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 Function, f, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_i
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals, Forall, Functions
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult

# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = Interval(one, n)
sub_expr3 = Qmult(Ket(v_i), Bra(v_i))
expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Equals(Function(f, [VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr3), domain = sub_expr2)]), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Function(f, [a_i]), sub_expr3), domain = sub_expr2)), domain = Functions(Complex, Complex)), domain = Complex)

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_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\left[\forall_{f \in \left[\mathbb{C} \rightarrow \mathbb{C}\right]}~\left(f\left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right) = \left(\sum_{i=1}^{n} \left(f\left(a_{i}\right) \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 64 end_index: 65
6	Operation	operator: 9 operand: 13
7	Operation	operator: 11 operands: 12
8	Lambda	parameter: 37 body: 28
9	Literal
10	ExprTuple	13
11	Literal
12	ExprTuple	14
13	Lambda	parameter: 52 body: 16
14	ExprRange	lambda_map: 17 start_index: 64 end_index: 65
15	ExprTuple	52
16	Conditional	value: 18 condition: 19
17	Lambda	parameter: 37 body: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 50 operands: 23
20	Operation	operator: 50 operands: 24
21	Literal
22	ExprTuple	25, 26
23	ExprTuple	52, 27
24	ExprTuple	28, 36
25	Operation	operator: 52 operand: 34
26	Operation	operator: 38 operand: 35
27	Operation	operator: 31 operands: 32
28	IndexedVar	variable: 61 index: 37
29	ExprTuple	34
30	ExprTuple	35
31	Literal
32	ExprTuple	36, 36
33	ExprTuple	37
34	Operation	operator: 38 operand: 41
35	Lambda	parameter: 72 body: 40
36	Literal
37	Variable
38	Literal
39	ExprTuple	41
40	Conditional	value: 42 condition: 46
41	Lambda	parameter: 72 body: 43
42	Operation	operator: 48 operands: 44
43	Conditional	value: 45 condition: 46
44	ExprTuple	47, 54
45	Operation	operator: 48 operands: 49
46	Operation	operator: 50 operands: 51
47	Operation	operator: 52 operand: 56
48	Literal
49	ExprTuple	56, 54
50	Literal
51	ExprTuple	72, 55
52	Variable
53	ExprTuple	56
54	Operation	operator: 57 operands: 58
55	Operation	operator: 59 operands: 60
56	IndexedVar	variable: 61 index: 72
57	Literal
58	ExprTuple	62, 63
59	Literal
60	ExprTuple	64, 65
61	Variable
62	Operation	operator: 66 operand: 69
63	Operation	operator: 67 operand: 69
64	Literal
65	Variable
66	Literal
67	Literal
68	ExprTuple	69
69	IndexedVar	variable: 70 index: 72
70	Variable
71	ExprTuple	72
72	Variable