9a0c1ab2d0
This PR migrates the ⌜p⌝ notation for embedding pure p : Prop into SPred σs to expand into a simple, first-order expression SPred.pure p that can be supported by e-matching in grind. Doing so deprives ⌜p⌝ notation of its idiom-bracket-like support for #selector and ‹Nat›ₛ syntax which is thus removed.
144 lines
3.5 KiB
Text
144 lines
3.5 KiB
Text
/-
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Copyright (c) 2022 Lars König. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Lars König, Mario Carneiro, Sebastian Graf
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-/
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import Std.Do.SPred.Notation
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open Std.Do
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variable
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(φ : Prop)
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(P Q R : SPred [Nat,Char,Bool])
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(Ψ : Nat → SPred [Nat, Char, Bool])
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(Φ : Nat → Nat → SPred [Nat, Char, Bool])
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/-- info: P ⊢ₛ Q : Prop -/
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#guard_msgs in
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#check P ⊢ₛ Q
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/-- info: P ⊢ₛ Q : Prop -/
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#guard_msgs in
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#check P ⊢ₛ Q
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/-- info: P ⊣⊢ₛ Q : Prop -/
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#guard_msgs in
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#check P ⊣⊢ₛ Q
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/-- info: ⌜φ⌝ : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check (⌜φ⌝ : SPred [Nat,Char,Bool])
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/-- info: spred(P ∧ Q) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P ∧ Q)
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/-- info: spred(P ∧ Q ∧ R) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P ∧ (Q ∧ R))
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/-- info: spred(P ∨ Q) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P ∨ Q)
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/-- info: spred(P ∨ Q ∨ R) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P ∨ (Q ∨ R))
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/-- info: spred(¬P) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(¬ P)
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/-- info: spred(¬(P ∨ Q)) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(¬ (P ∨ Q))
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/-- info: spred(P → Q) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P → Q)
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/-- info: spred(P → Q → R) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P → (Q → R))
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/-- info: spred(P ∧ Q → R) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred((P ∧ Q) → R)
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/-- info: spred(P → Q ∧ R) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P → (Q ∧ R))
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/-- info: spred(P ↔ Q) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(P ↔ Q)
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/-- info: spred(P ∧ Q ↔ Q ∧ P) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred((P ∧ Q) ↔ (Q ∧ P))
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/-- info: ∀ (x : Nat), x = 0 : Prop -/
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#guard_msgs in
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#check ∀ (x : Nat), x = 0
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/-- info: spred(∀ x, Ψ x) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∀ x, Ψ x)
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/-- info: spred(∀ x, Ψ x) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∀ (x : Nat), Ψ x)
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/-- info: spred(∀ x, ∀ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∀ x y, Φ x y)
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/-- info: spred(∀ x, ∀ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∀ (x : Nat) (y : Nat), Φ x y)
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/-- info: spred(∀ x, ∀ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∀ (x y : Nat), Φ x y)
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/-- info: spred(∃ x, Ψ x) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∃ x, Ψ x)
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/-- info: spred(∃ x, Ψ x) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∃ (x : Nat), Ψ x)
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/-- info: spred(∃ x, ∃ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∃ x y, Φ x y)
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/-- info: spred(∃ x, ∃ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∃ (x : Nat) (y : Nat), Φ x y)
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/-- info: spred(∃ x, ∃ y, Φ x y) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(∃ (x y : Nat), Φ x y)
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/--
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info: if true = true then
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match (1, 2) with
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| (x, y) => Φ x y
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else ⌜False⌝ : SPred [Nat, Char, Bool]
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-/
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#guard_msgs in
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#check spred(if true then term(match (1,2) with | (x,y) => Φ x y) else ⌜False⌝)
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/-- info: Ψ (1 + 1) : SPred [Nat, Char, Bool] -/
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#guard_msgs in
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#check spred(Ψ (1 + 1))
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example (P Q : SPred [Nat, Bool]): SPred [Char, Nat, Bool] :=
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spred(fun _ => ((∀ y, if y = 4 then (fun theNat => ⌜y = theNat⌝) ∧ P else Q) ∧ Q) → (∃ x, P → if (x : Bool) then Q else P))
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