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 Norm, ScalarMult, VecAdd
from proveit.logic import And, Equals, Forall, Implies
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import 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 = Equals(Norm(ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), one)
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 \| = 1\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 \| = 1\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 \| = 1\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 \| = 1\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: 25 end_index: 46
11	Conditional	value: 14 condition: 13
12	Lambda	parameter: 69 body: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	69, 19
17	Literal
18	ExprTuple	20, 56
19	Operation	operator: 21 operands: 22
20	Operation	operator: 23 operand: 26
21	Literal
22	ExprTuple	25, 46
23	Literal
24	ExprTuple	26
25	Operation	operator: 27 operands: 28
26	Operation	operator: 43 operands: 29
27	Literal
28	ExprTuple	30, 56
29	ExprTuple	31, 32
30	Operation	operator: 67 operand: 37
31	Operation	operator: 49 operands: 34
32	Operation	operator: 35 operands: 36
33	ExprTuple	37
34	ExprTuple	56, 38
35	Literal
36	ExprTuple	39, 40
37	Variable
38	Operation	operator: 63 operands: 41
39	Operation	operator: 52 operand: 46
40	Operation	operator: 43 operands: 44
41	ExprTuple	65, 45
42	ExprTuple	46
43	Literal
44	ExprTuple	47, 48
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