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, j, n
from proveit.core_expr_types import v_1_to_n, v_i, v_j
from proveit.linear_algebra import Adj, MatrixSpace, OrthoNormBases, Unitary
from proveit.logic import CartExp, Equals, Forall, Implies, InSet, Set
from proveit.numbers import Complex, Interval, KroneckerDelta, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [v_1_to_n], instance_expr = Implies(Forall(instance_param_or_params = [i, j], instance_expr = Equals(Qmult(Bra(v_i), Adj(U), U, Ket(v_j)), KroneckerDelta(i, j)), domain = Interval(one, n)), InSet(U, Unitary(n))), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(CartExp(Complex, n)))).with_wrapping(), domain = MatrixSpace(field = Complex, rows = n, columns = 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 \mathbb{C}^{n \times n}}~\left[\begin{array}{l}\forall_{v_{1}, v_{2}, \ldots, v_{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[\forall_{i, j \in \{1~\ldotp \ldotp~n\}}~\left(\left(\langle v_{i} \rvert \thinspace U^{\dagger} \thinspace U \thinspace \lvert v_{j} \rangle\right) = \delta_{i, j}\right)\right] \Rightarrow \left(U \in \textrm{U}\left(n\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: 87 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 30 operand: 6
3	Operation	operator: 66 operands: 5
4	ExprTuple	6
5	ExprTuple	87, 7
6	Lambda	parameter: 84 body: 8
7	Literal
8	Conditional	value: 9 condition: 10
9	Operation	operator: 30 operand: 13
10	Operation	operator: 66 operands: 12
11	ExprTuple	13
12	ExprTuple	84, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	19
16	Conditional	value: 20 condition: 21
17	Literal
18	NamedExprs	field: 50 rows: 87 columns: 87
19	ExprRange	lambda_map: 22 start_index: 86 end_index: 87
20	Operation	operator: 23 operands: 24
21	Operation	operator: 66 operands: 25
22	Lambda	parameter: 88 body: 73
23	Literal
24	ExprTuple	26, 27
25	ExprTuple	28, 29
26	Operation	operator: 30 operand: 36
27	Operation	operator: 66 operands: 32
28	Operation	operator: 43 operand: 38
29	Operation	operator: 34 operand: 39
30	Literal
31	ExprTuple	36
32	ExprTuple	84, 37
33	ExprTuple	38
34	Literal
35	ExprTuple	39
36	Lambda	parameters: 64 body: 40
37	Operation	operator: 41 operand: 87
38	Operation	operator: 43 operands: 44
39	Operation	operator: 45 operands: 46
40	Conditional	value: 47 condition: 48
41	Literal
42	ExprTuple	87
43	Literal
44	ExprTuple	49
45	Literal
46	ExprTuple	50, 87
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operands: 54
49	ExprRange	lambda_map: 55 start_index: 86 end_index: 87
50	Literal
51	Literal
52	ExprTuple	56, 57
53	Literal
54	ExprTuple	58, 59
55	Lambda	parameter: 88 body: 60
56	Operation	operator: 61 operands: 62
57	Operation	operator: 63 operands: 64
58	Operation	operator: 66 operands: 65
59	Operation	operator: 66 operands: 67
60	Operation	operator: 78 operand: 73
61	Literal
62	ExprTuple	69, 70, 84, 71
63	Literal
64	ExprTuple	92, 93
65	ExprTuple	92, 72
66	Literal
67	ExprTuple	93, 72
68	ExprTuple	73
69	Operation	operator: 74 operand: 83
70	Operation	operator: 76 operand: 84
71	Operation	operator: 78 operand: 85
72	Operation	operator: 80 operands: 81
73	IndexedVar	variable: 90 index: 88
74	Literal
75	ExprTuple	83
76	Literal
77	ExprTuple	84
78	Literal
79	ExprTuple	85
80	Literal
81	ExprTuple	86, 87
82	ExprTuple	88
83	IndexedVar	variable: 90 index: 92
84	Variable
85	IndexedVar	variable: 90 index: 93
86	Literal
87	Variable
88	Variable
89	ExprTuple	92
90	Variable
91	ExprTuple	93
92	Variable
93	Variable