Expression of type ExprTuple

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, ExprTuple, 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 = ExprTuple(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())

\left(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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameter: 88 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 31 operand: 7
4	Operation	operator: 67 operands: 6
5	ExprTuple	7
6	ExprTuple	88, 8
7	Lambda	parameter: 85 body: 9
8	Literal
9	Conditional	value: 10 condition: 11
10	Operation	operator: 31 operand: 14
11	Operation	operator: 67 operands: 13
12	ExprTuple	14
13	ExprTuple	85, 15
14	Lambda	parameters: 16 body: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	20
17	Conditional	value: 21 condition: 22
18	Literal
19	NamedExprs	field: 51 rows: 88 columns: 88
20	ExprRange	lambda_map: 23 start_index: 87 end_index: 88
21	Operation	operator: 24 operands: 25
22	Operation	operator: 67 operands: 26
23	Lambda	parameter: 89 body: 74
24	Literal
25	ExprTuple	27, 28
26	ExprTuple	29, 30
27	Operation	operator: 31 operand: 37
28	Operation	operator: 67 operands: 33
29	Operation	operator: 44 operand: 39
30	Operation	operator: 35 operand: 40
31	Literal
32	ExprTuple	37
33	ExprTuple	85, 38
34	ExprTuple	39
35	Literal
36	ExprTuple	40
37	Lambda	parameters: 65 body: 41
38	Operation	operator: 42 operand: 88
39	Operation	operator: 44 operands: 45
40	Operation	operator: 46 operands: 47
41	Conditional	value: 48 condition: 49
42	Literal
43	ExprTuple	88
44	Literal
45	ExprTuple	50
46	Literal
47	ExprTuple	51, 88
48	Operation	operator: 52 operands: 53
49	Operation	operator: 54 operands: 55
50	ExprRange	lambda_map: 56 start_index: 87 end_index: 88
51	Literal
52	Literal
53	ExprTuple	57, 58
54	Literal
55	ExprTuple	59, 60
56	Lambda	parameter: 89 body: 61
57	Operation	operator: 62 operands: 63
58	Operation	operator: 64 operands: 65
59	Operation	operator: 67 operands: 66
60	Operation	operator: 67 operands: 68
61	Operation	operator: 79 operand: 74
62	Literal
63	ExprTuple	70, 71, 85, 72
64	Literal
65	ExprTuple	93, 94
66	ExprTuple	93, 73
67	Literal
68	ExprTuple	94, 73
69	ExprTuple	74
70	Operation	operator: 75 operand: 84
71	Operation	operator: 77 operand: 85
72	Operation	operator: 79 operand: 86
73	Operation	operator: 81 operands: 82
74	IndexedVar	variable: 91 index: 89
75	Literal
76	ExprTuple	84
77	Literal
78	ExprTuple	85
79	Literal
80	ExprTuple	86
81	Literal
82	ExprTuple	87, 88
83	ExprTuple	89
84	IndexedVar	variable: 91 index: 93
85	Variable
86	IndexedVar	variable: 91 index: 94
87	Literal
88	Variable
89	Variable
90	ExprTuple	93
91	Variable
92	ExprTuple	94
93	Variable
94	Variable