Expression of type Conditional

from the theory of proveit.logic.booleans.quantification.universality

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, P, Q, R, m, n
from proveit.logic import And, Forall, Implies, InSet
from proveit.logic.booleans.quantification import general_bundled_forall_Pxy_if_Qx, general_nested_forall_Pxy_if_Qx
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [P, Q, R], instance_expr = Implies(general_nested_forall_Pxy_if_Qx, general_bundled_forall_Pxy_if_Qx).with_wrapping_at(1)), And(InSet(m, NaturalPos), InSet(n, 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\{\forall_{P, Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{m}~|~Q\left(x_{1}, x_{2}, \ldots, x_{m}\right)}~\left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~R\left(x_{1}, x_{2}, \ldots, x_{m},y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{m},y_{1}, y_{2}, \ldots, y_{n}\right)\right]\right] \\  \Rightarrow \left[\forall_{x_{1}, x_{2}, \ldots, x_{m}, y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{m}\right), R\left(x_{1}, x_{2}, \ldots, x_{m},y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{m},y_{1}, y_{2}, \ldots, y_{n}\right)\right] \end{array} \end{array}\right) \textrm{ if } m \in \mathbb{N}^+ ,  n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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