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 ExprRange, IndexedVar, V, Variable, b, j
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import TensorProd, VecAdd
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = TensorProd(a_1_to_i, VecAdd(b_1_to_j), c_1_to_k)
expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Implies(InSet(sub_expr2, V), Equals(VecAdd(ExprRange(sub_expr1, TensorProd(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j)), sub_expr2).with_wrapping_at(2)).with_wrapping_at(2)).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_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{1}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{2}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \ldots +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{j}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right) =  \\ \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\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}

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	30, 25, 32
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	16, 14
12	Literal
13	ExprTuple	15, 16
14	Variable
15	Operation	operator: 22 operands: 17
16	Operation	operator: 26 operands: 18
17	ExprTuple	19
18	ExprTuple	30, 20, 32
19	ExprRange	lambda_map: 21 start_index: 38 end_index: 29
20	Operation	operator: 22 operands: 23
21	Lambda	parameter: 42 body: 24
22	Literal
23	ExprTuple	25
24	Operation	operator: 26 operands: 27
25	ExprRange	lambda_map: 28 start_index: 38 end_index: 29
26	Literal
27	ExprTuple	30, 31, 32
28	Lambda	parameter: 47 body: 33
29	Variable
30	ExprRange	lambda_map: 34 start_index: 38 end_index: 35
31	IndexedVar	variable: 40 index: 42
32	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
33	IndexedVar	variable: 40 index: 47
34	Lambda	parameter: 47 body: 41
35	Variable
36	ExprTuple	42
37	Lambda	parameter: 47 body: 43
38	Literal
39	Variable
40	Variable
41	IndexedVar	variable: 44 index: 47
42	Variable
43	IndexedVar	variable: 45 index: 47
44	Variable
45	Variable
46	ExprTuple	47
47	Variable