Expression of type And

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
from proveit.logic import And, Equals, Forall
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)
expr = 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))).with_wrapping())

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[\forall_{a_{1}, b_{1}~|~a_{1} = b_{1}}~\left(\left(a_{1}\right) = \left(b_{1}\right)\right)\right] \land \left[\begin{array}{l}\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]\end{array}\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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 49 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 30 operand: 6
3	Operation	operator: 30 operand: 7
4	ExprTuple	6
5	ExprTuple	7
6	Lambda	parameters: 16 body: 8
7	Lambda	parameter: 63 body: 10
8	Conditional	value: 11 condition: 12
9	ExprTuple	63
10	Conditional	value: 13 condition: 14
11	Operation	operator: 65 operands: 15
12	Operation	operator: 65 operands: 16
13	Operation	operator: 30 operand: 21
14	Operation	operator: 49 operands: 18
15	ExprTuple	19, 20
16	ExprTuple	24, 25
17	ExprTuple	21
18	ExprTuple	22, 23
19	ExprTuple	24
20	ExprTuple	25
21	Lambda	parameters: 26 body: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 30 operand: 36
24	IndexedVar	variable: 69 index: 62
25	IndexedVar	variable: 70 index: 62
26	ExprTuple	46, 47
27	Conditional	value: 33 condition: 34
28	Literal
29	ExprTuple	63, 35
30	Literal
31	ExprTuple	36
32	ExprTuple	62
33	Operation	operator: 65 operands: 37
34	Operation	operator: 49 operands: 38
35	Literal
36	Lambda	parameters: 39 body: 40
37	ExprTuple	41, 42
38	ExprTuple	43
39	ExprTuple	57, 58
40	Conditional	value: 44 condition: 45
41	ExprTuple	46
42	ExprTuple	47
43	ExprRange	lambda_map: 59 start_index: 62 end_index: 51
44	Operation	operator: 65 operands: 48
45	Operation	operator: 49 operands: 50
46	ExprRange	lambda_map: 60 start_index: 62 end_index: 51
47	ExprRange	lambda_map: 61 start_index: 62 end_index: 51
48	ExprTuple	52, 53
49	Literal
50	ExprTuple	54
51	Operation	operator: 55 operands: 56
52	ExprTuple	57
53	ExprTuple	58
54	ExprRange	lambda_map: 59 start_index: 62 end_index: 63
55	Literal
56	ExprTuple	63, 62
57	ExprRange	lambda_map: 60 start_index: 62 end_index: 63
58	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
59	Lambda	parameter: 72 body: 64
60	Lambda	parameter: 72 body: 67
61	Lambda	parameter: 72 body: 68
62	Literal
63	Variable
64	Operation	operator: 65 operands: 66
65	Literal
66	ExprTuple	67, 68
67	IndexedVar	variable: 69 index: 72
68	IndexedVar	variable: 70 index: 72
69	Variable
70	Variable
71	ExprTuple	72
72	Variable