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 HilbertSpaces, Hspace, OrthoNormBases, OrthoProj, VecSum
from proveit.logic import Equals, Forall, InSet, SubsetEq
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 = [Hspace], instance_expr = Forall(instance_param_or_params = [X], instance_expr = Forall(instance_param_or_params = [n], instance_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(), domain = NaturalPos), condition = SubsetEq(X, Hspace)), domain = HilbertSpaces)

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_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}}~\left[\forall_{X~|~X \subseteq \mathcal{H}}~\left[\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(\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}\right]\right]\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: 23 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 52 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 23 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	52, 10
9	Lambda	parameter: 49 body: 12
10	Literal
11	ExprTuple	49
12	Conditional	value: 13 condition: 14
13	Operation	operator: 23 operand: 18
14	Operation	operator: 16 operands: 17
15	ExprTuple	18
16	Literal
17	ExprTuple	49, 52
18	Lambda	parameter: 75 body: 20
19	ExprTuple	75
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operand: 26
22	Operation	operator: 60 operands: 25
23	Literal
24	ExprTuple	26
25	ExprTuple	75, 27
26	Lambda	parameters: 28 body: 29
27	Literal
28	ExprTuple	30
29	Conditional	value: 31 condition: 32
30	ExprRange	lambda_map: 33 start_index: 74 end_index: 75
31	Operation	operator: 34 operands: 35
32	Operation	operator: 60 operands: 36
33	Lambda	parameter: 76 body: 66
34	Literal
35	ExprTuple	37, 38
36	ExprTuple	39, 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operand: 50
39	Operation	operator: 45 operands: 46
40	Operation	operator: 47 operand: 52
41	Literal
42	ExprTuple	52, 49
43	Literal
44	ExprTuple	50
45	Literal
46	ExprTuple	51
47	Literal
48	ExprTuple	52
49	Variable
50	Lambda	parameter: 79 body: 53
51	ExprRange	lambda_map: 54 start_index: 74 end_index: 75
52	Variable
53	Conditional	value: 55 condition: 56
54	Lambda	parameter: 76 body: 57
55	Operation	operator: 58 operands: 59
56	Operation	operator: 60 operands: 61
57	Operation	operator: 67 operand: 66
58	Literal
59	ExprTuple	63, 64
60	Literal
61	ExprTuple	79, 65
62	ExprTuple	66
63	Operation	operator: 67 operand: 73
64	Operation	operator: 68 operand: 73
65	Operation	operator: 70 operands: 71
66	IndexedVar	variable: 77 index: 76
67	Literal
68	Literal
69	ExprTuple	73
70	Literal
71	ExprTuple	74, 75
72	ExprTuple	76
73	IndexedVar	variable: 77 index: 79
74	Literal
75	Variable
76	Variable
77	Variable
78	ExprTuple	79
79	Variable