Expression of type Implies

from the theory of proveit.logic.booleans.disjunction

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 A, ExprRange, IndexedVar, Variable, m, n
from proveit.core_expr_types import A_1_to_m
from proveit.logic import And, Boolean, Forall, Implies, InSet, Or
from proveit.numbers import Add, NaturalPos, one, two

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(A, one)
sub_expr3 = IndexedVar(A, sub_expr1)
sub_expr4 = Add(m, one)
sub_expr5 = ExprRange(sub_expr1, sub_expr3, one, n)
expr = Implies(And(Forall(instance_param_or_params = [sub_expr2], instance_expr = InSet(sub_expr2, Boolean), domain = Boolean), Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [ExprRange(sub_expr1, sub_expr3, one, Add(m, two))], instance_expr = InSet(Or(ExprRange(sub_expr1, sub_expr3, one, sub_expr4)), Boolean), condition = ExprRange(sub_expr1, InSet(sub_expr3, Boolean), one, sub_expr4)), domain = NaturalPos, condition = Forall(instance_param_or_params = [A_1_to_m], instance_expr = InSet(Or(A_1_to_m), Boolean), domain = Boolean))), Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [sub_expr5], instance_expr = InSet(Or(sub_expr5), Boolean), domain = Boolean), domain = NaturalPos))

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(\left[\forall_{A_{1} \in \mathbb{B}}~\left(A_{1} \in \mathbb{B}\right)\right] \land \left[\forall_{m \in \mathbb{N}^+~|~\forall_{A_{1}, A_{2}, \ldots, A_{m} \in \mathbb{B}}~\left(\left(A_{1} \lor  A_{2} \lor  \ldots \lor  A_{m}\right) \in \mathbb{B}\right)}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{m + 2}~|~\left(A_{1} \in \mathbb{B}\right), \left(A_{2} \in \mathbb{B}\right), \ldots, \left(A_{m + 1} \in \mathbb{B}\right)}~\left(\left(A_{1} \lor  A_{2} \lor  \ldots \lor  A_{m + 1}\right) \in \mathbb{B}\right)\right]\right]\right) \Rightarrow \left[\forall_{n \in \mathbb{N}^+}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{n} \in \mathbb{B}}~\left(\left(A_{1} \lor  A_{2} \lor  \ldots \lor  A_{n}\right) \in \mathbb{B}\right)\right]\right]

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.	()	()	('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: 67 operands: 5
4	Operation	operator: 42 operand: 9
5	ExprTuple	7, 8
6	ExprTuple	9
7	Operation	operator: 42 operand: 14
8	Operation	operator: 42 operand: 15
9	Lambda	parameter: 69 body: 13
10	ExprTuple	14
11	ExprTuple	15
12	ExprTuple	69
13	Conditional	value: 16 condition: 17
14	Lambda	parameter: 32 body: 19
15	Lambda	parameter: 83 body: 21
16	Operation	operator: 42 operand: 27
17	Operation	operator: 84 operands: 23
18	ExprTuple	32
19	Conditional	value: 24 condition: 24
20	ExprTuple	83
21	Conditional	value: 25 condition: 26
22	ExprTuple	27
23	ExprTuple	69, 49
24	Operation	operator: 84 operands: 28
25	Operation	operator: 42 operand: 33
26	Operation	operator: 67 operands: 30
27	Lambda	parameters: 57 body: 31
28	ExprTuple	32, 87
29	ExprTuple	33
30	ExprTuple	34, 35
31	Conditional	value: 36 condition: 37
32	IndexedVar	variable: 88 index: 82
33	Lambda	parameters: 39 body: 40
34	Operation	operator: 84 operands: 41
35	Operation	operator: 42 operand: 50
36	Operation	operator: 84 operands: 44
37	Operation	operator: 67 operands: 45
38	ExprTuple	82
39	ExprTuple	46
40	Conditional	value: 47 condition: 48
41	ExprTuple	83, 49
42	Literal
43	ExprTuple	50
44	ExprTuple	51, 87
45	ExprTuple	52
46	ExprRange	lambda_map: 81 start_index: 82 end_index: 53
47	Operation	operator: 84 operands: 54
48	Operation	operator: 67 operands: 55
49	Literal
50	Lambda	parameters: 75 body: 56
51	Operation	operator: 74 operands: 57
52	ExprRange	lambda_map: 76 start_index: 82 end_index: 69
53	Operation	operator: 77 operands: 58
54	ExprTuple	59, 87
55	ExprTuple	60
56	Conditional	value: 61 condition: 62
57	ExprTuple	63
58	ExprTuple	83, 64
59	Operation	operator: 74 operands: 65
60	ExprRange	lambda_map: 76 start_index: 82 end_index: 73
61	Operation	operator: 84 operands: 66
62	Operation	operator: 67 operands: 68
63	ExprRange	lambda_map: 81 start_index: 82 end_index: 69
64	Literal
65	ExprTuple	70
66	ExprTuple	71, 87
67	Literal
68	ExprTuple	72
69	Variable
70	ExprRange	lambda_map: 81 start_index: 82 end_index: 73
71	Operation	operator: 74 operands: 75
72	ExprRange	lambda_map: 76 start_index: 82 end_index: 83
73	Operation	operator: 77 operands: 78
74	Literal
75	ExprTuple	79
76	Lambda	parameter: 90 body: 80
77	Literal
78	ExprTuple	83, 82
79	ExprRange	lambda_map: 81 start_index: 82 end_index: 83
80	Operation	operator: 84 operands: 85
81	Lambda	parameter: 90 body: 86
82	Literal
83	Variable
84	Literal
85	ExprTuple	86, 87
86	IndexedVar	variable: 88 index: 90
87	Literal
88	Variable
89	ExprTuple	90
90	Variable