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 k, m
from proveit.logic import Equals, Implies, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
expr = Implies(InSet(k, _m_domain), Equals(Mult(sub_expr1, Mult(sub_expr2, sub_expr3)), Mult(sub_expr1, sub_expr2, sub_expr3)))

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(k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right) \Rightarrow \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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: 5 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	53, 9
7	Literal
8	ExprTuple	10, 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 47 operands: 14
11	Operation	operator: 47 operands: 15
12	Literal
13	ExprTuple	16, 17
14	ExprTuple	19, 18
15	ExprTuple	19, 25, 26
16	Literal
17	Operation	operator: 20 operands: 21
18	Operation	operator: 47 operands: 22
19	Operation	operator: 49 operands: 23
20	Literal
21	ExprTuple	46, 24
22	ExprTuple	25, 26
23	ExprTuple	34, 27
24	Operation	operator: 38 operand: 32
25	Operation	operator: 43 operands: 29
26	Operation	operator: 49 operands: 30
27	Operation	operator: 47 operands: 31
28	ExprTuple	32
29	ExprTuple	32, 33
30	ExprTuple	34, 35
31	ExprTuple	55, 51, 52, 36, 53
32	Literal
33	Operation	operator: 49 operands: 37
34	Literal
35	Operation	operator: 38 operand: 41
36	Literal
37	ExprTuple	55, 40
38	Literal
39	ExprTuple	41
40	Operation	operator: 43 operands: 42
41	Operation	operator: 43 operands: 44
42	ExprTuple	56, 55
43	Literal
44	ExprTuple	45, 46
45	Operation	operator: 47 operands: 48
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	55, 51, 52, 53, 54
49	Literal
50	ExprTuple	55, 56
51	Literal
52	Literal
53	Variable
54	Variable
55	Literal
56	Literal