Expression of type Forall

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, X
from proveit.linear_algebra import HilbertSpaces, Hspace, ScalarMult
from proveit.logic import Equals, Forall, InSet
from proveit.physics.quantum import Qmult, bra_varphi, ket_varphi, var_ket_psi, var_ket_v, varphi

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [Hspace, X], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [varphi], instance_expr = Equals(Qmult(var_ket_psi, bra_varphi), Lambda(var_ket_v, Conditional(ScalarMult(Qmult(bra_varphi, var_ket_v), var_ket_psi), InSet(var_ket_v, Hspace)))), condition = InSet(ket_varphi, Hspace)), domain = X), domain = HilbertSpaces)

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_{\mathcal{H}, X \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}}~\left[\forall_{\lvert \psi \rangle \in X}~\left[\forall_{\varphi~|~\lvert \varphi \rangle \in \mathcal{H}}~\left(\left(\lvert \psi \rangle \thinspace \langle \varphi \rvert\right) = \left[\lvert v \rangle \mapsto \left\{\left(\langle \varphi \rvert \thinspace \lvert v \rangle\right) \cdot \lvert \psi \rangle \textrm{ if } \lvert v \rangle \in \mathcal{H}\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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	47, 25
4	Conditional	value: 5 condition: 6
5	Operation	operator: 21 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	11, 12
10	Lambda	parameter: 46 body: 14
11	Operation	operator: 16 operands: 15
12	Operation	operator: 16 operands: 17
13	ExprTuple	46
14	Conditional	value: 18 condition: 19
15	ExprTuple	47, 20
16	Literal
17	ExprTuple	25, 20
18	Operation	operator: 21 operand: 24
19	Operation	operator: 43 operands: 23
20	Literal
21	Literal
22	ExprTuple	24
23	ExprTuple	46, 25
24	Lambda	parameter: 54 body: 26
25	Variable
26	Conditional	value: 27 condition: 28
27	Operation	operator: 29 operands: 30
28	Operation	operator: 43 operands: 31
29	Literal
30	ExprTuple	32, 33
31	ExprTuple	34, 47
32	Operation	operator: 48 operands: 35
33	Lambda	parameter: 51 body: 37
34	Operation	operator: 38 operand: 54
35	ExprTuple	46, 50
36	ExprTuple	51
37	Conditional	value: 39 condition: 40
38	Literal
39	Operation	operator: 41 operands: 42
40	Operation	operator: 43 operands: 44
41	Literal
42	ExprTuple	45, 46
43	Literal
44	ExprTuple	51, 47
45	Operation	operator: 48 operands: 49
46	Variable
47	Variable
48	Literal
49	ExprTuple	50, 51
50	Operation	operator: 52 operand: 54
51	Variable
52	Literal
53	ExprTuple	54
54	Variable