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, alpha, b, i, k
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k
from proveit.linear_algebra import ScalarMult, TensorProd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V, W_1_to_k]
expr = Forall(instance_param_or_params = [alpha], instance_expr = Forall(instance_param_or_params = [i, k], instance_expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, ScalarMult(alpha, b), c_1_to_k), ScalarMult(alpha, TensorProd(a_1_to_i, b, c_1_to_k))).with_wrapping_at(1), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = Natural), domain = K)

# 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_{\alpha \in K}~\left[\forall_{i, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V, W_{1}, W_{2}, \ldots, W_{k} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{\left(a_{1} \in U_{1}\right), \left(a_{2} \in U_{2}\right), \ldots, \left(a_{i} \in U_{i}\right), b \in V,\left(c_{1} \in W_{1}\right), \left(c_{2} \in W_{2}\right), \ldots, \left(c_{k} \in W_{k}\right)}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(\alpha \cdot b\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left(\alpha \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\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: 30 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 78 body: 4
3	ExprTuple	78
4	Conditional	value: 5 condition: 6
5	Operation	operator: 30 operand: 9
6	Operation	operator: 76 operands: 8
7	ExprTuple	9
8	ExprTuple	78, 70
9	Lambda	parameters: 10 body: 11
10	ExprTuple	85, 88
11	Conditional	value: 12 condition: 13
12	Operation	operator: 30 operand: 16
13	Operation	operator: 47 operands: 15
14	ExprTuple	16
15	ExprTuple	17, 18
16	Lambda	parameters: 19 body: 20
17	Operation	operator: 76 operands: 21
18	Operation	operator: 76 operands: 22
19	ExprTuple	23, 68, 24
20	Conditional	value: 25 condition: 26
21	ExprTuple	85, 27
22	ExprTuple	88, 27
23	ExprRange	lambda_map: 28 start_index: 87 end_index: 85
24	ExprRange	lambda_map: 29 start_index: 87 end_index: 88
25	Operation	operator: 30 operand: 33
26	Operation	operator: 47 operands: 32
27	Literal
28	Lambda	parameter: 96 body: 82
29	Lambda	parameter: 96 body: 83
30	Literal
31	ExprTuple	33
32	ExprTuple	34, 35, 36
33	Lambda	parameters: 74 body: 37
34	ExprRange	lambda_map: 38 start_index: 87 end_index: 85
35	Operation	operator: 50 operands: 39
36	ExprRange	lambda_map: 40 start_index: 87 end_index: 88
37	Conditional	value: 41 condition: 42
38	Lambda	parameter: 96 body: 43
39	ExprTuple	68, 57
40	Lambda	parameter: 96 body: 44
41	Operation	operator: 45 operands: 46
42	Operation	operator: 47 operands: 48
43	Operation	operator: 50 operands: 49
44	Operation	operator: 50 operands: 51
45	Literal
46	ExprTuple	52, 53
47	Literal
48	ExprTuple	54, 55, 56
49	ExprTuple	82, 57
50	Literal
51	ExprTuple	83, 57
52	Operation	operator: 73 operands: 58
53	Operation	operator: 71 operands: 59
54	ExprRange	lambda_map: 60 start_index: 87 end_index: 85
55	Operation	operator: 76 operands: 61
56	ExprRange	lambda_map: 62 start_index: 87 end_index: 88
57	Operation	operator: 63 operand: 70
58	ExprTuple	79, 65, 81
59	ExprTuple	78, 66
60	Lambda	parameter: 96 body: 67
61	ExprTuple	80, 68
62	Lambda	parameter: 96 body: 69
63	Literal
64	ExprTuple	70
65	Operation	operator: 71 operands: 72
66	Operation	operator: 73 operands: 74
67	Operation	operator: 76 operands: 75
68	Variable
69	Operation	operator: 76 operands: 77
70	Variable
71	Literal
72	ExprTuple	78, 80
73	Literal
74	ExprTuple	79, 80, 81
75	ExprTuple	91, 82
76	Literal
77	ExprTuple	92, 83
78	Variable
79	ExprRange	lambda_map: 84 start_index: 87 end_index: 85
80	Variable
81	ExprRange	lambda_map: 86 start_index: 87 end_index: 88
82	IndexedVar	variable: 89 index: 96
83	IndexedVar	variable: 90 index: 96
84	Lambda	parameter: 96 body: 91
85	Variable
86	Lambda	parameter: 96 body: 92
87	Literal
88	Variable
89	Variable
90	Variable
91	IndexedVar	variable: 93 index: 96
92	IndexedVar	variable: 94 index: 96
93	Variable
94	Variable
95	ExprTuple	96
96	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 ¶