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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
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