Expression of type Lambda

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, Lambda, Variable, a, f, i, n
from proveit.core_expr_types import a_1_to_n, 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 = Lambda([a_1_to_n], 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(a_{1}, a_{2}, \ldots, a_{n}\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	3
2	Conditional	value: 4 condition: 5
3	ExprRange	lambda_map: 6 start_index: 62 end_index: 63
4	Operation	operator: 7 operand: 11
5	Operation	operator: 9 operands: 10
6	Lambda	parameter: 35 body: 26
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12
11	Lambda	parameter: 50 body: 14
12	ExprRange	lambda_map: 15 start_index: 62 end_index: 63
13	ExprTuple	50
14	Conditional	value: 16 condition: 17
15	Lambda	parameter: 35 body: 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 48 operands: 21
18	Operation	operator: 48 operands: 22
19	Literal
20	ExprTuple	23, 24
21	ExprTuple	50, 25
22	ExprTuple	26, 34
23	Operation	operator: 50 operand: 32
24	Operation	operator: 36 operand: 33
25	Operation	operator: 29 operands: 30
26	IndexedVar	variable: 59 index: 35
27	ExprTuple	32
28	ExprTuple	33
29	Literal
30	ExprTuple	34, 34
31	ExprTuple	35
32	Operation	operator: 36 operand: 39
33	Lambda	parameter: 70 body: 38
34	Literal
35	Variable
36	Literal
37	ExprTuple	39
38	Conditional	value: 40 condition: 44
39	Lambda	parameter: 70 body: 41
40	Operation	operator: 46 operands: 42
41	Conditional	value: 43 condition: 44
42	ExprTuple	45, 52
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operand: 54
46	Literal
47	ExprTuple	54, 52
48	Literal
49	ExprTuple	70, 53
50	Variable
51	ExprTuple	54
52	Operation	operator: 55 operands: 56
53	Operation	operator: 57 operands: 58
54	IndexedVar	variable: 59 index: 70
55	Literal
56	ExprTuple	60, 61
57	Literal
58	ExprTuple	62, 63
59	Variable
60	Operation	operator: 64 operand: 67
61	Operation	operator: 65 operand: 67
62	Literal
63	Variable
64	Literal
65	Literal
66	ExprTuple	67
67	IndexedVar	variable: 68 index: 70
68	Variable
69	ExprTuple	70
70	Variable