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 CartExp, 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 = [v_1_to_n, w_1_to_n], instance_expr = InSet(VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n)), Unitary(n)), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping(), 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\{\begin{array}{l}\forall_{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(\left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right) \in \textrm{U}\left(n\right)\right)\end{array} \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: 69 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 46 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	69, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	11, 12
10	Conditional	value: 13 condition: 14
11	ExprRange	lambda_map: 15 start_index: 68 end_index: 69
12	ExprRange	lambda_map: 16 start_index: 68 end_index: 69
13	Operation	operator: 46 operands: 17
14	Operation	operator: 18 operands: 19
15	Lambda	parameter: 82 body: 77
16	Lambda	parameter: 82 body: 78
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22, 23
20	Operation	operator: 24 operand: 30
21	Operation	operator: 26 operand: 69
22	Operation	operator: 46 operands: 28
23	Operation	operator: 46 operands: 29
24	Literal
25	ExprTuple	30
26	Literal
27	ExprTuple	69
28	ExprTuple	31, 33
29	ExprTuple	32, 33
30	Lambda	parameter: 76 body: 34
31	Operation	operator: 49 operand: 41
32	Operation	operator: 49 operand: 42
33	Operation	operator: 37 operand: 43
34	Conditional	value: 39 condition: 40
35	ExprTuple	41
36	ExprTuple	42
37	Literal
38	ExprTuple	43
39	Operation	operator: 44 operands: 45
40	Operation	operator: 46 operands: 47
41	Operation	operator: 49 operands: 48
42	Operation	operator: 49 operands: 50
43	Operation	operator: 51 operands: 52
44	Literal
45	ExprTuple	53, 54
46	Literal
47	ExprTuple	76, 55
48	ExprTuple	56
49	Literal
50	ExprTuple	57
51	Literal
52	ExprTuple	58, 69
53	Operation	operator: 74 operand: 66
54	Operation	operator: 60 operand: 67
55	Operation	operator: 62 operands: 63
56	ExprRange	lambda_map: 64 start_index: 68 end_index: 69
57	ExprRange	lambda_map: 65 start_index: 68 end_index: 69
58	Literal
59	ExprTuple	66
60	Literal
61	ExprTuple	67
62	Literal
63	ExprTuple	68, 69
64	Lambda	parameter: 82 body: 70
65	Lambda	parameter: 82 body: 71
66	IndexedVar	variable: 79 index: 76
67	IndexedVar	variable: 80 index: 76
68	Literal
69	Variable
70	Operation	operator: 74 operand: 77
71	Operation	operator: 74 operand: 78
72	ExprTuple	76
73	ExprTuple	77
74	Literal
75	ExprTuple	78
76	Variable
77	IndexedVar	variable: 79 index: 82
78	IndexedVar	variable: 80 index: 82
79	Variable
80	Variable
81	ExprTuple	82
82	Variable