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 U, i, j, n
from proveit.core_expr_types import v_i, v_j
from proveit.linear_algebra import Adj, Unitary
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Interval, KroneckerDelta, one
from proveit.physics.quantum import Bra, Ket, Qmult

# build up the expression from sub-expressions
expr = Implies(Forall(instance_param_or_params = [i, j], instance_expr = Equals(Qmult(Bra(v_i), Adj(U), U, Ket(v_j)), KroneckerDelta(i, j)), domain = Interval(one, n)), 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())

\left[\forall_{i, j \in \{1~\ldotp \ldotp~n\}}~\left(\left(\langle v_{i} \rvert \thinspace U^{\dagger} \thinspace U \thinspace \lvert v_{j} \rangle\right) = \delta_{i, j}\right)\right] \Rightarrow \left(U \in \textrm{U}\left(n\right)\right)

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.	()	()	('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 operand: 8
4	Operation	operator: 28 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	43, 9
8	Lambda	parameters: 26 body: 10
9	Operation	operator: 11 operand: 46
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	46
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 28 operands: 27
22	Operation	operator: 28 operands: 29
23	Literal
24	ExprTuple	30, 31, 43, 32
25	Literal
26	ExprTuple	50, 51
27	ExprTuple	50, 33
28	Literal
29	ExprTuple	51, 33
30	Operation	operator: 34 operand: 42
31	Operation	operator: 36 operand: 43
32	Operation	operator: 38 operand: 44
33	Operation	operator: 40 operands: 41
34	Literal
35	ExprTuple	42
36	Literal
37	ExprTuple	43
38	Literal
39	ExprTuple	44
40	Literal
41	ExprTuple	45, 46
42	IndexedVar	variable: 48 index: 50
43	Variable
44	IndexedVar	variable: 48 index: 51
45	Literal
46	Variable
47	ExprTuple	50
48	Variable
49	ExprTuple	51
50	Variable
51	Variable