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, RealNonNeg
from proveit.physics.quantum import Qmult, var_ket_psi

# build up the expression from sub-expressions
expr = Lambda(U, Conditional(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), InSet(U, Unitary(n))))

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())

U \mapsto \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] \textrm{ if } U \in \textrm{U}\left(n\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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