Expression of type Implies ¶

from the theory of proveit.core_expr_types.tuples ¶

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 ExprRange, IndexedVar, Variable, a, b, m, n
from proveit.logic import And, Equals, Forall, Implies
from proveit.numbers import Add, NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = IndexedVar(a, one)
sub_expr3 = IndexedVar(b, one)
sub_expr4 = IndexedVar(a, sub_expr1)
sub_expr5 = Add(m, one)
sub_expr6 = IndexedVar(b, sub_expr1)
sub_expr7 = ExprRange(sub_expr1, sub_expr4, one, sub_expr5)
sub_expr8 = ExprRange(sub_expr1, sub_expr6, one, sub_expr5)
sub_expr9 = Equals(sub_expr4, sub_expr6)
sub_expr10 = ExprRange(sub_expr1, sub_expr4, one, m)
sub_expr11 = ExprRange(sub_expr1, sub_expr6, one, m)
sub_expr12 = ExprRange(sub_expr1, sub_expr4, one, n)
sub_expr13 = ExprRange(sub_expr1, sub_expr6, one, n)
expr = Implies(And(Forall(instance_param_or_params = [sub_expr2, sub_expr3], instance_expr = Equals([sub_expr2], [sub_expr3]), condition = Equals(sub_expr2, sub_expr3)), Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [sub_expr7, sub_expr8], instance_expr = Equals([sub_expr7], [sub_expr8]), condition = ExprRange(sub_expr1, sub_expr9, one, sub_expr5)), domain = NaturalPos, condition = Forall(instance_param_or_params = [sub_expr10, sub_expr11], instance_expr = Equals([sub_expr10], [sub_expr11]), condition = ExprRange(sub_expr1, sub_expr9, one, m)))), Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [sub_expr12, sub_expr13], instance_expr = Equals([sub_expr12], [sub_expr13]), condition = ExprRange(sub_expr1, sub_expr9, one, n)), domain = NaturalPos))

# 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}, b_{1}~|~a_{1} = b_{1}}~\left(\left(a_{1}\right) = \left(b_{1}\right)\right)\right] \land \left[\forall_{m \in \mathbb{N}^+~|~\forall_{a_{1}, a_{2}, \ldots, a_{m}, b_{1}, b_{2}, \ldots, b_{m}~|~\left(a_{1} = b_{1}\right), \left(a_{2} = b_{2}\right), \ldots, \left(a_{m} = b_{m}\right)}~\left(\left(a_{1}, a_{2}, \ldots, a_{m}\right) = \left(b_{1}, b_{2}, \ldots, b_{m}\right)\right)}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m + 1}, b_{1}, b_{2}, \ldots, b_{m + 1}~|~\left(a_{1} = b_{1}\right), \left(a_{2} = b_{2}\right), \ldots, \left(a_{m + 1} = b_{m + 1}\right)}~\left(\left(a_{1}, a_{2}, \ldots, a_{m + 1}\right) = \left(b_{1}, b_{2}, \ldots, b_{m + 1}\right)\right)\right]\right]\right) \Rightarrow \left[\forall_{n \in \mathbb{N}^+}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}~|~\left(a_{1} = b_{1}\right), \left(a_{2} = b_{2}\right), \ldots, \left(a_{n} = b_{n}\right)}~\left(\left(a_{1}, a_{2}, \ldots, a_{n}\right) = \left(b_{1}, b_{2}, \ldots, b_{n}\right)\right)\right]\right]

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.	()	()	('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: 74 operands: 5
4	Operation	operator: 47 operand: 9
5	ExprTuple	7, 8
6	ExprTuple	9
7	Operation	operator: 47 operand: 14
8	Operation	operator: 47 operand: 15
9	Lambda	parameter: 70 body: 13
10	ExprTuple	14
11	ExprTuple	15
12	ExprTuple	70
13	Conditional	value: 16 condition: 17
14	Lambda	parameters: 29 body: 18
15	Lambda	parameter: 88 body: 20
16	Operation	operator: 47 operand: 27
17	Operation	operator: 45 operands: 22
18	Conditional	value: 23 condition: 24
19	ExprTuple	88
20	Conditional	value: 25 condition: 26
21	ExprTuple	27
22	ExprTuple	70, 54
23	Operation	operator: 90 operands: 28
24	Operation	operator: 90 operands: 29
25	Operation	operator: 47 operand: 36
26	Operation	operator: 74 operands: 31
27	Lambda	parameters: 32 body: 33
28	ExprTuple	34, 35
29	ExprTuple	41, 42
30	ExprTuple	36
31	ExprTuple	37, 38
32	ExprTuple	63, 64
33	Conditional	value: 39 condition: 40
34	ExprTuple	41
35	ExprTuple	42
36	Lambda	parameters: 43 body: 44
37	Operation	operator: 45 operands: 46
38	Operation	operator: 47 operand: 55
39	Operation	operator: 90 operands: 49
40	Operation	operator: 74 operands: 50
41	IndexedVar	variable: 94 index: 87
42	IndexedVar	variable: 95 index: 87
43	ExprTuple	71, 72
44	Conditional	value: 52 condition: 53
45	Literal
46	ExprTuple	88, 54
47	Literal
48	ExprTuple	55
49	ExprTuple	56, 57
50	ExprTuple	58
51	ExprTuple	87
52	Operation	operator: 90 operands: 59
53	Operation	operator: 74 operands: 60
54	Literal
55	Lambda	parameters: 61 body: 62
56	ExprTuple	63
57	ExprTuple	64
58	ExprRange	lambda_map: 84 start_index: 87 end_index: 70
59	ExprTuple	65, 66
60	ExprTuple	67
61	ExprTuple	82, 83
62	Conditional	value: 68 condition: 69
63	ExprRange	lambda_map: 85 start_index: 87 end_index: 70
64	ExprRange	lambda_map: 86 start_index: 87 end_index: 70
65	ExprTuple	71
66	ExprTuple	72
67	ExprRange	lambda_map: 84 start_index: 87 end_index: 76
68	Operation	operator: 90 operands: 73
69	Operation	operator: 74 operands: 75
70	Variable
71	ExprRange	lambda_map: 85 start_index: 87 end_index: 76
72	ExprRange	lambda_map: 86 start_index: 87 end_index: 76
73	ExprTuple	77, 78
74	Literal
75	ExprTuple	79
76	Operation	operator: 80 operands: 81
77	ExprTuple	82
78	ExprTuple	83
79	ExprRange	lambda_map: 84 start_index: 87 end_index: 88
80	Literal
81	ExprTuple	88, 87
82	ExprRange	lambda_map: 85 start_index: 87 end_index: 88
83	ExprRange	lambda_map: 86 start_index: 87 end_index: 88
84	Lambda	parameter: 97 body: 89
85	Lambda	parameter: 97 body: 92
86	Lambda	parameter: 97 body: 93
87	Literal
88	Variable
89	Operation	operator: 90 operands: 91
90	Literal
91	ExprTuple	92, 93
92	IndexedVar	variable: 94 index: 97
93	IndexedVar	variable: 95 index: 97
94	Variable
95	Variable
96	ExprTuple	97
97	Variable

Expression of type Implies¶

from the theory of proveit.core_expr_types.tuples¶

Expression of type Implies ¶

from the theory of proveit.core_expr_types.tuples ¶