Expression of type Implies ¶

from the theory of proveit.numbers.multiplication ¶

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 Function, Q, f
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Mult, Sum
from proveit.numbers.summation import summation_b1toj_fQ

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(f, sub_expr1)
sub_expr3 = Function(Q, sub_expr1)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Complex), condition = sub_expr3), Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, summation_b1toj_fQ, c_1_to_k), Sum(index_or_indices = sub_expr1, summand = Mult(a_1_to_i, sub_expr2, c_1_to_k), condition = sub_expr3)).with_wrapping_at(2), domain = Complex).with_wrapping()).with_wrapping_at(2)

# 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_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}\right)\end{array}\right] \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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')

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameters: 53 body: 10
9	Lambda	parameters: 11 body: 12
10	Conditional	value: 13 condition: 45
11	ExprTuple	46, 48
12	Conditional	value: 14 condition: 15
13	Operation	operator: 37 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	47, 41
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 43 operands: 25
22	Operation	operator: 33 operand: 30
23	ExprRange	lambda_map: 27 start_index: 61 end_index: 51
24	ExprRange	lambda_map: 28 start_index: 61 end_index: 55
25	ExprTuple	46, 29, 48
26	ExprTuple	30
27	Lambda	parameter: 67 body: 31
28	Lambda	parameter: 67 body: 32
29	Operation	operator: 33 operand: 39
30	Lambda	parameters: 53 body: 35
31	Operation	operator: 37 operands: 36
32	Operation	operator: 37 operands: 38
33	Literal
34	ExprTuple	39
35	Conditional	value: 40 condition: 45
36	ExprTuple	56, 41
37	Literal
38	ExprTuple	58, 41
39	Lambda	parameters: 53 body: 42
40	Operation	operator: 43 operands: 44
41	Literal
42	Conditional	value: 47 condition: 45
43	Literal
44	ExprTuple	46, 47, 48
45	Operation	operator: 49 operands: 53
46	ExprRange	lambda_map: 50 start_index: 61 end_index: 51
47	Operation	operator: 52 operands: 53
48	ExprRange	lambda_map: 54 start_index: 61 end_index: 55
49	Variable
50	Lambda	parameter: 67 body: 56
51	Variable
52	Variable
53	ExprTuple	57
54	Lambda	parameter: 67 body: 58
55	Variable
56	IndexedVar	variable: 59 index: 67
57	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
58	IndexedVar	variable: 63 index: 67
59	Variable
60	Lambda	parameter: 67 body: 64
61	Literal
62	Variable
63	Variable
64	IndexedVar	variable: 65 index: 67
65	Variable
66	ExprTuple	67
67	Variable

Expression of type Implies¶

from the theory of proveit.numbers.multiplication¶

Expression of type Implies ¶

from the theory of proveit.numbers.multiplication ¶