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 = [m, n], instance_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_{m, n}~\left[\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)\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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	62, 65
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameters: 8 body: 9
8	ExprTuple	58, 50, 59, 60
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	16, 17
14	Literal
15	ExprTuple	18, 19, 20, 21, 22, 23, 24
16	Operation	operator: 41 operands: 25
17	Operation	operator: 54 operand: 35
18	ExprRange	lambda_map: 27 start_index: 64 end_index: 62
19	Operation	operator: 44 operands: 28
20	Operation	operator: 44 operands: 29
21	ExprRange	lambda_map: 30 start_index: 64 end_index: 65
22	Operation	operator: 47 operands: 31
23	ExprRange	lambda_map: 32 start_index: 64 end_index: 62
24	ExprRange	lambda_map: 33 start_index: 64 end_index: 65
25	ExprTuple	50, 34
26	ExprTuple	35
27	Lambda	parameter: 71 body: 36
28	ExprTuple	50, 52
29	ExprTuple	59, 52
30	Lambda	parameter: 71 body: 37
31	ExprTuple	59, 53
32	Lambda	parameter: 71 body: 38
33	Lambda	parameter: 71 body: 39
34	Operation	operator: 56 operands: 40
35	Operation	operator: 41 operands: 42
36	Operation	operator: 44 operands: 43
37	Operation	operator: 44 operands: 45
38	Operation	operator: 47 operands: 46
39	Operation	operator: 47 operands: 48
40	ExprTuple	58, 49, 60
41	Literal
42	ExprTuple	50, 51
43	ExprTuple	66, 52
44	Literal
45	ExprTuple	67, 52
46	ExprTuple	66, 53
47	Literal
48	ExprTuple	67, 53
49	Operation	operator: 54 operand: 59
50	Variable
51	Operation	operator: 56 operands: 57
52	Literal
53	Literal
54	Literal
55	ExprTuple	59
56	Literal
57	ExprTuple	58, 59, 60
58	ExprRange	lambda_map: 61 start_index: 64 end_index: 62
59	Variable
60	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
61	Lambda	parameter: 71 body: 66
62	Variable
63	Lambda	parameter: 71 body: 67
64	Literal
65	Variable
66	IndexedVar	variable: 68 index: 71
67	IndexedVar	variable: 69 index: 71
68	Variable
69	Variable
70	ExprTuple	71
71	Variable