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 Function, K, V, i, j, k, s
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 ScalarMult, TensorProd, VecSpaces, VecSum
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 = Function(s, sub_expr1)
sub_expr3 = ScalarMult(sub_expr2, f__b_1_to_j)
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(TensorProd(a_1_to_i, sub_expr3, c_1_to_k), V), condition = Q__b_1_to_j), Equals(TensorProd(a_1_to_i, VecSum(index_or_indices = sub_expr1, summand = sub_expr3, condition = Q__b_1_to_j), c_1_to_k), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, TensorProd(a_1_to_i, f__b_1_to_j, c_1_to_k)), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), 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[\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}, 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} \left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\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(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)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot \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)\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: 37 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	76, 87, 80
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: 56 body: 15
11	Operation	operator: 50 operands: 16
12	Operation	operator: 50 operands: 17
13	Operation	operator: 50 operands: 18
14	ExprTuple	56
15	Conditional	value: 19 condition: 20
16	ExprTuple	76, 22
17	ExprTuple	87, 21
18	ExprTuple	80, 22
19	Operation	operator: 37 operand: 26
20	Operation	operator: 24 operands: 25
21	Literal
22	Literal
23	ExprTuple	26
24	Literal
25	ExprTuple	56, 27
26	Lambda	parameters: 28 body: 29
27	Operation	operator: 30 operand: 34
28	ExprTuple	71, 73
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: 78 body: 44
42	Operation	operator: 68 operands: 45
43	Operation	operator: 52 operand: 49
44	Conditional	value: 47 condition: 63
45	ExprTuple	71, 48, 73
46	ExprTuple	49
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operand: 57
49	Lambda	parameters: 78 body: 54
50	Literal
51	ExprTuple	55, 56
52	Literal
53	ExprTuple	57
54	Conditional	value: 58 condition: 63
55	Operation	operator: 68 operands: 59
56	Variable
57	Lambda	parameters: 78 body: 60
58	Operation	operator: 65 operands: 61
59	ExprTuple	71, 62, 73
60	Conditional	value: 62 condition: 63
61	ExprTuple	70, 64
62	Operation	operator: 65 operands: 66
63	Operation	operator: 67 operands: 78
64	Operation	operator: 68 operands: 69
65	Literal
66	ExprTuple	70, 72
67	Variable
68	Literal
69	ExprTuple	71, 72, 73
70	Operation	operator: 74 operands: 78
71	ExprRange	lambda_map: 75 start_index: 86 end_index: 76
72	Operation	operator: 77 operands: 78
73	ExprRange	lambda_map: 79 start_index: 86 end_index: 80
74	Variable
75	Lambda	parameter: 92 body: 81
76	Variable
77	Variable
78	ExprTuple	82
79	Lambda	parameter: 92 body: 83
80	Variable
81	IndexedVar	variable: 84 index: 92
82	ExprRange	lambda_map: 85 start_index: 86 end_index: 87
83	IndexedVar	variable: 88 index: 92
84	Variable
85	Lambda	parameter: 92 body: 89
86	Literal
87	Variable
88	Variable
89	IndexedVar	variable: 90 index: 92
90	Variable
91	ExprTuple	92
92	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 ¶