forked from oysteikt/sf1-template
376 lines
10 KiB
Coq
376 lines
10 KiB
Coq
Set Warnings "-notation-overridden,-parsing".
|
|
From Stdlib Require Export String.
|
|
From LF Require Import ProofObjects.
|
|
|
|
Parameter MISSING: Type.
|
|
|
|
Module Check.
|
|
|
|
Ltac check_type A B :=
|
|
match type of A with
|
|
| context[MISSING] => idtac "Missing:" A
|
|
| ?T => first [unify T B; idtac "Type: ok" | idtac "Type: wrong - should be (" B ")"]
|
|
end.
|
|
|
|
Ltac print_manual_grade A :=
|
|
match eval compute in A with
|
|
| Some (_ ?S ?C) =>
|
|
idtac "Score:" S;
|
|
match eval compute in C with
|
|
| ""%string => idtac "Comment: None"
|
|
| _ => idtac "Comment:" C
|
|
end
|
|
| None =>
|
|
idtac "Score: Ungraded";
|
|
idtac "Comment: None"
|
|
end.
|
|
|
|
End Check.
|
|
|
|
From LF Require Import ProofObjects.
|
|
Import Check.
|
|
|
|
Goal True.
|
|
|
|
idtac "------------------- eight_is_even --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> ev_8".
|
|
idtac "Possible points: 1".
|
|
check_type @ev_8 ((ev 8)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions ev_8.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "#> ev_8'".
|
|
idtac "Possible points: 1".
|
|
check_type @ev_8' ((ev 8)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions ev_8'.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- conj_fact --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.conj_fact".
|
|
idtac "Possible points: 2".
|
|
check_type @Props.conj_fact (
|
|
(forall (P Q R : Prop) (_ : and P Q) (_ : and Q R), and P R)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.conj_fact.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- or_commut' --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.or_commut'".
|
|
idtac "Possible points: 2".
|
|
check_type @Props.or_commut' ((forall (P Q : Prop) (_ : or P Q), or Q P)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.or_commut'.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- ex_ev_Sn --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.ex_ev_Sn".
|
|
idtac "Possible points: 2".
|
|
check_type @Props.ex_ev_Sn ((@ex nat (fun n : nat => ev (S n)))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.ex_ev_Sn.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- ex_match --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.ex_match".
|
|
idtac "Possible points: 2".
|
|
check_type @Props.ex_match (
|
|
(forall (A : Type) (P Q : forall _ : A, Prop)
|
|
(_ : forall (x : A) (_ : P x), Q x) (_ : @ex A (fun x : A => P x)),
|
|
@ex A (fun x : A => Q x))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.ex_match.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- p_implies_true --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.p_implies_true".
|
|
idtac "Possible points: 1".
|
|
check_type @Props.p_implies_true ((forall (P : Type) (_ : P), Props.True)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.p_implies_true.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- ex_falso_quodlibet' --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> Props.ex_falso_quodlibet'".
|
|
idtac "Possible points: 1".
|
|
check_type @Props.ex_falso_quodlibet' ((forall (P : Type) (_ : Props.False), P)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions Props.ex_falso_quodlibet'.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- eq_cons --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> EqualityPlayground.eq_cons".
|
|
idtac "Possible points: 2".
|
|
check_type @EqualityPlayground.eq_cons (
|
|
(forall (X : Type) (h1 h2 : X) (t1 t2 : list X)
|
|
(_ : @EqualityPlayground.eq X h1 h2)
|
|
(_ : @EqualityPlayground.eq (list X) t1 t2),
|
|
@EqualityPlayground.eq (list X) (@cons X h1 t1) (@cons X h2 t2))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions EqualityPlayground.eq_cons.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- equality__leibniz_equality --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> EqualityPlayground.equality__leibniz_equality".
|
|
idtac "Possible points: 2".
|
|
check_type @EqualityPlayground.equality__leibniz_equality (
|
|
(forall (X : Type) (x y : X) (_ : @EqualityPlayground.eq X x y)
|
|
(P : forall _ : X, Prop) (_ : P x),
|
|
P y)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions EqualityPlayground.equality__leibniz_equality.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- equality__leibniz_equality_term --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> EqualityPlayground.equality__leibniz_equality_term".
|
|
idtac "Possible points: 2".
|
|
check_type @EqualityPlayground.equality__leibniz_equality_term (
|
|
(forall (X : Type) (x y : X) (_ : @EqualityPlayground.eq X x y)
|
|
(P : forall _ : X, Prop) (_ : P x),
|
|
P y)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions EqualityPlayground.equality__leibniz_equality_term.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- and_assoc --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> and_assoc".
|
|
idtac "Possible points: 2".
|
|
check_type @and_assoc ((forall (P Q R : Prop) (_ : and P (and Q R)), and (and P Q) R)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions and_assoc.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- or_distributes_over_and --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> or_distributes_over_and".
|
|
idtac "Possible points: 3".
|
|
check_type @or_distributes_over_and (
|
|
(forall P Q R : Prop, iff (or P (and Q R)) (and (or P Q) (or P R)))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions or_distributes_over_and.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- negations --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> double_neg".
|
|
idtac "Possible points: 1".
|
|
check_type @double_neg ((forall (P : Prop) (_ : P), not (not P))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions double_neg.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "#> contradiction_implies_anything".
|
|
idtac "Possible points: 1".
|
|
check_type @contradiction_implies_anything ((forall (P Q : Prop) (_ : and P (not P)), Q)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions contradiction_implies_anything.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "#> de_morgan_not_or".
|
|
idtac "Possible points: 1".
|
|
check_type @de_morgan_not_or (
|
|
(forall (P Q : Prop) (_ : not (or P Q)), and (not P) (not Q))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions de_morgan_not_or.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- currying --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> curry".
|
|
idtac "Possible points: 1".
|
|
check_type @curry (
|
|
(forall (P Q R : Prop) (_ : forall _ : and P Q, R) (_ : P) (_ : Q), R)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions curry.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "#> uncurry".
|
|
idtac "Possible points: 1".
|
|
check_type @uncurry (
|
|
(forall (P Q R : Prop) (_ : forall (_ : P) (_ : Q), R) (_ : and P Q), R)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions uncurry.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- pe_implies_or_eq --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> pe_implies_or_eq".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 1".
|
|
check_type @pe_implies_or_eq (
|
|
(forall (_ : propositional_extensionality) (P Q : Prop),
|
|
@eq Prop (or P Q) (or Q P))).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions pe_implies_or_eq.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- pe_implies_true_eq --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> pe_implies_true_eq".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 1".
|
|
check_type @pe_implies_true_eq (
|
|
(forall (_ : propositional_extensionality) (P : Prop) (_ : P),
|
|
@eq Prop True P)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions pe_implies_true_eq.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac "------------------- pe_implies_pi --------------------".
|
|
idtac " ".
|
|
|
|
idtac "#> pe_implies_pi".
|
|
idtac "Advanced".
|
|
idtac "Possible points: 3".
|
|
check_type @pe_implies_pi ((forall _ : propositional_extensionality, proof_irrelevance)).
|
|
idtac "Assumptions:".
|
|
Abort.
|
|
Print Assumptions pe_implies_pi.
|
|
Goal True.
|
|
idtac " ".
|
|
|
|
idtac " ".
|
|
|
|
idtac "Max points - standard: 28".
|
|
idtac "Max points - advanced: 33".
|
|
idtac "".
|
|
idtac "Allowed Axioms:".
|
|
idtac "functional_extensionality".
|
|
idtac "FunctionalExtensionality.functional_extensionality_dep".
|
|
idtac "plus_le".
|
|
idtac "le_trans".
|
|
idtac "le_plus_l".
|
|
idtac "add_le_cases".
|
|
idtac "Sn_le_Sm__n_le_m".
|
|
idtac "O_le_n".
|
|
idtac "".
|
|
idtac "".
|
|
idtac "********** Summary **********".
|
|
idtac "".
|
|
idtac "Below is a summary of the automatically graded exercises that are incomplete.".
|
|
idtac "".
|
|
idtac "The output for each exercise can be any of the following:".
|
|
idtac " - 'Closed under the global context', if it is complete".
|
|
idtac " - 'MANUAL', if it is manually graded".
|
|
idtac " - A list of pending axioms, containing unproven assumptions. In this case".
|
|
idtac " the exercise is considered complete, if the axioms are all allowed.".
|
|
idtac "".
|
|
idtac "********** Standard **********".
|
|
idtac "---------- ev_8 ---------".
|
|
Print Assumptions ev_8.
|
|
idtac "---------- ev_8' ---------".
|
|
Print Assumptions ev_8'.
|
|
idtac "---------- Props.conj_fact ---------".
|
|
Print Assumptions Props.conj_fact.
|
|
idtac "---------- Props.or_commut' ---------".
|
|
Print Assumptions Props.or_commut'.
|
|
idtac "---------- Props.ex_ev_Sn ---------".
|
|
Print Assumptions Props.ex_ev_Sn.
|
|
idtac "---------- Props.ex_match ---------".
|
|
Print Assumptions Props.ex_match.
|
|
idtac "---------- Props.p_implies_true ---------".
|
|
Print Assumptions Props.p_implies_true.
|
|
idtac "---------- Props.ex_falso_quodlibet' ---------".
|
|
Print Assumptions Props.ex_falso_quodlibet'.
|
|
idtac "---------- EqualityPlayground.eq_cons ---------".
|
|
Print Assumptions EqualityPlayground.eq_cons.
|
|
idtac "---------- EqualityPlayground.equality__leibniz_equality ---------".
|
|
Print Assumptions EqualityPlayground.equality__leibniz_equality.
|
|
idtac "---------- EqualityPlayground.equality__leibniz_equality_term ---------".
|
|
Print Assumptions EqualityPlayground.equality__leibniz_equality_term.
|
|
idtac "---------- and_assoc ---------".
|
|
Print Assumptions and_assoc.
|
|
idtac "---------- or_distributes_over_and ---------".
|
|
Print Assumptions or_distributes_over_and.
|
|
idtac "---------- double_neg ---------".
|
|
Print Assumptions double_neg.
|
|
idtac "---------- contradiction_implies_anything ---------".
|
|
Print Assumptions contradiction_implies_anything.
|
|
idtac "---------- de_morgan_not_or ---------".
|
|
Print Assumptions de_morgan_not_or.
|
|
idtac "---------- curry ---------".
|
|
Print Assumptions curry.
|
|
idtac "---------- uncurry ---------".
|
|
Print Assumptions uncurry.
|
|
idtac "".
|
|
idtac "********** Advanced **********".
|
|
idtac "---------- pe_implies_or_eq ---------".
|
|
Print Assumptions pe_implies_or_eq.
|
|
idtac "---------- pe_implies_true_eq ---------".
|
|
Print Assumptions pe_implies_true_eq.
|
|
idtac "---------- pe_implies_pi ---------".
|
|
Print Assumptions pe_implies_pi.
|
|
Abort.
|
|
|
|
(* 2026-01-07 13:18 *)
|
|
|
|
(* 2026-01-07 13:18 *)
|