Expression of type Forall

from the theory of proveit.core_expr_types.tuples

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 i, j, n
from proveit.core_expr_types import Len, f_1_to_n, i_to_j_len
from proveit.core_expr_types.tuples import f_i_to_j__1_to_n
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Mult, Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [f_1_to_n, i, j], instance_expr = Equals(Len(operands = [f_i_to_j__1_to_n]), Mult(n, i_to_j_len)).with_wrapping_at(1), condition = InSet(i_to_j_len, 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_{f_{1}, f_{2}, \ldots, f_{n}, i, j~|~\left(j - i + 1\right) \in \mathbb{N}}~\left(\begin{array}{c} \begin{array}{l} |\left(f_{1}\left(i\right), f_{1}\left(i + 1\right), \ldots, f_{1}\left(j\right), f_{2}\left(i\right), f_{2}\left(i + 1\right), \ldots, f_{2}\left(j\right), \ldots\ldots, f_{n}\left(i\right), f_{n}\left(i + 1\right), \ldots, f_{n}\left(j\right)\right)| \\  = \left(n \cdot \left(j - i + 1\right)\right) \end{array} \end{array}\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	6, 36, 32
5	Conditional	value: 7 condition: 8
6	ExprRange	lambda_map: 9 start_index: 30 end_index: 25
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	Lambda	parameter: 41 body: 14
10	Literal
11	ExprTuple	15, 16
12	Literal
13	ExprTuple	23, 17
14	IndexedVar	variable: 39 index: 41
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	Literal
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	25, 23
22	ExprRange	lambda_map: 24 start_index: 30 end_index: 25
23	Operation	operator: 26 operands: 27
24	Lambda	parameter: 42 body: 28
25	Variable
26	Literal
27	ExprTuple	32, 29, 30
28	ExprRange	lambda_map: 31 start_index: 36 end_index: 32
29	Operation	operator: 33 operand: 36
30	Literal
31	Lambda	parameter: 41 body: 35
32	Variable
33	Literal
34	ExprTuple	36
35	Operation	operator: 37 operand: 41
36	Variable
37	IndexedVar	variable: 39 index: 42
38	ExprTuple	41
39	Variable
40	ExprTuple	42
41	Variable
42	Variable