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 x_1_to_n, x_i
from proveit.linear_algebra import Hspace, Identity, OrthoNormBases, VecSum
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, NaturalPos, 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 = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(VecSum(index_or_indices = [i], summand = Qmult(Ket(x_i), Bra(x_i)), domain = Interval(one, n)), Identity(Hspace)), condition = InSet(x_1_to_n_kets, OrthoNormBases(Hspace))).with_wrapping(), domain = NaturalPos)

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())

\forall_{n \in \mathbb{N}^+}~\left[\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(\left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\right)\right) = I_{\mathcal{H}}\right)\end{array}\right]

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

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