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, Real, e, i, pi, two

# build up the expression from sub-expressions
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)

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_{\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]

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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	51, 12
11	Lambda	parameters: 26 body: 13
12	Literal
13	Conditional	value: 14 condition: 15
14	Operation	operator: 17 operands: 16
15	Operation	operator: 17 operands: 18
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 25 operands: 23
20	Operation	operator: 27 operands: 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	ExprTuple	29, 32
24	ExprTuple	30, 32
25	Literal
26	ExprTuple	38, 32
27	Literal
28	ExprTuple	31, 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 36 operands: 35
31	Operation	operator: 36 operands: 37
32	Variable
33	Literal
34	ExprTuple	38, 50
35	ExprTuple	40, 39
36	Literal
37	ExprTuple	40, 41
38	Variable
39	Operation	operator: 48 operands: 42
40	Literal
41	Operation	operator: 48 operands: 43
42	ExprTuple	45, 46, 47, 44
43	ExprTuple	45, 46, 47, 51
44	Operation	operator: 48 operands: 49
45	Literal
46	Literal
47	Literal
48	Literal
49	ExprTuple	50, 51
50	Variable
51	Variable