More cleanup around addition of RuleInduction

This commit is contained in:
Adam Chlipala 2021-03-01 12:15:34 -05:00
parent 890d7610d7
commit d3c7a85b49
25 changed files with 651 additions and 31 deletions

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 8: Abstract Interpretation and Dataflow Analysis * Chapter 9: Abstract Interpretation and Dataflow Analysis
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 9: Compiler Correctness * Chapter 10: Compiler Correctness
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 9: Compiler Correctness * Chapter 10: Compiler Correctness
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 18: Concurrent Separation Logic * Chapter 19: Concurrent Separation Logic
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 15: Connecting to Real-World Programming Languages and Platforms * Chapter 16: Connecting to Real-World Programming Languages and Platforms
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 13: Deep and Shallow Embeddings * Chapter 14: Deep and Shallow Embeddings
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 12: Hoare Logic: Verifying Imperative Programs * Chapter 13: Hoare Logic: Verifying Imperative Programs
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 10: Lambda Calculus and Simple Type Soundness * Chapter 11: Lambda Calculus and Simple Type Soundness
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 8: Lambda Calculus and Simple Type Soundness * Chapter 11: Lambda Calculus and Simple Type Soundness
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 19: Process Algebra and Behavioral Refinement * Chapter 20: Process Algebra and Behavioral Refinement
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 6: Model Checking * Chapter 7: Model Checking
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 5: Model Checking * Chapter 7: Model Checking
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 7: Operational Semantics * Chapter 8: Operational Semantics
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 6: Operational Semantics * Chapter 8: Operational Semantics
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 16: Deriving Programs from Specifications * Chapter 17: Deriving Programs from Specifications
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ * License: https://creativecommons.org/licenses/by-nc-nd/4.0/
* Some material borrowed from Fiat <http://plv.csail.mit.edu/fiat/> *) * Some material borrowed from Fiat <http://plv.csail.mit.edu/fiat/> *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* New chapter: inductive relations and rule induction * Chapter 5: inductive relations and rule induction
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

617
RuleInduction_template.v Normal file
View file

@ -0,0 +1,617 @@
Require Import Frap.
(** * Finite sets as inductive predicates *)
Inductive my_favorite_numbers : nat -> Prop :=
| ILike17 : my_favorite_numbers 17
| ILike23 : my_favorite_numbers 23
| ILike42 : my_favorite_numbers 42.
Check my_favorite_numbers_ind.
Theorem favorites_below_50 : forall n, my_favorite_numbers n -> n < 50.
Proof.
Admitted.
(** * Transitive closure of relations *)
Inductive tc {A} (R : A -> A -> Prop) : A -> A -> Prop :=
| TcBase : forall x y, R x y -> tc R x y
| TcTrans : forall x y z, tc R x y -> tc R y z -> tc R x z.
(** ** Less-than reimagined *)
Definition oneApart (n m : nat) : Prop :=
n + 1 = m.
Definition lt' : nat -> nat -> Prop := tc oneApart.
Theorem lt'_lt : forall n m, lt' n m -> n < m.
Proof.
Admitted.
Theorem lt_lt' : forall n m, n < m -> lt' n m.
Proof.
Admitted.
(** ** Transitive closure is idempotent. *)
Theorem tc_tc2 : forall A (R : A -> A -> Prop) x y, tc R x y -> tc (tc R) x y.
Proof.
Admitted.
Theorem tc2_tc : forall A (R : A -> A -> Prop) x y, tc (tc R) x y -> tc R x y.
Proof.
Admitted.
(** * Permutation *)
(* Lifted from the Coq standard library: *)
Inductive Permutation {A} : list A -> list A -> Prop :=
| perm_nil :
Permutation [] []
| perm_skip : forall x l l',
Permutation l l' -> Permutation (x::l) (x::l')
| perm_swap : forall x y l,
Permutation (y::x::l) (x::y::l)
| perm_trans : forall l l' l'',
Permutation l l' -> Permutation l' l'' -> Permutation l l''.
Theorem Permutation_rev : forall A (ls : list A),
Permutation ls (rev ls).
Proof.
Admitted.
Theorem Permutation_length : forall A (ls1 ls2 : list A),
Permutation ls1 ls2 -> length ls1 = length ls2.
Proof.
Admitted.
Theorem Permutation_app : forall A (ls1 ls1' ls2 ls2' : list A),
Permutation ls1 ls1'
-> Permutation ls2 ls2'
-> Permutation (ls1 ++ ls2) (ls1' ++ ls2').
Proof.
Admitted.
(** * Simple propositional logic *)
Inductive prop :=
| Truth
| Falsehood
| And (p1 p2 : prop)
| Or (p1 p2 : prop).
Inductive valid : prop -> Prop :=
| ValidTruth :
valid Truth
| ValidAnd : forall p1 p2,
valid p1
-> valid p2
-> valid (And p1 p2)
| ValidOr1 : forall p1 p2,
valid p1
-> valid (Or p1 p2)
| ValidOr2 : forall p1 p2,
valid p2
-> valid (Or p1 p2).
Fixpoint interp (p : prop) : Prop :=
match p with
| Truth => True
| Falsehood => False
| And p1 p2 => interp p1 /\ interp p2
| Or p1 p2 => interp p1 \/ interp p2
end.
Theorem interp_valid : forall p, interp p -> valid p.
Proof.
Admitted.
Theorem valid_interp : forall p, valid p -> interp p.
Proof.
Admitted.
Fixpoint commuter (p : prop) : prop :=
match p with
| Truth => Truth
| Falsehood => Falsehood
| And p1 p2 => And (commuter p2) (commuter p1)
| Or p1 p2 => Or (commuter p2) (commuter p1)
end.
Theorem valid_commuter_fwd : forall p, valid p -> valid (commuter p).
Proof.
Admitted.
Theorem valid_commuter_bwd : forall p, valid (commuter p) -> valid p.
Proof.
Admitted.
(* Proofs for an extension I hope we'll get to:
Fixpoint interp (vars : var -> Prop) (p : prop) : Prop :=
match p with
| Truth => True
| Falsehood => False
| Var x => vars x
| And p1 p2 => interp vars p1 /\ interp vars p2
| Or p1 p2 => interp vars p1 \/ interp vars p2
| Imply p1 p2 => interp vars p1 -> interp vars p2
end.
Theorem valid_interp : forall vars hyps p,
valid hyps p
-> (forall h, hyps h -> interp vars h)
-> interp vars p.
Proof.
induct 1; simplify.
apply H0.
assumption.
propositional.
propositional.
propositional.
propositional.
propositional.
propositional.
propositional.
propositional.
apply IHvalid2.
propositional.
equality.
apply H2.
assumption.
apply IHvalid3.
propositional.
equality.
apply H2.
assumption.
apply IHvalid.
propositional.
equality.
apply H0.
assumption.
propositional.
excluded_middle (interp vars p); propositional.
(* Note that use of excluded middle is a bit controversial in Coq,
* and we'll generally be trying to avoid it,
* but it helps enough with this example that we don't sweat the details. *)
Qed.
Lemma valid_weaken : forall hyps1 p,
valid hyps1 p
-> forall hyps2 : prop -> Prop,
(forall h, hyps1 h -> hyps2 h)
-> valid hyps2 p.
Proof.
induct 1; simplify.
apply ValidHyp.
apply H0.
assumption.
apply ValidTruthIntro.
apply ValidFalsehoodElim.
apply IHvalid.
assumption.
apply ValidAndIntro.
apply IHvalid1.
assumption.
apply IHvalid2.
assumption.
apply ValidAndElim1 with p2.
apply IHvalid.
assumption.
apply ValidAndElim2 with p1.
apply IHvalid.
assumption.
apply ValidOrIntro1.
apply IHvalid.
assumption.
apply ValidOrIntro2.
apply IHvalid.
assumption.
apply ValidOrElim with p1 p2.
apply IHvalid1.
assumption.
apply IHvalid2.
first_order.
apply IHvalid3.
first_order.
apply ValidImplyIntro.
apply IHvalid.
propositional.
right.
apply H0.
assumption.
apply ValidImplyElim with p1.
apply IHvalid1.
assumption.
apply IHvalid2.
assumption.
apply ValidExcludedMiddle.
Qed.
Lemma valid_cut : forall hyps1 p p',
valid hyps1 p
-> forall hyps2, valid hyps2 p'
-> (forall h, hyps1 h -> hyps2 h \/ h = p')
-> valid hyps2 p.
Proof.
induct 1; simplify.
apply H1 in H.
propositional.
apply ValidHyp.
assumption.
equality.
apply ValidTruthIntro.
apply ValidFalsehoodElim.
apply IHvalid; assumption.
apply ValidAndIntro.
apply IHvalid1; assumption.
apply IHvalid2; assumption.
apply ValidAndElim1 with p2.
apply IHvalid; assumption.
apply ValidAndElim2 with p1.
apply IHvalid; assumption.
apply ValidOrIntro1.
apply IHvalid; assumption.
apply ValidOrIntro2.
apply IHvalid; assumption.
apply ValidOrElim with p1 p2.
apply IHvalid1; assumption.
apply IHvalid2.
apply valid_weaken with hyps2.
assumption.
propositional.
first_order.
apply IHvalid3.
apply valid_weaken with hyps2.
assumption.
propositional.
first_order.
apply ValidImplyIntro.
apply IHvalid.
apply valid_weaken with hyps2.
assumption.
propositional.
first_order.
apply ValidImplyElim with p1.
apply IHvalid1; assumption.
apply IHvalid2; assumption.
apply ValidExcludedMiddle.
Qed.
Fixpoint varsOf (p : prop) : list var :=
match p with
| Truth
| Falsehood => []
| Var x => [x]
| And p1 p2
| Or p1 p2
| Imply p1 p2 => varsOf p1 ++ varsOf p2
end.
Lemma interp_valid'' : forall p hyps,
(forall x, In x (varsOf p) -> hyps (Var x) \/ hyps (Not (Var x)))
-> (forall x, hyps (Var x) -> ~hyps (Not (Var x)))
-> IF interp (fun x => hyps (Var x)) p
then valid hyps p
else valid hyps (Not p).
Proof.
induct p; unfold IF_then_else; simplify.
left; propositional.
apply ValidTruthIntro.
right; propositional.
apply ValidImplyIntro.
apply ValidHyp.
propositional.
specialize (H x); propositional.
left; propositional.
apply ValidHyp.
assumption.
right; first_order.
apply ValidHyp.
assumption.
excluded_middle (interp (fun x => hyps (Var x)) p1).
excluded_middle (interp (fun x => hyps (Var x)) p2).
left; propositional.
apply ValidAndIntro.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
right; propositional.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
apply ValidImplyIntro.
apply ValidImplyElim with p2.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidAndElim2 with p1.
apply ValidHyp.
propositional.
right; propositional.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H2; propositional.
apply ValidImplyIntro.
apply ValidImplyElim with p1.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidAndElim1 with p2.
apply ValidHyp.
propositional.
excluded_middle (interp (fun x => hyps (Var x)) p1).
left; propositional.
apply ValidOrIntro1.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H2; propositional.
excluded_middle (interp (fun x => hyps (Var x)) p2).
left; propositional.
apply ValidOrIntro2.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
right; propositional.
apply ValidImplyIntro.
apply ValidOrElim with p1 p2.
apply ValidHyp.
propositional.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
apply ValidImplyElim with p1.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidHyp.
propositional.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
apply ValidImplyElim with p2.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidHyp.
propositional.
excluded_middle (interp (fun x => hyps (Var x)) p1).
excluded_middle (interp (fun x => hyps (Var x)) p2).
left; propositional.
apply ValidImplyIntro.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
apply valid_weaken with hyps.
assumption.
propositional.
right; propositional.
apply ValidImplyIntro.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H3; propositional.
assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
apply IHp2; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H4; propositional.
apply ValidImplyElim with p2.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidImplyElim with p1.
apply ValidHyp.
propositional.
apply valid_weaken with hyps.
assumption.
propositional.
left; propositional.
apply ValidImplyIntro.
assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
apply IHp1; propositional.
apply H.
apply in_or_app; propositional.
unfold IF_then_else in H2; propositional.
apply ValidFalsehoodElim.
apply ValidImplyElim with p1.
apply valid_weaken with hyps.
assumption.
propositional.
apply ValidHyp.
propositional.
Qed.
Lemma interp_valid' : forall p leftToDo alreadySplit,
(forall x, In x (varsOf p) -> In x (alreadySplit ++ leftToDo))
-> forall hyps, (forall x, In x alreadySplit -> hyps (Var x) \/ hyps (Not (Var x)))
-> (forall x, hyps (Var x) \/ hyps (Not (Var x)) -> In x alreadySplit)
-> (forall x, hyps (Var x) -> ~hyps (Not (Var x)))
-> (forall vars : var -> Prop,
(forall x, hyps (Var x) -> vars x)
-> (forall x, hyps (Not (Var x)) -> ~vars x)
-> interp vars p)
-> valid hyps p.
Proof.
induct leftToDo; simplify.
rewrite app_nil_r in H.
assert (IF interp (fun x : var => hyps (Var x)) p then valid hyps p else valid hyps (Not p)).
apply interp_valid''; first_order.
unfold IF_then_else in H4; propositional.
exfalso.
apply H4.
apply H3.
propositional.
first_order.
excluded_middle (In a alreadySplit).
apply IHleftToDo with alreadySplit; simplify.
apply H in H5.
apply in_app_or in H5.
simplify.
apply in_or_app.
propositional; subst.
propositional.
first_order.
first_order.
first_order.
first_order.
apply ValidOrElim with (Var a) (Not (Var a)).
apply ValidExcludedMiddle.
apply IHleftToDo with (alreadySplit ++ [a]); simplify.
apply H in H5.
apply in_app_or in H5.
simplify.
apply in_or_app.
propositional; subst.
left; apply in_or_app; propositional.
left; apply in_or_app; simplify; propositional.
apply in_app_or in H5.
simplify.
propositional; subst.
apply H0 in H6.
propositional.
propositional.
propositional.
invert H5.
apply in_or_app.
simplify.
propositional.
apply in_or_app.
simplify.
first_order.
invert H5.
apply in_or_app.
simplify.
first_order.
propositional.
invert H5.
invert H7.
first_order.
invert H5.
first_order.
apply H3.
first_order.
first_order.
apply IHleftToDo with (alreadySplit ++ [a]); simplify.
apply H in H5.
apply in_app_or in H5.
simplify.
apply in_or_app.
propositional; subst.
left; apply in_or_app; propositional.
left; apply in_or_app; simplify; propositional.
apply in_app_or in H5.
simplify.
propositional; subst.
apply H0 in H6.
propositional.
propositional.
propositional.
invert H5.
apply in_or_app.
simplify.
first_order.
invert H5.
apply in_or_app.
simplify.
propositional.
apply in_or_app.
simplify.
first_order.
propositional.
invert H7.
invert H7.
invert H5.
first_order.
first_order.
apply H3.
first_order.
first_order.
Qed.
Theorem interp_valid : forall p,
(forall vars, interp vars p)
-> valid (fun _ => False) p.
Proof.
simplify.
apply interp_valid' with (varsOf p) []; simplify; first_order.
Qed.
*)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 14: Separation Logic * Chapter 15: Separation Logic
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 20: Session Types * Chapter 21: Session Types
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 17: Operational Semantics for Shared-Memory Concurrency * Chapter 18: Operational Semantics for Shared-Memory Concurrency
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 5: Transition Systems * Chapter 6: Transition Systems
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 4: Transition Systems * Chapter 6: Transition Systems
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -1,5 +1,5 @@
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/> (** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 11: Types and Mutation * Chapter 12: Types and Mutation
* Author: Adam Chlipala * Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *) * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

View file

@ -12,32 +12,33 @@ FrapWithoutSets.v
Frap.v Frap.v
BasicSyntax_template.v BasicSyntax_template.v
BasicSyntax.v BasicSyntax.v
Polymorphism.v
Polymorphism_template.v Polymorphism_template.v
DataAbstraction.v Polymorphism.v
DataAbstraction_template.v DataAbstraction_template.v
DataAbstraction.v
Interpreters_template.v Interpreters_template.v
Interpreters.v Interpreters.v
FirstClassFunctions_template.v FirstClassFunctions_template.v
FirstClassFunctions.v FirstClassFunctions.v
RuleInduction_template.v
RuleInduction.v RuleInduction.v
TransitionSystems_template.v TransitionSystems_template.v
TransitionSystems.v TransitionSystems.v
IntroToProofScripting.v
IntroToProofScripting_template.v IntroToProofScripting_template.v
IntroToProofScripting.v
ModelChecking_template.v ModelChecking_template.v
ModelChecking.v ModelChecking.v
ProofByReflection.v
ProofByReflection_template.v ProofByReflection_template.v
ProofByReflection.v
OperationalSemantics_template.v OperationalSemantics_template.v
OperationalSemantics.v OperationalSemantics.v
LogicProgramming.v
LogicProgramming_template.v LogicProgramming_template.v
LogicProgramming.v
AbstractInterpretation.v AbstractInterpretation.v
CompilerCorrectness.v
CompilerCorrectness_template.v CompilerCorrectness_template.v
SubsetTypes.v CompilerCorrectness.v
SubsetTypes_template.v SubsetTypes_template.v
SubsetTypes.v
LambdaCalculusAndTypeSoundness_template.v LambdaCalculusAndTypeSoundness_template.v
LambdaCalculusAndTypeSoundness.v LambdaCalculusAndTypeSoundness.v
DependentInductiveTypes_template.v DependentInductiveTypes_template.v

View file

@ -1269,8 +1269,10 @@ The most important extension to the interpreter is that it now takes in a valuat
\denote{\phi_1 \Rightarrow \phi_2}v &=& \denote{\phi_1}v \Rightarrow \denote{\phi_2}v \denote{\phi_1 \Rightarrow \phi_2}v &=& \denote{\phi_1}v \Rightarrow \denote{\phi_2}v
\end{eqnarray*} \end{eqnarray*}
To indicate that $\phi$ is a \emph{tautology}\index{tautology} (that is, true under any values of the variables), we write $\denote{\phi}$, as a kind of abuse of notation expanding to $\forall v. \; \denote{\phi}v$.
\begin{theorem}[Soundness] \begin{theorem}[Soundness]
If $\vdash \phi$, then $\denote{\phi}v$ for any $v$. If $\vdash \phi$, then $\denote{\phi}$.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
By appeal to Lemma \ref{valid_interp'}. By appeal to Lemma \ref{valid_interp'}.
@ -1287,7 +1289,7 @@ The other direction, completeness, is quite a bit more involved, and indeed its
The basic idea is to do a proof by exhaustive case analysis over the truth values of all propositional variables $p$ that appear in a formula. The basic idea is to do a proof by exhaustive case analysis over the truth values of all propositional variables $p$ that appear in a formula.
\begin{theorem}[Completeness] \begin{theorem}[Completeness]
If $\denote{\phi}v$ for all $v$, then $\vdash \phi$. If $\denote{\phi}$, then $\vdash \phi$.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
By appeal to Lemma \ref{interp_valid'}. By appeal to Lemma \ref{interp_valid'}.