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, NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_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)), domain = NaturalPos)

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_{n \in \mathbb{N}^+}~\left[\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)\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	parameter: 33 body: 4
3	ExprTuple	33
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 20 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	33, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14, 44, 40
13	Conditional	value: 15 condition: 16
14	ExprRange	lambda_map: 17 start_index: 38 end_index: 33
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	Lambda	parameter: 49 body: 22
18	Literal
19	ExprTuple	23, 24
20	Literal
21	ExprTuple	31, 25
22	IndexedVar	variable: 47 index: 49
23	Operation	operator: 26 operands: 27
24	Operation	operator: 28 operands: 29
25	Literal
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	33, 31
30	ExprRange	lambda_map: 32 start_index: 38 end_index: 33
31	Operation	operator: 34 operands: 35
32	Lambda	parameter: 50 body: 36
33	Variable
34	Literal
35	ExprTuple	40, 37, 38
36	ExprRange	lambda_map: 39 start_index: 44 end_index: 40
37	Operation	operator: 41 operand: 44
38	Literal
39	Lambda	parameter: 49 body: 43
40	Variable
41	Literal
42	ExprTuple	44
43	Operation	operator: 45 operand: 49
44	Variable
45	IndexedVar	variable: 47 index: 50
46	ExprTuple	49
47	Variable
48	ExprTuple	50
49	Variable
50	Variable