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 Function, U, Variable, s, t
from proveit.linear_algebra import Unitary
from proveit.logic import Forall
from proveit.numbers import NaturalPos
from proveit.physics.quantum import normalized_var_ket_u, var_ket_u
from proveit.physics.quantum.QPE import phase, s_ket_domain, two_pow_s

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [s, t], instance_expr = Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [var_ket_u], instance_expr = Forall(instance_param_or_params = [phase], instance_expr = Function(Variable("_b", latex_format = r"{_{-}b}"), [s, t, U, var_ket_u, phase])), domain = s_ket_domain, condition = normalized_var_ket_u), domain = Unitary(two_pow_s)), 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_{s, t \in \mathbb{N}^+}~\left[\forall_{U \in \textrm{U}\left(2^{s}\right)}~\left[\forall_{\lvert u \rangle \in \mathbb{C}^{2^{s}}~|~\left \|\lvert u \rangle\right \| = 1}~\left[\forall_{\varphi}~{_{-}b}\left(s, t, U, \lvert u \rangle, \varphi\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: 28 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	59, 46
4	Conditional	value: 5 condition: 6
5	Operation	operator: 28 operand: 9
6	Operation	operator: 30 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameter: 47 body: 13
10	Operation	operator: 37 operands: 14
11	Operation	operator: 37 operands: 15
12	ExprTuple	47
13	Conditional	value: 16 condition: 17
14	ExprTuple	59, 18
15	ExprTuple	46, 18
16	Operation	operator: 28 operand: 21
17	Operation	operator: 37 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	47, 22
21	Lambda	parameter: 55 body: 23
22	Operation	operator: 24 operand: 54
23	Conditional	value: 26 condition: 27
24	Literal
25	ExprTuple	54
26	Operation	operator: 28 operand: 32
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32
30	Literal
31	ExprTuple	33, 34
32	Lambda	parameter: 48 body: 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	ExprTuple	48
36	Operation	operator: 41 operands: 42
37	Literal
38	ExprTuple	55, 43
39	Literal
40	ExprTuple	44, 45
41	Variable
42	ExprTuple	59, 46, 47, 55, 48
43	Operation	operator: 49 operands: 50
44	Operation	operator: 51 operand: 55
45	Literal
46	Variable
47	Variable
48	Variable
49	Literal
50	ExprTuple	53, 54
51	Literal
52	ExprTuple	55
53	Literal
54	Operation	operator: 56 operands: 57
55	Variable
56	Literal
57	ExprTuple	58, 59
58	Literal
59	Variable