Expression of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶

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, IndexedVar, K, V, Variable, b, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import TensorProd, VecAdd, VecSpaces
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = TensorProd(a_1_to_i, VecAdd(b_1_to_j), c_1_to_k)
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Implies(InSet(sub_expr2, V), Equals(VecAdd(ExprRange(sub_expr1, TensorProd(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j)), sub_expr2).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = Natural))

# 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_{K}~\left[\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{1}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{2}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \ldots +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{j}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right) =  \\ \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\end{array}\right]\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)

	core type	sub-expressions
0	Operation	operator: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 37 body: 3
3	Operation	operator: 25 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 6 body: 7
6	ExprTuple	65, 59, 69
7	Conditional	value: 8 condition: 9
8	Operation	operator: 25 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	14, 15, 16
13	Lambda	parameter: 44 body: 18
14	Operation	operator: 40 operands: 19
15	Operation	operator: 40 operands: 20
16	Operation	operator: 40 operands: 21
17	ExprTuple	44
18	Conditional	value: 22 condition: 23
19	ExprTuple	65, 24
20	ExprTuple	59, 24
21	ExprTuple	69, 24
22	Operation	operator: 25 operand: 29
23	Operation	operator: 27 operands: 28
24	Literal
25	Literal
26	ExprTuple	29
27	Literal
28	ExprTuple	44, 30
29	Lambda	parameters: 31 body: 32
30	Operation	operator: 33 operand: 37
31	ExprTuple	60, 55, 62
32	Operation	operator: 35 operands: 36
33	Literal
34	ExprTuple	37
35	Literal
36	ExprTuple	38, 39
37	Variable
38	Operation	operator: 40 operands: 41
39	Operation	operator: 42 operands: 43
40	Literal
41	ExprTuple	46, 44
42	Literal
43	ExprTuple	45, 46
44	Variable
45	Operation	operator: 52 operands: 47
46	Operation	operator: 56 operands: 48
47	ExprTuple	49
48	ExprTuple	60, 50, 62
49	ExprRange	lambda_map: 51 start_index: 68 end_index: 59
50	Operation	operator: 52 operands: 53
51	Lambda	parameter: 72 body: 54
52	Literal
53	ExprTuple	55
54	Operation	operator: 56 operands: 57
55	ExprRange	lambda_map: 58 start_index: 68 end_index: 59
56	Literal
57	ExprTuple	60, 61, 62
58	Lambda	parameter: 77 body: 63
59	Variable
60	ExprRange	lambda_map: 64 start_index: 68 end_index: 65
61	IndexedVar	variable: 70 index: 72
62	ExprRange	lambda_map: 67 start_index: 68 end_index: 69
63	IndexedVar	variable: 70 index: 77
64	Lambda	parameter: 77 body: 71
65	Variable
66	ExprTuple	72
67	Lambda	parameter: 77 body: 73
68	Literal
69	Variable
70	Variable
71	IndexedVar	variable: 74 index: 77
72	Variable
73	IndexedVar	variable: 75 index: 77
74	Variable
75	Variable
76	ExprTuple	77
77	Variable

Expression of type Forall¶

from the theory of proveit.linear_algebra.tensors¶

Expression of type Forall ¶

from the theory of proveit.linear_algebra.tensors ¶