Expression of type Implies

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.logic import Implies
from proveit.logic.booleans.quantification import general_bundled_forall_Pxy_if_Qx, general_nested_forall_Pxy_if_Qx

# build up the expression from sub-expressions
expr = Implies(general_nested_forall_Pxy_if_Qx, general_bundled_forall_Pxy_if_Qx).with_wrapping_at(1)

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_{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}

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.	()	(1)	('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: 13 operand: 7
4	Operation	operator: 13 operand: 8
5	ExprTuple	7
6	ExprTuple	8
7	Lambda	parameters: 22 body: 9
8	Lambda	parameters: 27 body: 10
9	Conditional	value: 11 condition: 18
10	Conditional	value: 23 condition: 12
11	Operation	operator: 13 operand: 17
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17
15	Literal
16	ExprTuple	18, 24
17	Lambda	parameters: 19 body: 20
18	Operation	operator: 21 operands: 22
19	ExprTuple	29
20	Conditional	value: 23 condition: 24
21	Variable
22	ExprTuple	28
23	Operation	operator: 25 operands: 27
24	Operation	operator: 26 operands: 27
25	Variable
26	Variable
27	ExprTuple	28, 29
28	ExprRange	lambda_map: 30 start_index: 33 end_index: 31
29	ExprRange	lambda_map: 32 start_index: 33 end_index: 34
30	Lambda	parameter: 40 body: 35
31	Variable
32	Lambda	parameter: 40 body: 36
33	Literal
34	Variable
35	IndexedVar	variable: 37 index: 40
36	IndexedVar	variable: 38 index: 40
37	Variable
38	Variable
39	ExprTuple	40
40	Variable