Expression of type Implies

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 Conditional, Lambda, beta, gamma, i, x, y
from proveit.linear_algebra import ScalarMult, TensorProd
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Add, Interval, four, one, two

# build up the expression from sub-expressions
sub_expr1 = Interval(two, four)
sub_expr2 = InSet(i, sub_expr1)
sub_expr3 = ScalarMult(beta, ScalarMult(Add(i, one), TensorProd(x, y)))
sub_expr4 = ScalarMult(ScalarMult(gamma, i), sub_expr3)
sub_expr5 = ScalarMult(gamma, ScalarMult(i, sub_expr3))
expr = Implies(Forall(instance_param_or_params = [i], instance_expr = Equals(sub_expr5, sub_expr4), domain = sub_expr1), Equals(Lambda(i, Conditional(sub_expr5, sub_expr2)), Lambda(i, Conditional(sub_expr4, sub_expr2))).with_wrapping_at(2)).with_wrapping_at(2)

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}{c} \begin{array}{l} \left[\forall_{i \in \{2~\ldotp \ldotp~4\}}~\left(\left(\gamma \cdot \left(i \cdot \left(\beta \cdot \left(\left(i + 1\right) \cdot \left(x {\otimes} y\right)\right)\right)\right)\right) = \left(\left(\gamma \cdot i\right) \cdot \left(\beta \cdot \left(\left(i + 1\right) \cdot \left(x {\otimes} y\right)\right)\right)\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[i \mapsto \left\{\gamma \cdot \left(i \cdot \left(\beta \cdot \left(\left(i + 1\right) \cdot \left(x {\otimes} y\right)\right)\right)\right) \textrm{ if } i \in \{2~\ldotp \ldotp~4\}\right..\right] =  \\ \left[i \mapsto \left\{\left(\gamma \cdot i\right) \cdot \left(\beta \cdot \left(\left(i + 1\right) \cdot \left(x {\otimes} y\right)\right)\right) \textrm{ if } i \in \{2~\ldotp \ldotp~4\}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 47 body: 11
9	Lambda	parameter: 47 body: 12
10	Lambda	parameter: 47 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	47
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	47, 23
21	Operation	operator: 39 operands: 24
22	Operation	operator: 39 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	35, 28
25	ExprTuple	29, 34
26	Literal
27	ExprTuple	30, 31
28	Operation	operator: 39 operands: 32
29	Operation	operator: 39 operands: 33
30	Literal
31	Literal
32	ExprTuple	47, 34
33	ExprTuple	35, 47
34	Operation	operator: 39 operands: 36
35	Variable
36	ExprTuple	37, 38
37	Variable
38	Operation	operator: 39 operands: 40
39	Literal
40	ExprTuple	41, 42
41	Operation	operator: 43 operands: 44
42	Operation	operator: 45 operands: 46
43	Literal
44	ExprTuple	47, 48
45	Literal
46	ExprTuple	49, 50
47	Variable
48	Literal
49	Variable
50	Variable