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Change notation to remain compatible with multiple Coq versions
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5 changed files with 31 additions and 31 deletions
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@ -1219,7 +1219,7 @@ Transparent heq himp lift star exis ptsto.
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(* Guarded predicates *)
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Definition guarded (P : Prop) (p : hprop) : hprop :=
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fun h => IF P then p h else emp%sep h.
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fun h => IFF P then p h else emp%sep h.
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Infix "===>" := guarded : sep_scope.
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@ -1127,7 +1127,7 @@ Qed.
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Transparent heq himp lift star exis ptsto.
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Definition guarded (P : Prop) (p : hprop) : hprop :=
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fun h => IF P then p h else emp%sep h.
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fun h => IFF P then p h else emp%sep h.
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Infix "===>" := guarded : sep_scope.
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@ -458,4 +458,4 @@ Arguments N.add: simpl never.
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Definition IF_then_else (p q1 q2 : Prop) :=
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(p /\ q1) \/ (~p /\ q2).
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Notation "'IF' p 'then' q1 'else' q2" := (IF_then_else p q1 q2) (at level 95).
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Notation "'IFF' p 'then' q1 'else' q2" := (IF_then_else p q1 q2) (at level 95).
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@ -586,7 +586,7 @@ Module PropositionalWithImplication.
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Lemma interp_valid'' : forall p hyps,
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(forall x, In x (varsOf p) -> hyps (Var x) \/ hyps (Not (Var x)))
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-> (forall x, hyps (Var x) -> ~hyps (Not (Var x)))
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-> IF interp (fun x => hyps (Var x)) p
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-> IFF interp (fun x => hyps (Var x)) p
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then valid hyps p
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else valid hyps (Not p).
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Proof.
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@ -612,18 +612,18 @@ Module PropositionalWithImplication.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidAndIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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right; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -637,7 +637,7 @@ Module PropositionalWithImplication.
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apply ValidHyp.
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propositional.
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right; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -654,7 +654,7 @@ Module PropositionalWithImplication.
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excluded_middle (interp (fun x => hyps (Var x)) p1).
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left; propositional.
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apply ValidOrIntro1.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -662,7 +662,7 @@ Module PropositionalWithImplication.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidOrIntro2.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -672,7 +672,7 @@ Module PropositionalWithImplication.
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apply ValidOrElim with p1 p2.
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apply ValidHyp.
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propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -683,7 +683,7 @@ Module PropositionalWithImplication.
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propositional.
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apply ValidHyp.
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propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -699,7 +699,7 @@ Module PropositionalWithImplication.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -709,12 +709,12 @@ Module PropositionalWithImplication.
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propositional.
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right; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -731,7 +731,7 @@ Module PropositionalWithImplication.
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propositional.
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left; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -759,7 +759,7 @@ Module PropositionalWithImplication.
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induct leftToDo; simplify.
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rewrite app_nil_r in H.
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assert (IF interp (fun x : var => hyps (Var x)) p then valid hyps p else valid hyps (Not p)).
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assert (IFF interp (fun x : var => hyps (Var x)) p then valid hyps p else valid hyps (Not p)).
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apply interp_valid''; first_order.
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unfold IF_then_else in H4; propositional.
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exfalso.
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@ -433,7 +433,7 @@ Fixpoint varsOf (p : prop) : list var :=
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Lemma interp_valid'' : forall p hyps,
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(forall x, In x (varsOf p) -> hyps (Var x) \/ hyps (Not (Var x)))
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-> (forall x, hyps (Var x) -> ~hyps (Not (Var x)))
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-> IF interp (fun x => hyps (Var x)) p
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-> IFF interp (fun x => hyps (Var x)) p
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then valid hyps p
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else valid hyps (Not p).
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Proof.
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@ -459,18 +459,18 @@ Proof.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidAndIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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right; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -484,7 +484,7 @@ Proof.
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apply ValidHyp.
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propositional.
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right; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -501,7 +501,7 @@ Proof.
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excluded_middle (interp (fun x => hyps (Var x)) p1).
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left; propositional.
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apply ValidOrIntro1.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -509,7 +509,7 @@ Proof.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidOrIntro2.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -519,7 +519,7 @@ Proof.
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apply ValidOrElim with p1 p2.
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apply ValidHyp.
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propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -530,7 +530,7 @@ Proof.
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propositional.
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apply ValidHyp.
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propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -546,7 +546,7 @@ Proof.
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excluded_middle (interp (fun x => hyps (Var x)) p2).
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left; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -556,12 +556,12 @@ Proof.
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propositional.
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right; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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unfold IF_then_else in H3; propositional.
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assert (IF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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assert (IFF interp (fun x : var => hyps (Var x)) p2 then valid hyps p2 else valid hyps (Not p2)).
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apply IHp2; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -578,7 +578,7 @@ Proof.
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propositional.
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left; propositional.
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apply ValidImplyIntro.
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assert (IF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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assert (IFF interp (fun x : var => hyps (Var x)) p1 then valid hyps p1 else valid hyps (Not p1)).
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apply IHp1; propositional.
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apply H.
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apply in_or_app; propositional.
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@ -606,7 +606,7 @@ Proof.
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induct leftToDo; simplify.
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rewrite app_nil_r in H.
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assert (IF interp (fun x : var => hyps (Var x)) p then valid hyps p else valid hyps (Not p)).
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assert (IFF interp (fun x : var => hyps (Var x)) p then valid hyps p else valid hyps (Not p)).
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apply interp_valid''; first_order.
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unfold IF_then_else in H4; propositional.
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exfalso.
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