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, ket0
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(ket0, 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 0 \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: 56 body: 5
4	ExprTuple	56
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	56, 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	33, 23
19	Operation	operator: 24 operand: 33
20	Operation	operator: 26 operands: 27
21	Operation	operator: 29 operands: 28
22	Operation	operator: 29 operands: 30
23	Operation	operator: 31 operands: 32
24	Literal
25	ExprTuple	33
26	Literal
27	ExprTuple	34, 35
28	ExprTuple	36, 52
29	Literal
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	37, 38
33	Literal
34	Operation	operator: 42 operands: 39
35	Operation	operator: 40 operands: 41
36	Literal
37	Operation	operator: 42 operands: 43
38	Operation	operator: 44 operand: 49
39	ExprTuple	46, 47
40	Literal
41	ExprTuple	56, 48
42	Literal
43	ExprTuple	52, 48
44	Literal
45	ExprTuple	49
46	Literal
47	Operation	operator: 50 operands: 51
48	Variable
49	Literal
50	Literal
51	ExprTuple	52, 53, 54, 55, 56
52	Literal
53	Literal
54	Literal
55	Literal
56	Variable