Expression of type Conditional

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 Conditional, ExprRange, IndexedVar, Variable, a, c, d, k, m, n
from proveit.core_expr_types import b_1_to_n
from proveit.logic import And, Equals, Forall, InSet
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 = Conditional(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)), And(InSet(m, Natural), InSet(n, Natural), InSet(k, 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())

\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) \textrm{ if } m \in \mathbb{N} ,  n \in \mathbb{N} ,  k \in \mathbb{N}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 19 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8, 9
6	Lambda	parameters: 10 body: 11
7	Operation	operator: 50 operands: 12
8	Operation	operator: 50 operands: 13
9	Operation	operator: 50 operands: 14
10	ExprTuple	37, 57, 43, 38
11	Conditional	value: 15 condition: 16
12	ExprTuple	45, 17
13	ExprTuple	62, 17
14	ExprTuple	47, 17
15	Operation	operator: 35 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24, 25, 26, 27
21	Operation	operator: 29 operands: 28
22	Operation	operator: 29 operands: 30
23	ExprRange	lambda_map: 31 start_index: 61 end_index: 45
24	ExprRange	lambda_map: 32 start_index: 61 end_index: 62
25	Operation	operator: 50 operands: 33
26	ExprRange	lambda_map: 34 start_index: 61 end_index: 47
27	Operation	operator: 35 operands: 36
28	ExprTuple	37, 42, 38
29	Literal
30	ExprTuple	37, 43, 38
31	Lambda	parameter: 66 body: 39
32	Lambda	parameter: 66 body: 40
33	ExprTuple	43, 56
34	Lambda	parameter: 66 body: 41
35	Literal
36	ExprTuple	42, 43
37	ExprRange	lambda_map: 44 start_index: 61 end_index: 45
38	ExprRange	lambda_map: 46 start_index: 61 end_index: 47
39	Operation	operator: 50 operands: 48
40	Operation	operator: 50 operands: 49
41	Operation	operator: 50 operands: 51
42	Operation	operator: 52 operands: 53
43	Variable
44	Lambda	parameter: 66 body: 54
45	Variable
46	Lambda	parameter: 66 body: 55
47	Variable
48	ExprTuple	54, 56
49	ExprTuple	63, 56
50	Literal
51	ExprTuple	55, 56
52	Literal
53	ExprTuple	57
54	IndexedVar	variable: 58 index: 66
55	IndexedVar	variable: 59 index: 66
56	Literal
57	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
58	Variable
59	Variable
60	Lambda	parameter: 66 body: 63
61	Literal
62	Variable
63	IndexedVar	variable: 64 index: 66
64	Variable
65	ExprTuple	66
66	Variable