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, U, 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 CartExp, Equals, Exists, 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(U, Conditional(Exists(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = Equals(U, VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n))), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping(), InSet(U, Unitary(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())

U \mapsto \left\{\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}, w_{1}, w_{2}, \ldots, w_{n}~|~\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)}~\\
\left(U = \left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right)\right)\end{array} \textrm{ if } U \in \textrm{U}\left(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	parameter: 24 body: 2
1	ExprTuple	24
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 48 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	24, 9
8	Lambda	parameters: 10 body: 11
9	Operation	operator: 12 operand: 71
10	ExprTuple	14, 15
11	Conditional	value: 16 condition: 17
12	Literal
13	ExprTuple	71
14	ExprRange	lambda_map: 18 start_index: 70 end_index: 71
15	ExprRange	lambda_map: 19 start_index: 70 end_index: 71
16	Operation	operator: 20 operands: 21
17	Operation	operator: 22 operands: 23
18	Lambda	parameter: 84 body: 79
19	Lambda	parameter: 84 body: 80
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26, 27
24	Variable
25	Operation	operator: 28 operand: 32
26	Operation	operator: 48 operands: 30
27	Operation	operator: 48 operands: 31
28	Literal
29	ExprTuple	32
30	ExprTuple	33, 35
31	ExprTuple	34, 35
32	Lambda	parameter: 78 body: 36
33	Operation	operator: 51 operand: 43
34	Operation	operator: 51 operand: 44
35	Operation	operator: 39 operand: 45
36	Conditional	value: 41 condition: 42
37	ExprTuple	43
38	ExprTuple	44
39	Literal
40	ExprTuple	45
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 51 operands: 50
44	Operation	operator: 51 operands: 52
45	Operation	operator: 53 operands: 54
46	Literal
47	ExprTuple	55, 56
48	Literal
49	ExprTuple	78, 57
50	ExprTuple	58
51	Literal
52	ExprTuple	59
53	Literal
54	ExprTuple	60, 71
55	Operation	operator: 76 operand: 68
56	Operation	operator: 62 operand: 69
57	Operation	operator: 64 operands: 65
58	ExprRange	lambda_map: 66 start_index: 70 end_index: 71
59	ExprRange	lambda_map: 67 start_index: 70 end_index: 71
60	Literal
61	ExprTuple	68
62	Literal
63	ExprTuple	69
64	Literal
65	ExprTuple	70, 71
66	Lambda	parameter: 84 body: 72
67	Lambda	parameter: 84 body: 73
68	IndexedVar	variable: 81 index: 78
69	IndexedVar	variable: 82 index: 78
70	Literal
71	Variable
72	Operation	operator: 76 operand: 79
73	Operation	operator: 76 operand: 80
74	ExprTuple	78
75	ExprTuple	79
76	Literal
77	ExprTuple	80
78	Variable
79	IndexedVar	variable: 81 index: 84
80	IndexedVar	variable: 82 index: 84
81	Variable
82	Variable
83	ExprTuple	84
84	Variable