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
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

# 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 = [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(TensorProd(a_1_to_i, vec_summation_b1toj_fQ, c_1_to_k), VecSum(index_or_indices = sub_expr1, summand = sub_expr2, condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), domain = VecSpaces(K)).with_wrapping()

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())

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

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 39 body: 4
3	ExprTuple	39
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	39, 11
10	Lambda	parameters: 12 body: 13
11	Operation	operator: 14 operand: 18
12	ExprTuple	46, 48
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: 53 body: 28
26	Operation	operator: 43 operands: 29
27	Operation	operator: 36 operand: 33
28	Conditional	value: 31 condition: 45
29	ExprTuple	46, 32, 48
30	ExprTuple	33
31	Operation	operator: 34 operands: 35
32	Operation	operator: 36 operand: 40
33	Lambda	parameters: 53 body: 38
34	Literal
35	ExprTuple	41, 39
36	Literal
37	ExprTuple	40
38	Conditional	value: 41 condition: 45
39	Variable
40	Lambda	parameters: 53 body: 42
41	Operation	operator: 43 operands: 44
42	Conditional	value: 47 condition: 45
43	Literal
44	ExprTuple	46, 47, 48
45	Operation	operator: 49 operands: 53
46	ExprRange	lambda_map: 50 start_index: 61 end_index: 51
47	Operation	operator: 52 operands: 53
48	ExprRange	lambda_map: 54 start_index: 61 end_index: 55
49	Variable
50	Lambda	parameter: 67 body: 56
51	Variable
52	Variable
53	ExprTuple	57
54	Lambda	parameter: 67 body: 58
55	Variable
56	IndexedVar	variable: 59 index: 67
57	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
58	IndexedVar	variable: 63 index: 67
59	Variable
60	Lambda	parameter: 67 body: 64
61	Literal
62	Variable
63	Variable
64	IndexedVar	variable: 65 index: 67
65	Variable
66	ExprTuple	67
67	Variable