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| theory putnam_2013_a5 imports Complex_Main | ||
| "HOL-Analysis.Finite_Cartesian_Product" | ||
| "HOL-Analysis.Lebesgue_Measure" | ||
| "HOL-Analysis.Cross3" | ||
| begin | ||
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| (* uses (nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real), (nat \<Rightarrow> (real^2)), and (nat \<Rightarrow> (real^3)) instead of limiting inputs to (Fin m) and ensuring inputs are strictly increasing *) | ||
| (* uses (nat ⇒ nat ⇒ nat ⇒ real), (nat ⇒ (real^2)), and (nat ⇒ (real^3)) instead of limiting inputs to (Fin m) and ensuring inputs are strictly increasing *) | ||
| theorem putnam_2013_a5: | ||
| fixes m :: nat | ||
| and area2 :: "(real^2) \<Rightarrow> (real^2) \<Rightarrow> (real^2) \<Rightarrow> real" | ||
| and area3 :: "(real^3) \<Rightarrow> (real^3) \<Rightarrow> (real^3) \<Rightarrow> real" | ||
| and areadef2 :: "(nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> bool" | ||
| and areadef3 :: "(nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> bool" | ||
| assumes mge3: "m \<ge> 3" | ||
| and harea2: "\<forall>a b c::real^2. area2 a b c = emeasure lebesgue (convex hull {a, b, c})" | ||
| and harea3: "\<forall>a b c::real^3. area3 a b c = emeasure lebesgue (convex hull {a, b, c})" | ||
| and hareadef2: "\<forall>a::nat\<Rightarrow>nat\<Rightarrow>nat\<Rightarrow>real. areadef2 a = (\<forall>A::nat\<Rightarrow>(real^2). (\<Sum>i::nat\<in>{0..(m-1)}. \<Sum>j::nat\<in>{i<..(m-1)}. \<Sum>k::nat\<in>{j<..(m-1)}. (a i j k * area2 (A i) (A j) (A k))) \<ge> 0)" | ||
| and hareadef3: "\<forall>a::nat\<Rightarrow>nat\<Rightarrow>nat\<Rightarrow>real. areadef3 a = (\<forall>A::nat\<Rightarrow>(real^3). (\<Sum>i::nat\<in>{0..(m-1)}. \<Sum>j::nat\<in>{i<..(m-1)}. \<Sum>k::nat\<in>{j<..(m-1)}. (a i j k * area3 (A i) (A j) (A k))) \<ge> 0)" | ||
| shows "\<forall>a::nat\<Rightarrow>nat\<Rightarrow>nat\<Rightarrow>real. (areadef2 a \<longrightarrow> areadef3 a)" | ||
| and area2 :: "(real^2) ⇒ (real^2) ⇒ (real^2) ⇒ real" | ||
| and area3 :: "(real^3) ⇒ (real^3) ⇒ (real^3) ⇒ real" | ||
| and areadef2 :: "(nat ⇒ nat ⇒ nat ⇒ real) ⇒ bool" | ||
| and areadef3 :: "(nat ⇒ nat ⇒ nat ⇒ real) ⇒ bool" | ||
| and cross :: "(real^3) ⇒ (real^3) ⇒ (real^3) ⇒ (real^3)" | ||
| defines "cross ≡ λ a b c. sgn (cross3 (b - a) (c - a))" | ||
| assumes mge3: "m ≥ 3" | ||
| and harea2: "∀a b c::real^2. area2 a b c = emeasure lebesgue (convex hull {a, b, c})" | ||
| and harea3: "∀a b c::real^3. area3 a b c = emeasure lebesgue (convex hull {a, b, c, a + (cross a b c), b + (cross a b c), c + (cross a b c)})" | ||
| and hareadef2: "∀a::nat⇒nat⇒nat⇒real. areadef2 a = (∀A::nat⇒(real^2). (∑i::nat∈{0..(m-1)}. ∑j::nat∈{i<..(m-1)}. ∑k::nat∈{j<..(m-1)}. (a i j k * area2 (A i) (A j) (A k))) ≥ 0)" | ||
| and hareadef3: "∀a::nat⇒nat⇒nat⇒real. areadef3 a = (∀A::nat⇒(real^3). (∑i::nat∈{0..(m-1)}. ∑j::nat∈{i<..(m-1)}. ∑k::nat∈{j<..(m-1)}. (a i j k * area3 (A i) (A j) (A k))) ≥ 0)" | ||
| shows "∀a::nat⇒nat⇒nat⇒real. (areadef2 a ⟶ areadef3 a)" | ||
| sorry | ||
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| end |