Expression of type Forall

from the theory of proveit.logic.sets.cartesian_products

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 x
from proveit.core_expr_types import A_1_to_m, a_1_to_m
from proveit.logic import And, CartProd, Equals, Forall, InSet
from proveit.logic.sets.cartesian_products import a_in_A__1_to_m

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [x, A_1_to_m, a_1_to_m], instance_expr = Equals(InSet(x, CartProd(A_1_to_m)), And(Equals(x, [a_1_to_m]), a_in_A__1_to_m)).with_wrapping_at(2))

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_{x, A_{1}, A_{2}, \ldots, A_{m}, a_{1}, a_{2}, \ldots, a_{m}}~\left(\begin{array}{c} \begin{array}{l} \left(x \in \left(A_{1} \times  A_{2} \times  \ldots \times  A_{m}\right)\right) =  \\ \left(\left(x = \left(a_{1}, a_{2}, \ldots, a_{m}\right)\right)\land \left(a_{1} \in A_{1}\right) \land  \left(a_{2} \in A_{2}\right) \land  \ldots \land  \left(a_{m} \in A_{m}\right)\right) \end{array} \end{array}\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	parameters: 4 body: 5
4	ExprTuple	21, 20, 25
5	Operation	operator: 17 operands: 6
6	ExprTuple	7, 8
7	Operation	operator: 26 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	21, 12
10	Literal
11	ExprTuple	13, 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprRange	lambda_map: 19 start_index: 29 end_index: 30
15	Literal
16	ExprTuple	20
17	Literal
18	ExprTuple	21, 22
19	Lambda	parameter: 36 body: 23
20	ExprRange	lambda_map: 24 start_index: 29 end_index: 30
21	Variable
22	ExprTuple	25
23	Operation	operator: 26 operands: 27
24	Lambda	parameter: 36 body: 31
25	ExprRange	lambda_map: 28 start_index: 29 end_index: 30
26	Literal
27	ExprTuple	32, 31
28	Lambda	parameter: 36 body: 32
29	Literal
30	Variable
31	IndexedVar	variable: 33 index: 36
32	IndexedVar	variable: 34 index: 36
33	Variable
34	Variable
35	ExprTuple	36
36	Variable