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, m, n, w, x, y, z
from proveit.core_expr_types import w_1_to_m, z_1_to_n
from proveit.logic import Equals, Forall, NotEquals
from proveit.numbers import Complex, Mult, Neg, frac, one, zero

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [w_1_to_m, x, y, z_1_to_n], instance_expr = Equals(frac(x, Mult(w_1_to_m, Neg(y), z_1_to_n)), Neg(frac(x, Mult(w_1_to_m, y, z_1_to_n)))), domain = Complex, conditions = [NotEquals(y, zero), ExprRange(sub_expr1, NotEquals(IndexedVar(w, sub_expr1), zero), one, m), ExprRange(sub_expr1, NotEquals(IndexedVar(z, sub_expr1), zero), one, n)])

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_{w_{1}, w_{2}, \ldots, w_{m}, x, y, z_{1}, z_{2}, \ldots, z_{n} \in \mathbb{C}~|~y \neq 0,\left(w_{1} \neq 0\right), \left(w_{2} \neq 0\right), \ldots, \left(w_{m} \neq 0\right),\left(z_{1} \neq 0\right), \left(z_{2} \neq 0\right), \ldots, \left(z_{n} \neq 0\right)}~\left(\frac{x}{w_{1} \cdot  w_{2} \cdot  \ldots \cdot  w_{m} \cdot \left(-y\right)\cdot z_{1} \cdot  z_{2} \cdot  \ldots \cdot  z_{n}} = \left(-\frac{x}{w_{1} \cdot  w_{2} \cdot  \ldots \cdot  w_{m} \cdot y\cdot z_{1} \cdot  z_{2} \cdot  \ldots \cdot  z_{n}}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	54, 46, 55, 56
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15, 16, 17, 18, 19, 20
12	Operation	operator: 37 operands: 21
13	Operation	operator: 50 operand: 31
14	ExprRange	lambda_map: 23 start_index: 60 end_index: 58
15	Operation	operator: 40 operands: 24
16	Operation	operator: 40 operands: 25
17	ExprRange	lambda_map: 26 start_index: 60 end_index: 61
18	Operation	operator: 43 operands: 27
19	ExprRange	lambda_map: 28 start_index: 60 end_index: 58
20	ExprRange	lambda_map: 29 start_index: 60 end_index: 61
21	ExprTuple	46, 30
22	ExprTuple	31
23	Lambda	parameter: 67 body: 32
24	ExprTuple	46, 48
25	ExprTuple	55, 48
26	Lambda	parameter: 67 body: 33
27	ExprTuple	55, 49
28	Lambda	parameter: 67 body: 34
29	Lambda	parameter: 67 body: 35
30	Operation	operator: 52 operands: 36
31	Operation	operator: 37 operands: 38
32	Operation	operator: 40 operands: 39
33	Operation	operator: 40 operands: 41
34	Operation	operator: 43 operands: 42
35	Operation	operator: 43 operands: 44
36	ExprTuple	54, 45, 56
37	Literal
38	ExprTuple	46, 47
39	ExprTuple	62, 48
40	Literal
41	ExprTuple	63, 48
42	ExprTuple	62, 49
43	Literal
44	ExprTuple	63, 49
45	Operation	operator: 50 operand: 55
46	Variable
47	Operation	operator: 52 operands: 53
48	Literal
49	Literal
50	Literal
51	ExprTuple	55
52	Literal
53	ExprTuple	54, 55, 56
54	ExprRange	lambda_map: 57 start_index: 60 end_index: 58
55	Variable
56	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
57	Lambda	parameter: 67 body: 62
58	Variable
59	Lambda	parameter: 67 body: 63
60	Literal
61	Variable
62	IndexedVar	variable: 64 index: 67
63	IndexedVar	variable: 65 index: 67
64	Variable
65	Variable
66	ExprTuple	67
67	Variable