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, b, k, m
from proveit.logic import Equals, Forall
from proveit.numbers import Add, DecimalSequence, Digits, Less, nine, one, zero

# 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)
expr = Forall(instance_param_or_params = [sub_expr2, b], instance_expr = Equals(Add(DecimalSequence(sub_expr2, b, ExprRange(sub_expr1, nine, one, k)), one), DecimalSequence(sub_expr2, Add(b, one), ExprRange(sub_expr1, zero, one, k))), domain = Digits, condition = Less(b, nine))

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_{1}, a_{2}, \ldots, a_{m}, b \in \mathbb{N}^{\leq 9}~|~b < 9}~\left(\left(\texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ b ~ 9, 9, ..\left(k - 3\right) \times.., 9) + 1\right) = \texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ \left(b + 1\right) ~ 0, 0, ..\left(k - 3\right) \times.., 0)\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	34, 36
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15, 16
12	Operation	operator: 29 operands: 17
13	Operation	operator: 27 operands: 18
14	ExprRange	lambda_map: 19 start_index: 42 end_index: 40
15	Operation	operator: 32 operands: 20
16	Operation	operator: 21 operands: 22
17	ExprTuple	23, 42
18	ExprTuple	34, 24, 25
19	Lambda	parameter: 48 body: 26
20	ExprTuple	36, 38
21	Literal
22	ExprTuple	36, 45
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	ExprRange	lambda_map: 31 start_index: 42 end_index: 43
26	Operation	operator: 32 operands: 33
27	Literal
28	ExprTuple	34, 36, 35
29	Literal
30	ExprTuple	36, 42
31	Lambda	parameter: 48 body: 37
32	Literal
33	ExprTuple	44, 38
34	ExprRange	lambda_map: 39 start_index: 42 end_index: 40
35	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
36	Variable
37	Literal
38	Literal
39	Lambda	parameter: 48 body: 44
40	Variable
41	Lambda	parameter: 48 body: 45
42	Literal
43	Variable
44	IndexedVar	variable: 46 index: 48
45	Literal
46	Variable
47	ExprTuple	48
48	Variable