Expression of type Lambda

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, Lambda, Variable, a, b, k, m
from proveit.logic import And, Equals, Forall, InSet
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 = Lambda([m, k], Conditional(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)), And(InSet(m, 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(m, k\right) \mapsto \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) \textrm{ if } m \in \mathbb{N} ,  k \in \mathbb{N}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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