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, 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

# 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 = [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(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), TensorProd(a_1_to_i, VecSum(index_or_indices = sub_expr1, summand = sub_expr3, condition = Q__b_1_to_j), c_1_to_k)).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), domain = VecSpaces(K)).with_wrapping()

# 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())

\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[\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] \\  = \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) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\end{array}

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