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 ExprRange, IndexedVar, S, Variable, m, x
from proveit.core_expr_types import x_1_to_m
from proveit.logic import And, CartExp, Forall, InSet
from proveit.numbers import NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [x_1_to_m, S], instance_expr = And(ExprRange(sub_expr1, InSet(IndexedVar(x, sub_expr1), S), one, m)), condition = InSet([x_1_to_m], CartExp(S, m))), 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_{m \in \mathbb{N}^+}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{m}, S~|~\left(x_{1}, x_{2}, \ldots, x_{m}\right) \in S^{m}}~\left(\left(x_{1} \in S\right) \land  \left(x_{2} \in S\right) \land  \ldots \land  \left(x_{m} \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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 29 body: 4
3	ExprTuple	29
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 30 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	29, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	23, 33
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 30 operands: 18
16	Literal
17	ExprTuple	19
18	ExprTuple	20, 21
19	ExprRange	lambda_map: 22 start_index: 28 end_index: 29
20	ExprTuple	23
21	Operation	operator: 24 operands: 25
22	Lambda	parameter: 36 body: 26
23	ExprRange	lambda_map: 27 start_index: 28 end_index: 29
24	Literal
25	ExprTuple	33, 29
26	Operation	operator: 30 operands: 31
27	Lambda	parameter: 36 body: 32
28	Literal
29	Variable
30	Literal
31	ExprTuple	32, 33
32	IndexedVar	variable: 34 index: 36
33	Variable
34	Variable
35	ExprTuple	36
36	Variable