Expression of type And

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 MatrixExp, MatrixMult, ScalarMult
from proveit.logic import And, Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, i, one, pi, two, zero
from proveit.physics.quantum.QPE import _U, _ket_u, _phase

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Exp(two, Neg(sub_expr1))
expr = And(ExprRange(sub_expr1, Equals(MatrixMult(MatrixExp(_U, sub_expr2), _ket_u), ScalarMult(Exp(e, Mult(two, pi, i, Mult(sub_expr2, _phase))), _ket_u)), Add(Neg(t), one), zero))

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(U^{2^{-\left(-t + 1\right)}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-\left(-t + 1\right)} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\right) \land  \left(\left(U^{2^{-\left(-t + 2\right)}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-\left(-t + 2\right)} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\right) \land  \ldots \land  \left(\left(U^{2^{-0}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-0} \cdot \varphi\right)} \cdot \lvert u \rangle\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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3
3	ExprRange	lambda_map: 4 start_index: 5 end_index: 6
4	Lambda	parameter: 45 body: 7
5	Operation	operator: 8 operands: 9
6	Literal
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15
12	Operation	operator: 43 operand: 21
13	Literal
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21
17	Literal
18	ExprTuple	22, 24
19	Literal
20	ExprTuple	23, 24
21	Variable
22	Operation	operator: 25 operands: 26
23	Operation	operator: 39 operands: 27
24	Literal
25	Literal
26	ExprTuple	28, 37
27	ExprTuple	29, 30
28	Literal
29	Literal
30	Operation	operator: 35 operands: 31
31	ExprTuple	41, 32, 33, 34
32	Literal
33	Literal
34	Operation	operator: 35 operands: 36
35	Literal
36	ExprTuple	37, 38
37	Operation	operator: 39 operands: 40
38	Literal
39	Literal
40	ExprTuple	41, 42
41	Literal
42	Operation	operator: 43 operand: 45
43	Literal
44	ExprTuple	45
45	Variable