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 = Conditional(Mult(sub_expr1, sub_expr2, sub_expr3), sub_expr4)
sub_expr6 = Conditional(Mult(sub_expr1, Mult(sub_expr2, sub_expr3)), sub_expr4)
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr6, sub_expr5)), Equals(Lambda(k, sub_expr6), Lambda(k, sub_expr5)).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}~\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) \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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 \left(\frac{1}{2^{\frac{t}{2}}} \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] =  \\ \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] \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: 13 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 64 body: 11
9	Lambda	parameter: 64 body: 15
10	Lambda	parameter: 64 body: 16
11	Operation	operator: 13 operands: 14
12	ExprTuple	64
13	Literal
14	ExprTuple	15, 16
15	Conditional	value: 17 condition: 19
16	Conditional	value: 18 condition: 19
17	Operation	operator: 58 operands: 20
18	Operation	operator: 58 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	25, 24
21	ExprTuple	25, 31, 32
22	Literal
23	ExprTuple	64, 26
24	Operation	operator: 58 operands: 27
25	Operation	operator: 60 operands: 28
26	Operation	operator: 29 operands: 30
27	ExprTuple	31, 32
28	ExprTuple	42, 33
29	Literal
30	ExprTuple	34, 35
31	Operation	operator: 54 operands: 36
32	Operation	operator: 60 operands: 37
33	Operation	operator: 58 operands: 38
34	Literal
35	Operation	operator: 39 operands: 40
36	ExprTuple	52, 41
37	ExprTuple	42, 43
38	ExprTuple	66, 62, 63, 44, 64
39	Literal
40	ExprTuple	57, 45
41	Operation	operator: 60 operands: 46
42	Literal
43	Operation	operator: 48 operand: 51
44	Literal
45	Operation	operator: 48 operand: 52
46	ExprTuple	66, 50
47	ExprTuple	51
48	Literal
49	ExprTuple	52
50	Operation	operator: 54 operands: 53
51	Operation	operator: 54 operands: 55
52	Literal
53	ExprTuple	67, 66
54	Literal
55	ExprTuple	56, 57
56	Operation	operator: 58 operands: 59
57	Operation	operator: 60 operands: 61
58	Literal
59	ExprTuple	66, 62, 63, 64, 65
60	Literal
61	ExprTuple	66, 67
62	Literal
63	Literal
64	Variable
65	Variable
66	Literal
67	Literal