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, 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, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase

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

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(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right), \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right), \ldots, \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right)\right)| = |\left(1, 2, \ldots, t\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
6	Literal
7	ExprTuple	9
8	ExprRange	lambda_map: 10 start_index: 11 end_index: 33
9	ExprRange	lambda_map: 12 start_index: 43 end_index: 24
10	Lambda	parameter: 56 body: 13
11	Operation	operator: 14 operands: 15
12	Lambda	parameter: 56 body: 56
13	Operation	operator: 30 operands: 16
14	Literal
15	ExprTuple	17, 43
16	ExprTuple	18, 19
17	Operation	operator: 54 operand: 24
18	Operation	operator: 36 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	24
21	ExprTuple	43, 25
22	Literal
23	ExprTuple	26, 27
24	Variable
25	Operation	operator: 50 operands: 28
26	Operation	operator: 39 operand: 33
27	Operation	operator: 30 operands: 31
28	ExprTuple	52, 32
29	ExprTuple	33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operands: 37
33	Literal
34	Operation	operator: 50 operands: 38
35	Operation	operator: 39 operand: 43
36	Literal
37	ExprTuple	43, 52
38	ExprTuple	41, 42
39	Literal
40	ExprTuple	43
41	Literal
42	Operation	operator: 44 operands: 45
43	Literal
44	Literal
45	ExprTuple	52, 46, 47, 48, 49
46	Literal
47	Literal
48	Operation	operator: 50 operands: 51
49	Literal
50	Literal
51	ExprTuple	52, 53
52	Literal
53	Operation	operator: 54 operand: 56
54	Literal
55	ExprTuple	56
56	Variable