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, 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 = [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))

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_{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]

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: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 41 body: 4
3	ExprTuple	41
4	Conditional	value: 5 condition: 6
5	Operation	operator: 15 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	41, 44
10	Lambda	parameter: 67 body: 12
11	ExprTuple	67
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 18
14	Operation	operator: 52 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	67, 19
18	Lambda	parameters: 20 body: 21
19	Literal
20	ExprTuple	22
21	Conditional	value: 23 condition: 24
22	ExprRange	lambda_map: 25 start_index: 66 end_index: 67
23	Operation	operator: 26 operands: 27
24	Operation	operator: 52 operands: 28
25	Lambda	parameter: 68 body: 58
26	Literal
27	ExprTuple	29, 30
28	ExprTuple	31, 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operand: 42
31	Operation	operator: 37 operands: 38
32	Operation	operator: 39 operand: 44
33	Literal
34	ExprTuple	44, 41
35	Literal
36	ExprTuple	42
37	Literal
38	ExprTuple	43
39	Literal
40	ExprTuple	44
41	Variable
42	Lambda	parameter: 71 body: 45
43	ExprRange	lambda_map: 46 start_index: 66 end_index: 67
44	Variable
45	Conditional	value: 47 condition: 48
46	Lambda	parameter: 68 body: 49
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operands: 53
49	Operation	operator: 59 operand: 58
50	Literal
51	ExprTuple	55, 56
52	Literal
53	ExprTuple	71, 57
54	ExprTuple	58
55	Operation	operator: 59 operand: 65
56	Operation	operator: 60 operand: 65
57	Operation	operator: 62 operands: 63
58	IndexedVar	variable: 69 index: 68
59	Literal
60	Literal
61	ExprTuple	65
62	Literal
63	ExprTuple	66, 67
64	ExprTuple	68
65	IndexedVar	variable: 69 index: 71
66	Literal
67	Variable
68	Variable
69	Variable
70	ExprTuple	71
71	Variable