Theorem len_of_empty_range_of_ranges of type Forall

from the theory of proveit.core_expr_types.tuples

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.core_expr_types.tuples import len_of_empty_range_of_ranges

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

Passed sanity check: len_of_empty_range_of_ranges matches stored_expr

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

\forall_{m, n \in \mathbb{N}~|~\left(n + 1\right) = m}~\left[\forall_{f, i, j}~\left(\begin{array}{c} \begin{array}{l} |\left(f\left(m, i\left(m\right)\right), f\left(m, i\left(m\right) + 1\right), \ldots, f\left(m, j\left(m\right)\right), f\left(m + 1, i\left(m + 1\right)\right), f\left(m + 1, i\left(m + 1\right) + 1\right), \ldots, f\left(m + 1, j\left(m + 1\right)\right), \ldots\ldots, f\left(n, i\left(n\right)\right), f\left(n, i\left(n\right) + 1\right), \ldots, f\left(n, j\left(n\right)\right)\right)| \\  = 0 \end{array} \end{array}\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	parameters: 3 body: 4
3	ExprTuple	34, 35
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	12, 13, 14
11	Lambda	parameters: 15 body: 16
12	Operation	operator: 18 operands: 17
13	Operation	operator: 18 operands: 19
14	Operation	operator: 21 operands: 20
15	ExprTuple	45, 42, 43
16	Operation	operator: 21 operands: 22
17	ExprTuple	34, 23
18	Literal
19	ExprTuple	35, 23
20	ExprTuple	24, 34
21	Literal
22	ExprTuple	25, 26
23	Literal
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operands: 30
26	Literal
27	Literal
28	ExprTuple	35, 31
29	Literal
30	ExprTuple	32
31	Literal
32	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
33	Lambda	parameter: 47 body: 36
34	Variable
35	Variable
36	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
37	Lambda	parameter: 48 body: 41
38	Operation	operator: 42 operand: 47
39	Operation	operator: 43 operand: 47
40	ExprTuple	48
41	Operation	operator: 45 operands: 46
42	Variable
43	Variable
44	ExprTuple	47
45	Variable
46	ExprTuple	47, 48
47	Variable
48	Variable