Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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 k, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr2 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one)))
expr = Equals(TensorProd(ScalarMult(sub_expr1, ket1), sub_expr2), ScalarMult(sub_expr1, TensorProd(ket1, sub_expr2))).with_wrapping_at(1)

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}{c} \begin{array}{l} \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right) {\otimes} \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) \\  = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\lvert 1 \rangle {\otimes} \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(1)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 10 operands: 5
4	Operation	operator: 26 operands: 6
5	ExprTuple	7, 14
6	ExprTuple	12, 8
7	Operation	operator: 26 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	13, 14
12	Operation	operator: 52 operands: 15
13	Operation	operator: 16 operand: 58
14	Operation	operator: 17 operand: 20
15	ExprTuple	38, 19
16	Literal
17	Literal
18	ExprTuple	20
19	Operation	operator: 42 operands: 21
20	Lambda	parameter: 49 body: 23
21	ExprTuple	56, 46, 47, 48, 50
22	ExprTuple	49
23	Conditional	value: 24 condition: 25
24	Operation	operator: 26 operands: 27
25	Operation	operator: 28 operands: 29
26	Literal
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	49, 32
30	Operation	operator: 52 operands: 33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operands: 37
33	ExprTuple	38, 39
34	Literal
35	ExprTuple	49, 57
36	Literal
37	ExprTuple	40, 41
38	Literal
39	Operation	operator: 42 operands: 43
40	Literal
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	56, 46, 47, 48, 49
44	Literal
45	ExprTuple	50, 51
46	Literal
47	Literal
48	Literal
49	Variable
50	Operation	operator: 52 operands: 53
51	Operation	operator: 54 operand: 58
52	Literal
53	ExprTuple	56, 57
54	Literal
55	ExprTuple	58
56	Literal
57	Variable
58	Literal