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, e
from proveit.core_expr_types import a_1_to_i, c_1_to_k, d_1_to_i, f_1_to_k
from proveit.linear_algebra import TensorProd, VecZero
from proveit.logic import Equals, Forall, Implies, NotEquals

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [b, e], instance_expr = Implies(Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, e, f_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, f_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V, conditions = [Equals(b, e), NotEquals(b, VecZero(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, e \in V~|~b = e, b \neq \vec{0}\left(V\right)}~\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} 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} e{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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	parameters: 23 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13, 14, 15, 16
11	Operation	operator: 22 operands: 17
12	Operation	operator: 22 operands: 18
13	Operation	operator: 20 operands: 19
14	Operation	operator: 20 operands: 21
15	Operation	operator: 22 operands: 23
16	Operation	operator: 24 operands: 25
17	ExprTuple	26, 27
18	ExprTuple	28, 29
19	ExprTuple	38, 44
20	Literal
21	ExprTuple	39, 44
22	Literal
23	ExprTuple	38, 39
24	Literal
25	ExprTuple	38, 30
26	Operation	operator: 34 operands: 31
27	Operation	operator: 34 operands: 32
28	Operation	operator: 34 operands: 33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 36 operand: 44
31	ExprTuple	40, 38, 41
32	ExprTuple	42, 39, 43
33	ExprTuple	40, 41
34	Literal
35	ExprTuple	42, 43
36	Literal
37	ExprTuple	44
38	Variable
39	Variable
40	ExprRange	lambda_map: 45 start_index: 50 end_index: 48
41	ExprRange	lambda_map: 46 start_index: 50 end_index: 51
42	ExprRange	lambda_map: 47 start_index: 50 end_index: 48
43	ExprRange	lambda_map: 49 start_index: 50 end_index: 51
44	Variable
45	Lambda	parameter: 61 body: 52
46	Lambda	parameter: 61 body: 53
47	Lambda	parameter: 61 body: 54
48	Variable
49	Lambda	parameter: 61 body: 55
50	Literal
51	Variable
52	IndexedVar	variable: 56 index: 61
53	IndexedVar	variable: 57 index: 61
54	IndexedVar	variable: 58 index: 61
55	IndexedVar	variable: 59 index: 61
56	Variable
57	Variable
58	Variable
59	Variable
60	ExprTuple	61
61	Variable