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import GameServer.Commands
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-- Wird im Level "Implication 11" ohne Beweis angenommen.
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LemmaDoc not_not as not_not in "Logic"
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"
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### Aussage
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`¬¬A ↔ A`
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### Annahmen
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`(A : Prop)`
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"
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-- Wird im Level "Implication 10" ohne Beweis angenommen.
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LemmaDoc not_or_of_imp as not_or_of_imp in "Logic"
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"
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### Aussage
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`¬A ∨ B`
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### Annahmen
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`(A B : Prop)`\\
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`(h : A → B)`
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"
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-- Wird im Level "Implication 12" bewiesen.
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LemmaDoc imp_iff_not_or as imp_iff_not_or in "Logic"
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"
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### Aussage
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`(A → B) ↔ ¬A ∨ B`
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### Annahmen
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`(A B : Prop)`
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"
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LemmaDoc Nat.succ_pos as Nat.succ_pos in "Nat"
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"
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"
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LemmaDoc Nat.pos_iff_ne_zero as Nat.pos_iff_ne_zero in "Nat"
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"
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"
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LemmaDoc zero_add as zero_add in "Addition"
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"This lemma says `∀ a : ℕ, 0 + a = a`."
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LemmaDoc add_zero as add_zero in "Addition"
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"This lemma says `∀ a : ℕ, a + 0 = a`."
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LemmaDoc add_succ as add_succ in "Addition"
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"This lemma says `∀ a b : ℕ, a + succ b = succ (a + b)`."
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LemmaDoc not_forall as not_forall in "Logic"
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"`∀ (A : Prop), ¬(∀ x, A) ↔ ∃x, (¬A)`."
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LemmaDoc not_exists as not_exists in "Logic"
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"`∀ (A : Prop), ¬(∃ x, A) ↔ ∀x, (¬A)`."
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DefinitionDoc Even
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"
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`even n` ist definiert als `∃ r, a = 2 * r`.
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Die Definition kann man mit `unfold even at *` einsetzen.
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"
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DefinitionDoc Odd
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"
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`odd n` ist definiert als `∃ r, a = 2 * r + 1`.
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Die Definition kann man mit `unfold odd at *` einsetzen.
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"
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LemmaDoc not_odd as not_odd in "Nat"
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"`¬ (odd n) ↔ even n`"
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LemmaDoc not_even as not_even in "Nat"
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"`¬ (even n) ↔ odd n`"
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LemmaDoc even_square as even_square in "Nat"
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"`∀ (n : ℕ), even n → even (n ^ 2)`"
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LemmaDoc mem_univ as mem_univ in "Set"
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"x ∈ @univ α"
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LemmaDoc not_mem_empty as not_mem_empty in "Set"
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""
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LemmaDoc empty_subset as empty_subset in "Set"
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""
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LemmaDoc Subset.antisymm_iff as Subset.antisymm_iff in "Set"
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""
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LemmaDoc Nat.prime_def_lt'' as Nat.prime_def_lt'' in "Nat"
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""
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LemmaDoc Finset.sum_add_distrib as Finset.sum_add_distrib in "Sum"
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""
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LemmaDoc Fin.sum_univ_castSucc as Fin.sum_univ_castSucc in "Sum"
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""
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LemmaDoc Nat.succ_eq_add_one as Nat.succ_eq_add_one in "Sum"
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""
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LemmaDoc add_comm as add_comm in "Nat"
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""
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LemmaDoc mul_add as mul_add in "Nat"
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""
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LemmaDoc add_mul as add_mul in "Nat"
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""
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LemmaDoc arithmetic_sum as arithmetic_sum in "Sum"
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""
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LemmaDoc add_pow_two as add_pow_two in "Nat"
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""
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LemmaDoc Finset.sum_comm as Finset.sum_comm in "Sum"
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""
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