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 k, t
from proveit.linear_algebra import ScalarMult, TensorProd
from proveit.logic import CartExp, Forall, InSet
from proveit.numbers import Complex, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, QubitSpace, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [k], instance_expr = InSet(TensorProd(ket1, ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t))), TensorProd(QubitSpace, CartExp(Complex, two_pow_t))), domain = Interval(zero, subtract(two_pow_t, one)))

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_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\left(\lvert 1 \rangle {\otimes} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\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: 55 body: 5
4	ExprTuple	55
5	Conditional	value: 6 condition: 7
6	Operation	operator: 9 operands: 8
7	Operation	operator: 9 operands: 10
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	55, 13
11	Operation	operator: 15 operands: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprTuple	19, 20
15	Literal
16	ExprTuple	21, 22
17	Literal
18	ExprTuple	23, 24
19	Operation	operator: 25 operand: 48
20	Operation	operator: 26 operands: 27
21	Operation	operator: 29 operands: 28
22	Operation	operator: 29 operands: 30
23	Literal
24	Operation	operator: 31 operands: 32
25	Literal
26	Literal
27	ExprTuple	33, 34
28	ExprTuple	35, 51
29	Literal
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	36, 37
33	Operation	operator: 41 operands: 38
34	Operation	operator: 39 operands: 40
35	Literal
36	Operation	operator: 41 operands: 42
37	Operation	operator: 43 operand: 48
38	ExprTuple	45, 46
39	Literal
40	ExprTuple	55, 47
41	Literal
42	ExprTuple	51, 47
43	Literal
44	ExprTuple	48
45	Literal
46	Operation	operator: 49 operands: 50
47	Variable
48	Literal
49	Literal
50	ExprTuple	51, 52, 53, 54, 55
51	Literal
52	Literal
53	Literal
54	Literal
55	Variable