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 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 = 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	69, 71
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	68, 66, 56, 57
14	Conditional	value: 17 condition: 18
15	ExprTuple	69, 19
16	ExprTuple	71, 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: 68 end_index: 66
29	ExprRange	lambda_map: 36 start_index: 56 end_index: 57
30	Literal
31	ExprTuple	37, 38
32	ExprTuple	39, 40
33	Literal
34	ExprTuple	41, 42
35	Lambda	parameter: 70 body: 43
36	Lambda	parameter: 70 body: 45
37	Operation	operator: 64 operands: 46
38	Literal
39	Operation	operator: 64 operands: 47
40	Operation	operator: 64 operands: 48
41	Operation	operator: 64 operands: 49
42	Operation	operator: 64 operands: 50
43	Operation	operator: 52 operand: 58
44	ExprTuple	70
45	Operation	operator: 52 operand: 59
46	ExprTuple	54, 55
47	ExprTuple	68, 69
48	ExprTuple	56, 71
49	ExprTuple	66, 69
50	ExprTuple	57, 71
51	ExprTuple	58
52	Variable
53	ExprTuple	59
54	Operation	operator: 64 operands: 60
55	Operation	operator: 61 operand: 68
56	Variable
57	Variable
58	Operation	operator: 64 operands: 63
59	Operation	operator: 64 operands: 65
60	ExprTuple	66, 67
61	Literal
62	ExprTuple	68
63	ExprTuple	70, 69
64	Literal
65	ExprTuple	70, 71
66	Variable
67	Literal
68	Variable
69	Variable
70	Variable
71	Variable