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|
Minlog loaded successfully
> loading natinf.scm ...
; ok, algebra nat added
; ok, program constant NatPlus: nat=>nat=>nat
; of t-degree 1 and arity 2 added
; warning: theorem NatPlusTotal stating totality missing
; ok, computation rule nat+++0 -> nat added
; ok, computation rule nat1+++Succ nat2 -> Succ(nat1+++nat2) added
; ok, RewriteGA0 has been added as a new global assumption.
; ok, rewrite rule 0+++nat -> nat added
; ok, RewriteGA1 has been added as a new global assumption.
; ok, rewrite rule Succ nat1+++nat2 -> Succ(nat1+++nat2) added
; ok, RewriteGA2 has been added as a new global assumption.
; ok, rewrite rule nat1+++(nat2+++nat3) -> nat1+++nat2+++nat3 added
; ok, program constant NatLt: nat=>nat=>boole
; of t-degree 1 and arity 2 added
; warning: theorem NatLtTotal stating totality missing
; ok, computation rule nat<<0 -> False added
; ok, computation rule 0<<Succ nat -> True added
; ok, computation rule Succ nat1<<Succ nat2 -> nat1<<nat2 added
; ok, RewriteGA3 has been added as a new global assumption.
; ok, rewrite rule nat<<nat -> False added
; ok, RewriteGA4 has been added as a new global assumption.
; ok, rewrite rule nat<<Succ nat -> True added
; ok, program constant Pred: nat=>nat
; of t-degree 1 and arity 1 added
; warning: theorem PredTotal stating totality missing
; ok, computation rule Pred 0 -> 0 added
; ok, computation rule Pred(Succ nat) -> nat added
; ok, program constant NatMinus: nat=>nat=>nat
; of t-degree 1 and arity 2 added
; warning: theorem NatMinusTotal stating totality missing
; ok, computation rule nat-0 -> nat added
; ok, computation rule nat1-Succ nat2 -> Pred(nat1-nat2) added
; ok, RewriteGA5 has been added as a new global assumption.
; ok, rewrite rule nat-nat -> 0 added
; ok, RewriteGA6 has been added as a new global assumption.
; ok, rewrite rule 0-nat -> 0 added
; ok, RewriteGA7 has been added as a new global assumption.
; ok, rewrite rule Pred(Succ nat1-nat2) -> nat1-nat2 added
; ok, RewriteGA8 has been added as a new global assumption.
; ok, rewrite rule nat1+++nat2-nat1 -> nat2 added
; ok, RewriteGA9 has been added as a new global assumption.
; ok, rewrite rule nat1+++nat2-nat2 -> nat1 added
; ok, RewriteGA10 has been added as a new global assumption.
; ok, rewrite rule nat1+++(nat2-nat3) -> nat1+++nat2-nat3 added
; ok, variable n: nat added
; ok, variable m: nat added
; ok, variable k: nat added
; ok, algebra yplus added
; ok, variable x: nat yplus unit added
; ok, variable y: nat yplus unit added
; ok, variable z: nat yplus unit added
; ok, program constant NatinfPlus: nat yplus unit=>nat yplus unit=>nat yplus unit
; of t-degree 1 and arity 2 added
; warning: theorem NatinfPlusTotal stating totality missing
; ok, computation rule x+Inl 0 -> x added
; ok, computation rule Inl n+Inl(Succ m) -> Inl(Succ(n+++m)) added
; ok, computation rule Inr Dummy+Inl(Succ m) -> Inr Dummy added
; ok, computation rule x+Inr Dummy -> Inr Dummy added
; ok, RewriteGA11 has been added as a new global assumption.
; ok, rewrite rule Inl 0+x -> x added
; ok, RewriteGA12 has been added as a new global assumption.
; ok, rewrite rule Inl(Succ n)+Inl m -> Inl(Succ(n+++m)) added
; ok, RewriteGA13 has been added as a new global assumption.
; ok, rewrite rule x1+(x2+x3) -> x1+x2+x3 added
; ok, program constant NatinfLe: nat yplus unit=>nat yplus unit=>boole
; of t-degree 1 and arity 2 added
; warning: theorem NatinfLeTotal stating totality missing
; ok, computation rule Inl 0<=y -> True added
; ok, computation rule Inl(Succ n)<=Inl 0 -> False added
; ok, computation rule Inl(Succ n)<=Inl(Succ m) -> Inl n<=Inl m added
; ok, computation rule Inl(Succ n)<=Inr Dummy -> True added
; ok, computation rule Inr Dummy<=Inl m -> False added
; ok, computation rule Inr Dummy<=Inr Dummy -> True added
; ok, RewriteGA14 has been added as a new global assumption.
; ok, rewrite rule x<=x -> True added
; ok, RewriteGA15 has been added as a new global assumption.
; ok, rewrite rule x<=Inr Dummy -> True added
; ok, RewriteGA16 has been added as a new global assumption.
; ok, rewrite rule x<=x+y -> True added
; ok, program constant NatinfLt: nat yplus unit=>nat yplus unit=>boole
; of t-degree 1 and arity 2 added
; warning: theorem NatinfLtTotal stating totality missing
; ok, computation rule x<Inl 0 -> False added
; ok, computation rule Inl 0<Inl(Succ m) -> True added
; ok, computation rule Inl(Succ n)<Inl(Succ m) -> Inl n<Inl m added
; ok, computation rule Inr Dummy<Inl(Succ m) -> False added
; ok, computation rule Inl n<Inr Dummy -> True added
; ok, computation rule Inr Dummy<Inr Dummy -> False added
; ok, RewriteGA17 has been added as a new global assumption.
; ok, rewrite rule x<x -> False added
; ok, RewriteGA18 has been added as a new global assumption.
; ok, rewrite rule x+y<x -> False added
; ok, RewriteGA19 has been added as a new global assumption.
; ok, rewrite rule Inr unit<y -> False added
; ok, RewriteGA20 has been added as a new global assumption.
; ok, rewrite rule Inl m<Inl(Succ m) -> True added
; ok, program constant NatinfMin: nat yplus unit=>nat yplus unit=>nat yplus unit
; of t-degree 1 and arity 2 added
; warning: theorem NatinfMinTotal stating totality missing
; ok, computation rule Inl 0 min y -> Inl 0 added
; ok, computation rule Inl(Succ n)min Inl 0 -> Inl 0 added
; ok, computation rule Inl(Succ n)min Inl(Succ m) -> Inl n min Inl m+Inl 1 added
; ok, computation rule Inl(Succ n)min Inr Dummy -> Inl(Succ n) added
; ok, computation rule Inr Dummy min y -> y added
; ok, RewriteGA21 has been added as a new global assumption.
; ok, rewrite rule x min Inl 0 -> Inl 0 added
; ok, RewriteGA22 has been added as a new global assumption.
; ok, rewrite rule x min x -> x added
; ok, RewriteGA23 has been added as a new global assumption.
; ok, rewrite rule (x+y)min x -> x added
; ok, RewriteGA24 has been added as a new global assumption.
; ok, rewrite rule x min(x+y) -> x added
; ok, RewriteGA25 has been added as a new global assumption.
; ok, rewrite rule x min y<=x -> True added
; ok, RewriteGA26 has been added as a new global assumption.
; ok, rewrite rule x min y<=y -> True added
; ok, RewriteGA27 has been added as a new global assumption.
; ok, rewrite rule x min y<x -> True added
; ok, RewriteGA28 has been added as a new global assumption.
; ok, rewrite rule x min y<y -> True added
> ; ok, variable cp: nat=>boole added
> ; ok, program constant Adjoin: (nat=>boole)=>nat=>nat=>boole
; of t-degree 1 and arity 3 added
; warning: theorem AdjoinTotal stating totality missing
> ; ok, computation rule Adjoin cp m n -> [if (n=m) True (cp n)] added
> ; ok, program constant Card: (nat=>boole)=>nat=>nat
; of t-degree 1 and arity 2 added
; warning: theorem CardTotal stating totality missing
> ; ok, computation rule Card cp 0 -> 0 added
> ; ok, computation rule Card cp(Succ n) -> Card cp n+++[if (cp n) 1 0] added
> ; ?_1: all cp,n Card cp n<<Succ n
> ; ok, we now have the new goal
; ?_2: all n Card cp n<<Succ n from
; cp
> ; ok, ?_2 can be obtained from
; ?_4: all n89.Card cp n89<<Succ n89 -> Card cp(Succ n89)<<Succ(Succ n89) from
; cp n87
; ?_3: Card cp 0<<1 from
; cp n87
> ; ok, the normalized goal is
; ?_5: T from
; cp n87
> ; ok, ?_5 is proved. The active goal now is
; ?_4: all n89.Card cp n89<<Succ n89 -> Card cp(Succ n89)<<Succ(Succ n89) from
; cp n87
> ; ok, we now have the new goal
; ?_6: Card cp(Succ n)<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
> ; ok, the normalized goal is
; ?_7: Card cp n+++[if (cp n) 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
> ; ok, ?_7 can be obtained from
; ?_9: (cp n -> F) -> Card cp n+++[if False 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; ?_8: cp n -> Card cp n+++[if True 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
> ; ok, we now have the new goal
; ?_10: Card cp n+++[if True 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; cp n:cp n
> ; ok, the normalized goal is
; ?_11: Card cp n<<Succ n from
; cp n87 n IH:Card cp n<<Succ n
; cp n:cp n
> ; ok, ?_11 is proved. The active goal now is
; ?_9: (cp n -> F) -> Card cp n+++[if False 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
> ; ok, we now have the new goal
; ?_12: Card cp n+++[if False 1 0]<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; cp n -> F:cp n -> F
> ; ok, the normalized goal is
; ?_13: Card cp n<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; cp n -> F:cp n -> F
> ; ok, Trans-<< has been added as a new global assumption.
> ; ok, ?_13 can be obtained from
; ?_15: Succ n<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; cp n -> F:cp n -> F
; ?_14: Card cp n<<Succ n from
; cp n87 n IH:Card cp n<<Succ n
; cp n -> F:cp n -> F
> ; ok, ?_14 is proved. The active goal now is
; ?_15: Succ n<<Succ(Succ n) from
; cp n87 n IH:Card cp n<<Succ n
; cp n -> F:cp n -> F
> ; ok, ?_15 is proved. Proof finished.
> ; ok, CardBound has been added as a new theorem.
> ; ?_1: all cp,n,m.Card cp n=n -> m<<n -> cp m
> ; ok, we now have the new goal
; ?_2: all n,m.Card cp n=n -> m<<n -> cp m from
; cp
> ; ok, ?_2 can be obtained from
; ?_4: all n98.
; (all m.Card cp n98=n98 -> m<<n98 -> cp m) ->
; all m.Card cp(Succ n98)=Succ n98 -> m<<Succ n98 -> cp m from
; cp n96
; ?_3: all m.Card cp 0=0 -> m<<0 -> cp m from
; cp n96
> ; ok, the normalized goal is
; ?_5: all m.T -> F -> cp m from
; cp n96
> ; ok, we now have the new goal
; ?_6: cp m from
; cp n96 m 1:T
; 2:F
> ; Not provable in minimal propositional logic.
; ok, ?_6 is proved in intuitionistic propositional logic.
; The active goal now is
; ?_4: all n98.
; (all m.Card cp n98=n98 -> m<<n98 -> cp m) ->
; all m.Card cp(Succ n98)=Succ n98 -> m<<Succ n98 -> cp m from
; cp n96
> ; ok, we now have the new goal
; ?_7: Card cp(Succ n)=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m
> ; ok, the normalized goal is
; ?_8: Card cp n+++[if (cp n) 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m
> ; ok, ?_8 can be obtained from
; ?_10: (cp n -> F) -> Card cp n+++[if False 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m
; ?_9: cp n -> Card cp n+++[if True 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m
> ; ok, we now have the new goal
; ?_11: Card cp n+++[if True 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
> ; ok, the normalized goal is
; ?_12: Card cp n=n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
> ; ok, we now have the new goal
; ?_13: cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
> ; ok, NatLtSuccElim has been added as a new global assumption.
; ok, program constant cNatLtSuccElim: nat=>nat=>alpha4=>alpha4=>alpha4
; of t-degree 1 and arity 0 added
; warning: theorem cNatLtSuccElimTotal stating totality missing
> ; ok, ?_13 can be obtained from
; ?_16: m=n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
; ?_15: m<<n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
; ?_14: m<<Succ n from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
> ; ok, ?_14 is proved. The active goal now is
; ?_15: m<<n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
> ; ok, ?_15 can be obtained from
; ?_17: Card cp n=n from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
> ; ok, ?_17 is proved. The active goal now is
; ?_16: m=n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
> ; ok, we now have the new goal
; ?_18: cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
; m=n:m=n
> ; ok, ?_18 can be obtained from
; ?_19: cp n from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n:cp n
; Card cp n=n:Card cp n=n
; m<<Succ n:m<<Succ n
; m=n:m=n
> ; ok, ?_19 is proved. The active goal now is
; ?_10: (cp n -> F) -> Card cp n+++[if False 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m
> ; ok, we now have the new goal
; ?_20: Card cp n+++[if False 1 0]=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
> ; ok, the normalized goal is
; ?_21: Card cp n=Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
> ; ok, we now have the new goal
; ?_22: m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
; Card cp n=Succ n:Card cp n=Succ n
> ; ok, ?_22 can be obtained from
; ?_24: Card cp n<<Succ n from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
; Card cp n=Succ n:Card cp n=Succ n
; ?_23: Card cp n<<Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
; Card cp n=Succ n:Card cp n=Succ n
> ; ok, ?_23 can be obtained from
; ?_25: Succ n<<Succ n -> m<<Succ n -> cp m from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
; Card cp n=Succ n:Card cp n=Succ n
> ; Not provable in minimal propositional logic.
; ok, ?_25 is proved in intuitionistic propositional logic.
; The active goal now is
; ?_24: Card cp n<<Succ n from
; cp n96 n IH:all m.Card cp n=n -> m<<n -> cp m
; m cp n -> F:cp n -> F
; Card cp n=Succ n:Card cp n=Succ n
> ; ok, ?_24 is proved. Proof finished.
> ; ok, CardMaxImpAll has been added as a new theorem.
> ; ?_1: all cp,m,n.n<<Succ m -> Card(Adjoin cp m)n=Card cp n
> ; ok, we now have the new goal
; ?_2: all n.n<<Succ m -> Card(Adjoin cp m)n=Card cp n from
; cp m
> ; ok, ?_2 can be obtained from
; ?_4: all n134.
; (n134<<Succ m -> Card(Adjoin cp m)n134=Card cp n134) ->
; Succ n134<<Succ m -> Card(Adjoin cp m)(Succ n134)=Card cp(Succ n134) from
; cp m n132
; ?_3: 0<<Succ m -> Card(Adjoin cp m)0=Card cp 0 from
; cp m n132
> ; ok, the normalized goal is
; ?_5: T -> T from
; cp m n132
> ; ok, ?_5 is proved in minimal propositional logic. The active goal now is
; ?_4: all n134.
; (n134<<Succ m -> Card(Adjoin cp m)n134=Card cp n134) ->
; Succ n134<<Succ m -> Card(Adjoin cp m)(Succ n134)=Card cp(Succ n134) from
; cp m n132
> ; ok, we now have the new goal
; ?_6: Succ n<<Succ m -> Card(Adjoin cp m)(Succ n)=Card cp(Succ n) from
; cp m n132 n IH:n<<Succ m -> Card(Adjoin cp m)n=Card cp n
> ; ok, the normalized goal is
; ?_7: n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if (cp n) 1 0]]=
; Card cp n+++[if (cp n) 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
> ; ok, ?_7 can be obtained from
; ?_9: (cp n -> F) ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; ?_8: cp n ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if True 1 0]]=
; Card cp n+++[if True 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
> ; ok, we now have the new goal
; ?_10: n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if True 1 0]]=
; Card cp n+++[if True 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
> ; ok, the normalized goal is
; ?_11: n<<m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
> ; ok, we now have the new goal
; ?_12: Card([n0][if (n0=m) True (cp n0)])n=Card cp n from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
; n<<m:n<<m
> ; ok, ?_12 can be obtained from
; ?_13: n<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
; n<<m:n<<m
> ; ok, ?_13 can be obtained from
; ?_15: m<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
; n<<m:n<<m
; ?_14: n<<m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
; n<<m:n<<m
> ; ok, ?_14 is proved. The active goal now is
; ?_15: m<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n:cp n
; n<<m:n<<m
> ; ok, ?_15 is proved. The active goal now is
; ?_9: (cp n -> F) ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
> ; ok, we now have the new goal
; ?_16: n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if (n=m) 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
> ; ok, ?_16 can be obtained from
; ?_18: (n=m -> F) ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if False 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; ?_17: n=m ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if True 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
> ; ok, we now have the new goal
; ?_19: n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if True 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m:n=m
> ; ok, ?_19 can be obtained from
; ?_20: m<<m ->
; Card([n135][if (n135=m) True (cp n135)])m+++[if True 1 [if False 1 0]]=
; Card cp m+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m:n=m
> ; ok, the normalized goal is
; ?_21: F -> Succ(Card([n0][if (n0=m) True (cp n0)])m)=Card cp m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m:n=m
> ; Not provable in minimal propositional logic.
; ok, ?_21 is proved in intuitionistic propositional logic.
; The active goal now is
; ?_18: (n=m -> F) ->
; n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if False 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
> ; ok, we now have the new goal
; ?_22: n<<m ->
; Card([n135][if (n135=m) True (cp n135)])n+++[if False 1 [if False 1 0]]=
; Card cp n+++[if False 1 0] from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
> ; ok, the normalized goal is
; ?_23: n<<m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
> ; ok, we now have the new goal
; ?_24: Card([n0][if (n0=m) True (cp n0)])n=Card cp n from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
; n<<m:n<<m
> ; ok, ?_24 can be obtained from
; ?_25: n<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
; n<<m:n<<m
> ; ok, ?_25 can be obtained from
; ?_27: m<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
; n<<m:n<<m
; ?_26: n<<m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
; n<<m:n<<m
> ; ok, ?_26 is proved. The active goal now is
; ?_27: m<<Succ m from
; cp m n132 n IH:n<<Succ m -> Card([n0][if (n0=m) True (cp n0)])n=Card cp n
; cp n -> F:cp n -> F
; n=m -> F:n=m -> F
; n<<m:n<<m
> ; ok, ?_27 is proved. Proof finished.
> ; ok, CountExtern has been added as a new theorem.
> ; ?_1: all cp,m.(cp m -> F) -> all n.m<<n -> Card(Adjoin cp m)n=Succ(Card cp n)
> ; ok, we now have the new goal
; ?_2: all n.m<<n -> Card(Adjoin cp m)n=Succ(Card cp n) from
; cp m cp m -> F:cp m -> F
> ; ok, ?_2 can be obtained from
; ?_4: all n161.
; (m<<n161 -> Card(Adjoin cp m)n161=Succ(Card cp n161)) ->
; m<<Succ n161 ->
; Card(Adjoin cp m)(Succ n161)=Succ(Card cp(Succ n161)) from
; cp m cp m -> F:cp m -> F
; n159
; ?_3: m<<0 -> Card(Adjoin cp m)0=Succ(Card cp 0) from
; cp m cp m -> F:cp m -> F
; n159
> ; ok, ?_3 is proved in minimal propositional logic. The active goal now is
; ?_4: all n161.
; (m<<n161 -> Card(Adjoin cp m)n161=Succ(Card cp n161)) ->
; m<<Succ n161 ->
; Card(Adjoin cp m)(Succ n161)=Succ(Card cp(Succ n161)) from
; cp m cp m -> F:cp m -> F
; n159
> ; ok, we now have the new goal
; ?_5: m<<Succ n -> Card(Adjoin cp m)(Succ n)=Succ(Card cp(Succ n)) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card(Adjoin cp m)n=Succ(Card cp n)
> ; ok, we now have the new goal
; ?_6: Card(Adjoin cp m)(Succ n)=Succ(Card cp(Succ n)) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card(Adjoin cp m)n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
> ; ok, the normalized goal is
; ?_7: Card([n162][if (n162=m) True (cp n162)])n+++[if (n=m) 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
> ; ok, ?_7 can be obtained from
; ?_9: (n=m -> F) ->
; Card([n162][if (n162=m) True (cp n162)])n+++[if False 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; ?_8: n=m ->
; Card([n162][if (n162=m) True (cp n162)])n+++[if True 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
> ; ok, we now have the new goal
; ?_10: Card([n162][if (n162=m) True (cp n162)])n+++[if True 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, ?_10 can be obtained from
; ?_11: Card([n162][if (n162=m) True (cp n162)])m+++[if True 1 [if (cp m) 1 0]]=
; Succ(Card cp m+++[if (cp m) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, the normalized goal is
; ?_12: Card([n0][if (n0=m) True (cp n0)])m=Card cp m+++[if (cp m) 1 0] from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, ?_12 can be obtained from
; ?_13: Card([n0][if (n0=m) True (cp n0)])m=Card cp m+++[if False 1 0] from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, the normalized goal is
; ?_14: Card([n0][if (n0=m) True (cp n0)])m=Card cp m from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, ?_14 can be obtained from
; ?_15: m<<Succ m from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m:n=m
> ; ok, ?_15 is proved. The active goal now is
; ?_9: (n=m -> F) ->
; Card([n162][if (n162=m) True (cp n162)])n+++[if False 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
> ; ok, we now have the new goal
; ?_16: Card([n162][if (n162=m) True (cp n162)])n+++
; [if False 1 [if (cp n) 1 0]]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
> ; ok, the normalized goal is
; ?_17: Card([n0][if (n0=m) True (cp n0)])n+++[if (cp n) 1 0]=
; Succ(Card cp n+++[if (cp n) 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
> ; ok, ?_17 can be obtained from
; ?_19: (cp n -> F) ->
; Card([n0][if (n0=m) True (cp n0)])n+++[if False 1 0]=
; Succ(Card cp n+++[if False 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; ?_18: cp n ->
; Card([n0][if (n0=m) True (cp n0)])n+++[if True 1 0]=
; Succ(Card cp n+++[if True 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
> ; ok, we now have the new goal
; ?_20: Card([n0][if (n0=m) True (cp n0)])n+++[if True 1 0]=
; Succ(Card cp n+++[if True 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
> ; ok, the normalized goal is
; ?_21: Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
> ; ok, ?_21 can be obtained from
; ?_22: m<<n from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
> ; ok, Succ-Lemma has been added as a new global assumption.
> ; ok, ?_22 can be obtained from
; ?_24: n=m -> F from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
; ?_23: m<<Succ n from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
> ; ok, ?_23 is proved. The active goal now is
; ?_24: n=m -> F from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n:cp n
> ; ok, ?_24 is proved. The active goal now is
; ?_19: (cp n -> F) ->
; Card([n0][if (n0=m) True (cp n0)])n+++[if False 1 0]=
; Succ(Card cp n+++[if False 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
> ; ok, we now have the new goal
; ?_25: Card([n0][if (n0=m) True (cp n0)])n+++[if False 1 0]=
; Succ(Card cp n+++[if False 1 0]) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
> ; ok, the normalized goal is
; ?_26: Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n) from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
> ; ok, ?_26 can be obtained from
; ?_27: m<<n from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
> ; ok, ?_27 can be obtained from
; ?_29: n=m -> F from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
; ?_28: m<<Succ n from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
> ; ok, ?_28 is proved. The active goal now is
; ?_29: n=m -> F from
; cp m cp m -> F:cp m -> F
; n159 n IH:m<<n -> Card([n0][if (n0=m) True (cp n0)])n=Succ(Card cp n)
; m<<Succ n:m<<Succ n
; n=m -> F:n=m -> F
; cp n -> F:cp n -> F
> ; ok, ?_29 is proved. Proof finished.
> ; ok, CountIntern has been added as a new theorem.
> ; ?_1: all cp,n.(all m.m<<n -> cp m) -> Card cp n=n
> ; ok, we now have the new goal
; ?_2: all n.(all m.m<<n -> cp m) -> Card cp n=n from
; cp
> ; ok, ?_2 can be obtained from
; ?_4: all n200.
; ((all m.m<<n200 -> cp m) -> Card cp n200=n200) ->
; (all m.m<<Succ n200 -> cp m) -> Card cp(Succ n200)=Succ n200 from
; cp n198
; ?_3: (all m.m<<0 -> cp m) -> Card cp 0=0 from
; cp n198
> ; ok, the normalized goal is
; ?_5: (all m.F -> cp m) -> T from
; cp n198
> ; ok, we now have the new goal
; ?_6: T from
; cp n198 1:all m.F -> cp m
> ; ok, ?_6 is proved. The active goal now is
; ?_4: all n200.
; ((all m.m<<n200 -> cp m) -> Card cp n200=n200) ->
; (all m.m<<Succ n200 -> cp m) -> Card cp(Succ n200)=Succ n200 from
; cp n198
> ; ok, we now have the new goal
; ?_7: (all m.m<<Succ n -> cp m) -> Card cp(Succ n)=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
> ; ok, the normalized goal is
; ?_8: (all m.m<<Succ n -> cp m) -> Card cp n+++[if (cp n) 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
> ; ok, ?_8 can be obtained from
; ?_10: (cp n -> F) ->
; (all m.m<<Succ n -> cp m) -> Card cp n+++[if False 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; ?_9: cp n -> (all m.m<<Succ n -> cp m) -> Card cp n+++[if True 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
> ; ok, we now have the new goal
; ?_11: (all m.m<<Succ n -> cp m) -> Card cp n+++[if True 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
> ; ok, the normalized goal is
; ?_12: (all m.m<<Succ n -> cp m) -> Card cp n=n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
> ; ok, we now have the new goal
; ?_13: Card cp n=n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; ok, ?_13 can be obtained from
; ?_14: all m.m<<n -> cp m from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; ok, we now have the new goal
; ?_15: cp m from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; m m<<n:m<<n
> ; ok, ?_15 can be obtained from
; ?_16: m<<Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; m m<<n:m<<n
> ; ok, ?_16 can be obtained from
; ?_18: n<<Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; m m<<n:m<<n
; ?_17: m<<n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; m m<<n:m<<n
> ; ok, ?_17 is proved. The active goal now is
; ?_18: n<<Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n:cp n
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; m m<<n:m<<n
> ; ok, ?_18 is proved. The active goal now is
; ?_10: (cp n -> F) ->
; (all m.m<<Succ n -> cp m) -> Card cp n+++[if False 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
> ; ok, we now have the new goal
; ?_19: (all m.m<<Succ n -> cp m) -> Card cp n+++[if False 1 0]=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
> ; ok, the normalized goal is
; ?_20: (all m.m<<Succ n -> cp m) -> Card cp n=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
> ; ok, we now have the new goal
; ?_21: Card cp n=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; ok, ?_21 can be obtained from
; ?_23: cp n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
; ?_22: cp n -> Card cp n=Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; Not provable in minimal propositional logic.
; ok, ?_22 is proved in intuitionistic propositional logic.
; The active goal now is
; ?_23: cp n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; ok, ?_23 can be obtained from
; ?_24: n<<Succ n from
; cp n198 n IH:(all m.m<<n -> cp m) -> Card cp n=n
; cp n -> F:cp n -> F
; all m.m<<Succ n -> cp m:
; all m.m<<Succ n -> cp m
> ; ok, ?_24 is proved. Proof finished.
> ; ok, CardAllImpMax has been added as a new theorem.
>
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