Expression of type Forall

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

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 P, Q, R
from proveit.logic import Forall, Implies
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy, general_exists_Rz_st_Qz, general_forall_st_Qx__Px_implies_Rx

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q, R], instance_expr = Implies(general_forall_st_Qx__Px_implies_Rx, Implies(general_exists_Py_st_Qy, general_exists_Rz_st_Qz).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())

\forall_{P, Q, R}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~\left(P\left(x_{1}, x_{2}, \ldots, x_{n}\right) \Rightarrow R\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right] \Rightarrow  \\ \left[\exists_{z_{1}, z_{2}, \ldots, z_{n}~|~Q\left(z_{1}, z_{2}, \ldots, z_{n}\right)}~R\left(z_{1}, z_{2}, \ldots, z_{n}\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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