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, c, d, k, m, n
from proveit.logic import Equals, Forall
from proveit.numbers import DecimalSequence, Digits, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = ExprRange(sub_expr1, b, one, n)
sub_expr3 = ExprRange(sub_expr1, IndexedVar(a, sub_expr1), one, m)
sub_expr4 = ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, n)
sub_expr5 = ExprRange(sub_expr1, IndexedVar(d, sub_expr1), one, k)
expr = Forall(instance_param_or_params = [sub_expr3, b, sub_expr4, sub_expr5], instance_expr = Equals(DecimalSequence(sub_expr3, sub_expr2, sub_expr5), DecimalSequence(sub_expr3, sub_expr4, sub_expr5)), domain = Digits, condition = Equals([sub_expr2], [sub_expr4]))

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, c_{1}, c_{2}, \ldots, c_{n}, d_{1}, d_{2}, \ldots, d_{k} \in \mathbb{N}^{\leq 9}~|~\left(b, b, ..\left(n - 3\right) \times.., b\right) = \left(c_{1}, c_{2}, \ldots, c_{n}\right)}~\left(\texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ b, b, ..\left(n - 3\right) \times.., b ~ d_{1}, d_{2}, \ldots, d_{k}) = \texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ c_{1}, c_{2}, \ldots, c_{n} ~ d_{1}, d_{2}, \ldots, d_{k})\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	27, 53, 43, 28
5	Conditional	value: 6 condition: 7
6	Operation	operator: 25 operands: 8
7	Operation	operator: 9 operands: 10
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13, 14, 15, 16, 17
11	Operation	operator: 19 operands: 18
12	Operation	operator: 19 operands: 20
13	ExprRange	lambda_map: 21 start_index: 49 end_index: 35
14	Operation	operator: 40 operands: 22
15	ExprRange	lambda_map: 23 start_index: 49 end_index: 50
16	ExprRange	lambda_map: 24 start_index: 49 end_index: 37
17	Operation	operator: 25 operands: 26
18	ExprTuple	27, 42, 28
19	Literal
20	ExprTuple	27, 43, 28
21	Lambda	parameter: 57 body: 29
22	ExprTuple	53, 46
23	Lambda	parameter: 57 body: 30
24	Lambda	parameter: 57 body: 31
25	Literal
26	ExprTuple	32, 33
27	ExprRange	lambda_map: 34 start_index: 49 end_index: 35
28	ExprRange	lambda_map: 36 start_index: 49 end_index: 37
29	Operation	operator: 40 operands: 38
30	Operation	operator: 40 operands: 39
31	Operation	operator: 40 operands: 41
32	ExprTuple	42
33	ExprTuple	43
34	Lambda	parameter: 57 body: 44
35	Variable
36	Lambda	parameter: 57 body: 45
37	Variable
38	ExprTuple	44, 46
39	ExprTuple	54, 46
40	Literal
41	ExprTuple	45, 46
42	ExprRange	lambda_map: 47 start_index: 49 end_index: 50
43	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
44	IndexedVar	variable: 51 index: 57
45	IndexedVar	variable: 52 index: 57
46	Literal
47	Lambda	parameter: 57 body: 53
48	Lambda	parameter: 57 body: 54
49	Literal
50	Variable
51	Variable
52	Variable
53	Variable
54	IndexedVar	variable: 55 index: 57
55	Variable
56	ExprTuple	57
57	Variable