Expression of type Forall

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 Variable, t
from proveit.linear_algebra import MatrixExp, MatrixMult, ScalarMult
from proveit.logic import Equals, Forall
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))
expr = Forall(instance_param_or_params = [sub_expr1], instance_expr = Equals(MatrixMult(MatrixExp(_U, sub_expr2), _ket_u), ScalarMult(Exp(e, Mult(two, pi, i, Mult(sub_expr2, _phase))), _ket_u)), domain = Interval(Add(Neg(t), one), 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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 51 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	51, 13
11	Operation	operator: 14 operands: 15
12	Operation	operator: 16 operands: 17
13	Operation	operator: 18 operands: 19
14	Literal
15	ExprTuple	20, 22
16	Literal
17	ExprTuple	21, 22
18	Literal
19	ExprTuple	23, 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 45 operands: 27
22	Literal
23	Operation	operator: 28 operands: 29
24	Literal
25	Literal
26	ExprTuple	30, 43
27	ExprTuple	31, 32
28	Literal
29	ExprTuple	33, 34
30	Literal
31	Literal
32	Operation	operator: 41 operands: 35
33	Operation	operator: 49 operand: 40
34	Literal
35	ExprTuple	47, 37, 38, 39
36	ExprTuple	40
37	Literal
38	Literal
39	Operation	operator: 41 operands: 42
40	Variable
41	Literal
42	ExprTuple	43, 44
43	Operation	operator: 45 operands: 46
44	Literal
45	Literal
46	ExprTuple	47, 48
47	Literal
48	Operation	operator: 49 operand: 51
49	Literal
50	ExprTuple	51
51	Variable