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 A, B, P, Q
from proveit.logic import Forall, Implies, SubsetEq
from proveit.logic.booleans.quantification import general_exists_in_A_Px_st_Qx, general_exists_in_B_Py_st_Qy

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q, A, B], instance_expr = Implies(general_exists_in_A_Px_st_Qx, general_exists_in_B_Py_st_Qy).with_wrapping_at(2), condition = SubsetEq(A, B))

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, A, B~|~A \subseteq B}~\left(\begin{array}{c} \begin{array}{l} \left[\exists_{x_{1}, x_{2}, \ldots, x_{n} \in A~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left[\exists_{y_{1}, y_{2}, \ldots, y_{n} \in B~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	26, 36, 49, 51
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	49, 51
12	Operation	operator: 15 operand: 17
13	Operation	operator: 15 operand: 18
14	ExprTuple	17
15	Literal
16	ExprTuple	18
17	Lambda	parameters: 34 body: 19
18	Lambda	parameters: 37 body: 20
19	Conditional	value: 21 condition: 22
20	Conditional	value: 23 condition: 24
21	Operation	operator: 26 operands: 34
22	Operation	operator: 27 operands: 25
23	Operation	operator: 26 operands: 37
24	Operation	operator: 27 operands: 28
25	ExprTuple	29, 30
26	Variable
27	Literal
28	ExprTuple	31, 32
29	ExprRange	lambda_map: 33 start_index: 47 end_index: 48
30	Operation	operator: 36 operands: 34
31	ExprRange	lambda_map: 35 start_index: 47 end_index: 48
32	Operation	operator: 36 operands: 37
33	Lambda	parameter: 56 body: 38
34	ExprTuple	39
35	Lambda	parameter: 56 body: 40
36	Variable
37	ExprTuple	41
38	Operation	operator: 44 operands: 42
39	ExprRange	lambda_map: 43 start_index: 47 end_index: 48
40	Operation	operator: 44 operands: 45
41	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
42	ExprTuple	50, 49
43	Lambda	parameter: 56 body: 50
44	Literal
45	ExprTuple	52, 51
46	Lambda	parameter: 56 body: 52
47	Literal
48	Variable
49	Variable
50	IndexedVar	variable: 53 index: 56
51	Variable
52	IndexedVar	variable: 54 index: 56
53	Variable
54	Variable
55	ExprTuple	56
56	Variable