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
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(Mult(sub_expr1, sub_expr2), Mult(sub_expr2, sub_expr1)), 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(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\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: 40 body: 5
4	ExprTuple	40
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	40, 14
12	Operation	operator: 32 operands: 15
13	Operation	operator: 32 operands: 16
14	Operation	operator: 17 operands: 18
15	ExprTuple	20, 19
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 42 operands: 23
20	Operation	operator: 42 operands: 24
21	Literal
22	Operation	operator: 25 operands: 26
23	ExprTuple	28, 27
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	36, 30
27	Operation	operator: 32 operands: 31
28	Literal
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operand: 41
31	ExprTuple	44, 37, 38, 39, 36
32	Literal
33	ExprTuple	44, 37, 38, 39, 40
34	Literal
35	ExprTuple	41
36	Operation	operator: 42 operands: 43
37	Literal
38	Literal
39	Literal
40	Variable
41	Literal
42	Literal
43	ExprTuple	44, 45
44	Literal
45	Variable