Expression of type Lambda

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, 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 = Lambda([j, m, n], 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(j, m, n\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	61, 50, 54
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11, 12
9	Lambda	parameters: 13 body: 14
10	Operation	operator: 17 operands: 15
11	Operation	operator: 17 operands: 16
12	Operation	operator: 17 operands: 18
13	ExprTuple	45, 47, 51, 48
14	Operation	operator: 19 operands: 20
15	ExprTuple	61, 21
16	ExprTuple	50, 22
17	Literal
18	ExprTuple	54, 22
19	Literal
20	ExprTuple	23, 24
21	Literal
22	Literal
23	Operation	operator: 25 operands: 26
24	Operation	operator: 27 operands: 28
25	Literal
26	ExprTuple	30, 29
27	Literal
28	ExprTuple	30, 31
29	Literal
30	Operation	operator: 42 operands: 32
31	Operation	operator: 36 operand: 35
32	ExprTuple	45, 34, 47
33	ExprTuple	35
34	Operation	operator: 36 operand: 39
35	Lambda	parameters: 52 body: 38
36	Literal
37	ExprTuple	39
38	Conditional	value: 40 condition: 44
39	Lambda	parameters: 52 body: 41
40	Operation	operator: 42 operands: 43
41	Conditional	value: 46 condition: 44
42	Literal
43	ExprTuple	45, 46, 47
44	Operation	operator: 48 operands: 52
45	ExprRange	lambda_map: 49 start_index: 60 end_index: 50
46	Operation	operator: 51 operands: 52
47	ExprRange	lambda_map: 53 start_index: 60 end_index: 54
48	Variable
49	Lambda	parameter: 66 body: 55
50	Variable
51	Variable
52	ExprTuple	56
53	Lambda	parameter: 66 body: 57
54	Variable
55	IndexedVar	variable: 58 index: 66
56	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
57	IndexedVar	variable: 62 index: 66
58	Variable
59	Lambda	parameter: 66 body: 63
60	Literal
61	Variable
62	Variable
63	IndexedVar	variable: 64 index: 66
64	Variable
65	ExprTuple	66
66	Variable