	41
26	Conditional	value: 28 condition: 29
27	Operation	operator: 31 operands: 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	41, 36
33	Literal
34	ExprTuple	41, 37
35	Operation	operator: 38 operands: 39
36	Variable
37	Variable
38	Literal
39	ExprTuple	40, 41
40	Variable
41	Variable