-sandwich.ma ordered_uniform.ma
property_sigma.ma ordered_uniform.ma russell_support.ma
-uniform.ma supremum.ma
bishop_set.ma ordered_set.ma
-sequence.ma nat/nat.ma
ordered_uniform.ma uniform.ma
-supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
-property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
bishop_set_rewrite.ma bishop_set.ma
-cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
+sequence.ma nat/nat.ma
nat_ordered_set.ma bishop_set.ma nat/compare.ma
lebesgue.ma property_exhaustivity.ma sandwich.ma
+property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
+cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
ordered_set.ma cprop_connectives.ma
+sandwich.ma ordered_uniform.ma
russell_support.ma cprop_connectives.ma nat/nat.ma
-models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+uniform.ma supremum.ma
+supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/q_support.ma Q/q/q.ma
-models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
-models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma
-models/q_function.ma models/q_bars.ma
models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
-models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
-models/list_support.ma list/list.ma
+models/q_support.ma Q/q/q.ma cprop_connectives.ma
models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
+models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+models/list_support.ma list/list.ma
+models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
+models/q_function.ma models/q_bars.ma nat_ordered_set.ma
+models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
Q/q/q.ma
datatypes/constructors.ma
list/list.ma
[ O ⇒ OQ
| S m ⇒
match l with
- [ nil ⇒ sum_bases l m + Qpos one
+ [ nil ⇒ sum_bases [] m + Qpos one
| cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
axiom sum_bases_empty_nat_of_q_ge_OQ:
intro; elim l; simplify; intros;
[1: elim n; [apply q_eq_to_le;reflexivity] simplify;
apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
-|2: cases n; [apply q_eq_to_le;reflexivity] simplify;
+|2: cases n; [apply q_eq_to_le;reflexivity] simplify;
apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
qed.
+alias symbol "leq" = "Q less or equal than".
lemma sum_bases_O:
∀l.∀x.sum_bases l x ≤ OQ → x = O.
intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
apply (sum_bases_ge_OQ l1);
qed.
-lemma sum_bases_increasing:
- ∀l,x.sum_bases l x < sum_bases l (S x).
-intro; elim l;
-[1: elim x;
- [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_pos_lt_OQ;
- |2: simplify in H ⊢ %;
- apply q_lt_plus; rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_OQ;
- assumption;]
-|2: elim x;
- [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_pos_lt_OQ;
- |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;
- apply q_lt_plus; rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
-qed.
-
-lemma sum_bases_lt_canc:
- ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
-intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
-generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
-intros 2;
-[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
- apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
-|2: cases (?:False); simplify in H2;
- apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
- apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
-|1: cases n in H2; intro;
- [1: cases (?:False); apply (q_lt_corefl ? H2);
- |2: simplify; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
- apply q_pos_lt_OQ;]]
-qed.
-lemma sum_bars_increasing2:
+lemma sum_bases_increasing:
∀l.∀n1,n2:nat.n1<n2→sum_bases l n1 < sum_bases l n2.
intro; elim l 0;
[1: intros 2; apply (cic:/matita/dama/nat_ordered_set/nat_elim2.con ???? n1 n2);
apply le_S_S_to_le; apply H2;]]
qed.
+
definition eject1 ≝
λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
coercion eject1.
definition value :
∀f:q_f.∀i:ℚ.∃p:nat × ℚ.
Or4
- (And3 (i ≤ start f) (\fst p = O) (\snd p = OQ))
+ (And3 (i < start f) (\fst p = O) (\snd p = OQ))
(And3
(start f + sum_bases (bars f) (len (bars f)) ≤ i)
(\fst p = O) (\snd p = OQ))
(And3 (bars f = []) (\fst p = O) (\snd p = OQ))
(And4
- (start f ≤ i ∧ i < start f + sum_bases (bars f) (len (bars f)))
+ (And3 (bars f ≠ []) (start f ≤ i) (i < start f + sum_bases (bars f) (len (bars f))))
(\fst p ≤ (len (bars f)))
(\snd p = \snd (nth (bars f) ▭ (\fst p)))
(sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f] ∧
let rc ≝ value (p - Qpos (\fst x)) tl in
〈S (\fst rc),\snd rc〉]]
in value :
- ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
- And3
+ ∀acc,l.∃p:nat × ℚ.OQ ≤ acc →
+ Or
+ (And3 (l = []) (\fst p = nat_of_q acc) (\snd p = OQ))
+ (And3
(sum_bases l (\fst p) ≤ acc)
(acc < sum_bases l (S (\fst p)))
- (\snd p = \snd (nth l ▭ (\fst p))));
+ (\snd p = \snd (nth l ▭ (\fst p)))));
[5: clearbody value;
cases (q_cmp i (start f));
[2: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption;
try reflexivity; apply q_lt_to_le; assumption;
- |1: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption;
- try reflexivity; apply q_eq_to_le; assumption;
+ |1: cases (bars f); [exists [apply 〈O,OQ〉] simplify; constructor 3; split;try assumption;reflexivity;]
+ cases (value ⅆ[i,start f] (b::l)) (p Hp);
+ cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H1; destruct H2]
+ cases H1; clear H1; lapply (sum_bases_O (b::l) (\fst p)) as H1;
+ [2: apply (q_le_trans ??? H2); rewrite > H; apply q_eq_to_le;
+ rewrite > q_d_x_x; reflexivity;
+ |1: exists [apply p] simplify; constructor 4; rewrite > H1; split;
+ try split; try rewrite > q_d_x_x; try autobatch depth=2;
+ [1: rewrite > H; rewrite > q_plus_sym; apply q_lt_plus;
+ rewrite > q_plus_minus; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_pos_lt_OQ;
+ |2: rewrite > H; rewrite > q_d_x_x; apply q_eq_to_le; reflexivity;
+ |3: rewrite > H; rewrite > q_d_x_x; apply q_lt_plus_trans;
+ try apply sum_bases_ge_OQ; apply q_pos_lt_OQ;]]
|3: cases (q_cmp i (start f+sum_bases (bars f) (len (bars f))));
[1: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption;
try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity;
intros;
[1: exists [apply 〈O,OQ〉] simplify; constructor 3; split; reflexivity;
|2: cases (value ⅆ[i,start f] (b::l)) (p Hp);
- cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value;
+ cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H3;destruct H4]
+ cases H3; clear H3;
exists [apply p]; constructor 4; split; try split; try assumption;
- [1: apply q_lt_to_le; assumption;
- |2: rewrite < H2; assumption;
- |3: cases (cmp_nat (\fst p) (len (bars f)));
- [1:apply lt_to_le;rewrite <H2; assumption|rewrite > H6;rewrite < H2;apply le_n]
+ [1: intro X; destruct X;
+ |2: apply q_lt_to_le; assumption;
+ |3: rewrite < H2; assumption;
+ |4: cases (cmp_nat (\fst p) (len (bars f)));
+ [1:apply lt_to_le;rewrite <H2; assumption|rewrite > H3;rewrite < H2;apply le_n]
cases (?:False); cases (\fst p) in H3 H4 H6; clear H5;
- [1: intros; apply (not_le_Sn_O ? H5);
- |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption]
+ [1: intros; apply (not_le_Sn_O ? H5);
+ |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption]
intros; lapply (q_lt_inj_plus_r ?? (Qopp (start f)) H1); clear H1;
generalize in match Hletin;
rewrite > (q_plus_sym (start f)); rewrite < q_plus_assoc;
do 2 rewrite < q_elim_minus; rewrite > q_plus_minus;
rewrite > q_plus_OQ; intro K; apply (q_lt_corefl (i-start f));
apply (q_lt_le_trans ???? H3); rewrite < H2;
- apply (q_lt_trans ??? K); apply sum_bars_increasing2;
+ apply (q_lt_trans ??? K); apply sum_bases_increasing;
assumption;]]]]]
-|1,3: intros; split;
- [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
+|1,3: intros; right; split;
+ [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- simplify; apply q_le_minus; assumption;
+ [1: intro; apply q_lt_to_le;assumption;
+ |3: simplify; cases H4; apply q_le_minus; assumption;
+ |2,5: simplify; cases H4; rewrite > H5; rewrite > H6;
+ apply q_le_minus; apply sum_bases_empty_nat_of_q_le_q;
+ |4: intro X; rewrite > X; apply q_eq_to_le; reflexivity;
+ |*: simplify; apply q_le_minus; cases H4; assumption;]
|2,5: cases (value (q-Qpos (\fst b)) l1);
cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- clear H3 H2 value;
- change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
- apply q_lt_plus; assumption;
+ [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
+ |3,6: cases H5; simplify; change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
+ apply q_lt_plus; assumption;
+ |2,5: simplify; cases H5; rewrite > H6; simplify; rewrite > H7;
+ apply q_lt_plus; apply sum_bases_empty_nat_of_q_le_q_one;]
|*: cases (value (q-Qpos (\fst b)) l1); simplify;
cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
- [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
- assumption;]
-|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
+ [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption;
+ |3,6: cases H5; assumption;
+ |*: cases H5; rewrite > H6; rewrite > H8;
+ elim (\fst w); [1,3:reflexivity;] simplify; assumption;]]
+|2: clear value H2; simplify; intros; right; split; [assumption|3:reflexivity]
rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
-|4: simplify; intros; split;
- [1: apply sum_bases_empty_nat_of_q_le_q;
- |2: apply sum_bases_empty_nat_of_q_le_q_one;
- |3: elim (nat_of_q q); [reflexivity] simplify; assumption]]
+|4: intros; left; split; reflexivity;]
+qed.
+
+lemma value_OQ_l:
+ ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6);
+qed.
+
+lemma value_OQ_r:
+ ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H);
+qed.
+
+lemma value_OQ_e:
+ ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (H1 H);
+qed.
+
+lemma value_ok:
+ ∀f,i. bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) →
+ And4
+ (\fst (\fst (value f i)) ≤ (len (bars f)))
+ (\snd (\fst (value f i)) = \snd (nth (bars f) ▭ (\fst (\fst (value f i)))))
+ (sum_bases (bars f) (\fst (\fst (value f i))) ≤ ⅆ[i,start f])
+ (ⅆ[i, start f] < sum_bases (bars f) (S (\fst (\fst (value f i))))).
+intros; cases (value f i); cases H3; simplify; clear H3; cases H4;
+[1,2,3: cases (?:False);
+ [1: apply (q_lt_le_incompat ?? H3 H1);
+ |2: apply (q_lt_le_incompat ?? H2 H3);
+ |3: apply (H H3);]
+|4: split; cases H7; try assumption;]
qed.
-
+
definition same_values ≝
λl1,l2:q_f.
∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
cases (unpos (start l1-init) H1); intro input;
simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
-(*cases (value l1 input) (v2 Hv2); *)
-cases Hv1 (HV1 HV1 HV1 HV1); (* cases Hv2 (HV2 HV2 HV2 HV2); clear Hv1 Hv2; *)
-cases HV1 (Hi1 Hv11 Hv12); (*cases HV2 (Hi2 Hv21 Hv22);*) clear HV1 (*HV2*);
-(* simplify; *)
-rewrite > Hv12; (*rewrite > Hv22;*) try reflexivity;
-[1: simplify in Hi1; cases (?:False);
- apply (q_lt_corefl (start l1)); cases (Hi2);
- autobatch by Hi2, Hi1, q_le_trans, H4, H, q_le_lt_trans, q_lt_le_trans.
-|2: simplify in Hi1; cases (?:False);
- apply (q_lt_corefl (start l1+sum_bases (bars l1) (len (bars l1))));
- cases Hi2; apply (q_le_lt_trans ???? H5);
- apply (q_le_trans ???? Hi1);
- rewrite > H2; rewrite > (q_plus_sym ? (start l1-init));
- rewrite > q_plus_assoc; apply q_le_inj_plus_r;
- apply q_eq_to_le;
- rewrite > q_elim_minus; rewrite > (q_plus_sym (start l1));
- rewrite > q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- reflexivity;
+cases Hv1 (HV1 HV1 HV1 HV1); cases HV1 (Hi1 Hv11 Hv12); clear HV1 Hv1;
+[1: cut (input < start l1) as K;[2: apply (q_lt_trans ??? Hi1 H)]
+ rewrite > (value_OQ_l ?? K); simplify; symmetry; assumption;
+|2: cut (start l1 + sum_bases (bars l1) (len (bars l1)) ≤ input) as K;[2:
+ simplify in Hi1; apply (q_le_trans ???? Hi1); rewrite > H2;
+ rewrite > q_plus_sym in ⊢ (? ? (? ? %));
+ rewrite > q_plus_assoc; rewrite > q_elim_minus;
+ rewrite > q_plus_sym in ⊢ (? ? (? (? ? %) ?));
+ rewrite > q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus; rewrite > q_plus_sym in ⊢ (? ? (? % ?));
+ rewrite > q_plus_OQ; apply q_eq_to_le; reflexivity;]
+ rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
|3: simplify in Hi1; destruct Hi1;
-|4: simplify in Hi1 H3 Hv12 Hv11 ⊢ %; cases H3; clear H3;
- cases (\fst v1) in H4; [intros;reflexivity] intros;
- simplify; simplify in H3;
+|4:
-
-
-
-
-simplify in ⊢ (? ? ? (? ? ? %));
-cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
-whd in ⊢ (% → ?); simplify in H3;
-[1: intro; cases H4; clear H4; rewrite > H3;
- cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
- [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
- |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
- |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
- rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
- symmetry; apply le_n_O_to_eq;
- rewrite > (sum_bases_O (〈w,OQ〉::bars l1) (\fst w1)); [apply le_n]
- clear H6 w2; simplify in H5:(? ? (? ? %));
- destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
- cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
- [1: cases (?:False); clear w2 H4 w1 H2 w H1;
- apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
- |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
- |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
- apply (q_lt_trans ??? H3 H);]
-|3: intro; cases H4; clear H4;
- cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
- [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
- simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
- cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
- cut (\fst w2 = O); [2: clear H10;
- symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O (bars l1) (\fst w2)); [apply le_n]
- apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
- apply q_eq_to_le; reflexivity;]
- rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
- cut (ⅆ[input,init] = Qpos w) as E; [2:
- rewrite > H2; rewrite < H4; rewrite > q_d_sym;
- rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
- cases (\fst w1) in H5 H6; intros;
- [1: cases (?:False); clear H5; simplify in H6;
- apply (q_lt_corefl ⅆ[input,init]);
- rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
- rewrite > q_plus_sym; assumption;
- |2: cases n in H5 H6; [intros; reflexivity] intros;
- cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
- [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));]
- apply (q_le_S ??? (sum_bases_ge_OQ ? n1));[apply []|3:apply l]
- simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
- rewrite > q_elim_minus; apply q_le_minus_r;
- rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
- |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
- simplify in H5 H6 ⊢ %;
- cases (\fst w1) in H5 H6; [intros; reflexivity]
- cases (bars l1);
- [1: intros; simplify; elim n [reflexivity] simplify; assumption;
- |2: simplify; intros; cases (?:False); clear H6;
- apply (q_lt_le_incompat (input - init) (Qpos w) );
- [1: rewrite > H2; do 2 rewrite > q_elim_minus;
- apply q_lt_plus; rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
- |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
- rewrite > q_d_sym
-
- ; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
- |3: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
- simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
-
-axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
-
-
-lemma case1 :
- ∀init,st,input,l.
- init<st → st<input →
- ⅆ[input,init] < sum_bases l O + (st-init) → False.
-intros 6; rewrite > q_d_sym; rewrite > q_d_noabs; [2:
- apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
-intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
-simplify in Y; cases (?:False);
-apply (q_lt_corefl st); apply (q_lt_trans ??? H1);
-apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
-apply q_eq_to_le; reflexivity;
-qed.
-
-lemma case2:
- ∀a,l1,init,st,input,n.
- init < st → st < input →
- sum_bases (a::l1) n + (st-init) ≤ ⅆ[input,init] →
- ⅆ[input,st] < sum_bases l1 O + Qpos (\fst a) →
- n = O.
-intros; cut (input - st < Qpos (\fst a)) as H6';[2:
- rewrite < q_d_noabs;[2:apply q_lt_to_le; assumption]
- rewrite > q_d_sym; apply (q_lt_le_trans ??? H3);
- rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity] clear H3;
-generalize in match H2; rewrite > q_d_sym; rewrite > q_d_noabs;
- [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc; intro X;
-lapply (q_le_canc_plus_r ??? X) as Y; clear X;
-lapply (q_le_inj_plus_r ?? (Qopp st) Y) as X; clear Y;
-cut (input + Qopp st < Qpos (\fst a)) as H6'';
- [2: rewrite < q_elim_minus; assumption;] clear H6';
-generalize in match (q_le_lt_trans ??? X H6''); clear X H6'';
-rewrite < q_plus_assoc; rewrite < q_elim_minus;
-rewrite > q_plus_minus; rewrite > q_plus_OQ; cases n; intro X; [reflexivity]
-cases (?:False);
-apply (q_lt_le_incompat (sum_bases l1 n1) OQ);[2: apply sum_bases_ge_OQ;]
-apply (q_lt_canc_plus_r ?? (Qpos (\fst a)));
-rewrite >(q_plus_sym OQ); rewrite > q_plus_OQ; apply X;
-qed.
-
-lemma case3:
- ∀init,st,input,l1,a,n.
- init<st → st<input →
- ⅆ[input,init]<OQ+Qpos a+(st-init) →
- sum_bases l1 n+Qpos a≤ⅆ[input,st] → False.
-intros;
-cut (sum_bases l1 n - ⅆ[input,st] < Qopp ⅆ[input,init] + (st - init)); [2:
- cut (sum_bases l1 n≤ⅆ[input,st]-Qpos a) as H7';[2:
- apply (q_le_canc_plus_r ?? (Qpos a));
- apply (q_le_trans ??? H3); rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity;] clear H3;
- rewrite > q_elim_minus; apply (q_lt_canc_plus_r ?? ⅆ[input,st]);
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply (q_le_lt_trans ??? H7'); clear H7'; rewrite > q_elim_minus;
- rewrite > q_plus_sym; apply q_lt_inj_plus_r;
- rewrite > q_plus_sym; apply q_lt_plus; rewrite > q_elim_opp;
- rewrite > q_plus_sym; apply (q_lt_canc_plus_r ?? (Qpos a));
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply (q_lt_le_trans ??? H2); rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
- rewrite > q_plus_sym; apply q_eq_to_le; reflexivity;]
-generalize in match Hcut; clear H2 H3 Hcut;
-rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le; assumption]
-rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-rewrite < q_plus_sym; rewrite < q_elim_minus;
-rewrite > (q_elim_minus input init);
-rewrite > q_minus_distrib; rewrite > q_elim_opp;
-rewrite > (q_elim_minus input st);
-rewrite > q_minus_distrib; rewrite > q_elim_opp;
-repeat rewrite > q_elim_minus;
-rewrite < q_plus_assoc in ⊢ (??% → ?);
-rewrite > (q_plus_sym (Qopp input) init);
-rewrite > q_plus_assoc;
-rewrite < q_plus_assoc in ⊢ (??(?%?) → ?);
-rewrite > (q_plus_sym (Qopp init) init);
-rewrite < (q_elim_minus init); rewrite >q_plus_minus;
-rewrite > q_plus_OQ; rewrite > (q_plus_sym st);
-rewrite < q_plus_assoc;
-rewrite < (q_plus_OQ (Qopp input + st)) in ⊢ (??% → ?);
-rewrite > (q_plus_sym ? OQ); intro X;
-lapply (q_lt_canc_plus_r ??? X) as Y; clear X;
-apply (q_lt_le_incompat ?? Y); apply sum_bases_ge_OQ;
-qed.
-
-lemma key:
- ∀init,input,l1,w1,w2,w.
- Qpos w = start l1 - init →
- init < start l1 →
- start l1 < input →
- sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
- ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
- sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
- ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
- \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
-intros 3 (init input l); cases l (st l);
-change in match (start (mk_q_f st l)) with st;
-change in match (bars (mk_q_f st l)) with l;
-elim l; clear l;
-[1: rewrite > nth_nil; cases w1 in H4;
- [1: intro X; cases (case1 ?????? X); assumption;
- |2: intros; simplify; rewrite > nth_nil; reflexivity;]
-|2: cases w1 in H4 H5; clear w1;
- [1: intros (Y X); cases (case1 ?????? X); assumption;
- |2: intros; simplify in H4 H5 H7 ⊢ %;
- generalize in match H6; generalize in match H7;
- generalize in match H4; generalize in match H5; clear H4 H5 H6 H7;
- apply (nat_elim2 ???? w2 n); clear w2 n; intros;
- [1: rewrite > (case2 a l1 init st input n); [reflexivity]
- try rewrite < H1; assumption;
- |2: simplify in H4 H7; cases (case3 ???????? H4 H7); assumption;
- |3: (* dipende se vanno oltre la lunghezza di l1,
- forse dovevo gestire il caso prima dell'induzione *)
- simplify in ⊢ (? ? (? ? ? %) ?);
- rewrite > (H (S m) ? w); [reflexivity] try assumption;
STOP
qed.
projects / helm.git / commitdiff
