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, SubsetEq
from proveit.numbers import Add, Mod, Real

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [L], instance_expr = Forall(instance_param_or_params = [S], instance_expr = Forall(instance_param_or_params = [c], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(Add(x, c), L), domain = S)), Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(x, L), domain = S))), domain = Real), condition = SubsetEq(S, Real)), 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_{L \in \mathbb{R}}~\left[\forall_{S~|~S \subseteq \mathbb{R}}~\left[\forall_{c \in \mathbb{R}}~\left(|\left\{\left(x + c\right) ~\textup{mod}~ L\right\}_{x \in S}| = |\left\{x ~\textup{mod}~ L\right\}_{x \in S}|\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 9
6	Operation	operator: 48 operands: 8
7	ExprTuple	9
8	ExprTuple	51, 28
9	Lambda	parameter: 52 body: 11
10	ExprTuple	52
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 18
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18
16	Literal
17	ExprTuple	52, 28
18	Lambda	parameter: 56 body: 20
19	ExprTuple	56
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operands: 24
22	Operation	operator: 48 operands: 25
23	Literal
24	ExprTuple	26, 27
25	ExprTuple	56, 28
26	Operation	operator: 30 operand: 32
27	Operation	operator: 30 operand: 33
28	Literal
29	ExprTuple	32
30	Literal
31	ExprTuple	33
32	Operation	operator: 35 operand: 37
33	Operation	operator: 35 operand: 38
34	ExprTuple	37
35	Literal
36	ExprTuple	38
37	Lambda	parameter: 55 body: 39
38	Lambda	parameter: 55 body: 41
39	Conditional	value: 42 condition: 44
40	ExprTuple	55
41	Conditional	value: 43 condition: 44
42	Operation	operator: 46 operands: 45
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 49
45	ExprTuple	50, 51
46	Literal
47	ExprTuple	55, 51
48	Literal
49	ExprTuple	55, 52
50	Operation	operator: 53 operands: 54
51	Variable
52	Variable
53	Literal
54	ExprTuple	55, 56
55	Variable
56	Variable