lean2/examples/standard/constable.lean

-- Theorems/Exercises from "Logical Investigations, with the Nuprl Proof Assistant"
-- by Robert L. Constable and Anne Trostle
-- http://www.nuprl.org/MathLibrary/LogicalInvestigations/
import standard

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)

theorem thm4 {P Q : Prop} : ¬P → P → Q
:= assume Hnp Hp,
   absurd_elim Q 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_elim Q 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

theorem thm9 {P : Prop} : (P ∨ ¬P) → (¬¬P → P)
:= assume (em : P ∨ ¬P) (Hnn : ¬¬P),
   or_elim em
     (assume Hp, Hp)
     (assume Hn, absurd_elim P 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_elim Q 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)))

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_elim (P w) 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_elim C 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