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-/-
-Theorems/Exercises from "Logical Investigations, with the Nuprl Proof Assistant"
-by Robert L. Constable and Anne Trostle
-http://www.nuprl.org/MathLibrary/LogicalInvestigations/
--/
-import logic
-
--- 2. The Minimal Implicational Calculus
-theorem thm1 {A B : Prop} : A → B → A :=
-assume Ha Hb, Ha
-
-theorem thm2 {A B C : Prop} : (A → B) → (A → B → C) → (A → C) :=
-assume Hab Habc Ha,
-  Habc Ha (Hab Ha)
-
-theorem thm3 {A B C : Prop} : (A → B) → (B → C) → (A → C) :=
-assume Hab Hbc Ha,
-  Hbc (Hab Ha)
-
--- 3. False Propositions and Negation
-theorem thm4 {P Q : Prop} : ¬P → P → Q :=
-assume Hnp Hp,
-  absurd Hp Hnp
-
-theorem thm5 {P : Prop} : P → ¬¬P :=
-assume (Hp : P) (HnP : ¬P),
-  absurd Hp HnP
-
-theorem thm6 {P Q : Prop} : (P → Q) → (¬Q → ¬P) :=
-assume (Hpq : P → Q) (Hnq : ¬Q) (Hp : P),
-  have Hq : Q, from Hpq Hp,
-  show false, from absurd Hq Hnq
-
-theorem thm7 {P Q : Prop} : (P → ¬P) → (P → Q) :=
-assume Hpnp Hp,
-  absurd Hp (Hpnp Hp)
-
-theorem thm8 {P Q : Prop} : ¬(P → Q) → (P → ¬Q) :=
-assume (Hn : ¬(P → Q)) (Hp : P) (Hq : Q),
-  -- Rermak we don't even need the hypothesis Hp
-  have H : P → Q, from assume H', Hq,
-  absurd H Hn
-
--- 4. Conjunction and Disjunction
-theorem thm9 {P : Prop} : (P ∨ ¬P) → (¬¬P → P) :=
-assume (em : P ∨ ¬P) (Hnn : ¬¬P),
-  or_elim em
-    (assume Hp, Hp)
-    (assume Hn, absurd Hn Hnn)
-
-theorem thm10 {P : Prop} : ¬¬(P ∨ ¬P) :=
-assume Hnem : ¬(P ∨ ¬P),
-  have Hnp : ¬P, from
-    assume Hp : P,
-      have Hem : P ∨ ¬P, from or_inl Hp,
-      absurd Hem Hnem,
-  have Hem : P ∨ ¬P, from or_inr Hnp,
-  absurd Hem Hnem
-
-theorem thm11 {P Q : Prop} : ¬P ∨ ¬Q → ¬(P ∧ Q) :=
-assume (H : ¬P ∨ ¬Q) (Hn : P ∧ Q),
-  or_elim H
-    (assume Hnp : ¬P, absurd (and_elim_left Hn) Hnp)
-    (assume Hnq : ¬Q, absurd (and_elim_right Hn) Hnq)
-
-theorem thm12 {P Q : Prop} : ¬(P ∨ Q) → ¬P ∧ ¬Q :=
-assume H : ¬(P ∨ Q),
-  have Hnp : ¬P, from assume Hp : P, absurd (or_inl Hp) H,
-  have Hnq : ¬Q, from assume Hq : Q, absurd (or_inr Hq) H,
-  and_intro Hnp Hnq
-
-theorem thm13 {P Q : Prop} : ¬P ∧ ¬Q → ¬(P ∨ Q) :=
-assume (H : ¬P ∧ ¬Q) (Hn : P ∨ Q),
-  or_elim Hn
-    (assume Hp : P, absurd Hp (and_elim_left H))
-    (assume Hq : Q, absurd Hq (and_elim_right H))
-
-theorem thm14 {P Q : Prop} : ¬P ∨ Q → P → Q :=
-assume (Hor : ¬P ∨ Q) (Hp : P),
-  or_elim Hor
-    (assume Hnp : ¬P, absurd Hp Hnp)
-    (assume Hq : Q, Hq)
-
-theorem thm15 {P Q : Prop} : (P → Q) → ¬¬(¬P ∨ Q) :=
-assume (Hpq : P → Q) (Hn : ¬(¬P ∨ Q)),
-  have H1 : ¬¬P ∧ ¬Q, from thm12 Hn,
-  have Hnp : ¬P, from mt Hpq (and_elim_right H1),
-  absurd Hnp (and_elim_left H1)
-
-theorem thm16 {P Q : Prop} : (P → Q) ∧ ((P ∨ ¬P) ∨ (Q ∨ ¬Q)) → ¬P ∨ Q :=
-assume H : (P → Q) ∧ ((P ∨ ¬P) ∨ (Q ∨ ¬Q)),
-  have Hpq : P → Q, from and_elim_left H,
-  or_elim (and_elim_right H)
-    (assume Hem1 : P ∨ ¬P, or_elim Hem1
-      (assume Hp : P, or_inr (Hpq Hp))
-      (assume Hnp : ¬P, or_inl Hnp))
-    (assume Hem2 : Q ∨ ¬Q, or_elim Hem2
-      (assume Hq : Q, or_inr Hq)
-      (assume Hnq : ¬Q, or_inl (mt Hpq Hnq)))
-
--- 5. First-Order Logic: All and Exists
-section
-parameters {T : Type} {C : Prop} {P : T → Prop}
-theorem thm17a : (C → ∀x, P x) → (∀x, C → P x) :=
-assume H : C → ∀x, P x,
-  take x : T, assume Hc : C,
-  H Hc x
-
-theorem thm17b : (∀x, C → P x) → (C → ∀x, P x) :=
-assume (H : ∀x, C → P x) (Hc : C),
-  take x : T,
-  H x Hc
-
-theorem thm18a : ((∃x, P x) → C) → (∀x, P x → C) :=
-assume H : (∃x, P x) → C,
-  take x, assume Hp : P x,
-  have Hex : ∃x, P x, from exists_intro x Hp,
-  H Hex
-
-theorem thm18b : (∀x, P x → C) → (∃x, P x) → C :=
-assume (H1 : ∀x, P x → C) (H2 : ∃x, P x),
-  obtain (w : T) (Hw : P w), from H2,
-  H1 w Hw
-
-theorem thm19a : (C ∨ ¬C) → (∃x : T, true) → (C → (∃x, P x)) → (∃x, C → P x) :=
-assume (Hem : C ∨ ¬C) (Hin : ∃x : T, true) (H1 : C → ∃x, P x),
-  or_elim Hem
-    (assume Hc : C,
-      obtain (w : T) (Hw : P w), from H1 Hc,
-      have Hr : C → P w, from assume Hc, Hw,
-      exists_intro w Hr)
-    (assume Hnc : ¬C,
-      obtain (w : T) (Hw : true), from Hin,
-      have Hr : C → P w, from assume Hc, absurd Hc Hnc,
-      exists_intro w Hr)
-
-theorem thm19b : (∃x, C → P x) → C → (∃x, P x) :=
-assume (H : ∃x, C → P x) (Hc : C),
-  obtain (w : T) (Hw : C → P w), from H,
-  exists_intro w (Hw Hc)
-
-theorem thm20a : (C ∨ ¬C) → (∃x : T, true) → ((¬∀x, P x) → ∃x, ¬P x) → ((∀x, P x) → C) → (∃x, P x → C) :=
-assume Hem Hin Hnf H,
-  or_elim Hem
-    (assume Hc : C,
-      obtain (w : T) (Hw : true), from Hin,
-      exists_intro w (assume H : P w, Hc))
-    (assume Hnc : ¬C,
-      have H1 : ¬(∀x, P x), from mt H Hnc,
-      have H2 : ∃x, ¬P x, from Hnf H1,
-      obtain (w : T) (Hw : ¬P w), from H2,
-      exists_intro w (assume H : P w, absurd H Hw))
-
-theorem thm20b : (∃x, P x → C) → (∀ x, P x) → C :=
-assume Hex Hall,
-  obtain (w : T) (Hw : P w → C), from Hex,
-  Hw (Hall w)
-
-theorem thm21a : (∃x : T, true) → ((∃x, P x) ∨ C) → (∃x, P x ∨ C) :=
-assume Hin H,
-  or_elim H
-    (assume Hex : ∃x, P x,
-      obtain (w : T) (Hw : P w), from Hex,
-      exists_intro w (or_inl Hw))
-    (assume Hc  : C,
-      obtain (w : T) (Hw : true), from Hin,
-      exists_intro w (or_inr Hc))
-
-theorem thm21b : (∃x, P x ∨ C) → ((∃x, P x) ∨ C) :=
-assume H,
-  obtain (w : T) (Hw : P w ∨ C), from H,
-  or_elim Hw
-    (assume H : P w, or_inl (exists_intro w H))
-    (assume Hc : C, or_inr Hc)
-
-theorem thm22a : (∀x, P x) ∨ C → ∀x, P x ∨ C :=
-assume H, take x,
-  or_elim H
-    (assume Hl, or_inl (Hl x))
-    (assume Hr, or_inr Hr)
-
-theorem thm22b : (C ∨ ¬C) → (∀x, P x ∨ C) → ((∀x, P x) ∨ C) :=
-assume Hem H1,
-  or_elim Hem
-    (assume Hc : C,   or_inr Hc)
-    (assume Hnc : ¬C,
-      have Hx : ∀x, P x, from
-        take x,
-        have H1 : P x ∨ C, from H1 x,
-        resolve_left H1 Hnc,
-      or_inl Hx)
-
-theorem thm23a : (∃x, P x) ∧ C → (∃x, P x ∧ C) :=
-assume H,
-  have Hex : ∃x, P x, from and_elim_left H,
-  have Hc : C, from and_elim_right H,
-  obtain (w : T) (Hw : P w), from Hex,
-  exists_intro w (and_intro Hw Hc)
-
-theorem thm23b : (∃x, P x ∧ C) → (∃x, P x) ∧ C :=
-assume H,
-  obtain (w : T) (Hw : P w ∧ C), from H,
-  have Hex : ∃x, P x, from exists_intro w (and_elim_left Hw),
-  and_intro Hex (and_elim_right Hw)
-
-theorem thm24a : (∀x, P x) ∧ C → (∀x, P x ∧ C) :=
-assume H, take x,
-  and_intro (and_elim_left H x) (and_elim_right H)
-
-theorem thm24b : (∃x : T, true) → (∀x, P x ∧ C) → (∀x, P x) ∧ C :=
-assume Hin H,
-  obtain (w : T) (Hw : true), from Hin,
-  have Hc : C, from and_elim_right (H w),
-  have Hx : ∀x, P x, from take x, and_elim_left (H x),
-  and_intro Hx Hc
-
-end -- of section