Expression of type Forall

from the theory of proveit.linear_algebra.matrices.exponentiation

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 A, m, rho, x
from proveit.linear_algebra import MatrixExp, MatrixMult, ScalarMult
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Mult, NaturalPos, Real, e, i, pi, two

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [rho], instance_expr = Forall(instance_param_or_params = [A, x], instance_expr = Equals(MatrixMult(MatrixExp(A, m), x), ScalarMult(Exp(e, Mult(two, pi, i, Mult(m, rho))), x)), condition = Equals(MatrixMult(A, x), ScalarMult(Exp(e, Mult(two, pi, i, rho)), x))), domain = Real), 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_{m \in \mathbb{N}^+}~\left[\forall_{\rho \in \mathbb{R}}~\left[\forall_{A, x~|~\left(A \thinspace x\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \rho} \cdot x\right)}~\left(\left(A^{m} \thinspace x\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(m \cdot \rho\right)} \cdot x\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: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 58 body: 4
3	ExprTuple	58
4	Conditional	value: 5 condition: 6
5	Operation	operator: 15 operand: 9
6	Operation	operator: 17 operands: 8
7	ExprTuple	9
8	ExprTuple	58, 10
9	Lambda	parameter: 59 body: 12
10	Literal
11	ExprTuple	59
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 19
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19
17	Literal
18	ExprTuple	59, 20
19	Lambda	parameters: 34 body: 21
20	Literal
21	Conditional	value: 22 condition: 23
22	Operation	operator: 25 operands: 24
23	Operation	operator: 25 operands: 26
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29, 30
27	Operation	operator: 33 operands: 31
28	Operation	operator: 35 operands: 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operands: 36
31	ExprTuple	37, 40
32	ExprTuple	38, 40
33	Literal
34	ExprTuple	46, 40
35	Literal
36	ExprTuple	39, 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 44 operands: 43
39	Operation	operator: 44 operands: 45
40	Variable
41	Literal
42	ExprTuple	46, 58
43	ExprTuple	48, 47
44	Literal
45	ExprTuple	48, 49
46	Variable
47	Operation	operator: 56 operands: 50
48	Literal
49	Operation	operator: 56 operands: 51
50	ExprTuple	53, 54, 55, 52
51	ExprTuple	53, 54, 55, 59
52	Operation	operator: 56 operands: 57
53	Literal
54	Literal
55	Literal
56	Literal
57	ExprTuple	58, 59
58	Variable
59	Variable