type top. type (arrow A B) :- type A, type B. of (app M N) B :- of M (arrow A B), of N A. of (abs A R) (arrow A B) :- type A, pi x\ (of x A => of (R x) B). % We add [type A] in order to reflect typing during reasoning. % Without this we would have to worry about judgments like % of (abs bad (x\ x)) (arrow bad bad) % which doesn't make sense since bad isn't a valid type. % We add this since Girard's proof assumes we can always find % at least one element in the reducability relation of c A :- type A. step (app M N) (app M' N) :- step M M'. step (app M N) (app M N') :- step N N'. step (app (abs A R) M) (R M). step (abs A R) (abs A R') :- pi x\ step (R x) (R' x).
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%% Girard's proof of strong normalization of the simply-typed lambda-calculus %% %% This is based on the presentation in Girard's Proofs and Types. %% Conventions: %% Types: A, B, C %% Terms: M, N, P, R, U, V %% Contexts: L, K Define sn M := forall N, {step M N} -> sn N. Define neutral M := forall A R, M = abs A R -> false. %% Reduce is the logical relation used to prove normalization. %% %% This definition uses itself recursively in a negative context (the %% left side of an implication). In order to guarantee consistency of %% the system we must ensure this definition is stratified. Note the %% second argument to reduce is strictly smaller in the (negative) %% recursive occurrence, thus the definition is indeed stratified. Define reduce M top := {of M top} /\ sn M. Define reduce M (arrow A B) := {of M (arrow A B)} /\ (forall U, reduce U A -> reduce (app M U) B). Theorem reduce_of : forall A M, reduce M A -> {of M A}. [Show Proof][Hide Proof] intros. case H1. search. search. % ctx defines the context of "of M A" judgments. The judgment {type A} % ensures that all types in the context are well-formed, e.g. they do % not contain nominal constants. Define ctx nil. Define nabla x, ctx (of x A :: L) := {type A} /\ ctx L. Define nabla x, name x. Theorem ctx_member : forall E L, ctx L -> member E L -> exists X A, E = of X A /\ name X /\ {type A}. [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. case H2. search. apply IH to H4 H5. search. Theorem type_ignores_ctx : forall L A, ctx L -> {L |- type A} -> {type A}. [Show Proof][Hide Proof] induction on 2. intros. case H2. apply ctx_member to H1 H3. search. apply IH to H1 H3. apply IH to H1 H4. search. % Some nice inversion lemmas for typing judgements in a non-empty context Theorem case_of_app : forall L M N B, ctx L -> {L |- of (app M N) B} -> exists A, {L |- of M (arrow A B)} /\ {L |- of N A}. [Show Proof][Hide Proof] intros. case H2. apply ctx_member to H1 H3. case H4. search. Theorem case_of_abs : forall L M A B, ctx L -> {L |- of (abs A M) B} -> exists C, B = arrow A C /\ {type A} /\ nabla x, {L, of x A |- of (M x) C}. [Show Proof][Hide Proof] intros. case H2. apply ctx_member to H1 H3. case H4. apply type_ignores_ctx to H1 H3. search. % Subject reduction - generalized version Theorem of_step_ext : forall L M N A, ctx L -> {L |- of M A} -> {step M N} -> {L |- of N A}. [Show Proof][Hide Proof] induction on 3. intros. case H3. apply case_of_app to H1 H2. apply IH to H1 H5 H4. search. apply case_of_app to H1 H2. apply IH to H1 H6 H4. search. apply case_of_app to H1 H2. apply case_of_abs to H1 H4. inst H7 with n1 = M1. cut H8 with H5. search. apply case_of_abs to H1 H2. apply IH to _ H6 H4. search. Theorem of_step : forall M N A, {of M A} -> {step M N} -> {of N A}. [Show Proof][Hide Proof] intros. apply of_step_ext to _ H1 H2. search. Theorem sn_step : forall M N, sn M -> {step M N} -> sn N. [Show Proof][Hide Proof] intros. case H1. apply H3 to H2. search. % CR2 Theorem reduce_step : forall M N A, reduce M A -> {step M N} -> reduce N A. [Show Proof][Hide Proof] induction on 1. intros. case H1. apply of_step to H3 H2. apply sn_step to H4 H2. search. unfold. apply of_step to H3 H2. search. intros. apply H4 to H5. apply IH to H6 _. search. Theorem sn_app_c : forall M, sn (app M c) -> sn M. [Show Proof][Hide Proof] induction on 1. intros. case H1. unfold. intros. assert {step (app M c) (app N c)}. apply H2 to H4. apply IH to H5. search. % CR1 and CR3 are mutually recursive, Theorem cr1_cr3 : (forall A M, {type A} -> reduce M A -> sn M) /\ (forall A M, neutral M -> {of M A} -> {type A} -> (forall P, {step M P} -> reduce P A) -> reduce M A). [Show Proof][Hide Proof] induction on 1 3. split*. % CR1 intros. case H2. % Atomic case search. % Arrow case case H1. assert (neutral c). unfold. intros. case H7. assert {of c A1}. assert (forall P, {step c P} -> reduce P A1). intros. case H9. apply IH1 to H7 H8 H5 H9. apply H4 to H10. apply IH to H6 H11. apply sn_app_c to H12. search. % CR3 intros. case H4 (keep). % Atomic type unfold. search. unfold. intros. apply H5 to H6. apply H1 to H4 H7. search. % Arrow type unfold. search. intros. apply IH to H6 H8. % We want to induct on (sn U) so we have to abstract the % variable U out entirely. assert forall U, sn U -> reduce U A1 -> reduce (app M U) B. induction on 1. intros. case H10. assert forall P, {step (app M U1) P} -> reduce P B. intros. case H13. apply H5 to H14. case H15. apply H17 to H11. search. apply H12 to H14. apply reduce_step to H11 H14. apply IH2 to H15 H16. search. case H2. apply H14 to _. assert neutral (app M U1). unfold. intros. case H14. assert {of (app M U1) B}. apply reduce_of to H11. search. apply IH1 to H14 H15 H7 H13. search. apply H10 to H9 H8. search. % CR1 Theorem reduce_sn : forall A M, {type A} -> reduce M A -> sn M. [Show Proof][Hide Proof] apply cr1_cr3. search. % CR3 Theorem neutral_step_reduce : forall A M, neutral M -> {of M A} -> {type A} -> (forall P, {step M P} -> reduce P A) -> reduce M A. [Show Proof][Hide Proof] apply cr1_cr3. search. Theorem of_type_ext : forall L M A, ctx L -> {L |- of M A} -> {type A}. [Show Proof][Hide Proof] induction on 2. intros. case H2. apply ctx_member to H1 H3. search. apply IH to H1 H3. case H5. search. apply type_ignores_ctx to H1 H3. apply IH to _ H4. search. apply type_ignores_ctx to H1 H3. search. Theorem of_type : forall M A, {of M A} -> {type A}. [Show Proof][Hide Proof] intros. apply of_type_ext to _ H1. search. Theorem reduce_const : forall C, {type C} -> reduce c C. [Show Proof][Hide Proof] intros. assert (neutral c). unfold. intros. case H2. assert (forall P, {step c P} -> reduce P C). intros. case H3. apply neutral_step_reduce to H2 _ H1 H3. search. Theorem abs_step_reduce_lemma : forall U M A B, sn U -> sn (M c) -> reduce U A -> (forall V, reduce V A -> reduce (M V) B) -> {of (abs A M) (arrow A B)} -> reduce (app (abs A M) U) B. [Show Proof][Hide Proof] induction on 1. induction on 2. intros. assert (forall P, {step (app (abs A M) U) P} -> reduce P B). intros. case H6. % Case 1: (app (abs A M) U) -> (app M' U) case H7. inst H8 with n1 = c. case H2. apply H10 to H9. apply IH1 to H1 H11 H3 _ _ with M = R', B = B. apply of_step to H5 _. search. intros. apply H4 to H12. inst H8 with n1 = V. apply reduce_step to H13 H14. search. search. % Case 2: (app (abs A M) U) -> (app (abs A M) N') case H1. apply H8 to H7. apply reduce_step to H3 H7. apply IH to H9 H2 H10 H4 H5 with M = M. search. % Case 3: (app (abs A M) U) -> (M U) apply H4 to H3. search. assert neutral (app (abs A M) U). unfold. intros. case H7. assert {of (app (abs A M) U) B}. apply reduce_of to H3. search. apply of_type to H8. apply neutral_step_reduce to H7 H8 H9 H6. search. Theorem abs_step_reduce : forall M A B, {of (abs A M) (arrow A B)} -> (forall V, reduce V A -> reduce (M V) B) -> reduce (abs A M) (arrow A B). [Show Proof][Hide Proof] intros. unfold. search. intros. apply of_type to H1. case H4. apply reduce_const to H5. apply H2 to H7. apply reduce_sn to H5 H3. apply reduce_sn to H6 H8. apply abs_step_reduce_lemma to H9 H10 H3 H2 H1 with M = M. search. %% Properties about nominal constants, e.g. where they can and cannot occur. % A term is closed if it can be typed in the empty context Define closed M := exists A, {of M A}. Define nabla x, fresh x M. Theorem member_fresh : forall X L E, member E L -> fresh X L -> fresh X E. [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. search. assert fresh X L1. case H2. search. apply IH to H3 H4. search. % Types do not contain nominal constants, thus any dependencies on % nominal constants can be pruned Theorem prune_type : forall A, nabla x, {type (A x)} -> exists B, A = (x\ B). [Show Proof][Hide Proof] induction on 1. intros. case H1. search. apply IH to H2. apply IH to H3. search. % A term cannot contain a nominal variable which does not appear in % its typing context L. Theorem prune_of : forall L R A, nabla x, ctx L -> {L |- of (R x) (A x)} -> exists M, R = (x\ M). [Show Proof][Hide Proof] induction on 2. intros. case H2. apply member_fresh to H3 _. case H4. search. apply IH to H1 H3. apply IH to H1 H4. search. apply type_ignores_ctx to H1 H3. apply prune_type to H5. apply IH to _ H4. search. search. % Closed terms cannot contain any nominal constants Theorem prune_closed : forall R, nabla x, closed (R x) -> exists M, R = (x\ M). [Show Proof][Hide Proof] intros. case H1. apply prune_of to _ H2. search. Theorem reduce_closed : forall M A, reduce M A -> closed M. [Show Proof][Hide Proof] intros. apply reduce_of to H1. search. Theorem prune_reduce : forall R A, nabla x, reduce (R x) A -> exists M, R = (x\ M). [Show Proof][Hide Proof] intros. apply reduce_closed to H1. apply prune_closed to H2. search. %% Now we need to state the generalize theorem. This will require reasoning %% about all possible closed instantiations of an open term. % Suppose that {L |- of M A} is true. Then M is an open term with nominal % constants that are listed in L. The judgment "subst L M N" holds for all % instantiations N of the nominal constants in M with terms that satisfy % the reduce relation for their respective types. The key to defining this % judgment is the use of nabla in the head of a definition which extracts % a nominal constant from L and M. The definition of subst then substitutes % for this nominal constant and continues processing the list L. Define subst nil M M. Define nabla x, subst (of x A :: L) (R x) M := exists U, reduce U A /\ subst L (R U) M. % Subst on a closed term has no effect Theorem closed_subst : forall L M N, closed M -> subst L M N -> M = N. [Show Proof][Hide Proof] induction on 2. intros. case H2. search. apply prune_closed to H1. apply IH to H1 H4. search. Theorem subst_const : forall L M, subst L c M -> M = c. [Show Proof][Hide Proof] induction on 1. intros. case H1. search. apply IH to H3. search. Theorem subst_var : forall L M N A, ctx L -> member (of M A) L -> subst L M N -> reduce N A. [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. case H2. case H3. apply reduce_closed to H6. apply closed_subst to H8 H7. search. case H3. apply member_fresh to H6 _. case H9. apply IH to H5 H6 H8. search. Theorem subst_app : forall L M N R, ctx L -> subst L (app M N) R -> exists MR NR, R = app MR NR /\ subst L M MR /\ subst L N NR. [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. search. case H2. apply IH to H4 H6. search. % This theorem has a {type A} hypothesis to ensure that no nominal variables % can appear in A and thus substitutions do not affect A Theorem subst_abs : forall L M R A, ctx L -> subst L (abs A M) R -> {type A} -> exists MR, R = abs A MR /\ (forall U, reduce U A -> nabla x, subst (of x A :: L) (M x) (MR U)). [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. exists M. split. search. intros. search. case H2. apply prune_type to H3. apply IH to H5 H7 H3. exists MR. split. search. intros. apply prune_reduce to H9. apply H8 to H9. case H10. search. Theorem subst_preserves_type : forall L M N A, ctx L -> subst L M N -> {L |- of M A} -> {of N A}. [Show Proof][Hide Proof] induction on 1. intros. case H1. case H2. search. case H2. apply reduce_of to H6. inst H3 with n1 = U. cut H9 with H8. apply of_type_ext to _ H3. apply prune_type to H11. apply IH to H5 H7 H10. search. Theorem strong_norm_ext : forall L M R A, ctx L -> {L |- of M A} -> subst L M R -> reduce R A. [Show Proof][Hide Proof] induction on 2. intros. case H2 (keep). apply subst_var to H1 H4 H3. search. apply subst_app to H1 H3. apply IH to H1 H4 H6. apply IH to H1 H5 H7. case H8. apply H11 to H9. search. apply type_ignores_ctx to H1 H4. apply subst_abs to H1 H3 H6. apply subst_preserves_type to H1 H3 H2. apply abs_step_reduce to H8 _. intros. apply type_ignores_ctx to H1 H4. apply H7 to H9. apply IH to _ H5 H11. search. search. apply subst_const to H3. apply type_ignores_ctx to H1 H4. apply reduce_const to H5. search. Theorem strong_norm : forall M A, {of M A} -> sn M. [Show Proof][Hide Proof] intros. apply strong_norm_ext to _ H1 _. apply of_type to H1. apply reduce_sn to _ H2. search.