Theorem leftward_commutation of type Forall

from the theory of proveit.logic.booleans.conjunction

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.booleans.conjunction import leftward_commutation

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

Passed sanity check: leftward_commutation matches stored_expr

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

\forall_{l, m, n \in \mathbb{N}}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{l}, B_{1}, B_{2}, \ldots, B_{m}, C, D_{1}, D_{2}, \ldots, D_{n} \in \mathbb{B}}~\left(\begin{array}{c} \begin{array}{l} \left(A_{1} \land  A_{2} \land  \ldots \land  A_{l}\land B_{1} \land  B_{2} \land  \ldots \land  B_{m} \land C\land D_{1} \land  D_{2} \land  \ldots \land  D_{n}\right) =  \\ \left(A_{1} \land  A_{2} \land  \ldots \land  A_{l} \land C\land B_{1} \land  B_{2} \land  \ldots \land  B_{m}\land D_{1} \land  D_{2} \land  \ldots \land  D_{n}\right) \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	45, 47, 50
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 31 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13
10	Lambda	parameters: 30 body: 14
11	Operation	operator: 53 operands: 15
12	Operation	operator: 53 operands: 16
13	Operation	operator: 53 operands: 17
14	Conditional	value: 18 condition: 19
15	ExprTuple	45, 20
16	ExprTuple	47, 20
17	ExprTuple	50, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 31 operands: 23
20	Literal
21	Literal
22	ExprTuple	24, 25
23	ExprTuple	26, 27, 28, 29
24	Operation	operator: 31 operands: 30
25	Operation	operator: 31 operands: 32
26	ExprRange	lambda_map: 33 start_index: 49 end_index: 45
27	ExprRange	lambda_map: 34 start_index: 49 end_index: 47
28	Operation	operator: 53 operands: 35
29	ExprRange	lambda_map: 36 start_index: 49 end_index: 50
30	ExprTuple	37, 38, 42, 39
31	Literal
32	ExprTuple	37, 42, 38, 39
33	Lambda	parameter: 63 body: 40
34	Lambda	parameter: 63 body: 41
35	ExprTuple	42, 58
36	Lambda	parameter: 63 body: 43
37	ExprRange	lambda_map: 44 start_index: 49 end_index: 45
38	ExprRange	lambda_map: 46 start_index: 49 end_index: 47
39	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
40	Operation	operator: 53 operands: 51
41	Operation	operator: 53 operands: 52
42	Variable
43	Operation	operator: 53 operands: 54
44	Lambda	parameter: 63 body: 55
45	Variable
46	Lambda	parameter: 63 body: 56
47	Variable
48	Lambda	parameter: 63 body: 57
49	Literal
50	Variable
51	ExprTuple	55, 58
52	ExprTuple	56, 58
53	Literal
54	ExprTuple	57, 58
55	IndexedVar	variable: 59 index: 63
56	IndexedVar	variable: 60 index: 63
57	IndexedVar	variable: 61 index: 63
58	Literal
59	Variable
60	Variable
61	Variable
62	ExprTuple	63
63	Variable