108 lines
2.2 KiB
Standard ML
108 lines
2.2 KiB
Standard ML
open HolKernel Parse boolLib bossLib;
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open rich_listTheory arithmeticTheory
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val _ = new_theory "e3";
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``!l. APPEND l [] = l``
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Induct >| [
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REWRITE_TAC [APPEND],
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ASM_REWRITE_TAC [APPEN]
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])
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open listTheory
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val``!l1 l2. REV l1 l2 = REVERSE l1 ++ l2``
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Induct >| [
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REWRITE_TAC [REV_DEF, REVERSE, APPEND],
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ASM_REWRITE_TAC [REVERSE, REV_DEF, APPEND_SNOC1]
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])
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``!l. REVERSE l = REV l []``
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Induct >| [
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REWRITE_TAC [REV_DEF, REVERSE],
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ASM_REWRITE_TAC [REVERSE, REV_DEF]
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])
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``!l1 l2. LENGTH (DROP (LENGTH l2) (l1 ++ l2)) = LENGTH l1``
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Induct >| [
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REWRITE_TAC[LENGTH, APPEND, DROP_LENGTH_NIL],
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Cases_on `l2` |> [
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REWRITE_TAC[APPEND_NIL, LENGTH, DROP],
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ASM_REWRITE_TAC[APPEND, LENGTH, DROP]
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``!l1 l2. LENGTH (DROP (LENGTH l2) (l1 ++ l2)) = LENGTH l1``
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REPEAT GEN_TAC >>
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SPEC_TAC (``l1:'a list``, ``l1:'a list``) THEN
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Induct_on `l2` >| [
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REWRITE_TAC[LENGTH, DROP, APPEND_NIL],
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Cases_on `l1` >| [
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REWRITE_TAC[LENGTH, DROP, APPEND, DROP_LENGTH_NIL],
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REPEAT GEN_TAC >>
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REWRITE_TAC[LENGTH, DROP, APPEND] THEN
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`t ++ h' :: l2 = (t ++ [h']) ++ l2` by REWRITE_TAC[APPEND, GSYM APPEND_ASSOC] >>
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ASM_REWRITE_TAC[] >>
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REWRITE_TAC[LENGTH_APPEND, LENGTH, ADD_CLAUSES]
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]
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])
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val MAKE_CHANGE_def = Define `
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(MAKE_CHANGE [] a = if (a = 0) then [[]] else []) /\
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(MAKE_CHANGE (c::cs) a = (
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(if (c <= a /\ 0 < a) then
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(MAP (\l. c::l) (MAKE_CHANGE cs (a - c)))
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else []) ++ (MAKE_CHANGE cs a)))`
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val MAKE_CHANGE_PROP = prove(``!cs. MAKE_CHANGE cs 0 = [[]]``,
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Induct >| [
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REWRITE_TAC[MAKE_CHANGE_def],
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ASM_REWRITE_TAC[MAKE_CHANGE_def, prim_recTheory.LESS_REFL, APPEND]
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])
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val MAKE_CHANGE_PROP1 = prove (``!cs a l.
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MEM l (MAKE_CHANGE cs a) ==> (SUM l = a)``,
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Induct >| [
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REPEAT GEN_TAC >>
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REWRITE_TAC[MAKE_CHANGE_def] >>
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Cases_on `a = 0` >> (
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ASM_REWRITE_TAC[MEM] THEN
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REPEAT STRIP_TAC THEN
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ASM_REWRITE_TAC[SUM]
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),
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REPEAT GEN_TAC >>
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ASM_REWRITE_TAC [MAKE_CHANGE_def, MEM_APPEND, DISJ_IMP_THM] >>
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Cases_on `h <= a /\ 0 < a` >| [
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ASM_REWRITE_TAC[MEM_MAP] THEN
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BETA_TAC THEN
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REPEAT STRIP_TAC THEN
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`SUM y = (a - h)` by METIS_TAC[] >>
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ASM_REWRITE_TAC [SUM] >>
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DECIDE_TAC,
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ASM_REWRITE_TAC[MEM]
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]
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]);
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val _ = export_theory();
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