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, n
from proveit.core_expr_types import Len, f_1_to_n
from proveit.core_expr_types.tuples import f_1_to_i__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], instance_expr = Equals(Len(operands = [f_1_to_i__1_to_n]), Mult(n, i)).with_wrapping_at(1), condition = InSet(i, 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~|~i \in \mathbb{N}}~\left(\begin{array}{c} \begin{array}{l} |\left(f_{1}\left(1\right), f_{1}\left(2\right), \ldots, f_{1}\left(i\right), f_{2}\left(1\right), f_{2}\left(2\right), \ldots, f_{2}\left(i\right), \ldots\ldots, f_{n}\left(1\right), f_{n}\left(2\right), \ldots, f_{n}\left(i\right)\right)| \\  = \left(n \cdot i\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, 28
5	Conditional	value: 7 condition: 8
6	ExprRange	lambda_map: 9 start_index: 27 end_index: 24
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	Lambda	parameter: 34 body: 14
10	Literal
11	ExprTuple	15, 16
12	Literal
13	ExprTuple	28, 17
14	IndexedVar	variable: 32 index: 34
15	Operation	operator: 18 operands: 19
16	Operation	operator: 20 operands: 21
17	Literal
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	24, 28
22	ExprRange	lambda_map: 23 start_index: 27 end_index: 24
23	Lambda	parameter: 35 body: 25
24	Variable
25	ExprRange	lambda_map: 26 start_index: 27 end_index: 28
26	Lambda	parameter: 34 body: 29
27	Literal
28	Variable
29	Operation	operator: 30 operand: 34
30	IndexedVar	variable: 32 index: 35
31	ExprTuple	34
32	Variable
33	ExprTuple	35
34	Variable
35	Variable