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 Conditional, Lambda, k, m
from proveit.logic import Equals, Forall, 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)))
sub_expr4 = InSet(k, _m_domain)
sub_expr5 = Mult(sub_expr1, sub_expr2, sub_expr3)
sub_expr6 = Mult(sub_expr2, Mult(sub_expr1, sub_expr3))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr5, sub_expr6), domain = _m_domain), Equals(Lambda(k, Conditional(sub_expr5, sub_expr4)), Lambda(k, Conditional(sub_expr6, sub_expr4))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\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) = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[k \mapsto \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}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\frac{1}{2^{\frac{t}{2}}} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(2)	('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 operand: 8
4	Operation	operator: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 65 body: 11
9	Lambda	parameter: 65 body: 12
10	Lambda	parameter: 65 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	65
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	65, 23
21	Operation	operator: 59 operands: 24
22	Operation	operator: 59 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	37, 28, 38
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 55 operands: 32
29	Operation	operator: 59 operands: 33
30	Literal
31	Operation	operator: 34 operands: 35
32	ExprTuple	48, 36
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	58, 39
36	Operation	operator: 61 operands: 40
37	Operation	operator: 61 operands: 41
38	Operation	operator: 61 operands: 42
39	Operation	operator: 51 operand: 48
40	ExprTuple	67, 44
41	ExprTuple	46, 45
42	ExprTuple	46, 47
43	ExprTuple	48
44	Operation	operator: 55 operands: 49
45	Operation	operator: 59 operands: 50
46	Literal
47	Operation	operator: 51 operand: 54
48	Literal
49	ExprTuple	68, 67
50	ExprTuple	67, 63, 64, 53, 65
51	Literal
52	ExprTuple	54
53	Literal
54	Operation	operator: 55 operands: 56
55	Literal
56	ExprTuple	57, 58
57	Operation	operator: 59 operands: 60
58	Operation	operator: 61 operands: 62
59	Literal
60	ExprTuple	67, 63, 64, 65, 66
61	Literal
62	ExprTuple	67, 68
63	Literal
64	Literal
65	Variable
66	Variable
67	Literal
68	Literal