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 = [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_{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]

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