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, Forall, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, 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(n, Conditional(Forall(instance_param_or_params = [U], instance_expr = 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(), domain = Unitary(n)), InSet(n, 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())

n \mapsto \left\{\forall_{U \in \textrm{U}\left(n\right)}~\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}\right] \textrm{ if } n \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: 79 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 56 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	79, 8
7	Lambda	parameter: 32 body: 10
8	Literal
9	ExprTuple	32
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operand: 16
12	Operation	operator: 56 operands: 15
13	Literal
14	ExprTuple	16
15	ExprTuple	32, 17
16	Lambda	parameters: 18 body: 19
17	Operation	operator: 20 operand: 79
18	ExprTuple	22, 23
19	Conditional	value: 24 condition: 25
20	Literal
21	ExprTuple	79
22	ExprRange	lambda_map: 26 start_index: 78 end_index: 79
23	ExprRange	lambda_map: 27 start_index: 78 end_index: 79
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	Lambda	parameter: 92 body: 87
27	Lambda	parameter: 92 body: 88
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Variable
33	Operation	operator: 36 operand: 40
34	Operation	operator: 56 operands: 38
35	Operation	operator: 56 operands: 39
36	Literal
37	ExprTuple	40
38	ExprTuple	41, 43
39	ExprTuple	42, 43
40	Lambda	parameter: 86 body: 44
41	Operation	operator: 59 operand: 51
42	Operation	operator: 59 operand: 52
43	Operation	operator: 47 operand: 53
44	Conditional	value: 49 condition: 50
45	ExprTuple	51
46	ExprTuple	52
47	Literal
48	ExprTuple	53
49	Operation	operator: 54 operands: 55
50	Operation	operator: 56 operands: 57
51	Operation	operator: 59 operands: 58
52	Operation	operator: 59 operands: 60
53	Operation	operator: 61 operands: 62
54	Literal
55	ExprTuple	63, 64
56	Literal
57	ExprTuple	86, 65
58	ExprTuple	66
59	Literal
60	ExprTuple	67
61	Literal
62	ExprTuple	68, 79
63	Operation	operator: 84 operand: 76
64	Operation	operator: 70 operand: 77
65	Operation	operator: 72 operands: 73
66	ExprRange	lambda_map: 74 start_index: 78 end_index: 79
67	ExprRange	lambda_map: 75 start_index: 78 end_index: 79
68	Literal
69	ExprTuple	76
70	Literal
71	ExprTuple	77
72	Literal
73	ExprTuple	78, 79
74	Lambda	parameter: 92 body: 80
75	Lambda	parameter: 92 body: 81
76	IndexedVar	variable: 89 index: 86
77	IndexedVar	variable: 90 index: 86
78	Literal
79	Variable
80	Operation	operator: 84 operand: 87
81	Operation	operator: 84 operand: 88
82	ExprTuple	86
83	ExprTuple	87
84	Literal
85	ExprTuple	88
86	Variable
87	IndexedVar	variable: 89 index: 92
88	IndexedVar	variable: 90 index: 92
89	Variable
90	Variable
91	ExprTuple	92
92	Variable