src/HOL/Library/Quotient_Sum.thy
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35221:5cb63cb4213f 35222:4f1fba00f66d
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`     1 (*  Title:      Quotient_Sum.thy`
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`     2     Author:     Cezary Kaliszyk and Christian Urban`
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`     3 *)`
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`     4 theory Quotient_Sum`
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`     5 imports Main Quotient_Syntax`
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`     6 begin`
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`     7 `
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`     8 section {* Quotient infrastructure for the sum type. *}`
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`     9 `
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`    10 fun`
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`    11   sum_rel`
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`    12 where`
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`    13   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"`
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`    14 | "sum_rel R1 R2 (Inl a1) (Inr b2) = False"`
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`    15 | "sum_rel R1 R2 (Inr a2) (Inl b1) = False"`
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`    16 | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"`
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`    17 `
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`    18 fun`
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`    19   sum_map`
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`    20 where`
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`    21   "sum_map f1 f2 (Inl a) = Inl (f1 a)"`
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`    22 | "sum_map f1 f2 (Inr a) = Inr (f2 a)"`
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`    23 `
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`    24 declare [[map "+" = (sum_map, sum_rel)]]`
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`    25 `
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`    26 `
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`    27 text {* should probably be in Sum_Type.thy *}`
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`    28 lemma split_sum_all:`
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`    29   shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"`
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`    30   apply(auto)`
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`    31   apply(case_tac x)`
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`    32   apply(simp_all)`
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`    33   done`
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`    34 `
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`    35 lemma sum_equivp[quot_equiv]:`
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`    36   assumes a: "equivp R1"`
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`    37   assumes b: "equivp R2"`
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`    38   shows "equivp (sum_rel R1 R2)"`
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`    39   apply(rule equivpI)`
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`    40   unfolding reflp_def symp_def transp_def`
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`    41   apply(simp_all add: split_sum_all)`
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`    42   apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])`
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`    43   apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])`
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`    44   apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])`
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`    45   done`
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`    46 `
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`    47 lemma sum_quotient[quot_thm]:`
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`    48   assumes q1: "Quotient R1 Abs1 Rep1"`
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`    49   assumes q2: "Quotient R2 Abs2 Rep2"`
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`    50   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"`
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`    51   unfolding Quotient_def`
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`    52   apply(simp add: split_sum_all)`
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`    53   apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])`
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`    54   apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])`
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`    55   using q1 q2`
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`    56   unfolding Quotient_def`
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`    57   apply(blast)+`
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`    58   done`
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`    59 `
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`    60 lemma sum_Inl_rsp[quot_respect]:`
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`    61   assumes q1: "Quotient R1 Abs1 Rep1"`
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`    62   assumes q2: "Quotient R2 Abs2 Rep2"`
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`    63   shows "(R1 ===> sum_rel R1 R2) Inl Inl"`
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`    64   by simp`
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`    65 `
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`    66 lemma sum_Inr_rsp[quot_respect]:`
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`    67   assumes q1: "Quotient R1 Abs1 Rep1"`
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`    68   assumes q2: "Quotient R2 Abs2 Rep2"`
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`    69   shows "(R2 ===> sum_rel R1 R2) Inr Inr"`
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`    70   by simp`
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`    71 `
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`    72 lemma sum_Inl_prs[quot_preserve]:`
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`    73   assumes q1: "Quotient R1 Abs1 Rep1"`
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`    74   assumes q2: "Quotient R2 Abs2 Rep2"`
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`    75   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"`
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`    76   apply(simp add: expand_fun_eq)`
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`    77   apply(simp add: Quotient_abs_rep[OF q1])`
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`    78   done`
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`    79 `
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`    80 lemma sum_Inr_prs[quot_preserve]:`
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`    81   assumes q1: "Quotient R1 Abs1 Rep1"`
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`    82   assumes q2: "Quotient R2 Abs2 Rep2"`
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`    83   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"`
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`    84   apply(simp add: expand_fun_eq)`
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`    85   apply(simp add: Quotient_abs_rep[OF q2])`
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`    86   done`
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`    87 `
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`    88 lemma sum_map_id[id_simps]:`
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`    89   shows "sum_map id id = id"`
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`    90   by (simp add: expand_fun_eq split_sum_all)`
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`    91 `
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`    92 lemma sum_rel_eq[id_simps]:`
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`    93   shows "sum_rel (op =) (op =) = (op =)"`
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`    94   by (simp add: expand_fun_eq split_sum_all)`
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`    95 `
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`    96 end`