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 import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Qmult(NumBra(m, _t), InverseFourierTransform(_t), NumKet(k, _t))
sub_expr2 = InSet(k, _m_domain)
sub_expr3 = Mult(frac(one, Exp(two, frac(_t, two))), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(sub_expr1, sub_expr3), domain = _m_domain), Equals(Lambda(k, Conditional(sub_expr1, sub_expr2)), Lambda(k, Conditional(sub_expr3, sub_expr2))).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({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right) = \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)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[k \mapsto \left\{{_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\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: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 68 body: 11
9	Lambda	parameter: 68 body: 12
10	Lambda	parameter: 68 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	68
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	68, 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 62 operands: 26
23	Operation	operator: 27 operands: 28
24	Literal
25	ExprTuple	29, 30, 31
26	ExprTuple	32, 33
27	Literal
28	ExprTuple	34, 35
29	Operation	operator: 36 operands: 37
30	Operation	operator: 38 operand: 71
31	Operation	operator: 40 operands: 41
32	Operation	operator: 58 operands: 42
33	Operation	operator: 64 operands: 43
34	Literal
35	Operation	operator: 44 operands: 45
36	Literal
37	ExprTuple	69, 71
38	Literal
39	ExprTuple	71
40	Literal
41	ExprTuple	68, 71
42	ExprTuple	56, 46
43	ExprTuple	47, 48
44	Literal
45	ExprTuple	61, 49
46	Operation	operator: 64 operands: 50
47	Literal
48	Operation	operator: 52 operand: 55
49	Operation	operator: 52 operand: 56
50	ExprTuple	70, 54
51	ExprTuple	55
52	Literal
53	ExprTuple	56
54	Operation	operator: 58 operands: 57
55	Operation	operator: 58 operands: 59
56	Literal
57	ExprTuple	71, 70
58	Literal
59	ExprTuple	60, 61
60	Operation	operator: 62 operands: 63
61	Operation	operator: 64 operands: 65
62	Literal
63	ExprTuple	70, 66, 67, 68, 69
64	Literal
65	ExprTuple	70, 71
66	Literal
67	Literal
68	Variable
69	Variable
70	Literal
71	Literal