Expression of type Lambda

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

# build up the expression from sub-expressions
expr = Lambda(m, Conditional(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), InSet(m, 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())

m \mapsto \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] \textrm{ if } m \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 56 body: 2
1	ExprTuple	56
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 15 operands: 6
5	ExprTuple	7
6	ExprTuple	56, 8
7	Lambda	parameter: 57 body: 10
8	Literal
9	ExprTuple	57
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operand: 17
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17
15	Literal
16	ExprTuple	57, 18
17	Lambda	parameters: 32 body: 19
18	Literal
19	Conditional	value: 20 condition: 21
20	Operation	operator: 23 operands: 22
21	Operation	operator: 23 operands: 24
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 31 operands: 29
26	Operation	operator: 33 operands: 30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprTuple	35, 38
30	ExprTuple	36, 38
31	Literal
32	ExprTuple	44, 38
33	Literal
34	ExprTuple	37, 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 42 operands: 41
37	Operation	operator: 42 operands: 43
38	Variable
39	Literal
40	ExprTuple	44, 56
41	ExprTuple	46, 45
42	Literal
43	ExprTuple	46, 47
44	Variable
45	Operation	operator: 54 operands: 48
46	Literal
47	Operation	operator: 54 operands: 49
48	ExprTuple	51, 52, 53, 50
49	ExprTuple	51, 52, 53, 57
50	Operation	operator: 54 operands: 55
51	Literal
52	Literal
53	Literal
54	Literal
55	ExprTuple	56, 57
56	Variable
57	Variable