Expression of type Implies

from the theory of proveit.physics.quantum.QPE

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, Variable, t
from proveit.logic import And, Forall, Implies, InSet
from proveit.numbers import Add, Interval, Natural, Neg, one, zero

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = InSet(Add(sub_expr1, t), Natural)
sub_expr3 = Add(Neg(t), one)
expr = Implies(Forall(instance_param_or_params = [sub_expr1], instance_expr = sub_expr2, domain = Interval(sub_expr3, zero)), And(ExprRange(sub_expr1, sub_expr2, sub_expr3, zero)))

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} \in \{-t + 1~\ldotp \ldotp~0\}}~\left(\left({_{-}a} + t\right) \in \mathbb{N}\right)\right] \Rightarrow \left(\left(\left(\left(-t + 1\right) + t\right) \in \mathbb{N}\right) \land  \left(\left(\left(-t + 2\right) + t\right) \in \mathbb{N}\right) \land  \ldots \land  \left(\left(0 + t\right) \in \mathbb{N}\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: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10
9	Lambda	parameter: 27 body: 11
10	ExprRange	lambda_map: 12 start_index: 25 end_index: 26
11	Conditional	value: 15 condition: 13
12	Lambda	parameter: 27 body: 15
13	Operation	operator: 17 operands: 16
14	ExprTuple	27
15	Operation	operator: 17 operands: 18
16	ExprTuple	27, 19
17	Literal
18	ExprTuple	20, 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 28 operands: 24
21	Literal
22	Literal
23	ExprTuple	25, 26
24	ExprTuple	27, 34
25	Operation	operator: 28 operands: 29
26	Literal
27	Variable
28	Literal
29	ExprTuple	30, 31
30	Operation	operator: 32 operand: 34
31	Literal
32	Literal
33	ExprTuple	34
34	Variable