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, Natural, 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 = [m, k], instance_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)), 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, k \in \mathbb{N}}~\left[\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)\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	52, 55
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 22 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12
10	Lambda	parameters: 13 body: 14
11	Operation	operator: 44 operands: 15
12	Operation	operator: 44 operands: 16
13	ExprTuple	46, 48
14	Conditional	value: 17 condition: 18
15	ExprTuple	52, 19
16	ExprTuple	55, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	Literal
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26, 27, 28
24	Operation	operator: 41 operands: 29
25	Operation	operator: 39 operands: 30
26	ExprRange	lambda_map: 31 start_index: 54 end_index: 52
27	Operation	operator: 44 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprTuple	35, 54
30	ExprTuple	46, 36, 37
31	Lambda	parameter: 60 body: 38
32	ExprTuple	48, 50
33	Literal
34	ExprTuple	48, 57
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operands: 42
37	ExprRange	lambda_map: 43 start_index: 54 end_index: 55
38	Operation	operator: 44 operands: 45
39	Literal
40	ExprTuple	46, 48, 47
41	Literal
42	ExprTuple	48, 54
43	Lambda	parameter: 60 body: 49
44	Literal
45	ExprTuple	56, 50
46	ExprRange	lambda_map: 51 start_index: 54 end_index: 52
47	ExprRange	lambda_map: 53 start_index: 54 end_index: 55
48	Variable
49	Literal
50	Literal
51	Lambda	parameter: 60 body: 56
52	Variable
53	Lambda	parameter: 60 body: 57
54	Literal
55	Variable
56	IndexedVar	variable: 58 index: 60
57	Literal
58	Variable
59	ExprTuple	60
60	Variable