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 B, ExprRange, IndexedVar, Variable, i, m, n
from proveit.core_expr_types import A_1_to_m, B_1_to_i, C_1_to_n
from proveit.linear_algebra import VecAdd
from proveit.logic import Equals, Forall, Implies, InClass
from proveit.numbers import Natural, NaturalPos, one
from proveit.physics.quantum import Qmult, QmultCodomain

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Qmult(A_1_to_m, VecAdd(B_1_to_i), C_1_to_n)
expr = Forall(instance_param_or_params = [i, m, n], instance_expr = Forall(instance_param_or_params = [A_1_to_m, B_1_to_i, C_1_to_n], instance_expr = Implies(InClass(sub_expr2, QmultCodomain), Equals(sub_expr2, VecAdd(ExprRange(sub_expr1, Qmult(A_1_to_m, IndexedVar(B, sub_expr1), C_1_to_n), one, i))).with_wrapping_at(2)).with_wrapping_at(2)), domains = [NaturalPos, Natural, 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())

\forall_{i \in \mathbb{N}^+, m \in \mathbb{N}, n \in \mathbb{N}}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{m}, B_{1}, B_{2}, \ldots, B_{i}, C_{1}, C_{2}, \ldots, C_{n}}~\left(\begin{array}{c} \begin{array}{l} \left(\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace \left(B_{1} +  B_{2} +  \ldots +  B_{i}\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(B_{1} +  B_{2} +  \ldots +  B_{i}\right)\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) =  \\ \left(\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace B_{1}\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) +  \left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace B_{2}\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) +  \ldots +  \left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace B_{i}\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right)\right) \end{array} \end{array}\right) \end{array} \end{array}\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	parameters: 3 body: 4
3	ExprTuple	44, 52, 57
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13, 14
11	Lambda	parameters: 15 body: 16
12	Operation	operator: 19 operands: 17
13	Operation	operator: 19 operands: 18
14	Operation	operator: 19 operands: 20
15	ExprTuple	48, 41, 50
16	Operation	operator: 21 operands: 22
17	ExprTuple	44, 23
18	ExprTuple	52, 24
19	Literal
20	ExprTuple	57, 24
21	Literal
22	ExprTuple	25, 26
23	Literal
24	Literal
25	Operation	operator: 27 operands: 28
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	32, 31
29	Literal
30	ExprTuple	32, 33
31	Literal
32	Operation	operator: 45 operands: 34
33	Operation	operator: 38 operands: 35
34	ExprTuple	48, 36, 50
35	ExprTuple	37
36	Operation	operator: 38 operands: 39
37	ExprRange	lambda_map: 40 start_index: 56 end_index: 44
38	Literal
39	ExprTuple	41
40	Lambda	parameter: 59 body: 42
41	ExprRange	lambda_map: 43 start_index: 56 end_index: 44
42	Operation	operator: 45 operands: 46
43	Lambda	parameter: 64 body: 47
44	Variable
45	Literal
46	ExprTuple	48, 49, 50
47	IndexedVar	variable: 53 index: 64
48	ExprRange	lambda_map: 51 start_index: 56 end_index: 52
49	IndexedVar	variable: 53 index: 59
50	ExprRange	lambda_map: 55 start_index: 56 end_index: 57
51	Lambda	parameter: 64 body: 58
52	Variable
53	Variable
54	ExprTuple	59
55	Lambda	parameter: 64 body: 60
56	Literal
57	Variable
58	IndexedVar	variable: 61 index: 64
59	Variable
60	IndexedVar	variable: 62 index: 64
61	Variable
62	Variable
63	ExprTuple	64
64	Variable