Expression of type Forall ¶

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 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 = Forall(instance_param_or_params = [n], instance_expr = 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)), domain = NaturalPos)

# 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())

\forall_{n \in \mathbb{N}^+}~\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]\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 32 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 89 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 32 operand: 8
5	Operation	operator: 68 operands: 7
6	ExprTuple	8
7	ExprTuple	89, 9
8	Lambda	parameter: 86 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 32 operand: 15
12	Operation	operator: 68 operands: 14
13	ExprTuple	15
14	ExprTuple	86, 16
15	Lambda	parameters: 17 body: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	21
18	Conditional	value: 22 condition: 23
19	Literal
20	NamedExprs	field: 52 rows: 89 columns: 89
21	ExprRange	lambda_map: 24 start_index: 88 end_index: 89
22	Operation	operator: 25 operands: 26
23	Operation	operator: 68 operands: 27
24	Lambda	parameter: 90 body: 75
25	Literal
26	ExprTuple	28, 29
27	ExprTuple	30, 31
28	Operation	operator: 32 operand: 38
29	Operation	operator: 68 operands: 34
30	Operation	operator: 45 operand: 40
31	Operation	operator: 36 operand: 41
32	Literal
33	ExprTuple	38
34	ExprTuple	86, 39
35	ExprTuple	40
36	Literal
37	ExprTuple	41
38	Lambda	parameters: 66 body: 42
39	Operation	operator: 43 operand: 89
40	Operation	operator: 45 operands: 46
41	Operation	operator: 47 operands: 48
42	Conditional	value: 49 condition: 50
43	Literal
44	ExprTuple	89
45	Literal
46	ExprTuple	51
47	Literal
48	ExprTuple	52, 89
49	Operation	operator: 53 operands: 54
50	Operation	operator: 55 operands: 56
51	ExprRange	lambda_map: 57 start_index: 88 end_index: 89
52	Literal
53	Literal
54	ExprTuple	58, 59
55	Literal
56	ExprTuple	60, 61
57	Lambda	parameter: 90 body: 62
58	Operation	operator: 63 operands: 64
59	Operation	operator: 65 operands: 66
60	Operation	operator: 68 operands: 67
61	Operation	operator: 68 operands: 69
62	Operation	operator: 80 operand: 75
63	Literal
64	ExprTuple	71, 72, 86, 73
65	Literal
66	ExprTuple	94, 95
67	ExprTuple	94, 74
68	Literal
69	ExprTuple	95, 74
70	ExprTuple	75
71	Operation	operator: 76 operand: 85
72	Operation	operator: 78 operand: 86
73	Operation	operator: 80 operand: 87
74	Operation	operator: 82 operands: 83
75	IndexedVar	variable: 92 index: 90
76	Literal
77	ExprTuple	85
78	Literal
79	ExprTuple	86
80	Literal
81	ExprTuple	87
82	Literal
83	ExprTuple	88, 89
84	ExprTuple	90
85	IndexedVar	variable: 92 index: 94
86	Variable
87	IndexedVar	variable: 92 index: 95
88	Literal
89	Variable
90	Variable
91	ExprTuple	94
92	Variable
93	ExprTuple	95
94	Variable
95	Variable

Expression of type Forall¶

from the theory of proveit.physics.quantum.algebra¶

Expression of type Forall ¶

from the theory of proveit.physics.quantum.algebra ¶