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, 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 = 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()

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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',)

# display the expression information
stored_expr.expr_info()

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