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chap3Script.sml
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chap3Script.sml
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open HolKernel Parse boolLib bossLib;
open boolSimps metisLib basic_swapTheory relationTheory listTheory hurdUtils;
local open pred_setLib in end;
open binderLib BasicProvers nomsetTheory termTheory chap2Theory appFOLDLTheory;
open horeductionTheory
val _ = new_theory "chap3";
val SUBSET_DEF = pred_setTheory.SUBSET_DEF
(* definition from p30 *)
val beta_def = Define`beta M N = ?x body arg. (M = LAM x body @@ arg) /\
(N = [arg/x]body)`;
val _ = Unicode.unicode_version {u = UnicodeChars.beta, tmnm = "beta"}
val beta_alt = store_thm(
"beta_alt",
``!X M N. FINITE X ==>
(beta M N = ?x body arg. (M = LAM x body @@ arg) /\
(N = [arg/x] body) /\
~(x IN X))``,
SRW_TAC [][beta_def, EQ_IMP_THM] THENL [
SRW_TAC [][LAM_eq_thm] THEN
Q_TAC (NEW_TAC "z") `x INSERT FV body UNION X` THEN
MAP_EVERY Q.EXISTS_TAC [`z`, `tpm [(x,z)] body`] THEN
SRW_TAC [][] THEN
Q_TAC SUFF_TAC `tpm [(x,z)] body = [VAR z/x]body`
THEN1 SRW_TAC [][lemma15a] THEN
SRW_TAC [][GSYM fresh_tpm_subst, pmact_flip_args],
METIS_TAC []
]);
val strong_bvc_term_ind = store_thm(
"strong_bvc_term_ind",
``!P fv. (!s x. P (VAR s) x) /\
(!M N x. (!x. P M x) /\ (!x. P N x) ==> P (M @@ N) x) /\
(!v x M. ~(v IN fv x) /\ (!x. P M x) ==> P (LAM v M) x) /\
(!x. FINITE (fv x)) ==>
!t x. P t x``,
REPEAT GEN_TAC THEN STRIP_TAC THEN
Q_TAC SUFF_TAC `!t p x. P (tpm p t) x` THEN1 METIS_TAC [pmact_nil] THEN
HO_MATCH_MP_TAC simple_induction THEN SRW_TAC [][] THEN
Q.ABBREV_TAC `u = lswapstr p v` THEN
Q.ABBREV_TAC `M = tpm p t` THEN
Q_TAC (NEW_TAC "z") `u INSERT FV M UNION fv x` THEN
`LAM u M = LAM z (tpm [(z, u)] M)` by SRW_TAC [][tpm_ALPHA] THEN
`tpm [(z,u)] M = tpm ((z,u)::p) t`
by SRW_TAC [][Abbr`M`, GSYM pmact_decompose] THEN
SRW_TAC [][])
val _ = set_fixity "-b->" (Infix(NONASSOC, 450))
val _ = overload_on("-b->", ``compat_closure beta``)
val _ = set_fixity "-b->*" (Infix(NONASSOC, 450))
val _ = overload_on ("-b->*", ``RTC (-b->)``)
val ubeta_arrow = "-" ^ UnicodeChars.beta ^ "->"
val _ = Unicode.unicode_version {u = ubeta_arrow, tmnm = "-b->"}
val _ = Unicode.unicode_version {u = ubeta_arrow^"*", tmnm = "-b->*"}
Theorem permutative_beta[simp]:
permutative beta
Proof
dsimp[permutative_def, beta_def, LAM_eq_thm, PULL_EXISTS, tpm_subst]
QED
Theorem beta_tpm[simp]:
beta (tpm π M) (tpm π N) ⇔ beta M N
Proof
metis_tac[permutative_def, permutative_beta, pmact_inverse]
QED
(* slightly strengthened in the beta-case compared to the naive application
of cc_gen_ind *)
Theorem ccbeta_gen_ind:
(!v M N X. v NOTIN FV N /\ v NOTIN fv X ==>
P ((LAM v M) @@ N) ([N/v]M) X) /\
(!M1 M2 N X. (!X. P M1 M2 X) ==> P (M1 @@ N) (M2 @@ N) X) /\
(!M N1 N2 X. (!X. P N1 N2 X) ==> P (M @@ N1) (M @@ N2) X) /\
(!v M1 M2 X. v NOTIN fv X /\ (!X. P M1 M2 X) ==>
P (LAM v M1) (LAM v M2) X) /\
(!X. FINITE (fv X)) ==>
!M N. M -b-> N ==> !X. P M N X
Proof
STRIP_TAC THEN
ho_match_mp_tac cc_gen_ind >> qexists ‘fv’ >> simp[] >>
rw[beta_def] >>
rename [‘P (LAM u M @@ N) ([N/u] M) X’] >>
Q_TAC (NEW_TAC "v") ‘u INSERT FV M ∪ fv X ∪ FV N’ >>
‘LAM u M = LAM v ([VAR v/u] M)’ by simp[SIMPLE_ALPHA] >>
‘[N/u] M = [N/v]([VAR v/u] M)’ by simp[lemma15a]>> simp[]
QED
Theorem ccbeta_ind =
ccbeta_gen_ind |> SPEC_ALL
|> Q.INST [‘fv’ |-> ‘\x. X’, ‘P’ |-> ‘\M N X. P' M N’]
|> Q.INST [‘P'’ |-> ‘P’]
|> SRULE[] |> Q.GENL [‘P’, ‘X’]
val beta_substitutive = store_thm(
"beta_substitutive",
``substitutive beta``,
SRW_TAC [][substitutive_def] THEN
Q.SPEC_THEN `v INSERT FV N` ASSUME_TAC beta_alt THEN
FULL_SIMP_TAC (srw_ss()) [] THEN
Q.EXISTS_TAC `x` THEN SRW_TAC [][SUB_THM, GSYM substitution_lemma]);
val cc_beta_subst = store_thm(
"cc_beta_subst",
``!M N. M -b-> N ==> !P v. [P/v]M -b-> [P/v]N``,
METIS_TAC [beta_substitutive, compat_closure_substitutive,
substitutive_def]);
val reduction_beta_subst = store_thm(
"reduction_beta_subst",
``!M N. M -b->* N ==> !P v. [P/v]M -b->* [P/v]N``,
METIS_TAC [beta_substitutive, reduction_substitutive, substitutive_def]);
val cc_beta_FV_SUBSET = store_thm(
"cc_beta_FV_SUBSET",
``!M N. M -b-> N ==> FV N SUBSET FV M``,
HO_MATCH_MP_TAC ccbeta_ind THEN Q.EXISTS_TAC `{}` THEN
SRW_TAC [][SUBSET_DEF, FV_SUB] THEN PROVE_TAC []);
Theorem cc_beta_tpm:
!M N. M -b-> N ==> !p. tpm p M -b-> tpm p N
Proof
HO_MATCH_MP_TAC ccbeta_ind THEN qexists ‘{}’ >> SRW_TAC [][tpm_subst] THEN
METIS_TAC [compat_closure_rules, beta_def]
QED
val cc_beta_tpm_eqn = store_thm(
"cc_beta_tpm_eqn",
``tpm pi M -b-> N <=> M -b-> tpm (REVERSE pi) N``,
METIS_TAC [pmact_inverse, cc_beta_tpm]);
val cc_beta_thm = store_thm(
"cc_beta_thm",
``(!s t. VAR s -b-> t <=> F) /\
(!M N P. M @@ N -b-> P <=>
(?v M0. (M = LAM v M0) /\ (P = [N/v]M0)) \/
(?M'. (P = M' @@ N) /\ M -b-> M') \/
(?N'. (P = M @@ N') /\ N -b-> N')) /\
(!v M N. LAM v M -b-> N <=> ?N0. (N = LAM v N0) /\ M -b-> N0)``,
REPEAT CONJ_TAC THEN
SIMP_TAC (srw_ss()) [beta_def, SimpLHS, Once compat_closure_cases] THEN
REPEAT STRIP_TAC THEN EQ_TAC THEN
SRW_TAC [][] THEN SRW_TAC [][] THENL [
PROVE_TAC [],
PROVE_TAC [],
REPEAT (POP_ASSUM MP_TAC) THEN
Q_TAC SUFF_TAC
`!v w M N P.
(LAM v M = LAM w N) ==>
compat_closure beta N P ==>
?M0. (LAM w P = LAM v M0) /\
compat_closure beta M M0` THEN1 PROVE_TAC [] THEN
SRW_TAC [][LAM_eq_thm] THEN Q.EXISTS_TAC `tpm [(v,w)] P` THEN
SRW_TAC [][pmact_flip_args, cc_beta_tpm_eqn] THEN
METIS_TAC [cc_beta_FV_SUBSET, SUBSET_DEF],
PROVE_TAC []
]);
val ccbeta_rwt = store_thm(
"ccbeta_rwt",
``(VAR s -b-> N <=> F) /\
(LAM x M -b-> N <=> ?N0. (N = LAM x N0) /\ M -b-> N0) /\
(LAM x M @@ N -b-> P <=>
(?M'. (P = LAM x M' @@ N) /\ M -b-> M') \/
(?N'. (P = LAM x M @@ N') /\ N -b-> N') \/
(P = [N/x]M)) /\
(~is_abs M ==>
(M @@ N -b-> P <=>
(?M'. (P = M' @@ N) /\ M -b-> M') \/
(?N'. (P = M @@ N') /\ N -b-> N')))``,
SRW_TAC [][cc_beta_thm] THENL [
SRW_TAC [][EQ_IMP_THM, LAM_eq_thm] THEN SRW_TAC [][] THENL [
METIS_TAC [fresh_tpm_subst, lemma15a],
SRW_TAC [boolSimps.DNF_ss][tpm_eqr]
],
Q_TAC SUFF_TAC `!v M'. M <> LAM v M'` THEN1 METIS_TAC[] THEN
Q.SPEC_THEN `M` FULL_STRUCT_CASES_TAC term_CASES THEN
FULL_SIMP_TAC (srw_ss()) []
]);
Theorem ccbeta_LAMl_rwt :
!vs M N. LAMl vs M -b-> N <=> ?M'. N = LAMl vs M' /\ M -b-> M'
Proof
Induct_on ‘vs’
>> rw [ccbeta_rwt] (* only one goal left *)
>> EQ_TAC >> rw []
>- (Q.EXISTS_TAC ‘M'’ >> art [])
>> Q.EXISTS_TAC ‘LAMl vs M'’ >> rw []
QED
val beta_normal_form_bnf = store_thm(
"beta_normal_form_bnf",
``normal_form beta = bnf``,
SIMP_TAC (srw_ss()) [FUN_EQ_THM, EQ_IMP_THM, normal_form_def,
FORALL_AND_THM] THEN
CONJ_TAC THENL [
Q_TAC SUFF_TAC `!t. ~bnf t ==> can_reduce beta t` THEN1 PROVE_TAC [] THEN
HO_MATCH_MP_TAC nc_INDUCTION2 THEN
Q.EXISTS_TAC `{}` THEN SRW_TAC [][] THENL [
PROVE_TAC [can_reduce_rules],
PROVE_TAC [can_reduce_rules],
Q_TAC SUFF_TAC `redex beta (t @@ t')` THEN1
PROVE_TAC [can_reduce_rules] THEN
SRW_TAC [][redex_def, beta_def] THEN PROVE_TAC [is_abs_thm, term_CASES],
PROVE_TAC [lemma14a, can_reduce_rules]
],
Q_TAC SUFF_TAC `!t. can_reduce beta t ==> ~bnf t` THEN1 PROVE_TAC [] THEN
HO_MATCH_MP_TAC can_reduce_ind THEN SRW_TAC [][redex_def, beta_def] THEN
SRW_TAC [][]
]);
val nf_of_def = Define`nf_of R M N <=> normal_form R N /\ conversion R M N`;
val prop3_10 = store_thm(
"prop3_10",
``!R M N.
compat_closure R M N = ?P Q c. one_hole_context c /\ (M = c P) /\
(N = c Q) /\ R P Q``,
GEN_TAC THEN SIMP_TAC (srw_ss()) [EQ_IMP_THM, FORALL_AND_THM] THEN
CONJ_TAC THENL [
HO_MATCH_MP_TAC compat_closure_ind THEN SRW_TAC [][] THENL [
MAP_EVERY Q.EXISTS_TAC [`M`,`N`,`\x.x`] THEN
SRW_TAC [][one_hole_context_rules],
MAP_EVERY Q.EXISTS_TAC [`P`,`Q`,`\t. z @@ c t`] THEN
SRW_TAC [][one_hole_context_rules],
MAP_EVERY Q.EXISTS_TAC [`P`,`Q`,`\t. c t @@ z`] THEN
SRW_TAC [][one_hole_context_rules],
MAP_EVERY Q.EXISTS_TAC [`P`,`Q`,`\t. LAM v (c t)`] THEN
SRW_TAC [][one_hole_context_rules]
],
PROVE_TAC [compat_closure_compatible, compatible_def,
compat_closure_rules]
]);
val corollary3_2_1 = store_thm(
"corollary3_2_1",
``!R M. normal_form R M ==> (!N. ~compat_closure R M N) /\
(!N. reduction R M N ==> (M = N))``,
SIMP_TAC (srw_ss()) [normal_form_def] THEN REPEAT GEN_TAC THEN
STRIP_TAC THEN
Q.SUBGOAL_THEN `!N. ~compat_closure R M N` ASSUME_TAC THENL [
GEN_TAC THEN POP_ASSUM MP_TAC THEN
CONV_TAC CONTRAPOS_CONV THEN SIMP_TAC (srw_ss())[] THEN
MAP_EVERY Q.ID_SPEC_TAC [`N`, `M`] THEN
HO_MATCH_MP_TAC compat_closure_ind THEN
PROVE_TAC [can_reduce_rules, redex_def],
ALL_TAC
] THEN ASM_SIMP_TAC (srw_ss()) [] THEN
PROVE_TAC [RTC_CASES1]);
val bnf_reduction_to_self = store_thm(
"bnf_reduction_to_self",
``bnf M ==> (M -b->* N <=> (N = M))``,
METIS_TAC [corollary3_2_1, beta_normal_form_bnf, RTC_RULES]);
local open relationTheory
in
val diamond_property_def = save_thm("diamond_property_def", diamond_def)
end
val _ = overload_on("diamond_property", ``relation$diamond``)
(* This is not the same CR as appears in relationTheory. There
CR R = diamond (RTC R)
Here,
CR R = diamond (RTC (compat_closure R))
In other words, this formulation allows us to write
CR beta
rather than having to write
CR (compat_closure beta)
*)
val CR_def = Define`CR R = diamond_property (reduction R)`;
val theorem3_13 = store_thm(
"theorem3_13",
``!R. CR R ==>
!M N. conversion R M N ==> ?Z. reduction R M Z /\ reduction R N Z``,
GEN_TAC THEN STRIP_TAC THEN SIMP_TAC (srw_ss()) [] THEN
HO_MATCH_MP_TAC equiv_closure_ind THEN REVERSE (SRW_TAC [][]) THEN1
(`?Z2. reduction R Z Z2 /\ reduction R Z' Z2` by
PROVE_TAC [CR_def, diamond_property_def] THEN
PROVE_TAC [reduction_rules]) THEN
PROVE_TAC [reduction_rules]);
val corollary3_3_1 = store_thm(
"corollary3_3_1",
``!R. CR R ==> (!M N. nf_of R M N ==> reduction R M N) /\
(!M N1 N2. nf_of R M N1 /\ nf_of R M N2 ==> (N1 = N2))``,
SRW_TAC [][nf_of_def] THENL [
PROVE_TAC [corollary3_2_1, theorem3_13],
`conversion R N1 N2` by
(FULL_SIMP_TAC (srw_ss()) [] THEN
PROVE_TAC [equiv_closure_rules]) THEN
`?Z. reduction R N1 Z /\ reduction R N2 Z` by
PROVE_TAC [theorem3_13] THEN
PROVE_TAC [corollary3_2_1]
]);
val diamond_TC = diamond_TC_diamond
val bvc_cases = store_thm(
"bvc_cases",
``!X. FINITE X ==>
!t. (?s. t = VAR s) \/ (?t1 t2. t = t1 @@ t2) \/
(?v t0. ~(v IN X) /\ (t = LAM v t0))``,
SRW_TAC [][] THEN
Q.SPEC_THEN `t` FULL_STRUCT_CASES_TAC term_CASES THEN
SRW_TAC [][LAM_eq_thm] THEN
SRW_TAC [boolSimps.DNF_ss][] THEN
SRW_TAC [][Once tpm_eqr] THEN
Q_TAC (NEW_TAC "z") `v INSERT X UNION FV t0` THEN
METIS_TAC []);
(* Definition 3.2.3 [1, p60] (one-step parallel beta-reduction) *)
Inductive grandbeta :
[~REFL[simp]:]
!M. grandbeta M M
[~ABS:]
!M M' x. grandbeta M M' ==> grandbeta (LAM x M) (LAM x M')
[~APP:]
!M N M' N'. grandbeta M M' /\ grandbeta N N' ==> grandbeta (M @@ N) (M' @@ N')
[~BETA:]
!M N M' N' x. grandbeta M M' /\ grandbeta N N' ==>
grandbeta ((LAM x M) @@ N) ([N'/x] M')
End
val _ = set_fixity "=b=>" (Infix(NONASSOC,450))
val _ = overload_on ("=b=>", ``grandbeta``)
val _ = set_fixity "=b=>*" (Infix(NONASSOC,450))
val _ = overload_on ("=b=>*", ``RTC grandbeta``)
val gbarrow = "=" ^ UnicodeChars.beta ^ "=>"
val _ = Unicode.unicode_version {u = gbarrow, tmnm = "=b=>"}
val _ = Unicode.unicode_version {u = gbarrow ^ "*", tmnm = "=b=>*"}
val grandbeta_bvc_gen_ind = store_thm(
"grandbeta_bvc_gen_ind",
``!P fv.
(!M x. P M M x) /\
(!v M1 M2 x. v NOTIN fv x /\ (!x. P M1 M2 x) ==>
P (LAM v M1) (LAM v M2) x) /\
(!M1 M2 N1 N2 x. (!x. P M1 M2 x) /\ (!x. P N1 N2 x) ==>
P (M1 @@ N1) (M2 @@ N2) x) /\
(!M1 M2 N1 N2 v x.
v NOTIN fv x /\ v NOTIN FV N1 /\ v NOTIN FV N2 /\
(!x. P M1 M2 x) /\ (!x. P N1 N2 x) ==>
P ((LAM v M1) @@ N1) ([N2/v]M2) x) /\
(!x. FINITE (fv x)) ==>
!M N. M =b=> N ==> !x. P M N x``,
REPEAT GEN_TAC THEN STRIP_TAC THEN
Q_TAC SUFF_TAC `!M N. grandbeta M N ==> !p x. P (tpm p M) (tpm p N) x`
THEN1 METIS_TAC [pmact_nil] THEN
HO_MATCH_MP_TAC grandbeta_ind THEN SRW_TAC [][] THENL [
Q.ABBREV_TAC `M' = tpm p M` THEN
Q.ABBREV_TAC `N' = tpm p N` THEN
Q.ABBREV_TAC `v = lswapstr p x` THEN
Q_TAC (NEW_TAC "z") `v INSERT fv x' UNION FV M' UNION FV N'` THEN
`(LAM v M' = LAM z (tpm ((z,v)::p) M)) /\
(LAM v N' = LAM z (tpm ((z,v)::p) N))`
by (ONCE_REWRITE_TAC [tpm_CONS] THEN
SRW_TAC [][Abbr`M'`, Abbr`N'`, tpm_ALPHA]) THEN
SRW_TAC [][],
Q.MATCH_ABBREV_TAC `P (LAM (lswapstr p v0) (tpm p ML) @@ tpm p MR)
(tpm p ([NR/v0] NL)) ctx` THEN
markerLib.RM_ALL_ABBREVS_TAC THEN
SRW_TAC [][tpm_subst] THEN
Q.ABBREV_TAC `v = lswapstr p v0` THEN
Q.ABBREV_TAC `M1 = tpm p ML` THEN
Q.ABBREV_TAC `N1 = tpm p MR` THEN
Q.ABBREV_TAC `M2 = tpm p NL` THEN
Q.ABBREV_TAC `N2 = tpm p NR` THEN
Q_TAC (NEW_TAC "z")
`v INSERT fv ctx UNION FV N1 UNION FV N2 UNION FV M1 UNION
FV M2` THEN
`LAM v M1 = LAM z (tpm [(z,v)] M1)` by SRW_TAC [][tpm_ALPHA] THEN
`[N2/v]M2 = [N2/z](tpm [(z,v)] M2)`
by SRW_TAC [][fresh_tpm_subst, lemma15a] THEN
SRW_TAC [][] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
SRW_TAC [][Abbr`N1`,Abbr`N2`] THEN
`(tpm [(z,v)] M1 = tpm ((z,v)::p) ML) /\
(tpm [(z,v)] M2 = tpm ((z,v)::p) NL)`
by SRW_TAC [][GSYM pmact_decompose, Abbr`M1`,Abbr`M2`] THEN
SRW_TAC [][]
]);
val grandbeta_bvc_ind = save_thm(
"grandbeta_bvc_ind",
(Q.GEN `P` o Q.GEN `X` o
SIMP_RULE bool_ss [] o
SPECL [``(\M:term N:term x:'a. P M N:bool)``, ``\x:'a. X:string -> bool``])
grandbeta_bvc_gen_ind);
val exercise3_3_1 = store_thm(
"exercise3_3_1",
``!M N. M -b-> N ==> M =b=> N``,
HO_MATCH_MP_TAC compat_closure_ind THEN SRW_TAC [][beta_def] THEN
PROVE_TAC [grandbeta_rules]);
val app_grandbeta = store_thm( (* property 3 on p. 37 *)
"app_grandbeta",
``!M N L. M @@ N =b=> L <=>
(?M' N'. M =b=> M' /\ N =b=> N' /\ (L = M' @@ N')) \/
(?x P P' N'. (M = LAM x P) /\ P =b=> P' /\
N =b=> N' /\ (L = [N'/x]P'))``,
REPEAT GEN_TAC THEN EQ_TAC THENL [
SIMP_TAC (srw_ss()) [SimpL ``(==>)``, Once grandbeta_cases] THEN
SIMP_TAC (srw_ss()) [DISJ_IMP_THM, GSYM LEFT_FORALL_IMP_THM,
grandbeta_rules] THEN PROVE_TAC [],
SRW_TAC [][] THEN PROVE_TAC [grandbeta_rules]
]);
val grandbeta_permutative = store_thm(
"grandbeta_permutative",
``!M N. M =b=> N ==> !pi. tpm pi M =b=> tpm pi N``,
HO_MATCH_MP_TAC grandbeta_ind THEN SRW_TAC [][tpm_subst] THEN
METIS_TAC [grandbeta_rules]);
val grandbeta_permutative_eqn = store_thm(
"grandbeta_permutative_eqn",
``tpm pi M =b=> tpm pi N <=> M =b=> N``,
METIS_TAC [pmact_inverse, grandbeta_permutative]);
val _ = export_rewrites ["grandbeta_permutative_eqn"]
val grandbeta_substitutive = store_thm(
"grandbeta_substitutive",
``!M N. M =b=> N ==> [P/x]M =b=> [P/x]N``,
HO_MATCH_MP_TAC grandbeta_bvc_ind THEN
Q.EXISTS_TAC `x INSERT FV P` THEN
SRW_TAC [][SUB_THM, grandbeta_rules] THEN
SRW_TAC [][lemma2_11, grandbeta_rules]);
val grandbeta_FV = store_thm(
"grandbeta_FV",
``!M N. M =b=> N ==> FV N SUBSET FV M``,
HO_MATCH_MP_TAC grandbeta_ind THEN
SRW_TAC [][SUBSET_DEF, FV_SUB] THEN
PROVE_TAC []);
val abs_grandbeta = store_thm(
"abs_grandbeta",
``!M N v. LAM v M =b=> N <=> ∃N0. N = LAM v N0 /\ M =b=> N0``,
REPEAT GEN_TAC THEN EQ_TAC THENL [
SIMP_TAC (srw_ss()) [Once grandbeta_cases, SimpL ``(==>)``] THEN
SIMP_TAC (srw_ss()) [DISJ_IMP_THM, grandbeta_rules] THEN
SRW_TAC [][LAM_eq_thm] THENL [
PROVE_TAC [],
SRW_TAC [][LAM_eq_thm, tpm_eqr, pmact_flip_args] THEN
PROVE_TAC [SUBSET_DEF, grandbeta_FV]
],
PROVE_TAC [grandbeta_rules]
]);
val lemma3_15 = save_thm("lemma3_15", abs_grandbeta);
val redex_grandbeta = store_thm(
"redex_grandbeta",
``LAM v M @@ N =b=> L <=>
(∃M' N'. M =b=> M' /\ N =b=> N' /\
(L = LAM v M' @@ N')) \/
(∃M' N'. M =b=> M' /\ N =b=> N' /\ (L = [N'/v]M'))``,
SRW_TAC [][app_grandbeta, EQ_IMP_THM] THENL [
PROVE_TAC [abs_grandbeta],
FULL_SIMP_TAC (srw_ss()) [LAM_eq_thm] THEN DISJ2_TAC THENL [
METIS_TAC [],
SRW_TAC [][] THEN
MAP_EVERY Q.EXISTS_TAC [`tpm [(v,x)] P'`, `N'`] THEN
SRW_TAC [][] THEN
`v NOTIN FV P'`
by METIS_TAC [grandbeta_FV, SUBSET_DEF] THEN
SRW_TAC [][fresh_tpm_subst, lemma15a]
],
METIS_TAC [grandbeta_rules],
METIS_TAC []
]);
val var_grandbeta = store_thm(
"var_grandbeta",
``!v N. VAR v =b=> N <=> (N = VAR v)``,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC [grandbeta_cases] THEN
SRW_TAC [][]);
val grandbeta_cosubstitutive = store_thm(
"grandbeta_cosubstitutive",
``!M. N =b=> N' ==> [N/x] M =b=> [N'/x] M``,
HO_MATCH_MP_TAC nc_INDUCTION2 THEN
Q.EXISTS_TAC `x INSERT FV N UNION FV N'` THEN
SRW_TAC [][grandbeta_rules, SUB_VAR]);
(* property 1 on p37, and Barendregt's lemma 3.2.4 *)
val grandbeta_subst = store_thm(
"grandbeta_subst",
``M =b=> M' /\ N =b=> N' ==> [N/x]M =b=> [N'/x]M'``,
Q_TAC SUFF_TAC
`!M M'. M =b=> M' ==> N =b=> N' ==>
[N/x] M =b=> [N'/x]M'` THEN1
METIS_TAC [] THEN
HO_MATCH_MP_TAC grandbeta_bvc_ind THEN
Q.EXISTS_TAC `x INSERT FV N UNION FV N'` THEN
SRW_TAC [][SUB_THM, grandbeta_rules] THENL [
METIS_TAC [grandbeta_cosubstitutive],
RES_TAC THEN
SRW_TAC [][lemma2_11, grandbeta_rules]
]);
val strong_grandbeta_ind =
IndDefLib.derive_strong_induction (grandbeta_rules, grandbeta_ind)
val strong_grandbeta_bvc_gen_ind =
(GEN_ALL o
SIMP_RULE (srw_ss()) [grandbeta_rules, FORALL_AND_THM,
GSYM CONJ_ASSOC] o
Q.SPEC `\M N x. P M N x /\ grandbeta M N`)
grandbeta_bvc_gen_ind
val lemma3_16 = store_thm( (* p. 37 *)
"lemma3_16",
``diamond_property grandbeta``,
Q_TAC SUFF_TAC `!M M1. M =b=> M1 ==>
!M2. M =b=> M2 ==>
?M3. M1 =b=> M3 /\ M2 =b=> M3` THEN1
PROVE_TAC [diamond_property_def] THEN
HO_MATCH_MP_TAC strong_grandbeta_bvc_gen_ind THEN Q.EXISTS_TAC `FV` THEN
SIMP_TAC (srw_ss()) [] THEN REPEAT CONJ_TAC THENL [
(* reflexive case *)
PROVE_TAC [grandbeta_rules],
(* lambda case *)
MAP_EVERY Q.X_GEN_TAC [`v`, `M`,`M1`, `M2`] THEN REPEAT STRIP_TAC THEN
`?P. (M2 = LAM v P) /\ M =b=> P` by PROVE_TAC [abs_grandbeta] THEN
SRW_TAC [][] THEN PROVE_TAC [grandbeta_rules],
(* app case *)
MAP_EVERY Q.X_GEN_TAC [`f`,`g`,`x`,`y`, `fx'`] THEN STRIP_TAC THEN
STRIP_TAC THEN
`(?f' x'. (fx' = f' @@ x') /\ f =b=> f' /\ x =b=> x') \/
(?v P P' x'. (f = LAM v P) /\ P =b=> P' /\ x =b=> x' /\
(fx' = [x'/v]P'))` by
(FULL_SIMP_TAC (srw_ss()) [app_grandbeta] THEN PROVE_TAC [])
THENL [
METIS_TAC [grandbeta_rules],
`?P2. (g = LAM v P2) /\ P =b=> P2` by
PROVE_TAC [abs_grandbeta] THEN
SRW_TAC [][] THEN
`?ff. LAM v P2 =b=> ff /\ LAM v P' =b=> ff` by
PROVE_TAC [grandbeta_rules] THEN
`?xx. y =b=> xx /\ x' =b=> xx` by PROVE_TAC [] THEN
`?PP. P' =b=> PP /\ (ff = LAM v PP)` by
PROVE_TAC [abs_grandbeta] THEN
SRW_TAC [][] THEN
`P2 =b=> PP` by PROVE_TAC [abs_grandbeta, term_11] THEN
PROVE_TAC [grandbeta_rules, grandbeta_subst]
],
(* beta-redex case *)
MAP_EVERY Q.X_GEN_TAC [`M`, `M'`, `N`, `N'`, `x`, `M2`] THEN
SRW_TAC [][redex_grandbeta] THENL [
(* other reduction didn't beta-reduce *)
`?Mfin. M' =b=> Mfin /\ M'' =b=> Mfin` by METIS_TAC [] THEN
`?Nfin. N' =b=> Nfin /\ N'' =b=> Nfin` by METIS_TAC [] THEN
Q.EXISTS_TAC `[Nfin/x]Mfin` THEN
METIS_TAC [grandbeta_rules, grandbeta_subst],
(* other reduction also beta-reduced *)
`?Mfin. M' =b=> Mfin /\ M'' =b=> Mfin` by METIS_TAC [] THEN
`?Nfin. N' =b=> Nfin /\ N'' =b=> Nfin` by METIS_TAC [] THEN
Q.EXISTS_TAC `[Nfin/x]Mfin` THEN
METIS_TAC [grandbeta_rules, grandbeta_subst]
]
]);
Theorem theorem3_17:
TC grandbeta = reduction beta
Proof
‘RTC grandbeta = reduction beta’ suffices_by
(disch_then (simp o single o GSYM) >> simp[cj 1 $ GSYM TC_RC_EQNS] >>
simp[FUN_EQ_THM, RC_DEF, EQ_IMP_THM, DISJ_IMP_THM] >>
metis_tac[reflexive_TC, reflexive_def, grandbeta_rules]) >>
irule RTC_BRACKETS_RTC_EQN >> conj_tac >~
[‘_ -β-> _ ⇒ _’] >- metis_tac[exercise3_3_1] >>
Induct_on ‘grandbeta’ >> metis_tac[reduction_rules, beta_def]
QED
Theorem RTC_grandbeta:
RTC grandbeta = reduction beta
Proof
simp[GSYM TC_RC_EQNS, theorem3_17]
QED
val beta_CR = store_thm(
"beta_CR",
``CR beta``,
PROVE_TAC [CR_def, lemma3_16, theorem3_17, diamond_TC]);
val betaCR_square = store_thm(
"betaCR_square",
``M -b->* N1 /\ M -b->* N2 ==> ?N. N1 -b->* N /\ N2 -b->* N``,
METIS_TAC [beta_CR, diamond_property_def, CR_def]);
val bnf_triangle = store_thm(
"bnf_triangle",
``M -b->* N /\ M -b->* N' /\ bnf N ==> N' -b->* N``,
METIS_TAC [betaCR_square, bnf_reduction_to_self]);
Theorem Omega_betaloops[simp] :
Omega -b-> N <=> N = Omega
Proof
FULL_SIMP_TAC (srw_ss()) [ccbeta_rwt, Omega_def]
QED
Theorem Omega_starloops[simp] :
Omega -b->* N <=> N = Omega
Proof
Suff ‘!M N. M -b->* N ==> (M = Omega) ==> (N = Omega)’
>- METIS_TAC [RTC_RULES]
>> HO_MATCH_MP_TAC RTC_INDUCT >> SRW_TAC [][]
>> FULL_SIMP_TAC std_ss [Omega_betaloops]
QED
Theorem Omega_app_betaloops[local] :
Omega @@ A -b-> N ==> ?A'. N = Omega @@ A'
Proof
‘~is_abs Omega’ by rw [Omega_def]
>> rw [ccbeta_rwt]
>- (Q.EXISTS_TAC ‘A’ >> rw [])
>> Q.EXISTS_TAC ‘N'’ >> rw []
QED
Theorem Omega_app_starloops :
Omega @@ A -β->* N ⇒ ∃A'. N = Omega @@ A'
Proof
Suff ‘!M N. M -b->* N ==> !A. M = Omega @@ A ==> ?A'. N = Omega @@ A'’
>- METIS_TAC [RTC_RULES]
>> HO_MATCH_MP_TAC RTC_INDUCT >> SRW_TAC [][]
>> ‘?A'. M' = Omega @@ A'’ by METIS_TAC [Omega_app_betaloops]
>> Q.PAT_X_ASSUM ‘!A. M' = Omega @@ A ==> P’ (MP_TAC o (Q.SPEC ‘A'’))
>> RW_TAC std_ss []
QED
Theorem Omega_appstar_betaloops[local] :
!N. Omega @* Ms -b-> N ==> ?Ms'. N = Omega @* Ms'
Proof
Induct_on ‘Ms’ using SNOC_INDUCT
>- (rw [] >> Q.EXISTS_TAC ‘[]’ >> rw [])
>> rw [appstar_SNOC]
>> ‘~is_abs (Omega @* Ms)’ by rw [Omega_def]
>> fs [ccbeta_rwt]
>- (Q.PAT_X_ASSUM ‘!N. P’ (MP_TAC o (Q.SPEC ‘M'’)) \\
RW_TAC std_ss [] \\
Q.EXISTS_TAC ‘SNOC x Ms'’ >> rw [])
>> Q.EXISTS_TAC ‘SNOC N' Ms’ >> rw []
QED
Theorem Omega_appstar_starloops :
Omega @* Ms -b->* N ==> ?Ms'. N = Omega @* Ms'
Proof
Suff ‘!M N. M -b->* N ==> !Ms. M = Omega @* Ms ==> ?Ms'. N = Omega @* Ms'’
>- METIS_TAC [RTC_RULES]
>> HO_MATCH_MP_TAC RTC_INDUCT >> SRW_TAC [][]
>- (Q.EXISTS_TAC ‘Ms’ >> rw [])
>> ‘?Ms'. M' = Omega @* Ms'’ by METIS_TAC [Omega_appstar_betaloops]
>> Q.PAT_X_ASSUM ‘!Ms. M' = Omega @* Ms ==> P’ (MP_TAC o (Q.SPEC ‘Ms'’))
>> RW_TAC std_ss []
QED
val lameq_betaconversion = store_thm(
"lameq_betaconversion",
``!M N. M == N <=> conversion beta M N``,
SIMP_TAC (srw_ss()) [EQ_IMP_THM, FORALL_AND_THM] THEN CONJ_TAC THENL [
HO_MATCH_MP_TAC lameq_ind THEN REPEAT STRIP_TAC THENL [
Q_TAC SUFF_TAC `beta (LAM x M @@ N) ([N/x] M)` THEN1
PROVE_TAC [conversion_rules] THEN
SRW_TAC [][beta_def] THEN PROVE_TAC [],
PROVE_TAC [conversion_rules],
PROVE_TAC [conversion_rules],
PROVE_TAC [conversion_rules],
PROVE_TAC [conversion_compatible, compatible_def, rightctxt,
rightctxt_thm],
PROVE_TAC [conversion_compatible, compatible_def, leftctxt],
PROVE_TAC [conversion_compatible, compatible_def, absctxt]
],
SIMP_TAC (srw_ss()) [] THEN
HO_MATCH_MP_TAC equiv_closure_ind THEN REPEAT CONJ_TAC THEN1
(HO_MATCH_MP_TAC compat_closure_ind THEN SRW_TAC [][beta_def] THEN
PROVE_TAC [lameq_rules]) THEN
PROVE_TAC [lameq_rules]
]);
val prop3_18 = save_thm("prop3_18", lameq_betaconversion);
(* |- !M N. M == N ==> ?Z. M -b->* Z /\ N -b->* Z *)
Theorem lameq_CR = REWRITE_RULE [GSYM lameq_betaconversion, beta_CR]
(Q.SPEC ‘beta’ theorem3_13)
val ccbeta_lameq = store_thm(
"ccbeta_lameq",
``!M N. M -b-> N ==> M == N``,
SRW_TAC [][lameq_betaconversion, EQC_R]);
val betastar_lameq = store_thm(
"betastar_lameq",
``!M N. M -b->* N ==> M == N``,
SRW_TAC [][lameq_betaconversion, RTC_EQC]);
val betastar_lameq_bnf = store_thm(
"betastar_lameq_bnf",
``bnf N ==> (M -b->* N <=> M == N)``,
METIS_TAC [theorem3_13, beta_CR, betastar_lameq, bnf_reduction_to_self,
lameq_betaconversion]);
(* moved here from churchnumScript.sml *)
Theorem lameq_triangle :
M == N ∧ M == P ∧ bnf N ∧ bnf P ⇒ (N = P)
Proof
METIS_TAC [betastar_lameq_bnf, lameq_rules, bnf_reduction_to_self]
QED
(* |- !M N. M =b=> N ==> M -b->* N *)
Theorem grandbeta_imp_betastar =
(REWRITE_RULE [theorem3_17] (Q.ISPEC ‘grandbeta’ TC_SUBSET))
|> (Q.SPECL [‘M’, ‘N’]) |> (Q.GENL [‘M’, ‘N’])
Theorem grandbeta_imp_lameq :
!M N. M =b=> N ==> M == N
Proof
rpt STRIP_TAC
>> MATCH_MP_TAC betastar_lameq
>> MATCH_MP_TAC grandbeta_imp_betastar >> art []
QED
(* |- !R x y z. R^+ x y /\ R^+ y z ==> R^+ x z *)
Theorem TC_TRANS[local] = REWRITE_RULE [transitive_def] TC_TRANSITIVE
Theorem abs_betastar :
!x M Z. LAM x M -b->* Z <=> ?N'. (Z = LAM x N') /\ M -b->* N'
Proof
rpt GEN_TAC
>> reverse EQ_TAC
>- (rw [] >> PROVE_TAC [reduction_rules])
(* stage work *)
>> REWRITE_TAC [SYM theorem3_17]
>> Q.ID_SPEC_TAC ‘Z’
>> HO_MATCH_MP_TAC (Q.ISPEC ‘grandbeta’ TC_INDUCT_ALT_RIGHT)
>> rpt STRIP_TAC
>- (FULL_SIMP_TAC std_ss [abs_grandbeta] \\
Q.EXISTS_TAC ‘N0’ >> art [] \\
MATCH_MP_TAC TC_SUBSET >> art [])
>> Q.PAT_X_ASSUM ‘Z = LAM x N'’ (FULL_SIMP_TAC std_ss o wrap)
>> FULL_SIMP_TAC std_ss [abs_grandbeta]
>> Q.EXISTS_TAC ‘N0’ >> art []
>> MATCH_MP_TAC TC_TRANS
>> Q.EXISTS_TAC ‘N'’ >> art []
>> MATCH_MP_TAC TC_SUBSET >> art []
QED
val lameq_consistent = store_thm(
"lameq_consistent",
``consistent $==``,
SRW_TAC [][consistent_def] THEN
MAP_EVERY Q.EXISTS_TAC [`S`,`K`] THEN STRIP_TAC THEN
`conversion beta S K` by PROVE_TAC [prop3_18] THEN
`?Z. reduction beta S Z /\ reduction beta K Z` by
PROVE_TAC [theorem3_13, beta_CR] THEN
`normal_form beta S` by PROVE_TAC [S_beta_normal, beta_normal_form_bnf] THEN
`normal_form beta K` by PROVE_TAC [K_beta_normal, beta_normal_form_bnf] THEN
`S = K` by PROVE_TAC [corollary3_2_1] THEN
FULL_SIMP_TAC (srw_ss()) [S_def, K_def]);
val has_bnf_thm = store_thm(
"has_bnf_thm",
``has_bnf M <=> ?N. M -b->* N /\ bnf N``,
EQ_TAC THENL [
METIS_TAC [lameq_betaconversion, chap2Theory.has_bnf_def, theorem3_13,
beta_CR, beta_normal_form_bnf, corollary3_2_1],
SRW_TAC [][chap2Theory.has_bnf_def, lameq_betaconversion] THEN
METIS_TAC [RTC_EQC]
]);
val Omega_reachable_no_bnf = store_thm(
"Omega_reachable_no_bnf",
``M -b->* Omega ==> ~has_bnf M``,
REPEAT STRIP_TAC THEN
FULL_SIMP_TAC (srw_ss()) [has_bnf_thm] THEN
`Omega -b->* N` by METIS_TAC [bnf_triangle] THEN
`N = Omega` by FULL_SIMP_TAC (srw_ss()) [] THEN
FULL_SIMP_TAC (srw_ss()) []);
val weak_diamond_def =
save_thm("weak_diamond_def", WCR_def)
val _ = overload_on("weak_diamond", ``relation$WCR``)
(* likewise, these definitions of WCR and SN, differ from those in
relation by wrapping the argument in a call to compat_closure
*)
val WCR_def = (* definition 3.20, p39 *) Define`
WCR R = weak_diamond (compat_closure R)
`;
val SN_def = Define`SN R = relation$SN (compat_closure R)`;
val newmans_lemma = store_thm( (* lemma3_22, p39 *)
"newmans_lemma",
``!R. SN R /\ WCR R ==> CR R``,
SIMP_TAC (srw_ss()) [SN_def, WCR_def, Newmans_lemma,
CR_def,
GSYM relationTheory.diamond_def,
GSYM relationTheory.CR_def]);
val commute_def = (* p43 *)
Define`commute R1 R2 = !x x1 x2. R1 x x1 /\ R2 x x2 ==>
?x3. R2 x1 x3 /\ R1 x2 x3`;
val commute_COMM = store_thm(
"commute_COMM",
``commute R1 R2 = commute R2 R1``,
PROVE_TAC [commute_def]);
val diamond_RC = diamond_RC_diamond
(* |- !R. diamond_property R ==> diamond_property (RC R) *)
val diamond_RTC = store_thm(
"diamond_RTC",
``!R. diamond_property R ==> diamond_property (RTC R)``,
PROVE_TAC [diamond_TC, diamond_RC, TC_RC_EQNS]);
val hr_lemma0 = prove(
``!R1 R2. diamond_property R1 /\ diamond_property R2 /\ commute R1 R2 ==>
diamond_property (RTC (R1 RUNION R2))``,
REPEAT STRIP_TAC THEN
Q_TAC SUFF_TAC `diamond_property (R1 RUNION R2)` THEN1
PROVE_TAC [diamond_RTC] THEN
FULL_SIMP_TAC (srw_ss()) [diamond_property_def, commute_def,
RUNION] THEN
PROVE_TAC []);
val CC_RUNION_MONOTONE = store_thm(
"CC_RUNION_MONOTONE",
``!R1 x y. compat_closure R1 x y ==> compat_closure (R1 RUNION R2) x y``,
GEN_TAC THEN HO_MATCH_MP_TAC compat_closure_ind THEN
PROVE_TAC [compat_closure_rules, RUNION]);
val CC_RUNION_DISTRIB = store_thm(
"CC_RUNION_DISTRIB",
``!R1 R2. compat_closure (R1 RUNION R2) =
compat_closure R1 RUNION compat_closure R2``,
REPEAT GEN_TAC THEN
Q_TAC SUFF_TAC
`(!x y. compat_closure (R1 RUNION R2) x y ==>
(compat_closure R1 RUNION compat_closure R2) x y) /\
(!x y. (compat_closure R1 RUNION compat_closure R2) x y ==>
compat_closure (R1 RUNION R2) x y)` THEN1
SIMP_TAC (srw_ss()) [FUN_EQ_THM, EQ_IMP_THM, FORALL_AND_THM] THEN
CONJ_TAC THENL [
HO_MATCH_MP_TAC compat_closure_ind THEN
PROVE_TAC [compat_closure_rules, RUNION],
SRW_TAC [][RUNION] THEN
PROVE_TAC [RUNION_COMM, CC_RUNION_MONOTONE]
]);
val hindley_rosen_lemma = store_thm( (* p43 *)
"hindley_rosen_lemma",
``(!R1 R2. diamond_property R1 /\ diamond_property R2 /\ commute R1 R2 ==>
diamond_property (RTC (R1 RUNION R2))) /\
(!R1 R2. CR R1 /\ CR R2 /\ commute (reduction R1) (reduction R2) ==>
CR (R1 RUNION R2))``,
CONJ_TAC THENL [
MATCH_ACCEPT_TAC hr_lemma0,
SRW_TAC [][CR_def] THEN
`diamond_property (RTC (RTC (compat_closure R1) RUNION
RTC (compat_closure R2)))`
by PROVE_TAC [hr_lemma0] THEN
FULL_SIMP_TAC (srw_ss()) [RTC_RUNION, CC_RUNION_DISTRIB]
]);
val eta_def =
Define`eta M N <=> ∃v. M = LAM v (N @@ VAR v) ∧ v ∉ FV N`;
val _ = Unicode.unicode_version {u = UnicodeChars.eta, tmnm = "eta"}
val eta_normal_form_enf = store_thm(
"eta_normal_form_enf",
``normal_form eta = enf``,
Q_TAC SUFF_TAC `(!x. ~enf x ==> can_reduce eta x) /\
(!x. can_reduce eta x ==> ~enf x)` THEN1
(SIMP_TAC (srw_ss())[normal_form_def, FUN_EQ_THM, EQ_IMP_THM,
FORALL_AND_THM] THEN PROVE_TAC []) THEN
CONJ_TAC THENL [
HO_MATCH_MP_TAC nc_INDUCTION2 THEN Q.EXISTS_TAC `{}` THEN
SRW_TAC [][] THENL [
PROVE_TAC [can_reduce_rules],
PROVE_TAC [can_reduce_rules],
PROVE_TAC [can_reduce_rules, lemma14a],
Q_TAC SUFF_TAC `?u. eta (LAM y x) u` THEN1
PROVE_TAC [can_reduce_rules, redex_def] THEN
FULL_SIMP_TAC (srw_ss()) [is_comb_APP_EXISTS, eta_def] THEN
SRW_TAC [][] THEN
FULL_SIMP_TAC (srw_ss()) [rand_thm, rator_thm] THEN PROVE_TAC []
],
HO_MATCH_MP_TAC can_reduce_ind THEN
SRW_TAC [][redex_def, eta_def] THEN
SRW_TAC [][]
]);
val no_eta_thm = store_thm(
"no_eta_thm",
``(!s t. ~(eta (VAR s) t)) /\
(!t u v. ~(eta (t @@ u) v))``,
SRW_TAC [][eta_def]);
val cc_eta_thm = store_thm(
"cc_eta_thm",
``(!s t. compat_closure eta (VAR s) t <=> F) /\
(!t u v. compat_closure eta (t @@ u) v <=>
(?t'. (v = t' @@ u) /\ compat_closure eta t t') \/
(?u'. (v = t @@ u') /\ compat_closure eta u u'))``,
REPEAT CONJ_TAC THEN
SIMP_TAC (srw_ss()) [SimpLHS, Once compat_closure_cases] THEN
SIMP_TAC (srw_ss()) [no_eta_thm, EQ_IMP_THM, DISJ_IMP_THM,
GSYM LEFT_FORALL_IMP_THM, RIGHT_AND_OVER_OR,
LEFT_AND_OVER_OR, FORALL_AND_THM,
is_comb_APP_EXISTS, GSYM LEFT_EXISTS_AND_THM]);
val eta_substitutive = store_thm(
"eta_substitutive",
``substitutive eta``,
SRW_TAC [][substitutive_def, eta_def] THEN
Q_TAC (NEW_TAC "z") `{v;v'} UNION FV M' UNION FV N` THEN
`LAM v (M' @@ VAR v) = LAM z ([VAR z/v] (M' @@ VAR v))`
by SRW_TAC [][SIMPLE_ALPHA] THEN
` _ = LAM z ([VAR z/v] M' @@ VAR z)` by SRW_TAC [][SUB_THM] THEN
ASM_SIMP_TAC (srw_ss()) [SUB_THM, lemma14b] THEN
Q.EXISTS_TAC `z` THEN SRW_TAC [][FV_SUB]);
val cc_eta_subst = store_thm(
"cc_eta_subst",
``!M N. compat_closure eta M N ==>
!P v. compat_closure eta ([P/v] M) ([P/v] N)``,
METIS_TAC [eta_substitutive, compat_closure_substitutive, substitutive_def]);
val cc_eta_tpm = store_thm(
"cc_eta_tpm",
``!M N. compat_closure eta M N ==>
compat_closure eta (tpm pi M) (tpm pi N)``,
METIS_TAC [compat_closure_permutative, substitutive_implies_permutative,
eta_substitutive, permutative_def])
val cc_eta_tpm_eqn = store_thm(
"cc_eta_tpm_eqn",
``compat_closure eta (tpm pi M) N =
compat_closure eta M (tpm (REVERSE pi) N)``,
METIS_TAC [cc_eta_tpm, pmact_inverse]);
val eta_deterministic = store_thm(
"eta_deterministic",
``!M N1 N2. eta M N1 /\ eta M N2 ==> (N1 = N2)``,
SIMP_TAC (srw_ss() ++ boolSimps.DNF_ss) [eta_def, LAM_eq_thm, tpm_fresh]);
val cc_eta_FV_SUBSET = store_thm(
"cc_eta_FV_SUBSET",
``!M N. compat_closure eta M N ==> FV N SUBSET FV M``,
HO_MATCH_MP_TAC compat_closure_ind THEN
SIMP_TAC (srw_ss()) [SUBSET_DEF] THEN
Q_TAC SUFF_TAC `!M N. eta M N ==> !s. s IN FV N ==> s IN FV M` THEN1
PROVE_TAC [] THEN
SIMP_TAC (srw_ss()) [eta_def, GSYM LEFT_FORALL_IMP_THM]);
val cc_eta_LAM = store_thm(
"cc_eta_LAM",
``!t v u. compat_closure eta (LAM v t) u <=>
(?t'. (u = LAM v t') /\ compat_closure eta t t') \/
eta (LAM v t) u``,
SIMP_TAC (srw_ss()) [Once compat_closure_cases, SimpLHS] THEN
SIMP_TAC (srw_ss() ++ boolSimps.DNF_ss)[LAM_eq_thm, EQ_IMP_THM, tpm_eqr,
cc_eta_tpm_eqn, pmact_flip_args] THEN
REPEAT STRIP_TAC THEN DISJ1_TAC THEN
Q_TAC SUFF_TAC `~(v' IN FV (tpm [(v,v')] y))` THEN1 SRW_TAC [][] THEN
METIS_TAC [cc_eta_FV_SUBSET, SUBSET_DEF]);
val eta_LAM = store_thm(
"eta_LAM",
``!v t u. eta (LAM v t) u <=> t = u @@ VAR v ∧ v ∉ FV u``,
SRW_TAC [][eta_def, LAM_eq_thm, EQ_IMP_THM] THEN SRW_TAC [][tpm_fresh] THEN
SRW_TAC [boolSimps.DNF_ss][]);
val CR_eta_lemma = prove(
``!M M1 M2. eta M M1 /\ compat_closure eta M M2 /\ ~(M1 = M2) ==>
?M3. compat_closure eta M1 M3 /\ compat_closure eta M2 M3``,
REPEAT STRIP_TAC THEN
`?v. (M = LAM v (M1 @@ VAR v)) /\ ~(v IN FV M1)` by PROVE_TAC [eta_def] THEN
FULL_SIMP_TAC (srw_ss()) [cc_eta_LAM, cc_eta_thm] THENL [
Q.EXISTS_TAC `rator t'` THEN
SRW_TAC [][eta_LAM] THEN
METIS_TAC [cc_eta_FV_SUBSET, SUBSET_DEF],
FULL_SIMP_TAC (srw_ss()) [eta_LAM]
]);
val cc_strong_ind =
IndDefLib.derive_strong_induction (compat_closure_rules, compat_closure_ind)