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 i, n
from proveit.core_expr_types import v_1_to_n, v_i, w_1_to_n, w_i
from proveit.linear_algebra import OrthoNormBases, Unitary, VecSum
from proveit.logic import CartExp, Forall, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = Forall(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = InSet(VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n)), Unitary(n)), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).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}, w_{1}, w_{2}, \ldots, w_{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\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left(\left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right) \in \textrm{U}\left(n\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	6, 7
5	Conditional	value: 8 condition: 9
6	ExprRange	lambda_map: 10 start_index: 63 end_index: 64
7	ExprRange	lambda_map: 11 start_index: 63 end_index: 64
8	Operation	operator: 41 operands: 12
9	Operation	operator: 13 operands: 14
10	Lambda	parameter: 77 body: 72
11	Lambda	parameter: 77 body: 73
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	17, 18
15	Operation	operator: 19 operand: 25
16	Operation	operator: 21 operand: 64
17	Operation	operator: 41 operands: 23
18	Operation	operator: 41 operands: 24
19	Literal
20	ExprTuple	25
21	Literal
22	ExprTuple	64
23	ExprTuple	26, 28
24	ExprTuple	27, 28
25	Lambda	parameter: 71 body: 29
26	Operation	operator: 44 operand: 36
27	Operation	operator: 44 operand: 37
28	Operation	operator: 32 operand: 38
29	Conditional	value: 34 condition: 35
30	ExprTuple	36
31	ExprTuple	37
32	Literal
33	ExprTuple	38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operands: 42
36	Operation	operator: 44 operands: 43
37	Operation	operator: 44 operands: 45
38	Operation	operator: 46 operands: 47
39	Literal
40	ExprTuple	48, 49
41	Literal
42	ExprTuple	71, 50
43	ExprTuple	51
44	Literal
45	ExprTuple	52
46	Literal
47	ExprTuple	53, 64
48	Operation	operator: 69 operand: 61
49	Operation	operator: 55 operand: 62
50	Operation	operator: 57 operands: 58
51	ExprRange	lambda_map: 59 start_index: 63 end_index: 64
52	ExprRange	lambda_map: 60 start_index: 63 end_index: 64
53	Literal
54	ExprTuple	61
55	Literal
56	ExprTuple	62
57	Literal
58	ExprTuple	63, 64
59	Lambda	parameter: 77 body: 65
60	Lambda	parameter: 77 body: 66
61	IndexedVar	variable: 74 index: 71
62	IndexedVar	variable: 75 index: 71
63	Literal
64	Variable
65	Operation	operator: 69 operand: 72
66	Operation	operator: 69 operand: 73
67	ExprTuple	71
68	ExprTuple	72
69	Literal
70	ExprTuple	73
71	Variable
72	IndexedVar	variable: 74 index: 77
73	IndexedVar	variable: 75 index: 77
74	Variable
75	Variable
76	ExprTuple	77
77	Variable