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 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 = [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)

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_{\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]

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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 32 body: 4
3	ExprTuple	32
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 29 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	32, 11
10	Lambda	parameter: 40 body: 12
11	Variable
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 29 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	20, 33
18	Operation	operator: 34 operands: 21
19	Lambda	parameter: 37 body: 23
20	Operation	operator: 24 operand: 40
21	ExprTuple	32, 36
22	ExprTuple	37
23	Conditional	value: 25 condition: 26
24	Literal
25	Operation	operator: 27 operands: 28
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	31, 32
29	Literal
30	ExprTuple	37, 33
31	Operation	operator: 34 operands: 35
32	Variable
33	Variable
34	Literal
35	ExprTuple	36, 37
36	Operation	operator: 38 operand: 40
37	Variable
38	Literal
39	ExprTuple	40
40	Variable