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