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.linear_algebra import ScalarMult, TensorProd, 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 _ket_u, _phase, _psi_t_ket

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Equals(TensorProd(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(), _ket_u), TensorProd(_psi_t_ket, _ket_u))

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