diff --git a/theories/algorithmic.v b/theories/algorithmic.v index 6fe5fad..96b7cb5 100644 --- a/theories/algorithmic.v +++ b/theories/algorithmic.v @@ -657,18 +657,6 @@ Proof. hauto lq:on use:Sub.bind_univ_noconf. Qed. -Lemma T_AbsUniv_Imp' n Γ (a : PTm (S n)) i : - Γ ⊢ PAbs a ∈ PUniv i -> False. -Proof. - hauto lq:on use:synsub_to_usub, Sub.bind_univ_noconf, Abs_Inv. -Qed. - -Lemma T_PairUniv_Imp' n Γ (a b : PTm n) i : - Γ ⊢ PPair a b ∈ PUniv i -> False. -Proof. - hauto lq:on use:synsub_to_usub, Sub.bind_univ_noconf, Pair_Inv. -Qed. - Lemma T_AbsBind_Imp n Γ a p A0 B0 (U : PTm n) : Γ ⊢ PAbs a ∈ U -> Γ ⊢ PBind p A0 B0 ∈ U -> @@ -946,7 +934,6 @@ Proof. repeat split => //=; sfirstorder b:on use:ne_nf. Qed. - Lemma algo_metric_bind n k p0 (A0 : PTm n) B0 p1 A1 B1 : algo_metric k (PBind p0 A0 B0) (PBind p1 A1 B1) -> p0 = p1 /\ exists j, j < k /\ algo_metric j A0 A1 /\ algo_metric j B0 B1. @@ -972,16 +959,6 @@ Lemma T_Univ_Raise n Γ (a : PTm n) i j : Γ ⊢ a ∈ PUniv j. Proof. hauto lq:on rew:off use:T_Conv, Su_Univ, wff_mutual. Qed. -Lemma Bind_Univ_Inv n Γ p (A : PTm n) B i : - Γ ⊢ PBind p A B ∈ PUniv i -> - Γ ⊢ A ∈ PUniv i /\ funcomp (ren_PTm shift) (scons A Γ) ⊢ B ∈ PUniv i. -Proof. - move /Bind_Inv. - move => [i0][hA][hB]h. - move /synsub_to_usub : h => [_ [_ /Sub.univ_inj ? ]]. - sfirstorder use:T_Univ_Raise. -Qed. - Lemma Abs_Pi_Inv n Γ (a : PTm (S n)) A B : Γ ⊢ PAbs a ∈ PBind PPi A B -> funcomp (ren_PTm shift) (scons A Γ) ⊢ a ∈ B. @@ -1401,248 +1378,3 @@ with CoqLEq_R {n} : PTm n -> PTm n -> Prop := (* ----------------------- *) a ≪ b where "a ≪ b" := (CoqLEq_R a b) and "a ⋖ b" := (CoqLEq a b). - -Scheme coqleq_ind := Induction for CoqLEq Sort Prop - with coqleq_r_ind := Induction for CoqLEq_R Sort Prop. - -Combined Scheme coqleq_mutual from coqleq_ind, coqleq_r_ind. - -Definition salgo_metric {n} k (a b : PTm n) := - exists i j va vb, nsteps LoRed.R i a va /\ nsteps LoRed.R j b vb /\ nf va /\ nf vb /\ ESub.R va vb /\ size_PTm va + size_PTm vb + i + j <= k. - -Lemma salgo_metric_algo_metric n k (a b : PTm n) : - ishne a \/ ishne b -> - salgo_metric k a b -> - algo_metric k a b. -Proof. - move => h. - move => [i][j][va][vb][hva][hvb][nva][nvb][hS]sz. - rewrite/ESub.R in hS. - move : hS => [va'][vb'][h0][h1]h2. - suff : va' = vb' by sauto lq:on. - have {}hva : rtc RERed.R a va by hauto lq:on use:@relations.rtc_nsteps, REReds.FromRReds, LoReds.ToRReds. - have {}hvb : rtc RERed.R b vb by hauto lq:on use:@relations.rtc_nsteps, REReds.FromRReds, LoReds.ToRReds. - apply REReds.FromEReds in h0, h1. - have : ishne va' \/ ishne vb' by - hauto lq:on rew:off use:@relations.rtc_transitive, REReds.hne_preservation. - hauto lq:on use:Sub1.hne_refl. -Qed. - -Lemma coqleq_sound_mutual : forall n, - (forall (a b : PTm n), a ⋖ b -> forall Γ i, Γ ⊢ a ∈ PUniv i -> Γ ⊢ b ∈ PUniv i -> Γ ⊢ a ≲ b ) /\ - (forall (a b : PTm n), a ≪ b -> forall Γ i, Γ ⊢ a ∈ PUniv i -> Γ ⊢ b ∈ PUniv i -> Γ ⊢ a ≲ b ). -Proof. - apply coqleq_mutual. - - hauto lq:on use:wff_mutual ctrs:LEq. - - move => n A0 A1 B0 B1 hA ihA hB ihB Γ i. - move /Bind_Univ_Inv => [hA0]hB0 /Bind_Univ_Inv [hA1]hB1. - have hlA : Γ ⊢ A1 ≲ A0 by sfirstorder. - have hΓ : ⊢ Γ by sfirstorder use:wff_mutual. - apply Su_Transitive with (B := PBind PPi A1 B0). - by apply : Su_Pi; eauto using E_Refl, Su_Eq. - apply : Su_Pi; eauto using E_Refl, Su_Eq. - apply : ihB; eauto using ctx_eq_subst_one. - - move => n A0 A1 B0 B1 hA ihA hB ihB Γ i. - move /Bind_Univ_Inv => [hA0]hB0 /Bind_Univ_Inv [hA1]hB1. - have hlA : Γ ⊢ A0 ≲ A1 by sfirstorder. - have hΓ : ⊢ Γ by sfirstorder use:wff_mutual. - apply Su_Transitive with (B := PBind PSig A0 B1). - apply : Su_Sig; eauto using E_Refl, Su_Eq. - apply : ihB; by eauto using ctx_eq_subst_one. - apply : Su_Sig; eauto using E_Refl, Su_Eq. - - sauto lq:on use:coqeq_sound_mutual, Su_Eq. - - move => n a a' b b' ? ? ? ih Γ i ha hb. - have /Su_Eq ? : Γ ⊢ a ≡ a' ∈ PUniv i by sfirstorder use:HReds.ToEq. - have /E_Symmetric /Su_Eq ? : Γ ⊢ b ≡ b' ∈ PUniv i by sfirstorder use:HReds.ToEq. - suff : Γ ⊢ a' ≲ b' by eauto using Su_Transitive. - eauto using HReds.preservation. -Qed. - -Lemma salgo_metric_case n k (a b : PTm n) : - salgo_metric k a b -> - (ishf a \/ ishne a) \/ exists k' a', HRed.R a a' /\ salgo_metric k' a' b /\ k' < k. -Proof. - move=>[i][j][va][vb][h0] [h1][h2][h3][[v [h4 h5]]] h6. - case : a h0 => //=; try firstorder. - - inversion h0 as [|A B C D E F]; subst. - hauto qb:on use:ne_hne. - inversion E; subst => /=. - + hauto lq:on use:HRed.AppAbs unfold:algo_metric solve+:lia. - + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:algo_metric solve+:lia. - + sfirstorder qb:on use:ne_hne. - - inversion h0 as [|A B C D E F]; subst. - hauto qb:on use:ne_hne. - inversion E; subst => /=. - + hauto lq:on use:HRed.ProjPair unfold:algo_metric solve+:lia. - + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:algo_metric solve+:lia. -Qed. - -Lemma CLE_HRedL n (a a' b : PTm n) : - HRed.R a a' -> - a' ≪ b -> - a ≪ b. -Proof. - hauto lq:on ctrs:rtc, CoqLEq_R inv:CoqLEq_R. -Qed. - -Lemma CLE_HRedR n (a a' b : PTm n) : - HRed.R a a' -> - b ≪ a' -> - b ≪ a. -Proof. - hauto lq:on ctrs:rtc, CoqLEq_R inv:CoqLEq_R. -Qed. - - -Lemma algo_metric_caseR n k (a b : PTm n) : - salgo_metric k a b -> - (ishf b \/ ishne b) \/ exists k' b', HRed.R b b' /\ salgo_metric k' a b' /\ k' < k. -Proof. - move=>[i][j][va][vb][h0] [h1][h2][h3][[v [h4 h5]]] h6. - case : b h1 => //=; try by firstorder. - - inversion 1 as [|A B C D E F]; subst. - hauto qb:on use:ne_hne. - inversion E; subst => /=. - + hauto q:on use:HRed.AppAbs unfold:salgo_metric solve+:lia. - + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:salgo_metric solve+:lia. - + sfirstorder qb:on use:ne_hne. - - inversion 1 as [|A B C D E F]; subst. - hauto qb:on use:ne_hne. - inversion E; subst => /=. - + hauto lq:on use:HRed.ProjPair unfold:algo_metric solve+:lia. - + hauto q:on ctrs:HRed.R use: hf_hred_lored unfold:algo_metric solve+:lia. -Qed. - -Lemma salgo_metric_sub n k (a b : PTm n) : - salgo_metric k a b -> - Sub.R a b. -Proof. - rewrite /algo_metric. - move => [i][j][va][vb][h0][h1][h2][h3][[va' [vb' [hva [hvb hS]]]]]h5. - have {}h0 : rtc LoRed.R a va by hauto lq:on use:@relations.rtc_nsteps. - have {}h1 : rtc LoRed.R b vb by hauto lq:on use:@relations.rtc_nsteps. - apply REReds.FromEReds in hva,hvb. - apply LoReds.ToRReds in h0,h1. - apply REReds.FromRReds in h0,h1. - rewrite /Sub.R. exists va', vb'. sfirstorder use:@relations.rtc_transitive. -Qed. - -Lemma salgo_metric_pi n k (A0 : PTm n) B0 A1 B1 : - salgo_metric k (PBind PPi A0 B0) (PBind PPi A1 B1) -> - exists j, j < k /\ salgo_metric j A1 A0 /\ salgo_metric j B0 B1. -Proof. - move => [i][j][va][vb][h0][h1][h2][h3][h4]h5. - move : lored_nsteps_bind_inv h0 => /[apply]. - move => [i0][i1][a2][b2][?][?][?][ha02]hb02. subst. - move : lored_nsteps_bind_inv h1 => /[apply]. - move => [j0][j1][a3][b3][?][?][?][ha13]hb13. subst. - move /ESub.pi_inj : h4 => [? ?]. - hauto qb:on solve+:lia. -Qed. - -Lemma salgo_metric_sig n k (A0 : PTm n) B0 A1 B1 : - salgo_metric k (PBind PSig A0 B0) (PBind PSig A1 B1) -> - exists j, j < k /\ salgo_metric j A0 A1 /\ salgo_metric j B0 B1. -Proof. - move => [i][j][va][vb][h0][h1][h2][h3][h4]h5. - move : lored_nsteps_bind_inv h0 => /[apply]. - move => [i0][i1][a2][b2][?][?][?][ha02]hb02. subst. - move : lored_nsteps_bind_inv h1 => /[apply]. - move => [j0][j1][a3][b3][?][?][?][ha13]hb13. subst. - move /ESub.sig_inj : h4 => [? ?]. - hauto qb:on solve+:lia. -Qed. - -Lemma coqleq_complete' n k (a b : PTm n) : - salgo_metric k a b -> (forall Γ i, Γ ⊢ a ∈ PUniv i -> Γ ⊢ b ∈ PUniv i -> a ≪ b). -Proof. - move : k n a b. - elim /Wf_nat.lt_wf_ind. - move => n ih. - move => k a b /[dup] h /salgo_metric_case. - (* Cases where a and b can take steps *) - case; cycle 1. - move : k a b h. - qauto l:on use:HRed.preservation, CLE_HRedL, hred_hne. - case /algo_metric_caseR : (h); cycle 1. - qauto l:on use:HRed.preservation, CLE_HRedR, hred_hne. - (* Cases where neither a nor b can take steps *) - case => fb; case => fa. - - case : a fa h => //=; try hauto depth:1 lq:on use:T_AbsUniv_Imp', T_PairUniv_Imp'. - + case : b fb => //=; try hauto depth:1 lq:on use:T_AbsUniv_Imp', T_PairUniv_Imp'. - * move => p0 A0 B0 _ p1 A1 B1 _. - move => h. - have ? : p1 = p0 by - hauto lq:on rew:off use:salgo_metric_sub, Sub.bind_inj. - subst. - case : p0 h => //=. - ** move /salgo_metric_pi. - move => [j [hj [hA hB]]] Γ i. - move /Bind_Univ_Inv => [hA1 hB1] /Bind_Univ_Inv [hA0 hB0]. - have ihA : A0 ≪ A1 by hauto l:on. - econstructor; eauto using E_Refl; constructor=> //. - have ihA' : Γ ⊢ A0 ≲ A1 by hauto l:on use:coqleq_sound_mutual. - suff : funcomp (ren_PTm shift) (scons A0 Γ) ⊢ B1 ∈ PUniv i - by hauto l:on. - eauto using ctx_eq_subst_one. - ** move /salgo_metric_sig. - move => [j [hj [hA hB]]] Γ i. - move /Bind_Univ_Inv => [hA1 hB1] /Bind_Univ_Inv [hA0 hB0]. - have ihA : A1 ≪ A0 by hauto l:on. - econstructor; eauto using E_Refl; constructor=> //. - have ihA' : Γ ⊢ A1 ≲ A0 by hauto l:on use:coqleq_sound_mutual. - suff : funcomp (ren_PTm shift) (scons A1 Γ) ⊢ B0 ∈ PUniv i - by hauto l:on. - eauto using ctx_eq_subst_one. - * hauto lq:on use:salgo_metric_sub, Sub.bind_univ_noconf. - + case : b fb => //=; try hauto depth:1 lq:on use:T_AbsUniv_Imp', T_PairUniv_Imp'. - * hauto lq:on use:salgo_metric_sub, Sub.univ_bind_noconf. - * move => *. econstructor; eauto using rtc_refl. - hauto lq:on use:salgo_metric_sub, Sub.univ_inj, CLE_UnivCong. - (* Both cases are impossible *) - - have {}h : DJoin.R a b by - hauto lq:on use:salgo_metric_algo_metric, algo_metric_join. - case : b fb h => //=; try hauto depth:1 lq:on use:T_AbsUniv_Imp', T_PairUniv_Imp'. - + hauto lq:on use:DJoin.hne_bind_noconf. - + hauto lq:on use:DJoin.hne_univ_noconf. - - have {}h : DJoin.R b a by - hauto lq:on use:salgo_metric_algo_metric, algo_metric_join, DJoin.symmetric. - case : a fa h => //=; try hauto depth:1 lq:on use:T_AbsUniv_Imp', T_PairUniv_Imp'. - + hauto lq:on use:DJoin.hne_bind_noconf. - + hauto lq:on use:DJoin.hne_univ_noconf. - - move => Γ i ha hb. - econstructor; eauto using rtc_refl. - apply CLE_NeuNeu. move {ih}. - have {}h : algo_metric n a b by - hauto lq:on use:salgo_metric_algo_metric. - eapply coqeq_complete'; eauto. -Qed. - -Lemma coqleq_complete n Γ (a b : PTm n) : - Γ ⊢ a ≲ b -> a ≪ b. -Proof. - move => h. - suff : exists k, salgo_metric k a b by hauto lq:on use:coqleq_complete', regularity. - eapply fundamental_theorem in h. - move /logrel.SemLEq_SN_Sub : h. - move => h. - have : exists va vb : PTm n, - rtc LoRed.R a va /\ - rtc LoRed.R b vb /\ nf va /\ nf vb /\ ESub.R va vb - by hauto l:on use:Sub.standardization_lo. - move => [va][vb][hva][hvb][nva][nvb]hj. - move /relations.rtc_nsteps : hva => [i hva]. - move /relations.rtc_nsteps : hvb => [j hvb]. - exists (i + j + size_PTm va + size_PTm vb). - hauto lq:on solve+:lia. -Qed. - -Lemma coqleq_sound : forall n Γ (a b : PTm n) i j, - Γ ⊢ a ∈ PUniv i -> Γ ⊢ b ∈ PUniv j -> a ≪ b -> Γ ⊢ a ≲ b. -Proof. - move => n Γ a b i j. - have [*] : i <= i + j /\ j <= i + j by lia. - have : Γ ⊢ a ∈ PUniv (i + j) /\ Γ ⊢ b ∈ PUniv (i + j) - by sfirstorder use:T_Univ_Raise. - sfirstorder use:coqleq_sound_mutual. -Qed. diff --git a/theories/fp_red.v b/theories/fp_red.v index e89c191..b8bcc41 100644 --- a/theories/fp_red.v +++ b/theories/fp_red.v @@ -2337,17 +2337,6 @@ Module DJoin. case : c => //=. Qed. - Lemma hne_bind_noconf n (a b : PTm n) : - R a b -> ishne a -> isbind b -> False. - Proof. - move => [c [h0 h1]] h2 h3. - have {h0 h1 h2 h3} : ishne c /\ isbind c by - hauto l:on use:REReds.hne_preservation, - REReds.bind_preservation. - move => []. - case : c => //=. - Qed. - Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 : DJoin.R (PBind p0 A0 B0) (PBind p1 A1 B1) -> p0 = p1 /\ DJoin.R A0 A1 /\ DJoin.R B0 B1. @@ -2631,53 +2620,14 @@ Module Sub1. R a b -> R (subst_PTm ρ a) (subst_PTm ρ b). Proof. move => h. move : m ρ. elim : n a b /h; hauto lq:on ctrs:R. Qed. - Lemma hne_refl n (a b : PTm n) : - ishne a \/ ishne b -> R a b -> a = b. - Proof. hauto q:on inv:R. Qed. - End Sub1. -Module ESub. - Definition R {n} (a b : PTm n) := exists c0 c1, rtc ERed.R a c0 /\ rtc ERed.R b c1 /\ Sub1.R c0 c1. - - Lemma pi_inj n (A0 A1 : PTm n) B0 B1 : - R (PBind PPi A0 B0) (PBind PPi A1 B1) -> - R A1 A0 /\ R B0 B1. - Proof. - move => [u0 [u1 [h0 [h1 h2]]]]. - move /EReds.bind_inv : h0 => [A2][B2][?][h3]h4. subst. - move /EReds.bind_inv : h1 => [A3][B3][?][h5]h6. subst. - sauto lq:on rew:off inv:Sub1.R. - Qed. - - Lemma sig_inj n (A0 A1 : PTm n) B0 B1 : - R (PBind PSig A0 B0) (PBind PSig A1 B1) -> - R A0 A1 /\ R B0 B1. - Proof. - move => [u0 [u1 [h0 [h1 h2]]]]. - move /EReds.bind_inv : h0 => [A2][B2][?][h3]h4. subst. - move /EReds.bind_inv : h1 => [A3][B3][?][h5]h6. subst. - sauto lq:on rew:off inv:Sub1.R. - Qed. - -End ESub. - Module Sub. Definition R {n} (a b : PTm n) := exists c d, rtc RERed.R a c /\ rtc RERed.R b d /\ Sub1.R c d. Lemma refl n (a : PTm n) : R a a. Proof. sfirstorder use:@rtc_refl unfold:R. Qed. - Lemma ToJoin n (a b : PTm n) : - ishne a \/ ishne b -> - R a b -> - DJoin.R a b. - Proof. - move => h [c][d][h0][h1]h2. - have : ishne c \/ ishne d by hauto q:on use:REReds.hne_preservation. - hauto lq:on rew:off use:Sub1.hne_refl. - Qed. - Lemma transitive n (a b c : PTm n) : SN b -> R a b -> R b c -> R a c. Proof. rewrite /R. @@ -2829,41 +2779,4 @@ Module Sub. move => [a0][b0][h0][h1]h2. hauto ctrs:rtc use:REReds.cong', Sub1.substing. Qed. - - Lemma ToESub n (a b : PTm n) : nf a -> nf b -> R a b -> ESub.R a b. - Proof. hauto q:on use:REReds.ToEReds. Qed. - - Lemma standardization n (a b : PTm n) : - SN a -> SN b -> R a b -> - exists va vb, rtc RRed.R a va /\ rtc RRed.R b vb /\ nf va /\ nf vb /\ ESub.R va vb. - Proof. - move => h0 h1 hS. - have : exists v, rtc RRed.R a v /\ nf v by sfirstorder use:LoReds.FromSN, LoReds.ToRReds. - move => [v [hv2 hv3]]. - have : exists v, rtc RRed.R b v /\ nf v by sfirstorder use:LoReds.FromSN, LoReds.ToRReds. - move => [v' [hv2' hv3']]. - move : (hv2) (hv2') => *. - apply DJoin.FromRReds in hv2, hv2'. - move/DJoin.symmetric in hv2'. - apply FromJoin in hv2, hv2'. - have hv : R v v' by eauto using transitive. - have {}hv : ESub.R v v' by hauto l:on use:ToESub. - hauto lq:on. - Qed. - - Lemma standardization_lo n (a b : PTm n) : - SN a -> SN b -> R a b -> - exists va vb, rtc LoRed.R a va /\ rtc LoRed.R b vb /\ nf va /\ nf vb /\ ESub.R va vb. - Proof. - move => /[dup] sna + /[dup] snb. - move : standardization; repeat move/[apply]. - move => [va][vb][hva][hvb][nfva][nfvb]hj. - move /LoReds.FromSN : sna => [va' [hva' hva'0]]. - move /LoReds.FromSN : snb => [vb' [hvb' hvb'0]]. - exists va', vb'. - repeat split => //=. - have : va = va' /\ vb = vb' by sfirstorder use:red_uniquenf, LoReds.ToRReds. - case; congruence. - Qed. - End Sub.