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
from proveit.core_expr_types import Len
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, sqrt, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

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

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^{t} \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)\right)| = |\left(1, \ldots, 2\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
6	Literal
7	ExprTuple	10
8	Lambda	parameter: 15 body: 12
9	Lambda	parameter: 65 body: 13
10	ExprRange	lambda_map: 14 start_index: 48 end_index: 61
11	ExprTuple	15
12	Operation	operator: 32 operands: 16
13	Operation	operator: 32 operands: 17
14	Lambda	parameter: 65 body: 65
15	Variable
16	ExprTuple	19, 18
17	ExprTuple	19, 20
18	Operation	operator: 23 operands: 21
19	Operation	operator: 40 operands: 22
20	Operation	operator: 23 operands: 24
21	ExprTuple	27, 25
22	ExprTuple	48, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 32 operands: 29
26	Operation	operator: 58 operands: 30
27	Operation	operator: 43 operand: 36
28	Operation	operator: 32 operands: 33
29	ExprTuple	34, 38
30	ExprTuple	61, 35
31	ExprTuple	36
32	Literal
33	ExprTuple	37, 38
34	Operation	operator: 58 operands: 39
35	Operation	operator: 40 operands: 41
36	Literal
37	Operation	operator: 58 operands: 42
38	Operation	operator: 43 operand: 48
39	ExprTuple	46, 45
40	Literal
41	ExprTuple	48, 61
42	ExprTuple	46, 47
43	Literal
44	ExprTuple	48
45	Operation	operator: 50 operands: 49
46	Literal
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	61, 53, 54, 52, 56
50	Literal
51	ExprTuple	61, 53, 54, 55, 56
52	Operation	operator: 58 operands: 57
53	Literal
54	Literal
55	Operation	operator: 58 operands: 59
56	Literal
57	ExprTuple	61, 60
58	Literal
59	ExprTuple	61, 62
60	Variable
61	Literal
62	Operation	operator: 63 operand: 65
63	Literal
64	ExprTuple	65
65	Variable