Theorem true_for_all_then_true_for_each of type Forall

from the theory of proveit.logic.sets.enumeration

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.logic.sets.enumeration import true_for_all_then_true_for_each

# check that the built expression is the same as the stored expression
assert true_for_all_then_true_for_each.expr == stored_expr
assert true_for_all_then_true_for_each.expr._style_id == stored_expr._style_id
print("Passed sanity check: true_for_all_then_true_for_each matches stored_expr")

Passed sanity check: true_for_all_then_true_for_each matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n}}~\left[\forall_{P~|~\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\right)}~\left(P\left(a_{1}\right) \land  P\left(a_{2}\right) \land  \ldots \land  P\left(a_{n}\right)\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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 41 body: 4
3	ExprTuple	41
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 9
6	Operation	operator: 32 operands: 8
7	ExprTuple	9
8	ExprTuple	41, 10
9	Lambda	parameters: 37 body: 11
10	Literal
11	Operation	operator: 20 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 30 body: 15
14	ExprTuple	30
15	Conditional	value: 16 condition: 17
16	Operation	operator: 18 operands: 19
17	Operation	operator: 20 operand: 23
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23
22	ExprRange	lambda_map: 24 start_index: 40 end_index: 41
23	Lambda	parameter: 34 body: 25
24	Lambda	parameter: 45 body: 26
25	Conditional	value: 27 condition: 28
26	Operation	operator: 30 operand: 42
27	Operation	operator: 30 operand: 34
28	Operation	operator: 32 operands: 33
29	ExprTuple	42
30	Variable
31	ExprTuple	34
32	Literal
33	ExprTuple	34, 35
34	Variable
35	Operation	operator: 36 operands: 37
36	Literal
37	ExprTuple	38
38	ExprRange	lambda_map: 39 start_index: 40 end_index: 41
39	Lambda	parameter: 45 body: 42
40	Literal
41	Variable
42	IndexedVar	variable: 43 index: 45
43	Variable
44	ExprTuple	45
45	Variable