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 P, Py, S, y, z
from proveit.logic import Equals, Forall, Implies, InSet, NotEquals
from proveit.numbers import Complex, Sum, frac, zero

# build up the expression from sub-expressions
sub_expr1 = [y]
expr = Forall(instance_param_or_params = [P, S], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(Py, Complex), domain = S), Forall(instance_param_or_params = [z], instance_expr = Equals(frac(Sum(index_or_indices = sub_expr1, summand = Py, domain = S), z), Sum(index_or_indices = sub_expr1, summand = frac(Py, z), domain = S)), domain = Complex, condition = NotEquals(z, zero))))

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_{P, S}~\left(\left[\forall_{y \in S}~\left(P\left(y\right) \in \mathbb{C}\right)\right] \Rightarrow \left[\forall_{z \in \mathbb{C}~|~z \neq 0}~\left(\frac{\sum_{y \in S}~P\left(y\right)}{z} = \left[\sum_{y \in S}~\frac{P\left(y\right)}{z}\right]\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: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	51, 53
4	Operation	operator: 5 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Operation	operator: 10 operand: 12
8	Operation	operator: 10 operand: 13
9	ExprTuple	12
10	Literal
11	ExprTuple	13
12	Lambda	parameter: 54 body: 14
13	Lambda	parameter: 48 body: 16
14	Conditional	value: 17 condition: 46
15	ExprTuple	48
16	Conditional	value: 18 condition: 19
17	Operation	operator: 49 operands: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	ExprTuple	47, 36
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 44 operands: 29
26	Operation	operator: 38 operand: 35
27	Operation	operator: 49 operands: 31
28	Operation	operator: 32 operands: 33
29	ExprTuple	34, 48
30	ExprTuple	35
31	ExprTuple	48, 36
32	Literal
33	ExprTuple	48, 37
34	Operation	operator: 38 operand: 41
35	Lambda	parameter: 54 body: 40
36	Literal
37	Literal
38	Literal
39	ExprTuple	41
40	Conditional	value: 42 condition: 46
41	Lambda	parameter: 54 body: 43
42	Operation	operator: 44 operands: 45
43	Conditional	value: 47 condition: 46
44	Literal
45	ExprTuple	47, 48
46	Operation	operator: 49 operands: 50
47	Operation	operator: 51 operand: 54
48	Variable
49	Literal
50	ExprTuple	54, 53
51	Variable
52	ExprTuple	54
53	Variable
54	Variable