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_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 = [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))

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_{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)

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 35 body: 5
4	ExprTuple	35
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 33 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	35, 13
11	Operation	operator: 35 operand: 18
12	Operation	operator: 21 operand: 19
13	Operation	operator: 16 operands: 17
14	ExprTuple	18
15	ExprTuple	19
16	Literal
17	ExprTuple	20, 20
18	Operation	operator: 21 operand: 24
19	Lambda	parameter: 55 body: 23
20	Literal
21	Literal
22	ExprTuple	24
23	Conditional	value: 25 condition: 29
24	Lambda	parameter: 55 body: 26
25	Operation	operator: 31 operands: 27
26	Conditional	value: 28 condition: 29
27	ExprTuple	30, 37
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 35 operand: 39
31	Literal
32	ExprTuple	39, 37
33	Literal
34	ExprTuple	55, 38
35	Variable
36	ExprTuple	39
37	Operation	operator: 40 operands: 41
38	Operation	operator: 42 operands: 43
39	IndexedVar	variable: 44 index: 55
40	Literal
41	ExprTuple	45, 46
42	Literal
43	ExprTuple	47, 48
44	Variable
45	Operation	operator: 49 operand: 52
46	Operation	operator: 50 operand: 52
47	Literal
48	Variable
49	Literal
50	Literal
51	ExprTuple	52
52	IndexedVar	variable: 53 index: 55
53	Variable
54	ExprTuple	55
55	Variable