Expression of type Lambda

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 Conditional, Lambda, i, j
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Complex, Mult, Natural, zero

# build up the expression from sub-expressions
expr = Lambda([i, j], Conditional(Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = Equals(Mult(a_1_to_i, zero, b_1_to_j), zero), domain = Complex), And(InSet(i, Natural), InSet(j, Natural))))

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())

\left(i, j\right) \mapsto \left\{\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot 0\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right) = 0\right) \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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