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 U
from proveit.linear_algebra import MatrixMult, ScalarMult, Unitary
from proveit.logic import And, Equals, InSet
from proveit.numbers import Exp, IntervalCO, Mult, Real, e, i, one, pi, two, zero
from proveit.physics.quantum import normalized_var_ket_u, var_ket_u
from proveit.physics.quantum.QPE import phase, s_ket_domain, two_pow_s

# build up the expression from sub-expressions
expr = And(InSet(U, Unitary(two_pow_s)), InSet(var_ket_u, s_ket_domain), InSet(phase, Real), InSet(phase, IntervalCO(zero, one)), normalized_var_ket_u, Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_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(U \in \textrm{U}\left(2^{s}\right)\right) \land \left(\lvert u \rangle \in \mathbb{C}^{2^{s}}\right) \land \left(\varphi \in \mathbb{R}\right) \land \left(\varphi \in \left[0,1\right)\right) \land \left(\left \|\lvert u \rangle\right \| = 1\right) \land \left(\left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \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, 4, 5, 6, 7, 8
3	Operation	operator: 12 operands: 9
4	Operation	operator: 12 operands: 10
5	Operation	operator: 12 operands: 11
6	Operation	operator: 12 operands: 13
7	Operation	operator: 15 operands: 14
8	Operation	operator: 15 operands: 16
9	ExprTuple	40, 17
10	ExprTuple	42, 18
11	ExprTuple	54, 19
12	Literal
13	ExprTuple	54, 20
14	ExprTuple	21, 39
15	Literal
16	ExprTuple	22, 23
17	Operation	operator: 24 operand: 37
18	Operation	operator: 26 operands: 27
19	Literal
20	Operation	operator: 28 operands: 29
21	Operation	operator: 30 operand: 42
22	Operation	operator: 32 operands: 33
23	Operation	operator: 34 operands: 35
24	Literal
25	ExprTuple	37
26	Literal
27	ExprTuple	36, 37
28	Literal
29	ExprTuple	38, 39
30	Literal
31	ExprTuple	42
32	Literal
33	ExprTuple	40, 42
34	Literal
35	ExprTuple	41, 42
36	Literal
37	Operation	operator: 44 operands: 43
38	Literal
39	Literal
40	Variable
41	Operation	operator: 44 operands: 45
42	Variable
43	ExprTuple	51, 46
44	Literal
45	ExprTuple	47, 48
46	Variable
47	Literal
48	Operation	operator: 49 operands: 50
49	Literal
50	ExprTuple	51, 52, 53, 54
51	Literal
52	Literal
53	Literal
54	Variable