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 X, i, n
from proveit.core_expr_types import x_1_to_n, x_i
from proveit.linear_algebra import Hspace, OrthoNormBases, OrthoProj, 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 x_1_to_n_kets

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(OrthoProj(Hspace, X), VecSum(index_or_indices = [i], summand = Qmult(Ket(x_i), Bra(x_i)), domain = Interval(one, n))), condition = InSet(x_1_to_n_kets, OrthoNormBases(Hspace))).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_{x_{1}, x_{2}, \ldots, x_{n}~|~\left\{\lvert x_{1} \rangle, \lvert x_{2} \rangle, \ldots, \lvert x_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left(\textrm{OrthoProj}\left(\mathcal{H}, X\right) = \left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\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: 50 end_index: 51
7	Operation	operator: 10 operands: 11
8	Operation	operator: 36 operands: 12
9	Lambda	parameter: 52 body: 42
10	Literal
11	ExprTuple	13, 14
12	ExprTuple	15, 16
13	Operation	operator: 17 operands: 18
14	Operation	operator: 19 operand: 26
15	Operation	operator: 21 operands: 22
16	Operation	operator: 23 operand: 28
17	Literal
18	ExprTuple	28, 25
19	Literal
20	ExprTuple	26
21	Literal
22	ExprTuple	27
23	Literal
24	ExprTuple	28
25	Variable
26	Lambda	parameter: 55 body: 29
27	ExprRange	lambda_map: 30 start_index: 50 end_index: 51
28	Variable
29	Conditional	value: 31 condition: 32
30	Lambda	parameter: 52 body: 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 43 operand: 42
34	Literal
35	ExprTuple	39, 40
36	Literal
37	ExprTuple	55, 41
38	ExprTuple	42
39	Operation	operator: 43 operand: 49
40	Operation	operator: 44 operand: 49
41	Operation	operator: 46 operands: 47
42	IndexedVar	variable: 53 index: 52
43	Literal
44	Literal
45	ExprTuple	49
46	Literal
47	ExprTuple	50, 51
48	ExprTuple	52
49	IndexedVar	variable: 53 index: 55
50	Literal
51	Variable
52	Variable
53	Variable
54	ExprTuple	55
55	Variable