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 ExprRange, Variable, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Exp, Mult, NaturalPos, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase, _psi_t_ket

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [t], instance_expr = Equals(_psi_t_ket, TensorProd(ExprRange(sub_expr1, ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Add(Neg(t), one), zero).with_decreasing_order())), domain = NaturalPos)

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_{t \in \mathbb{N}^+}~\left(\lvert \psi_{t} \rangle = \left(\left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes}  \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right) {\otimes}  \ldots {\otimes}  \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)\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: 31 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	31, 13
11	Operation	operator: 14 operand: 31
12	Operation	operator: 15 operands: 16
13	Literal
14	Literal
15	Literal
16	ExprTuple	17
17	ExprRange	lambda_map: 18 start_index: 19 end_index: 40
18	Lambda	parameter: 63 body: 20
19	Operation	operator: 21 operands: 22
20	Operation	operator: 37 operands: 23
21	Literal
22	ExprTuple	24, 50
23	ExprTuple	25, 26
24	Operation	operator: 61 operand: 31
25	Operation	operator: 43 operands: 28
26	Operation	operator: 29 operands: 30
27	ExprTuple	31
28	ExprTuple	50, 32
29	Literal
30	ExprTuple	33, 34
31	Variable
32	Operation	operator: 57 operands: 35
33	Operation	operator: 46 operand: 40
34	Operation	operator: 37 operands: 38
35	ExprTuple	59, 39
36	ExprTuple	40
37	Literal
38	ExprTuple	41, 42
39	Operation	operator: 43 operands: 44
40	Literal
41	Operation	operator: 57 operands: 45
42	Operation	operator: 46 operand: 50
43	Literal
44	ExprTuple	50, 59
45	ExprTuple	48, 49
46	Literal
47	ExprTuple	50
48	Literal
49	Operation	operator: 51 operands: 52
50	Literal
51	Literal
52	ExprTuple	59, 53, 54, 55, 56
53	Literal
54	Literal
55	Operation	operator: 57 operands: 58
56	Literal
57	Literal
58	ExprTuple	59, 60
59	Literal
60	Operation	operator: 61 operand: 63
61	Literal
62	ExprTuple	63
63	Variable