Expression of type Implies

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 ExprRange, Function, IndexedVar, Q, Variable, alpha, beta, c, delta, f, gamma, j
from proveit.linear_algebra import VecSum
from proveit.logic import Equals, Implies, InClass
from proveit.numbers import one
from proveit.physics.quantum import Qmult, QmultCodomain, bra_varphi, ket_psi

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, j)]
sub_expr3 = Function(f, sub_expr2)
sub_expr4 = Function(Q, sub_expr2)
sub_expr5 = Qmult(bra_varphi, alpha, VecSum(index_or_indices = sub_expr2, summand = sub_expr3, condition = sub_expr4), beta, ket_psi, gamma, bra_varphi, delta)
expr = Implies(InClass(sub_expr5, QmultCodomain), Equals(sub_expr5, VecSum(index_or_indices = sub_expr2, summand = Qmult(bra_varphi, alpha, sub_expr3, beta, ket_psi, gamma, bra_varphi, delta), condition = sub_expr4)).with_wrapping_at(1)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\left(\langle \varphi \rvert \thinspace \alpha \thinspace \left[\sum_{c_{1}, c_{2}, \ldots, c_{j}~|~Q\left(c_{1}, c_{2}, \ldots, c_{j}\right)}~f\left(c_{1}, c_{2}, \ldots, c_{j}\right)\right] \thinspace \beta \thinspace \lvert \psi \rangle \thinspace \gamma \thinspace \langle \varphi \rvert \thinspace \delta\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\langle \varphi \rvert \thinspace \alpha \thinspace \left[\sum_{c_{1}, c_{2}, \ldots, c_{j}~|~Q\left(c_{1}, c_{2}, \ldots, c_{j}\right)}~f\left(c_{1}, c_{2}, \ldots, c_{j}\right)\right] \thinspace \beta \thinspace \lvert \psi \rangle \thinspace \gamma \thinspace \langle \varphi \rvert \thinspace \delta\right) \\  = \left[\sum_{c_{1}, c_{2}, \ldots, c_{j}~|~Q\left(c_{1}, c_{2}, \ldots, c_{j}\right)}~\left(\langle \varphi \rvert \thinspace \alpha \thinspace f\left(c_{1}, c_{2}, \ldots, c_{j}\right) \thinspace \beta \thinspace \lvert \psi \rangle \thinspace \gamma \thinspace \langle \varphi \rvert \thinspace \delta\right)\right] \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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