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 = [S], instance_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), condition = SubsetEq(S, 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_{S~|~S \subseteq \mathbb{R}}~\left[\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)\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: 45 body: 4
3	ExprTuple	45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	45, 21
11	Lambda	parameter: 48 body: 13
12	ExprTuple	48
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 41 operands: 18
16	Literal
17	ExprTuple	19, 20
18	ExprTuple	48, 21
19	Operation	operator: 23 operand: 25
20	Operation	operator: 23 operand: 26
21	Literal
22	ExprTuple	25
23	Literal
24	ExprTuple	26
25	Operation	operator: 28 operand: 30
26	Operation	operator: 28 operand: 31
27	ExprTuple	30
28	Literal
29	ExprTuple	31
30	Lambda	parameter: 49 body: 32
31	Lambda	parameter: 49 body: 34
32	Conditional	value: 35 condition: 37
33	ExprTuple	49
34	Conditional	value: 36 condition: 37
35	Operation	operator: 39 operands: 38
36	Operation	operator: 39 operands: 40
37	Operation	operator: 41 operands: 42
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	49, 44
41	Literal
42	ExprTuple	49, 45
43	Operation	operator: 46 operands: 47
44	Variable
45	Variable
46	Literal
47	ExprTuple	48, 49
48	Variable
49	Variable