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, 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(U, Conditional(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(), InSet(U, MatrixSpace(field = Complex, rows = n, columns = 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())

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