Expression of type Conditional

from the theory of proveit.physics.quantum.algebra

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 Conditional, ExprRange, Function, IndexedVar, Variable, a, f, i, n
from proveit.core_expr_types import a_i, v_i
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import And, Equals, Forall, Functions, InSet
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [i]
sub_expr3 = Interval(one, n)
sub_expr4 = Qmult(Ket(v_i), Bra(v_i))
expr = Conditional(Forall(instance_param_or_params = [f], instance_expr = Equals(Function(f, [VecSum(index_or_indices = sub_expr2, summand = ScalarMult(a_i, sub_expr4), domain = sub_expr3)]), VecSum(index_or_indices = sub_expr2, summand = ScalarMult(Function(f, [a_i]), sub_expr4), domain = sub_expr3)), domain = Functions(Complex, Complex)), And(ExprRange(sub_expr1, InSet(IndexedVar(a, sub_expr1), Complex), one, n)))

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())

\left\{\forall_{f \in \left[\mathbb{C} \rightarrow \mathbb{C}\right]}~\left(f\left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right) = \left(\sum_{i=1}^{n} \left(f\left(a_{i}\right) \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \textrm{ if } \left(a_{1} \in \mathbb{C}\right) \land  \left(a_{2} \in \mathbb{C}\right) \land  \ldots \land  \left(a_{n} \in \mathbb{C}\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8
7	Lambda	parameter: 46 body: 10
8	ExprRange	lambda_map: 11 start_index: 58 end_index: 59
9	ExprTuple	46
10	Conditional	value: 12 condition: 13
11	Lambda	parameter: 31 body: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 44 operands: 17
14	Operation	operator: 44 operands: 18
15	Literal
16	ExprTuple	19, 20
17	ExprTuple	46, 21
18	ExprTuple	22, 30
19	Operation	operator: 46 operand: 28
20	Operation	operator: 32 operand: 29
21	Operation	operator: 25 operands: 26
22	IndexedVar	variable: 55 index: 31
23	ExprTuple	28
24	ExprTuple	29
25	Literal
26	ExprTuple	30, 30
27	ExprTuple	31
28	Operation	operator: 32 operand: 35
29	Lambda	parameter: 66 body: 34
30	Literal
31	Variable
32	Literal
33	ExprTuple	35
34	Conditional	value: 36 condition: 40
35	Lambda	parameter: 66 body: 37
36	Operation	operator: 42 operands: 38
37	Conditional	value: 39 condition: 40
38	ExprTuple	41, 48
39	Operation	operator: 42 operands: 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operand: 50
42	Literal
43	ExprTuple	50, 48
44	Literal
45	ExprTuple	66, 49
46	Variable
47	ExprTuple	50
48	Operation	operator: 51 operands: 52
49	Operation	operator: 53 operands: 54
50	IndexedVar	variable: 55 index: 66
51	Literal
52	ExprTuple	56, 57
53	Literal
54	ExprTuple	58, 59
55	Variable
56	Operation	operator: 60 operand: 63
57	Operation	operator: 61 operand: 63
58	Literal
59	Variable
60	Literal
61	Literal
62	ExprTuple	63
63	IndexedVar	variable: 64 index: 66
64	Variable
65	ExprTuple	66
66	Variable