Expression of type Lambda

from the theory of proveit.numbers.summation

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, Function, Lambda, R, S, a, f, fa, fx, g, gy, n, x, y
from proveit.logic import Card, Equals, Forall, Implies, InSet, SetOfAll
from proveit.numbers import Complex, Natural, Sum

# build up the expression from sub-expressions
sub_expr1 = [y]
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [f, g, R, S], instance_expr = Implies(Forall(instance_param_or_params = [a], instance_expr = InSet(fa, Complex), domain = R), Equals(Sum(index_or_indices = [x], summand = fx, domain = R), Sum(index_or_indices = sub_expr1, summand = Function(f, [gy]), domain = S))), conditions = [Equals(SetOfAll(instance_param_or_params = sub_expr1, instance_element = gy, domain = S), R), Equals(Card(S), n), Equals(Card(R), n)]), InSet(n, 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())

n \mapsto \left\{\forall_{f, g, R, S~|~\left\{g\left(y\right)\right\}_{y \in S} = R, |S| = n, |R| = n}~\left(\left[\forall_{a \in R}~\left(f\left(a\right) \in \mathbb{C}\right)\right] \Rightarrow \left(\left[\sum_{x \in R}~f\left(x\right)\right] = \left[\sum_{y \in S}~f\left(g\left(y\right)\right)\right]\right)\right) \textrm{ if } n \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	parameter: 35 body: 2
1	ExprTuple	35
2	Conditional	value: 3 condition: 4
3	Operation	operator: 22 operand: 7
4	Operation	operator: 66 operands: 6
5	ExprTuple	7
6	ExprTuple	35, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	64, 73, 70, 72
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19, 20, 21
17	Operation	operator: 22 operand: 29
18	Operation	operator: 27 operands: 24
19	Operation	operator: 27 operands: 25
20	Operation	operator: 27 operands: 26
21	Operation	operator: 27 operands: 28
22	Literal
23	ExprTuple	29
24	ExprTuple	30, 31
25	ExprTuple	32, 70
26	ExprTuple	33, 35
27	Literal
28	ExprTuple	34, 35
29	Lambda	parameter: 68 body: 36
30	Operation	operator: 38 operand: 47
31	Operation	operator: 38 operand: 48
32	Operation	operator: 40 operand: 49
33	Operation	operator: 43 operand: 72
34	Operation	operator: 43 operand: 70
35	Variable
36	Conditional	value: 45 condition: 46
37	ExprTuple	47
38	Literal
39	ExprTuple	48
40	Literal
41	ExprTuple	49
42	ExprTuple	72
43	Literal
44	ExprTuple	70
45	Operation	operator: 66 operands: 50
46	Operation	operator: 66 operands: 51
47	Lambda	parameter: 69 body: 52
48	Lambda	parameter: 75 body: 53
49	Lambda	parameter: 75 body: 54
50	ExprTuple	55, 56
51	ExprTuple	68, 70
52	Conditional	value: 57 condition: 58
53	Conditional	value: 59 condition: 60
54	Conditional	value: 71 condition: 60
55	Operation	operator: 64 operand: 68
56	Literal
57	Operation	operator: 64 operand: 69
58	Operation	operator: 66 operands: 63
59	Operation	operator: 64 operand: 71
60	Operation	operator: 66 operands: 67
61	ExprTuple	68
62	ExprTuple	69
63	ExprTuple	69, 70
64	Variable
65	ExprTuple	71
66	Literal
67	ExprTuple	75, 72
68	Variable
69	Variable
70	Variable
71	Operation	operator: 73 operand: 75
72	Variable
73	Variable
74	ExprTuple	75
75	Variable