Expression of type Lambda

from the theory of proveit.linear_algebra.matrices

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, b, k, x
from proveit.linear_algebra import MatrixMult, ScalarMult
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, NaturalPos

# build up the expression from sub-expressions
expr = Lambda(k, Conditional(Forall(instance_param_or_params = [b], instance_expr = Forall(instance_param_or_params = [A, x], instance_expr = Implies(Equals(MatrixMult(A, x), ScalarMult(b, x)), Equals(MatrixMult(Exp(A, k), x), ScalarMult(Exp(b, k), x)))), domain = Complex), InSet(k, 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())

k \mapsto \left\{\forall_{b \in \mathbb{C}}~\left[\forall_{A, x}~\left(\left(\left(A \thinspace x\right) = \left(b \cdot x\right)\right) \Rightarrow \left(\left(A^{k} \thinspace x\right) = \left(b^{k} \cdot x\right)\right)\right)\right] \textrm{ if } k \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: 45 body: 2
1	ExprTuple	45
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 15 operands: 6
5	ExprTuple	7
6	ExprTuple	45, 8
7	Lambda	parameter: 44 body: 10
8	Literal
9	ExprTuple	44
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	44, 18
17	Lambda	parameters: 31 body: 19
18	Literal
19	Operation	operator: 20 operands: 21
20	Literal
21	ExprTuple	22, 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	43, 39
32	ExprTuple	44, 39
33	Literal
34	ExprTuple	37, 39
35	Literal
36	ExprTuple	38, 39
37	Operation	operator: 41 operands: 40
38	Operation	operator: 41 operands: 42
39	Variable
40	ExprTuple	43, 45
41	Literal
42	ExprTuple	44, 45
43	Variable
44	Variable
45	Variable