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Expression of type Forall

from the theory of proveit.logic.booleans.disjunction

In [1]:
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, ExprRange, IndexedVar, Variable, m
from proveit.core_expr_types import A_1_to_m, B_1_to_m
from proveit.logic import Boolean, Forall, Implies, Or
from proveit.numbers import Natural, one
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [A_1_to_m, B_1_to_m], instance_expr = Or(B_1_to_m), domain = Boolean, conditions = [Or(A_1_to_m), ExprRange(sub_expr1, Implies(IndexedVar(A, sub_expr1), IndexedVar(B, sub_expr1)), one, m)]), domain = Natural)
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\forall_{m \in \mathbb{N}}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{m}, B_{1}, B_{2}, \ldots, B_{m} \in \mathbb{B}~|~A_{1} \lor  A_{2} \lor  \ldots \lor  A_{m},\left(A_{1} \Rightarrow B_{1}\right), \left(A_{2} \Rightarrow B_{2}\right), \ldots, \left(A_{m} \Rightarrow B_{m}\right)}~\left(B_{1} \lor  B_{2} \lor  \ldots \lor  B_{m}\right)\right]
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 39
body: 4
3ExprTuple39
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 35
operands: 9
7Literal
8ExprTuple10
9ExprTuple39, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple32, 19
13Conditionalvalue: 14
condition: 15
14Operationoperator: 27
operands: 16
15Operationoperator: 17
operands: 18
16ExprTuple19
17Literal
18ExprTuple20, 21, 22, 23
19ExprRangelambda_map: 24
start_index: 38
end_index: 39
20ExprRangelambda_map: 25
start_index: 38
end_index: 39
21ExprRangelambda_map: 26
start_index: 38
end_index: 39
22Operationoperator: 27
operands: 28
23ExprRangelambda_map: 29
start_index: 38
end_index: 39
24Lambdaparameter: 48
body: 44
25Lambdaparameter: 48
body: 30
26Lambdaparameter: 48
body: 31
27Literal
28ExprTuple32
29Lambdaparameter: 48
body: 33
30Operationoperator: 35
operands: 34
31Operationoperator: 35
operands: 36
32ExprRangelambda_map: 37
start_index: 38
end_index: 39
33Operationoperator: 40
operands: 41
34ExprTuple43, 42
35Literal
36ExprTuple44, 42
37Lambdaparameter: 48
body: 43
38Literal
39Variable
40Literal
41ExprTuple43, 44
42Literal
43IndexedVarvariable: 45
index: 48
44IndexedVarvariable: 46
index: 48
45Variable
46Variable
47ExprTuple48
48Variable