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 = [K], instance_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))

expr:

# 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_{K}~\left[\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]\right]

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 33 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 73 body: 3
3	Operation	operator: 33 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 81 body: 7
6	ExprTuple	81
7	Conditional	value: 8 condition: 9
8	Operation	operator: 33 operand: 12
9	Operation	operator: 79 operands: 11
10	ExprTuple	12
11	ExprTuple	81, 73
12	Lambda	parameters: 13 body: 14
13	ExprTuple	88, 91
14	Conditional	value: 15 condition: 16
15	Operation	operator: 33 operand: 19
16	Operation	operator: 50 operands: 18
17	ExprTuple	19
18	ExprTuple	20, 21
19	Lambda	parameters: 22 body: 23
20	Operation	operator: 79 operands: 24
21	Operation	operator: 79 operands: 25
22	ExprTuple	26, 71, 27
23	Conditional	value: 28 condition: 29
24	ExprTuple	88, 30
25	ExprTuple	91, 30
26	ExprRange	lambda_map: 31 start_index: 90 end_index: 88
27	ExprRange	lambda_map: 32 start_index: 90 end_index: 91
28	Operation	operator: 33 operand: 36
29	Operation	operator: 50 operands: 35
30	Literal
31	Lambda	parameter: 99 body: 85
32	Lambda	parameter: 99 body: 86
33	Literal
34	ExprTuple	36
35	ExprTuple	37, 38, 39
36	Lambda	parameters: 77 body: 40
37	ExprRange	lambda_map: 41 start_index: 90 end_index: 88
38	Operation	operator: 53 operands: 42
39	ExprRange	lambda_map: 43 start_index: 90 end_index: 91
40	Conditional	value: 44 condition: 45
41	Lambda	parameter: 99 body: 46
42	ExprTuple	71, 60
43	Lambda	parameter: 99 body: 47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operands: 51
46	Operation	operator: 53 operands: 52
47	Operation	operator: 53 operands: 54
48	Literal
49	ExprTuple	55, 56
50	Literal
51	ExprTuple	57, 58, 59
52	ExprTuple	85, 60
53	Literal
54	ExprTuple	86, 60
55	Operation	operator: 76 operands: 61
56	Operation	operator: 74 operands: 62
57	ExprRange	lambda_map: 63 start_index: 90 end_index: 88
58	Operation	operator: 79 operands: 64
59	ExprRange	lambda_map: 65 start_index: 90 end_index: 91
60	Operation	operator: 66 operand: 73
61	ExprTuple	82, 68, 84
62	ExprTuple	81, 69
63	Lambda	parameter: 99 body: 70
64	ExprTuple	83, 71
65	Lambda	parameter: 99 body: 72
66	Literal
67	ExprTuple	73
68	Operation	operator: 74 operands: 75
69	Operation	operator: 76 operands: 77
70	Operation	operator: 79 operands: 78
71	Variable
72	Operation	operator: 79 operands: 80
73	Variable
74	Literal
75	ExprTuple	81, 83
76	Literal
77	ExprTuple	82, 83, 84
78	ExprTuple	94, 85
79	Literal
80	ExprTuple	95, 86
81	Variable
82	ExprRange	lambda_map: 87 start_index: 90 end_index: 88
83	Variable
84	ExprRange	lambda_map: 89 start_index: 90 end_index: 91
85	IndexedVar	variable: 92 index: 99
86	IndexedVar	variable: 93 index: 99
87	Lambda	parameter: 99 body: 94
88	Variable
89	Lambda	parameter: 99 body: 95
90	Literal
91	Variable
92	Variable
93	Variable
94	IndexedVar	variable: 96 index: 99
95	IndexedVar	variable: 97 index: 99
96	Variable
97	Variable
98	ExprTuple	99
99	Variable