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, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = frac(t, two)
sub_expr2 = frac(one, Exp(two, sub_expr1))
sub_expr3 = frac(one, Exp(two, subtract(frac(Add(t, one), two), sub_expr1)))
sub_expr4 = TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), 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(ScalarMult(sub_expr3, ScalarMult(sub_expr2, sub_expr4)), ScalarMult(Mult(sub_expr3, sub_expr2), sub_expr4))

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())

\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\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)\right)\right) = \left(\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \frac{1}{2^{\frac{t}{2}}}\right) \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\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)\right)

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.	()	()	('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: 45 operands: 5
4	Operation	operator: 45 operands: 6
5	ExprTuple	12, 7
6	ExprTuple	8, 11
7	Operation	operator: 45 operands: 9
8	Operation	operator: 70 operands: 10
9	ExprTuple	13, 11
10	ExprTuple	12, 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 64 operands: 16
13	Operation	operator: 64 operands: 17
14	Literal
15	ExprTuple	18, 19
16	ExprTuple	86, 20
17	ExprTuple	86, 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 24 operand: 30
20	Operation	operator: 80 operands: 26
21	Operation	operator: 80 operands: 27
22	Literal
23	ExprTuple	28, 29
24	Literal
25	ExprTuple	30
26	ExprTuple	84, 31
27	ExprTuple	84, 56
28	Operation	operator: 44 operand: 68
29	Operation	operator: 45 operands: 33
30	Lambda	parameter: 77 body: 35
31	Operation	operator: 72 operands: 36
32	ExprTuple	68
33	ExprTuple	37, 38
34	ExprTuple	77
35	Conditional	value: 39 condition: 40
36	ExprTuple	41, 42
37	Operation	operator: 80 operands: 43
38	Operation	operator: 44 operand: 86
39	Operation	operator: 45 operands: 46
40	Operation	operator: 47 operands: 48
41	Operation	operator: 64 operands: 49
42	Operation	operator: 82 operand: 56
43	ExprTuple	66, 51
44	Literal
45	Literal
46	ExprTuple	52, 53
47	Literal
48	ExprTuple	77, 54
49	ExprTuple	55, 84
50	ExprTuple	56
51	Operation	operator: 70 operands: 57
52	Operation	operator: 80 operands: 58
53	Operation	operator: 59 operands: 60
54	Operation	operator: 61 operands: 62
55	Operation	operator: 72 operands: 63
56	Operation	operator: 64 operands: 65
57	ExprTuple	84, 74, 75, 76, 78
58	ExprTuple	66, 67
59	Literal
60	ExprTuple	77, 85
61	Literal
62	ExprTuple	68, 69
63	ExprTuple	85, 86
64	Literal
65	ExprTuple	85, 84
66	Literal
67	Operation	operator: 70 operands: 71
68	Literal
69	Operation	operator: 72 operands: 73
70	Literal
71	ExprTuple	84, 74, 75, 76, 77
72	Literal
73	ExprTuple	78, 79
74	Literal
75	Literal
76	Literal
77	Variable
78	Operation	operator: 80 operands: 81
79	Operation	operator: 82 operand: 86
80	Literal
81	ExprTuple	84, 85
82	Literal
83	ExprTuple	86
84	Literal
85	Variable
86	Literal