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 ExprRange, Lambda, Variable, t
from proveit.core_expr_types import Len
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, three, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _ket_u, _phase

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = frac(one, sqrt(two))
expr = Equals(Len(operands = [Lambda(sub_expr1, ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1)))), Lambda(sub_expr2, ScalarMult(sub_expr3, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr2)), _phase)), ket1)))), Lambda(sub_expr1, _ket_u)]), Len(operands = [ExprRange(sub_expr2, sub_expr2, one, three)]))

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({_{-}b} \mapsto \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right), {_{-}a} \mapsto \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right), {_{-}b} \mapsto \lvert u \rangle\right)| = |\left(1, 2, \ldots, 3\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9, 10
6	Literal
7	ExprTuple	11
8	Lambda	parameter: 20 body: 12
9	Lambda	parameter: 68 body: 13
10	Lambda	parameter: 20 body: 15
11	ExprRange	lambda_map: 16 start_index: 72 end_index: 17
12	Operation	operator: 35 operands: 18
13	Operation	operator: 35 operands: 19
14	ExprTuple	20
15	Literal
16	Lambda	parameter: 68 body: 68
17	Literal
18	ExprTuple	22, 21
19	ExprTuple	22, 23
20	Variable
21	Operation	operator: 26 operands: 24
22	Operation	operator: 43 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	30, 28
25	ExprTuple	72, 29
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 35 operands: 32
29	Operation	operator: 60 operands: 33
30	Operation	operator: 46 operand: 39
31	Operation	operator: 35 operands: 36
32	ExprTuple	37, 41
33	ExprTuple	63, 38
34	ExprTuple	39
35	Literal
36	ExprTuple	40, 41
37	Operation	operator: 60 operands: 42
38	Operation	operator: 43 operands: 44
39	Literal
40	Operation	operator: 60 operands: 45
41	Operation	operator: 46 operand: 72
42	ExprTuple	49, 48
43	Literal
44	ExprTuple	72, 63
45	ExprTuple	49, 50
46	Literal
47	ExprTuple	72
48	Operation	operator: 52 operands: 51
49	Literal
50	Operation	operator: 52 operands: 53
51	ExprTuple	63, 55, 56, 54, 58
52	Literal
53	ExprTuple	63, 55, 56, 57, 58
54	Operation	operator: 60 operands: 59
55	Literal
56	Literal
57	Operation	operator: 60 operands: 61
58	Literal
59	ExprTuple	63, 62
60	Literal
61	ExprTuple	63, 64
62	Operation	operator: 73 operand: 67
63	Literal
64	Operation	operator: 73 operand: 68
65	ExprTuple	67
66	ExprTuple	68
67	Operation	operator: 69 operands: 70
68	Variable
69	Literal
70	ExprTuple	71, 72
71	Operation	operator: 73 operand: 75
72	Literal
73	Literal
74	ExprTuple	75
75	Variable