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, 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(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()

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(\left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\right)\right) = I_{\mathcal{H}}\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: 48 end_index: 49
7	Operation	operator: 10 operands: 11
8	Operation	operator: 34 operands: 12
9	Lambda	parameter: 50 body: 40
10	Literal
11	ExprTuple	13, 14
12	ExprTuple	15, 16
13	Operation	operator: 17 operand: 24
14	Operation	operator: 19 operand: 26
15	Operation	operator: 20 operands: 21
16	Operation	operator: 22 operand: 26
17	Literal
18	ExprTuple	24
19	Literal
20	Literal
21	ExprTuple	25
22	Literal
23	ExprTuple	26
24	Lambda	parameter: 53 body: 27
25	ExprRange	lambda_map: 28 start_index: 48 end_index: 49
26	Variable
27	Conditional	value: 29 condition: 30
28	Lambda	parameter: 50 body: 31
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 41 operand: 40
32	Literal
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	53, 39
36	ExprTuple	40
37	Operation	operator: 41 operand: 47
38	Operation	operator: 42 operand: 47
39	Operation	operator: 44 operands: 45
40	IndexedVar	variable: 51 index: 50
41	Literal
42	Literal
43	ExprTuple	47
44	Literal
45	ExprTuple	48, 49
46	ExprTuple	50
47	IndexedVar	variable: 51 index: 53
48	Literal
49	Variable
50	Variable
51	Variable
52	ExprTuple	53
53	Variable