Expression of type Forall

from the theory of proveit.numbers.division

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, n, x, y
from proveit.core_expr_types import x_1_to_n
from proveit.logic import Equals, Forall, NotEquals
from proveit.numbers import Add, Complex, NaturalPos, frac, one, zero

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n, y], instance_expr = Equals(frac(Add(x_1_to_n), y), Add(ExprRange(sub_expr1, frac(IndexedVar(x, sub_expr1), y), one, n))), domain = Complex, condition = NotEquals(y, zero)), 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_{x_{1}, x_{2}, \ldots, x_{n}, y \in \mathbb{C}~|~y \neq 0}~\left(\frac{x_{1} +  x_{2} +  \ldots +  x_{n}}{y} = \left(\frac{x_{1}}{y} +  \frac{x_{2}}{y} +  \ldots +  \frac{x_{n}}{y}\right)\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: 45 body: 4
3	ExprTuple	45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 38 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	45, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	40, 49
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22, 23, 24
20	Operation	operator: 46 operands: 25
21	Operation	operator: 35 operands: 26
22	ExprRange	lambda_map: 27 start_index: 44 end_index: 45
23	Operation	operator: 38 operands: 28
24	Operation	operator: 29 operands: 30
25	ExprTuple	31, 49
26	ExprTuple	32
27	Lambda	parameter: 52 body: 33
28	ExprTuple	49, 42
29	Literal
30	ExprTuple	49, 34
31	Operation	operator: 35 operands: 36
32	ExprRange	lambda_map: 37 start_index: 44 end_index: 45
33	Operation	operator: 38 operands: 39
34	Literal
35	Literal
36	ExprTuple	40
37	Lambda	parameter: 52 body: 41
38	Literal
39	ExprTuple	48, 42
40	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
41	Operation	operator: 46 operands: 47
42	Literal
43	Lambda	parameter: 52 body: 48
44	Literal
45	Variable
46	Literal
47	ExprTuple	48, 49
48	IndexedVar	variable: 50 index: 52
49	Variable
50	Variable
51	ExprTuple	52
52	Variable