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, alpha, b
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
from proveit.logic import Equals, Forall

# build up the expression from sub-expressions
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 = [U_1_to_i, V, W_1_to_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_{\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}

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: 29 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
11	Operation	operator: 28 operands: 16
12	Operation	operator: 26 operands: 17
13	ExprRange	lambda_map: 18 start_index: 42 end_index: 40
14	Operation	operator: 31 operands: 19
15	ExprRange	lambda_map: 20 start_index: 42 end_index: 43
16	ExprTuple	34, 21, 36
17	ExprTuple	33, 22
18	Lambda	parameter: 51 body: 23
19	ExprTuple	35, 24
20	Lambda	parameter: 51 body: 25
21	Operation	operator: 26 operands: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 31 operands: 30
24	Variable
25	Operation	operator: 31 operands: 32
26	Literal
27	ExprTuple	33, 35
28	Literal
29	ExprTuple	34, 35, 36
30	ExprTuple	46, 37
31	Literal
32	ExprTuple	47, 38
33	Variable
34	ExprRange	lambda_map: 39 start_index: 42 end_index: 40
35	Variable
36	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
37	IndexedVar	variable: 44 index: 51
38	IndexedVar	variable: 45 index: 51
39	Lambda	parameter: 51 body: 46
40	Variable
41	Lambda	parameter: 51 body: 47
42	Literal
43	Variable
44	Variable
45	Variable
46	IndexedVar	variable: 48 index: 51
47	IndexedVar	variable: 49 index: 51
48	Variable
49	Variable
50	ExprTuple	51
51	Variable