Expression of type Forall

from the theory of proveit.numbers.numerals.decimals

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 ExprRange, IndexedVar, Variable, a, c, d, k, m, n
from proveit.core_expr_types import b_1_to_n
from proveit.logic import Equals, Forall
from proveit.numbers import Add, DecimalSequence, Digits, Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = ExprRange(sub_expr1, IndexedVar(a, sub_expr1), one, m)
sub_expr3 = ExprRange(sub_expr1, IndexedVar(d, sub_expr1), one, k)
sub_expr4 = Add(b_1_to_n)
expr = Forall(instance_param_or_params = [m, n, k], instance_expr = Forall(instance_param_or_params = [sub_expr2, b_1_to_n, c, sub_expr3], instance_expr = Equals(DecimalSequence(sub_expr2, sub_expr4, sub_expr3), DecimalSequence(sub_expr2, c, sub_expr3)), domain = Digits, condition = Equals(sub_expr4, c)), domain = Natural)

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_{m, n, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m}, b_{1}, b_{2}, \ldots, b_{n}, c, d_{1}, d_{2}, \ldots, d_{k} \in \mathbb{N}^{\leq 9}~|~\left(b_{1} +  b_{2} +  \ldots +  b_{n}\right) = c}~\left(\texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ \left(b_{1} +  b_{2} +  \ldots +  b_{n}\right) ~ d_{1}, d_{2}, \ldots, d_{k}) = \texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ c ~ d_{1}, d_{2}, \ldots, d_{k})\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	49, 66, 51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 23 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13
10	Lambda	parameters: 14 body: 15
11	Operation	operator: 54 operands: 16
12	Operation	operator: 54 operands: 17
13	Operation	operator: 54 operands: 18
14	ExprTuple	41, 61, 47, 42
15	Conditional	value: 19 condition: 20
16	ExprTuple	49, 21
17	ExprTuple	66, 21
18	ExprTuple	51, 21
19	Operation	operator: 39 operands: 22
20	Operation	operator: 23 operands: 24
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28, 29, 30, 31
25	Operation	operator: 33 operands: 32
26	Operation	operator: 33 operands: 34
27	ExprRange	lambda_map: 35 start_index: 65 end_index: 49
28	ExprRange	lambda_map: 36 start_index: 65 end_index: 66
29	Operation	operator: 54 operands: 37
30	ExprRange	lambda_map: 38 start_index: 65 end_index: 51
31	Operation	operator: 39 operands: 40
32	ExprTuple	41, 46, 42
33	Literal
34	ExprTuple	41, 47, 42
35	Lambda	parameter: 70 body: 43
36	Lambda	parameter: 70 body: 44
37	ExprTuple	47, 60
38	Lambda	parameter: 70 body: 45
39	Literal
40	ExprTuple	46, 47
41	ExprRange	lambda_map: 48 start_index: 65 end_index: 49
42	ExprRange	lambda_map: 50 start_index: 65 end_index: 51
43	Operation	operator: 54 operands: 52
44	Operation	operator: 54 operands: 53
45	Operation	operator: 54 operands: 55
46	Operation	operator: 56 operands: 57
47	Variable
48	Lambda	parameter: 70 body: 58
49	Variable
50	Lambda	parameter: 70 body: 59
51	Variable
52	ExprTuple	58, 60
53	ExprTuple	67, 60
54	Literal
55	ExprTuple	59, 60
56	Literal
57	ExprTuple	61
58	IndexedVar	variable: 62 index: 70
59	IndexedVar	variable: 63 index: 70
60	Literal
61	ExprRange	lambda_map: 64 start_index: 65 end_index: 66
62	Variable
63	Variable
64	Lambda	parameter: 70 body: 67
65	Literal
66	Variable
67	IndexedVar	variable: 68 index: 70
68	Variable
69	ExprTuple	70
70	Variable