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 A, W, i, j, m, n
from proveit.core_expr_types import v_i, w_1_to_n, w_j
from proveit.linear_algebra import OrthoNormBases, VecSum
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import w_1_to_n_kets

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