Expression of type Implies

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
from proveit.logic import And, Equals, Implies, InSet
from proveit.numbers import Exp, Interval, IntervalCO, Mult, Real, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import var_ket_u
from proveit.physics.quantum.QPE import phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = InSet(phase, Real)
sub_expr2 = InSet(Mult(two_pow_t, phase), Interval(zero, subtract(two_pow_t, one)))
sub_expr3 = Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))
expr = Implies(And(sub_expr1, sub_expr2, sub_expr3), And(sub_expr1, sub_expr2, sub_expr3, InSet(phase, IntervalCO(zero, one))))

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(\varphi \in \mathbb{R}\right) \land \left(\left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 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)\right) \Rightarrow \left(\left(\varphi \in \mathbb{R}\right) \land \left(\left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 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) \land \left(\varphi \in \left[0,1\right)\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',)
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, 10
6	Literal
7	ExprTuple	8, 9, 10, 11
8	Operation	operator: 16 operands: 12
9	Operation	operator: 16 operands: 13
10	Operation	operator: 14 operands: 15
11	Operation	operator: 16 operands: 17
12	ExprTuple	56, 18
13	ExprTuple	19, 20
14	Literal
15	ExprTuple	21, 22
16	Literal
17	ExprTuple	56, 23
18	Literal
19	Operation	operator: 49 operands: 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 27 operands: 28
22	Operation	operator: 29 operands: 30
23	Operation	operator: 31 operands: 32
24	ExprTuple	41, 56
25	Literal
26	ExprTuple	37, 33
27	Literal
28	ExprTuple	34, 36
29	Literal
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	37, 52
33	Operation	operator: 38 operands: 39
34	Variable
35	Operation	operator: 45 operands: 40
36	Variable
37	Literal
38	Literal
39	ExprTuple	41, 42
40	ExprTuple	43, 44
41	Operation	operator: 45 operands: 46
42	Operation	operator: 47 operand: 52
43	Literal
44	Operation	operator: 49 operands: 50
45	Literal
46	ExprTuple	53, 51
47	Literal
48	ExprTuple	52
49	Literal
50	ExprTuple	53, 54, 55, 56
51	Variable
52	Literal
53	Literal
54	Literal
55	Literal
56	Variable