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 V, b
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals, Forall, Implies

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [b], instance_expr = Implies(Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, e_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, b, e_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V)

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_{b \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \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) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i} {\otimes} b{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 30 body: 5
4	ExprTuple	30
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	30, 14
12	Operation	operator: 16 operands: 15
13	Operation	operator: 16 operands: 17
14	Variable
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21
18	Operation	operator: 25 operands: 22
19	Operation	operator: 25 operands: 23
20	Operation	operator: 25 operands: 24
21	Operation	operator: 25 operands: 26
22	ExprTuple	27, 28
23	ExprTuple	29, 31
24	ExprTuple	27, 30, 28
25	Literal
26	ExprTuple	29, 30, 31
27	ExprRange	lambda_map: 32 start_index: 37 end_index: 35
28	ExprRange	lambda_map: 33 start_index: 37 end_index: 38
29	ExprRange	lambda_map: 34 start_index: 37 end_index: 35
30	Variable
31	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
32	Lambda	parameter: 48 body: 39
33	Lambda	parameter: 48 body: 40
34	Lambda	parameter: 48 body: 41
35	Variable
36	Lambda	parameter: 48 body: 42
37	Literal
38	Variable
39	IndexedVar	variable: 43 index: 48
40	IndexedVar	variable: 44 index: 48
41	IndexedVar	variable: 45 index: 48
42	IndexedVar	variable: 46 index: 48
43	Variable
44	Variable
45	Variable
46	Variable
47	ExprTuple	48
48	Variable