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 K, V, i, j, k
from proveit.core_expr_types import Q__b_1_to_j, a_1_to_i, b_1_to_j, c_1_to_k, f__b_1_to_j
from proveit.linear_algebra import TensorProd, VecSpaces, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Natural, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = TensorProd(a_1_to_i, f__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, c_1_to_k], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, V), condition = Q__b_1_to_j), Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, condition = Q__b_1_to_j), TensorProd(a_1_to_i, vec_summation_b1toj_fQ, c_1_to_k)).with_wrapping_at(1)).with_wrapping_at(2)), domain = VecSpaces(K)), domains = [Natural, NaturalPos, 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 \in \mathbb{N}, j \in \mathbb{N}^+, k \in \mathbb{N}}~\left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\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[\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]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\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: 37 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	66, 79, 68
4	Conditional	value: 5 condition: 6
5	Operation	operator: 37 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	11, 12, 13
10	Lambda	parameter: 55 body: 15
11	Operation	operator: 50 operands: 16
12	Operation	operator: 50 operands: 17
13	Operation	operator: 50 operands: 18
14	ExprTuple	55
15	Conditional	value: 19 condition: 20
16	ExprTuple	66, 22
17	ExprTuple	79, 21
18	ExprTuple	68, 22
19	Operation	operator: 37 operand: 26
20	Operation	operator: 24 operands: 25
21	Literal
22	Literal
23	ExprTuple	26
24	Literal
25	ExprTuple	55, 27
26	Lambda	parameters: 28 body: 29
27	Operation	operator: 30 operand: 34
28	ExprTuple	61, 62
29	Operation	operator: 32 operands: 33
30	Literal
31	ExprTuple	34
32	Literal
33	ExprTuple	35, 36
34	Variable
35	Operation	operator: 37 operand: 41
36	Operation	operator: 39 operands: 40
37	Literal
38	ExprTuple	41
39	Literal
40	ExprTuple	42, 43
41	Lambda	parameters: 71 body: 44
42	Operation	operator: 53 operand: 48
43	Operation	operator: 58 operands: 46
44	Conditional	value: 47 condition: 64
45	ExprTuple	48
46	ExprTuple	61, 49, 62
47	Operation	operator: 50 operands: 51
48	Lambda	parameters: 71 body: 52
49	Operation	operator: 53 operand: 57
50	Literal
51	ExprTuple	56, 55
52	Conditional	value: 56 condition: 64
53	Literal
54	ExprTuple	57
55	Variable
56	Operation	operator: 58 operands: 59
57	Lambda	parameters: 71 body: 60
58	Literal
59	ExprTuple	61, 63, 62
60	Conditional	value: 63 condition: 64
61	ExprRange	lambda_map: 65 start_index: 78 end_index: 66
62	ExprRange	lambda_map: 67 start_index: 78 end_index: 68
63	Operation	operator: 69 operands: 71
64	Operation	operator: 70 operands: 71
65	Lambda	parameter: 83 body: 72
66	Variable
67	Lambda	parameter: 83 body: 73
68	Variable
69	Variable
70	Variable
71	ExprTuple	74
72	IndexedVar	variable: 75 index: 83
73	IndexedVar	variable: 76 index: 83
74	ExprRange	lambda_map: 77 start_index: 78 end_index: 79
75	Variable
76	Variable
77	Lambda	parameter: 83 body: 80
78	Literal
79	Variable
80	IndexedVar	variable: 81 index: 83
81	Variable
82	ExprTuple	83
83	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 ¶