Expression of type Conditional

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, Q, f, j, m, n
from proveit.core_expr_types import A_1_to_m, C_1_to_n, Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import And, Equals, Forall, Implies, InClass, InSet
from proveit.numbers import Natural, NaturalPos
from proveit.physics.quantum import Qmult, QmultCodomain

# build up the expression from sub-expressions
sub_expr1 = Qmult(A_1_to_m, vec_summation_b1toj_fQ, C_1_to_n)
expr = Conditional(Forall(instance_param_or_params = [A_1_to_m, C_1_to_n, f, Q], instance_expr = Implies(InClass(sub_expr1, QmultCodomain), Equals(sub_expr1, VecSum(index_or_indices = [b_1_to_j], summand = Qmult(A_1_to_m, f__b_1_to_j, C_1_to_n), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2)), And(InSet(j, NaturalPos), InSet(m, Natural), InSet(n, Natural)))

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_{A_{1}, A_{2}, \ldots, A_{m}, C_{1}, C_{2}, \ldots, C_{n}, f, Q}~\left(\begin{array}{c} \begin{array}{l} \left(\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right) \textrm{ if } j \in \mathbb{N}^+ ,  m \in \mathbb{N} ,  n \in \mathbb{N}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9, 10
7	Lambda	parameters: 11 body: 12
8	Operation	operator: 15 operands: 13
9	Operation	operator: 15 operands: 14
10	Operation	operator: 15 operands: 16
11	ExprTuple	43, 45, 49, 46
12	Operation	operator: 17 operands: 18
13	ExprTuple	59, 19
14	ExprTuple	48, 20
15	Literal
16	ExprTuple	52, 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	Literal
21	Operation	operator: 23 operands: 24
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	28, 27
25	Literal
26	ExprTuple	28, 29
27	Literal
28	Operation	operator: 40 operands: 30
29	Operation	operator: 34 operand: 33
30	ExprTuple	43, 32, 45
31	ExprTuple	33
32	Operation	operator: 34 operand: 37
33	Lambda	parameters: 50 body: 36
34	Literal
35	ExprTuple	37
36	Conditional	value: 38 condition: 42
37	Lambda	parameters: 50 body: 39
38	Operation	operator: 40 operands: 41
39	Conditional	value: 44 condition: 42
40	Literal
41	ExprTuple	43, 44, 45
42	Operation	operator: 46 operands: 50
43	ExprRange	lambda_map: 47 start_index: 58 end_index: 48
44	Operation	operator: 49 operands: 50
45	ExprRange	lambda_map: 51 start_index: 58 end_index: 52
46	Variable
47	Lambda	parameter: 64 body: 53
48	Variable
49	Variable
50	ExprTuple	54
51	Lambda	parameter: 64 body: 55
52	Variable
53	IndexedVar	variable: 56 index: 64
54	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
55	IndexedVar	variable: 60 index: 64
56	Variable
57	Lambda	parameter: 64 body: 61
58	Literal
59	Variable
60	Variable
61	IndexedVar	variable: 62 index: 64
62	Variable
63	ExprTuple	64
64	Variable