Proof of proveit.logic.sets.comprehension.fold theorem

import proveit
theory = proveit.Theory() # the theorem's theory
from proveit import defaults
from proveit.logic import Implies
from proveit.logic.sets.comprehension import comprehension_def

%proving fold

With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
fold:
(see dependencies)

defaults.assumptions = fold.conditions

defaults.assumptions:

comprehension_def

 ⊢  

comprehension_def_inst = comprehension_def.instantiate().instantiate()

comprehension_def_inst:  ⊢  

exists_eq_impl = Implies(comprehension_def_inst.rhs, comprehension_def_inst.rhs).prove()

exists_eq_impl:  ⊢  

exists_eq_impl.inner_expr().rhs.substitute(comprehension_def_inst.derive_reversed())

 ⊢  

fold may now be readily provable (assuming required theorems are usable).  Simply execute "%qed".

%qed

proveit.logic.sets.comprehension.fold has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	⊢
	: , : , :
2	theorem		⊢
	proveit.logic.equality.sub_left_side_into
3	instantiation	5	⊢
	:
4	instantiation	6, 7	⊢
	:
5	theorem		⊢
	proveit.logic.booleans.implication.self_implication
6	axiom		⊢
	proveit.logic.sets.comprehension.comprehension_def
7	assumption		⊢