Expression of type Forall

from the theory of proveit.core_expr_types.tuples

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 a, b, i, j, k, l
from proveit.core_expr_types.tuples import neg_shift_equiv_both
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Add, Integer, Natural, one, subtract

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, b], instance_expr = Forall(instance_param_or_params = [i, j, k, l], instance_expr = neg_shift_equiv_both, conditions = [InSet(subtract(Add(j, one), i), Natural), Equals(Add(i, a), Add(k, b)), Equals(Add(j, a), Add(l, b))]), domain = Integer)

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_{a, b \in \mathbb{Z}}~\left[\forall_{i, j, k, l~|~\left(\left(j + 1\right) - i\right) \in \mathbb{N}, \left(i + a\right) = \left(k + b\right), \left(j + a\right) = \left(l + b\right)}~\left(\left(f\left(i + a\right), f\left(\left(i - 1\right) + a\right), \ldots, f\left(j + a\right)\right) = \left(f\left(k + b\right), f\left(\left(k - 1\right) + b\right), \ldots, f\left(l + b\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	73, 75
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 21 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 13 body: 14
11	Operation	operator: 30 operands: 15
12	Operation	operator: 30 operands: 16
13	ExprTuple	72, 70, 61, 62
14	Conditional	value: 17 condition: 18
15	ExprTuple	73, 19
16	ExprTuple	75, 19
17	Operation	operator: 33 operands: 20
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25, 26, 27
23	ExprTuple	28
24	ExprTuple	29
25	Operation	operator: 30 operands: 31
26	Operation	operator: 33 operands: 32
27	Operation	operator: 33 operands: 34
28	ExprRange	lambda_map: 35 start_index: 60 end_index: 36
29	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
30	Literal
31	ExprTuple	40, 41
32	ExprTuple	42, 43
33	Literal
34	ExprTuple	44, 45
35	Lambda	parameter: 78 body: 46
36	Operation	operator: 76 operand: 70
37	Lambda	parameter: 78 body: 48
38	Operation	operator: 76 operand: 61
39	Operation	operator: 76 operand: 62
40	Operation	operator: 68 operands: 51
41	Literal
42	Operation	operator: 68 operands: 52
43	Operation	operator: 68 operands: 53
44	Operation	operator: 68 operands: 54
45	Operation	operator: 68 operands: 55
46	Operation	operator: 57 operand: 63
47	ExprTuple	70
48	Operation	operator: 57 operand: 64
49	ExprTuple	61
50	ExprTuple	62
51	ExprTuple	59, 60
52	ExprTuple	72, 73
53	ExprTuple	61, 75
54	ExprTuple	70, 73
55	ExprTuple	62, 75
56	ExprTuple	63
57	Variable
58	ExprTuple	64
59	Operation	operator: 68 operands: 65
60	Operation	operator: 76 operand: 72
61	Variable
62	Variable
63	Operation	operator: 68 operands: 67
64	Operation	operator: 68 operands: 69
65	ExprTuple	70, 71
66	ExprTuple	72
67	ExprTuple	74, 73
68	Literal
69	ExprTuple	74, 75
70	Variable
71	Literal
72	Variable
73	Variable
74	Operation	operator: 76 operand: 78
75	Variable
76	Literal
77	ExprTuple	78
78	Variable