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 MatrixExp, MatrixMult, ScalarMult
from proveit.logic import And, Equals, Forall, Implies
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, i, one, pi, two, zero
from proveit.physics.quantum.QPE import _U, _ket_u, _phase

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Exp(two, Neg(sub_expr1))
sub_expr3 = Add(Neg(t), one)
sub_expr4 = Equals(MatrixMult(MatrixExp(_U, sub_expr2), _ket_u), ScalarMult(Exp(e, Mult(two, pi, i, Mult(sub_expr2, _phase))), _ket_u))
expr = Implies(Forall(instance_param_or_params = [sub_expr1], instance_expr = sub_expr4, domain = Interval(sub_expr3, zero)), And(ExprRange(sub_expr1, sub_expr4, sub_expr3, 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(U^{2^{-{_{-}a}}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-{_{-}a}} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\right)\right] \Rightarrow \left(\left(\left(U^{2^{-\left(-t + 1\right)}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-\left(-t + 1\right)} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\right) \land  \left(\left(U^{2^{-\left(-t + 2\right)}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-\left(-t + 2\right)} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\right) \land  \ldots \land  \left(\left(U^{2^{-0}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-0} \cdot \varphi\right)} \cdot \lvert u \rangle\right)\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: 59 body: 11
10	ExprRange	lambda_map: 12 start_index: 28 end_index: 29
11	Conditional	value: 14 condition: 13
12	Lambda	parameter: 59 body: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	59, 19
17	Literal
18	ExprTuple	20, 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 26 operands: 27
22	Literal
23	ExprTuple	28, 29
24	Literal
25	ExprTuple	30, 32
26	Literal
27	ExprTuple	31, 32
28	Operation	operator: 33 operands: 34
29	Literal
30	Operation	operator: 35 operands: 36
31	Operation	operator: 53 operands: 37
32	Literal
33	Literal
34	ExprTuple	38, 39
35	Literal
36	ExprTuple	40, 51
37	ExprTuple	41, 42
38	Operation	operator: 57 operand: 45
39	Literal
40	Literal
41	Literal
42	Operation	operator: 49 operands: 44
43	ExprTuple	45
44	ExprTuple	55, 46, 47, 48
45	Variable
46	Literal
47	Literal
48	Operation	operator: 49 operands: 50
49	Literal
50	ExprTuple	51, 52
51	Operation	operator: 53 operands: 54
52	Literal
53	Literal
54	ExprTuple	55, 56
55	Literal
56	Operation	operator: 57 operand: 59
57	Literal
58	ExprTuple	59
59	Variable