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 ExprRange, Variable, t
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import And, Forall, Implies, InSet
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import QubitSpace, ket0, ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(Neg(t), one)
sub_expr3 = InSet(ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), QubitSpace)
expr = Implies(Forall(instance_param_or_params = [sub_expr1], instance_expr = sub_expr3, domain = Interval(sub_expr2, zero)), And(ExprRange(sub_expr1, sub_expr3, sub_expr2, zero)))

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[\forall_{{_{-}a} \in \{-t + 1~\ldotp \ldotp~0\}}~\left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) \in \mathbb{C}^{2}\right)\right] \Rightarrow \left(\left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) \in \mathbb{C}^{2}\right) \land  \left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 2\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) \in \mathbb{C}^{2}\right) \land  \ldots \land  \left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) \in \mathbb{C}^{2}\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 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10
9	Lambda	parameter: 69 body: 11
10	ExprRange	lambda_map: 12 start_index: 26 end_index: 46
11	Conditional	value: 14 condition: 13
12	Lambda	parameter: 69 body: 14
13	Operation	operator: 16 operands: 15
14	Operation	operator: 16 operands: 17
15	ExprTuple	69, 18
16	Literal
17	ExprTuple	19, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 42 operands: 23
20	Operation	operator: 24 operands: 25
21	Literal
22	ExprTuple	26, 46
23	ExprTuple	27, 28
24	Literal
25	ExprTuple	29, 65
26	Operation	operator: 30 operands: 31
27	Operation	operator: 49 operands: 32
28	Operation	operator: 33 operands: 34
29	Literal
30	Literal
31	ExprTuple	35, 56
32	ExprTuple	56, 36
33	Literal
34	ExprTuple	37, 38
35	Operation	operator: 67 operand: 44
36	Operation	operator: 63 operands: 40
37	Operation	operator: 52 operand: 46
38	Operation	operator: 42 operands: 43
39	ExprTuple	44
40	ExprTuple	65, 45
41	ExprTuple	46
42	Literal
43	ExprTuple	47, 48
44	Variable
45	Operation	operator: 49 operands: 50
46	Literal
47	Operation	operator: 63 operands: 51
48	Operation	operator: 52 operand: 56
49	Literal
50	ExprTuple	56, 65
51	ExprTuple	54, 55
52	Literal
53	ExprTuple	56
54	Literal
55	Operation	operator: 57 operands: 58
56	Literal
57	Literal
58	ExprTuple	65, 59, 60, 61, 62
59	Literal
60	Literal
61	Operation	operator: 63 operands: 64
62	Literal
63	Literal
64	ExprTuple	65, 66
65	Literal
66	Operation	operator: 67 operand: 69
67	Literal
68	ExprTuple	69
69	Variable