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 m, n, x, y
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, zero

# build up the expression from sub-expressions
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(Mult(w_1_to_m, Neg(x), z_1_to_n), y), Neg(frac(Mult(w_1_to_m, x, z_1_to_n), y))), domain = Complex, condition = NotEquals(y, 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_{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(\frac{w_{1} \cdot  w_{2} \cdot  \ldots \cdot  w_{m} \cdot \left(-x\right)\cdot z_{1} \cdot  z_{2} \cdot  \ldots \cdot  z_{n}}{y} = \left(-\frac{w_{1} \cdot  w_{2} \cdot  \ldots \cdot  w_{m} \cdot x\cdot z_{1} \cdot  z_{2} \cdot  \ldots \cdot  z_{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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	54, 57
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameters: 8 body: 9
8	ExprTuple	50, 51, 44, 52
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
16	Operation	operator: 37 operands: 23
17	Operation	operator: 46 operand: 32
18	ExprRange	lambda_map: 25 start_index: 56 end_index: 54
19	Operation	operator: 40 operands: 26
20	Operation	operator: 40 operands: 27
21	ExprRange	lambda_map: 28 start_index: 56 end_index: 57
22	Operation	operator: 29 operands: 30
23	ExprTuple	31, 44
24	ExprTuple	32
25	Lambda	parameter: 63 body: 33
26	ExprTuple	51, 45
27	ExprTuple	44, 45
28	Lambda	parameter: 63 body: 34
29	Literal
30	ExprTuple	44, 35
31	Operation	operator: 48 operands: 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 40 operands: 39
34	Operation	operator: 40 operands: 41
35	Literal
36	ExprTuple	50, 42, 52
37	Literal
38	ExprTuple	43, 44
39	ExprTuple	58, 45
40	Literal
41	ExprTuple	59, 45
42	Operation	operator: 46 operand: 51
43	Operation	operator: 48 operands: 49
44	Variable
45	Literal
46	Literal
47	ExprTuple	51
48	Literal
49	ExprTuple	50, 51, 52
50	ExprRange	lambda_map: 53 start_index: 56 end_index: 54
51	Variable
52	ExprRange	lambda_map: 55 start_index: 56 end_index: 57
53	Lambda	parameter: 63 body: 58
54	Variable
55	Lambda	parameter: 63 body: 59
56	Literal
57	Variable
58	IndexedVar	variable: 60 index: 63
59	IndexedVar	variable: 61 index: 63
60	Variable
61	Variable
62	ExprTuple	63
63	Variable