Expression of type Lambda

from the theory of proveit.physics.quantum.algebra

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 Conditional, Lambda, U, alpha, n
from proveit.linear_algebra import Norm, Unitary
from proveit.logic import CartExp, Equals, Forall, InSet
from proveit.numbers import Complex, NaturalPos, RealNonNeg
from proveit.physics.quantum import Qmult, var_ket_psi

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [alpha], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Equals(Norm(Qmult(U, var_ket_psi)), alpha), domain = CartExp(Complex, n), condition = Equals(Norm(var_ket_psi), alpha)), domain = RealNonNeg), domain = Unitary(n)), InSet(n, 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())

n \mapsto \left\{\forall_{U \in \textrm{U}\left(n\right)}~\left[\forall_{\alpha \in \mathbb{R}^{\ge 0}}~\left[\forall_{\lvert \psi \rangle \in \mathbb{C}^{n}~|~\left \|\lvert \psi \rangle\right \| = \alpha}~\left(\left \|U \thinspace \lvert \psi \rangle\right \| = \alpha\right)\right]\right] \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 53 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 22 operand: 6
3	Operation	operator: 37 operands: 5
4	ExprTuple	6
5	ExprTuple	53, 7
6	Lambda	parameter: 51 body: 9
7	Literal
8	ExprTuple	51
9	Conditional	value: 10 condition: 11
10	Operation	operator: 22 operand: 14
11	Operation	operator: 37 operands: 13
12	ExprTuple	14
13	ExprTuple	51, 15
14	Lambda	parameter: 44 body: 17
15	Operation	operator: 18 operand: 53
16	ExprTuple	44
17	Conditional	value: 20 condition: 21
18	Literal
19	ExprTuple	53
20	Operation	operator: 22 operand: 25
21	Operation	operator: 37 operands: 24
22	Literal
23	ExprTuple	25
24	ExprTuple	44, 26
25	Lambda	parameter: 54 body: 27
26	Literal
27	Conditional	value: 28 condition: 29
28	Operation	operator: 39 operands: 30
29	Operation	operator: 31 operands: 32
30	ExprTuple	33, 44
31	Literal
32	ExprTuple	34, 35
33	Operation	operator: 49 operand: 41
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	ExprTuple	41
37	Literal
38	ExprTuple	54, 42
39	Literal
40	ExprTuple	43, 44
41	Operation	operator: 45 operands: 46
42	Operation	operator: 47 operands: 48
43	Operation	operator: 49 operand: 54
44	Variable
45	Literal
46	ExprTuple	51, 54
47	Literal
48	ExprTuple	52, 53
49	Literal
50	ExprTuple	54
51	Variable
52	Literal
53	Variable
54	Variable