forked from Isabelle_DOF/Isabelle_DOF
BAC2017: tried proofs
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@ -9,6 +9,10 @@ section*[idir::Author,affiliation="''CentraleSupelec''",
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email="''idir.aitsadoune@centralesupelec.fr''"]
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email="''idir.aitsadoune@centralesupelec.fr''"]
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{*Idir AIT SADOUNE*}
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{*Idir AIT SADOUNE*}
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section*[keller::Author,affiliation="''LRI, Univ. Paris-Sud''",
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email="''Chantal.Keller@lri.fr''"]
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{*Chantal KELLER*}
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subsection*[header::Header,examSubject= "[algebra,geometry]",
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subsection*[header::Header,examSubject= "[algebra,geometry]",
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examTitle="''BACCALAUREAT GENERAL MATHEMATIQUES''",
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examTitle="''BACCALAUREAT GENERAL MATHEMATIQUES''",
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date="''21-06-2017''",
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date="''21-06-2017''",
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@ -24,7 +28,7 @@ timeAllowed="240::int"]
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*}
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*}
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subsubsection*[exo1 :: Exercise,Exercise.concerns= "{examiner,validator,student}",
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subsubsection*[exo1 :: Exercise,Exercise.concerns= "{examiner,validator,student}",
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Exercise.content="[q1::Task]"]
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Exercise.content="[q1::Task,q2,q3a]"]
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{*
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{*
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On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par : @{term "h(x) = x * exponent (-x)"}
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On considère la fonction h définie sur l’intervalle [0..+\<infinity>] par : @{term "h(x) = x * exponent (-x)"}
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*}
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*}
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@ -38,7 +42,7 @@ level="oneStar", mark="1::int", type="formal"]
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{* Déterminer la limite de la fonction @{term h} en +\<infinity>. *}
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{* Déterminer la limite de la fonction @{term h} en +\<infinity>. *}
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text*[a1::Answer_Formal_Step]
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text*[a1::Answer_Formal_Step]
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{* First Step: Fill in term and justification*}
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{* Fill in term and justification*}
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lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
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lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
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sorry
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sorry
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@ -47,25 +51,50 @@ lemma q1 : "(h \<longlongrightarrow> (0::real)) at_top"
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subsubsection*[q2::Task, Task.concerns= "{examiner,validator,student}",
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subsubsection*[q2::Task, Task.concerns= "{examiner,validator,student}",
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level="oneStar", mark="1::int", type="formal"]
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level="oneStar", mark="1::int", type="formal"]
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{* Étudier les variations de la fonction @{term h} sur l'intervalle [0..+\<infinity>] et dresser son tableau
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{* Étudier les variations de la fonction @{term h} sur l'intervalle [0..+\<infinity>] et dresser son tableau
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de variation} *}
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de variation *}
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find_theorems exp
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text*[a2::Answer_Formal_Step]
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text*[a2::Answer_Formal_Step]
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{* First Step: Fill in term and justification*}
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{* Fill in term and justification*}
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lemma q2_a : "DERIV h x :> (1 - x) * exp (- x)"
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definition h' :: "real \<Rightarrow> real"
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where "h' x \<equiv> (1 - x) * exp (- x)"
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lemma q2_a : "DERIV h x :> h' x"
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proof -
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proof -
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have * : "DERIV (exp \<circ> uminus) x :> - (exp (-x))"
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have * : "DERIV (exp \<circ> uminus) x :> - (exp (-x))"
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by (simp add: has_derivative_minus has_derivative_compose Transcendental.DERIV_exp)
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by (simp add: has_derivative_compose)
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have ** : "DERIV id x :> 1"
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have ** : "DERIV id x :> 1"
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by (metis DERIV_ident eq_id_iff)
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by (metis DERIV_ident eq_id_iff)
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have *** : "DERIV h x :> x * (- (exp (- x))) + 1 * (exp (- x))"
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have *** : "DERIV h x :> x * (- (exp (- x))) + 1 * (exp (- x))"
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by (simp add: * ** has_derivative_mult)
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by (simp add: * ** has_derivative_mult comp_def)
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show ?thesis
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show ?thesis
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by (metis "***" left_diff_distrib mult_minus_right uminus_add_conv_diff)
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by (metis "***" left_diff_distrib mult_minus_right uminus_add_conv_diff)
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qed
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qed
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lemma q2_b : "0 \<le> x \<and> x \<le> y \<and> y \<le> 1 \<Longrightarrow> h x \<le> h y"
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sorry
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lemma q2_c : "1 \<le> x \<and> x \<le> y \<Longrightarrow> h x \<ge> h y"
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sorry
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subsubsection*[q3a::Task, Task.concerns= "{examiner,validator,student}",
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level="oneStar", mark="1::int", type="formal"]
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{* Vérifier que pour tout nombre réel x appartenant à l'intervalle [0..+\<infinity>], on a :
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@term{h x = (exp (- x)) - (h' x)} *}
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lemma q3a : "h x = (exp (- x)) - (h' x)"
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by (simp add : h_def h'_def left_diff_distrib)
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text*[a3a::Answer_Formal_Step]
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{* Fill in term and justification*}
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close_monitor*[exam]
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close_monitor*[exam]
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end
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end
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