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, Lambda, i, n
from proveit.core_expr_types import v_1_to_n, v_i, w_1_to_n, w_i
from proveit.linear_algebra import OrthoNormBases, Unitary, VecSum
from proveit.logic import And, CartExp, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = Lambda([v_1_to_n, w_1_to_n], Conditional(InSet(VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n)), Unitary(n)), And(InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1))))

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(v_{1}, v_{2}, \ldots, v_{n}, w_{1}, w_{2}, \ldots, w_{n}\right) \mapsto \left\{\left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right) \in \textrm{U}\left(n\right) \textrm{ if } \left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right) ,  \left\{\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\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, 4
2	Conditional	value: 5 condition: 6
3	ExprRange	lambda_map: 7 start_index: 60 end_index: 61
4	ExprRange	lambda_map: 8 start_index: 60 end_index: 61
5	Operation	operator: 38 operands: 9
6	Operation	operator: 10 operands: 11
7	Lambda	parameter: 74 body: 69
8	Lambda	parameter: 74 body: 70
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15
12	Operation	operator: 16 operand: 22
13	Operation	operator: 18 operand: 61
14	Operation	operator: 38 operands: 20
15	Operation	operator: 38 operands: 21
16	Literal
17	ExprTuple	22
18	Literal
19	ExprTuple	61
20	ExprTuple	23, 25
21	ExprTuple	24, 25
22	Lambda	parameter: 68 body: 26
23	Operation	operator: 41 operand: 33
24	Operation	operator: 41 operand: 34
25	Operation	operator: 29 operand: 35
26	Conditional	value: 31 condition: 32
27	ExprTuple	33
28	ExprTuple	34
29	Literal
30	ExprTuple	35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 38 operands: 39
33	Operation	operator: 41 operands: 40
34	Operation	operator: 41 operands: 42
35	Operation	operator: 43 operands: 44
36	Literal
37	ExprTuple	45, 46
38	Literal
39	ExprTuple	68, 47
40	ExprTuple	48
41	Literal
42	ExprTuple	49
43	Literal
44	ExprTuple	50, 61
45	Operation	operator: 66 operand: 58
46	Operation	operator: 52 operand: 59
47	Operation	operator: 54 operands: 55
48	ExprRange	lambda_map: 56 start_index: 60 end_index: 61
49	ExprRange	lambda_map: 57 start_index: 60 end_index: 61
50	Literal
51	ExprTuple	58
52	Literal
53	ExprTuple	59
54	Literal
55	ExprTuple	60, 61
56	Lambda	parameter: 74 body: 62
57	Lambda	parameter: 74 body: 63
58	IndexedVar	variable: 71 index: 68
59	IndexedVar	variable: 72 index: 68
60	Literal
61	Variable
62	Operation	operator: 66 operand: 69
63	Operation	operator: 66 operand: 70
64	ExprTuple	68
65	ExprTuple	69
66	Literal
67	ExprTuple	70
68	Variable
69	IndexedVar	variable: 71 index: 74
70	IndexedVar	variable: 72 index: 74
71	Variable
72	Variable
73	ExprTuple	74
74	Variable