Expression of type Implies

from the theory of proveit.logic.equality

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, h, n, theta
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [n]
sub_expr2 = Function(h, sub_expr1)
sub_expr3 = Function(theta, sub_expr1)
sub_expr4 = InSet(n, Natural)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(sub_expr2, sub_expr3), domain = Natural), Equals(Lambda(n, Conditional(sub_expr2, sub_expr4)), Lambda(n, Conditional(sub_expr3, sub_expr4))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{n \in \mathbb{N}}~\left(h\left(n\right) = \theta\left(n\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[n \mapsto \left\{h\left(n\right) \textrm{ if } n \in \mathbb{N}\right..\right] =  \\ \left[n \mapsto \left\{\theta\left(n\right) \textrm{ if } n \in \mathbb{N}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(2)	('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: 5 operand: 8
4	Operation	operator: 16 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 26 body: 11
9	Lambda	parameter: 26 body: 12
10	Lambda	parameter: 26 body: 13
11	Conditional	value: 14 condition: 15
12	Conditional	value: 20 condition: 15
13	Conditional	value: 21 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	26, 22
20	Operation	operator: 23 operand: 26
21	Operation	operator: 24 operand: 26
22	Literal
23	Variable
24	Variable
25	ExprTuple	26
26	Variable