Theorem true_for_each_then_true_for_all_conditioned 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_each_then_true_for_all_conditioned

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

Passed sanity check: true_for_each_then_true_for_all_conditioned 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, Q~|~\left(Q\left(a_{1}\right) \Rightarrow P\left(a_{1}\right)\right), \left(Q\left(a_{2}\right) \Rightarrow P\left(a_{2}\right)\right), \ldots, \left(Q\left(a_{n}\right) \Rightarrow P\left(a_{n}\right)\right)}~\left[\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}~|~Q\left(x\right)}~P\left(x\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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 49 body: 4
3	ExprTuple	49
4	Conditional	value: 5 condition: 6
5	Operation	operator: 18 operand: 9
6	Operation	operator: 36 operands: 8
7	ExprTuple	9
8	ExprTuple	49, 10
9	Lambda	parameters: 45 body: 11
10	Literal
11	Operation	operator: 18 operand: 13
12	ExprTuple	13
13	Lambda	parameters: 14 body: 15
14	ExprTuple	40, 39
15	Conditional	value: 16 condition: 17
16	Operation	operator: 18 operand: 21
17	Operation	operator: 28 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	22
21	Lambda	parameter: 43 body: 23
22	ExprRange	lambda_map: 24 start_index: 48 end_index: 49
23	Conditional	value: 25 condition: 26
24	Lambda	parameter: 53 body: 27
25	Operation	operator: 40 operand: 43
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 39 operand: 43
34	Operation	operator: 39 operand: 50
35	Operation	operator: 40 operand: 50
36	Literal
37	ExprTuple	43, 42
38	ExprTuple	43
39	Variable
40	Variable
41	ExprTuple	50
42	Operation	operator: 44 operands: 45
43	Variable
44	Literal
45	ExprTuple	46
46	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
47	Lambda	parameter: 53 body: 50
48	Literal
49	Variable
50	IndexedVar	variable: 51 index: 53
51	Variable
52	ExprTuple	53
53	Variable