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, OrthoNormBases, Unitary
from proveit.logic import CartExp, Equals, Forall, Implies, InSet, Set
from proveit.numbers import Complex, Interval, KroneckerDelta, 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([v_1_to_n], Conditional(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))), InSet(Set(v_1_to_n_kets), OrthoNormBases(CartExp(Complex, 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())

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