(* *)
(**************************************************************************)
+include "nat_ordered_set.ma".
include "models/q_bars.ma".
+axiom le_le_eq: ∀x,y:Q. x ≤ y → y ≤ x → x = y.
+
lemma initial_shift_same_values:
∀l1:q_f.∀init.init < start l1 →
same_values l1
cut (\fst w2 = O); [2: clear H10;
symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
- left; reflexivity;]
+ 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] right; assumption;]
+ 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]);
|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 (mk_q_f init ?) n1));[apply [];|3:apply l]
- simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus w);
- apply q_le_minus_r; rewrite < q_minus_r;
- rewrite < E in ⊢ (??%); assumption;]
+ 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 ⊢ %;
- simplify in H5:(? ? (? ? %));
-
+ 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 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 4 (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;
+[1: rewrite > nth_nil; cases w1 in H4;
+ [1: 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 ??? H2);
+ apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ apply q_eq_to_le; reflexivity;
+ |2: intros; simplify; rewrite > nth_nil; reflexivity;]
+|2: FACTORIZE w1>0
+
+ (* interesting case: init < start < input *)
+ intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+ simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+ elim (\fst w2) in H9 H10;
+ [1: elim (\fst w1) in H5 H6;
+ [1: cases (?:False); clear H5 H8 H7;
+ apply (q_lt_antisym input (start l1)); [2: assumption]
+ rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6;
+ [2: apply q_lt_to_le; assumption]
+ rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6;
+ rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init));
+ do 2 rewrite < q_elim_minus; assumption;
+ |2:
+
+ cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+ cases (\fst w1) in H5 H6; intros; [1:
+ cases (?:False); clear H5 H9 H10;
+ apply (q_lt_antisym input (start l1)); [2: assumption]
+ rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6;
+ [2: apply q_lt_to_le; assumption]
+ rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6;
+ rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init));
+ do 2 rewrite < q_elim_minus; assumption;]
+ apply eq_f;
+ cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n));[2:
+ apply (q_le_lt_trans ??? H9);
+ apply (q_lt_trans ??? ? H6);
+ 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_lt_to_le;assumption]
+ do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
+ apply q_lt_plus; rewrite > q_plus_sym;
+ rewrite > q_elim_minus; rewrite < q_plus_assoc;
+ rewrite < q_elim_minus; rewrite > q_plus_minus;
+ rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
+ clear H9 H6;
+ cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
+ 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_lt_to_le;assumption]
+ rewrite > H2; rewrite > (q_elim_minus (start ?));
+ rewrite > q_minus_distrib; rewrite > q_elim_opp;
+ do 2 rewrite > q_elim_minus;
+ do 2 rewrite < q_plus_assoc;
+ rewrite > (q_plus_sym ? init);
+ rewrite > (q_plus_assoc ? init);
+ rewrite > (q_plus_sym ? init);
+ rewrite < (q_elim_minus init); rewrite > q_plus_minus;
+ rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+ rewrite < q_elim_minus; reflexivity;]
+ cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
+ apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
+ rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
+ assumption;] clear Hcut1 H5 H10;
+ generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2;
+ apply (nat_elim2 ???? n (\fst w2));
+ [3: intros (x y); apply eq_f; apply H5; clear H5;
+ [1: clear H7; apply sum_bases_lt_canc; assumption;
+ |2: clear H6; ]
+ |2: intros; cases (?:False); clear H6;
+ cases n1 in H5; intro;
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+ |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
+ elim n2 in H5 H6;
+
+
+ elim (bars l1) 0;
+ [1: intro; elim n1; [reflexivity] cases (?:False);
+
+
+ intros; clear H5;
+ elim n1 in H6; [reflexivity] cases (?:False);
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+qed.
+
alias symbol "pi2" = "pair pi2".
projects / helm.git / blobdiff