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header {* Lambda Cube Examples *}


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theory Example


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imports Cube


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begin


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text {*


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Examples taken from:


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H. Barendregt. Introduction to Generalised Type Systems.


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J. Functional Programming.


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*}


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method_setup depth_solve = {*

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Attrib.thms >> (fn thms => K (METHOD (fn facts =>

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(DEPTH_SOLVE (HEADGOAL (ares_tac (facts @ thms)))))))


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*}

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method_setup depth_solve1 = {*

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Attrib.thms >> (fn thms => K (METHOD (fn facts =>

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(DEPTH_SOLVE_1 (HEADGOAL (ares_tac (facts @ thms)))))))


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*}

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method_setup strip_asms = {*

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Attrib.thms >> (fn thms => K (METHOD (fn facts =>


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REPEAT (resolve_tac [@{thm strip_b}, @{thm strip_s}] 1 THEN


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DEPTH_SOLVE_1 (ares_tac (facts @ thms) 1)))))

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*}

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subsection {* Simple types *}


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schematic_lemma "A:*  A>A : ?T"

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by (depth_solve rules)


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schematic_lemma "A:*  Lam a:A. a : ?T"

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by (depth_solve rules)


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schematic_lemma "A:* B:* b:B  Lam x:A. b : ?T"

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by (depth_solve rules)


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schematic_lemma "A:* b:A  (Lam a:A. a)^b: ?T"

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by (depth_solve rules)


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schematic_lemma "A:* B:* c:A b:B  (Lam x:A. b)^ c: ?T"

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by (depth_solve rules)


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schematic_lemma "A:* B:*  Lam a:A. Lam b:B. a : ?T"

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by (depth_solve rules)


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subsection {* Secondorder types *}


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schematic_lemma (in L2) " Lam A:*. Lam a:A. a : ?T"

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by (depth_solve rules)


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schematic_lemma (in L2) "A:*  (Lam B:*.Lam b:B. b)^A : ?T"

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by (depth_solve rules)


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schematic_lemma (in L2) "A:* b:A  (Lam B:*.Lam b:B. b) ^ A ^ b: ?T"

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by (depth_solve rules)


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schematic_lemma (in L2) " Lam B:*.Lam a:(Pi A:*.A).a ^ ((Pi A:*.A)>B) ^ a: ?T"

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by (depth_solve rules)


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subsection {* Weakly higherorder propositional logic *}


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schematic_lemma (in Lomega) " Lam A:*.A>A : ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega) "B:*  (Lam A:*.A>A) ^ B : ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega) "B:* b:B  (Lam y:B. b): ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega) "A:* F:*>*  F^(F^A): ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega) "A:*  Lam F:*>*.F^(F^A): ?T"

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by (depth_solve rules)


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subsection {* LP *}


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schematic_lemma (in LP) "A:*  A > * : ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>* a:A  P^a: ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>A>* a:A  Pi a:A. P^a^a: ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>* Q:A>*  Pi a:A. P^a > Q^a: ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>*  Pi a:A. P^a > P^a: ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>*  Lam a:A. Lam x:P^a. x: ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>* Q:*  (Pi a:A. P^a>Q) > (Pi a:A. P^a) > Q : ?T"

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by (depth_solve rules)


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schematic_lemma (in LP) "A:* P:A>* Q:* a0:A 

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Lam x:Pi a:A. P^a>Q. Lam y:Pi a:A. P^a. x^a0^(y^a0): ?T"


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by (depth_solve rules)


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subsection {* Omegaorder types *}


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schematic_lemma (in L2) "A:* B:*  Pi C:*.(A>B>C)>C : ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega2) " Lam A:*.Lam B:*.Pi C:*.(A>B>C)>C : ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega2) " Lam A:*.Lam B:*.Lam x:A. Lam y:B. x : ?T"

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by (depth_solve rules)


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schematic_lemma (in Lomega2) "A:* B:*  ?p : (A>B) > ((B>Pi P:*.P)>(A>Pi P:*.P))"

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apply (strip_asms rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (rule lam_ss)


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apply assumption


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prefer 2


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apply (depth_solve1 rules)


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apply (erule pi_elim)


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apply assumption


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apply (erule pi_elim)


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apply assumption


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apply assumption


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done


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subsection {* Secondorder Predicate Logic *}


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schematic_lemma (in LP2) "A:* P:A>*  Lam a:A. P^a>(Pi A:*.A) : ?T"

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by (depth_solve rules)


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schematic_lemma (in LP2) "A:* P:A>A>* 

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(Pi a:A. Pi b:A. P^a^b>P^b^a>Pi P:*.P) > Pi a:A. P^a^a>Pi P:*.P : ?T"


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by (depth_solve rules)


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schematic_lemma (in LP2) "A:* P:A>A>* 

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?p: (Pi a:A. Pi b:A. P^a^b>P^b^a>Pi P:*.P) > Pi a:A. P^a^a>Pi P:*.P"


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 {* Antisymmetry implies irreflexivity: *}


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apply (strip_asms rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (rule lam_ss)


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apply assumption


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prefer 2


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apply (depth_solve1 rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (erule pi_elim, assumption, assumption?)+


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done


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subsection {* LPomega *}


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schematic_lemma (in LPomega) "A:*  Lam P:A>A>*.Lam a:A. P^a^a : ?T"

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by (depth_solve rules)


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schematic_lemma (in LPomega) " Lam A:*.Lam P:A>A>*.Lam a:A. P^a^a : ?T"

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by (depth_solve rules)


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subsection {* Constructions *}


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schematic_lemma (in CC) " Lam A:*.Lam P:A>*.Lam a:A. P^a>Pi P:*.P: ?T"

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by (depth_solve rules)


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schematic_lemma (in CC) " Lam A:*.Lam P:A>*.Pi a:A. P^a: ?T"

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by (depth_solve rules)


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schematic_lemma (in CC) "A:* P:A>* a:A  ?p : (Pi a:A. P^a)>P^a"

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apply (strip_asms rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (erule pi_elim, assumption, assumption)


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done


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subsection {* Some random examples *}


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schematic_lemma (in LP2) "A:* c:A f:A>A 

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Lam a:A. Pi P:A>*.P^c > (Pi x:A. P^x>P^(f^x)) > P^a : ?T"


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by (depth_solve rules)


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schematic_lemma (in CC) "Lam A:*.Lam c:A. Lam f:A>A.

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Lam a:A. Pi P:A>*.P^c > (Pi x:A. P^x>P^(f^x)) > P^a : ?T"


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by (depth_solve rules)


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schematic_lemma (in LP2)

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"A:* a:A b:A  ?p: (Pi P:A>*.P^a>P^b) > (Pi P:A>*.P^b>P^a)"


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 {* Symmetry of Leibnitz equality *}


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apply (strip_asms rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (erule_tac a = "Lam x:A. Pi Q:A>*.Q^x>Q^a" in pi_elim)


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apply (depth_solve1 rules)


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apply (unfold beta)


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apply (erule imp_elim)


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apply (rule lam_bs)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply (rule lam_ss)


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apply (depth_solve1 rules)


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prefer 2


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apply (depth_solve1 rules)


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apply assumption


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apply assumption


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done


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end
