-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
-nat_ordered_set.ma bishop_set.ma nat/compare.ma
+russell_support.ma cprop_connectives.ma nat/nat.ma
lebesgue.ma property_exhaustivity.ma sandwich.ma
ordered_set.ma cprop_connectives.ma
-russell_support.ma cprop_connectives.ma nat/nat.ma
-models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+bishop_set.ma ordered_set.ma
+bishop_set_rewrite.ma bishop_set.ma
+sandwich.ma ordered_uniform.ma
+nat_ordered_set.ma bishop_set.ma nat/compare.ma
+property_sigma.ma ordered_uniform.ma russell_support.ma
+sequence.ma nat/nat.ma
+property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
+supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
+uniform.ma supremum.ma
+models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma cprop_connectives.ma
+models/q_function.ma Q/q/qtimes.ma models/q_bars.ma
models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
+models/list_support.ma cprop_connectives.ma list/list.ma nat/le_arith.ma nat/minus.ma
models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
-models/q_function.ma Q/q/qtimes.ma models/q_shift.ma nat_ordered_set.ma
-models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
+models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma cprop_connectives.ma
+models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
+models/q_shift.ma
models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
-models/q_value_skip.ma models/q_shift.ma
-models/q_shift.ma models/q_bars.ma
-models/list_support.ma list/list.ma
-models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
+models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
+models/q_value_skip.ma
Q/q/qplus.ma
Q/q/qtimes.ma
datatypes/constructors.ma
logic/equality.ma
nat/compare.ma
nat/le_arith.ma
+nat/minus.ma
nat/nat.ma
nat/plus.ma
intros; elim l1; [reflexivity] simplify; rewrite < H; reflexivity;
qed.
-inductive non_empty_list (A:Type) : list A → Type :=
+coinductive non_empty_list (A:Type) : list A → Type :=
| show_head: ∀x,l. non_empty_list A (x::l).
lemma len_gt_non_empty :
destruct H6; simplify; assumption;]
qed.
-(*
-lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
-intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
-cases (bars_not_nil f); intros;
-cases (cmp_nat i (len l));
-[1: lapply (sorted_tail_bigger ?? H ? H2) as K; simplify in H1;
- rewrite > H1 in K; apply K;
-|2: rewrite > H2; simplify; elim l; simplify; [apply (q_pos_OQ one)]
- assumption;
-|3: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
- cases n in H3; intros; [cases (not_le_Sn_O ? H3)] apply (H2 n1);
- apply (le_S_S_to_le ?? H3);]
-qed.
-*)
-
(* move in nat/ *)
lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m.
intros; rewrite > sym_plus; apply (le_S_S n (m+n)); alias id "le_plus_n" = "cic:/matita/nat/le_arith/le_plus_n.con".
|2: apply H3; assumption]]
qed.
-inductive cases_bool (p:bool) : bool → CProp ≝
+coinductive cases_bool (p:bool) : bool → CProp ≝
| case_true : p = true → cases_bool p true
| cases_false : p = false → cases_bool p false.
intros; cases (f x);[left;|right] reflexivity;
qed.
+coinductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝
+| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l.
+
+lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l.
+intros 2; elim n;
+[1: elim l in H; [cases (not_le_Sn_O ? H)]
+ apply (break_to ?? ? [] a l1); reflexivity;
+|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros;
+ [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1;
+ rewrite > len_append in H1; rewrite > plus_n_SO in H1;
+ cases (not_le_Sn_n ? H1);
+ |2: apply (break_to ?? ? (l1@[x]) t l3);
+ [2: simplify; rewrite > associative_append; assumption;
+ |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]]
+qed.
+
include "cprop_connectives.ma".
definition eject_N ≝
definition inject_N ≝ λP.λp:nat.λh:P p. ex_introT ? P p h.
coercion inject_N with 0 1 nocomposites.
-inductive find_spec (T:Type) (P:T→bool) (l:list T) (d:T) (res : nat) : nat → CProp ≝
+coinductive find_spec (T:Type) (P:T→bool) (l:list T) (d:T) (res : nat) : nat → CProp ≝
| found:
∀i. i < \len l → P (\nth l d i) = true → res = i →
(∀j. j < i → P (\nth l d j) = false) → find_spec T P l d res i
include "models/list_support.ma".
include "cprop_connectives.ma".
-definition bar ≝ ℚ × ℚ.
+definition bar ≝ ℚ × (ℚ × ℚ).
notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
interpretation "Q x Q" 'q2 = (Prod Q Q).
-definition empty_bar : bar ≝ 〈Qpos one,OQ〉.
+definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
notation "\rect" with precedence 90 for @{'empty_bar}.
interpretation "q0" 'empty_bar = empty_bar.
bars: list bar;
bars_sorted : sorted q2_lt bars;
bars_begin_OQ : nth_base bars O = OQ;
- bars_end_OQ : nth_height bars (pred (\len bars)) = OQ
+ bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
}.
lemma len_bases_gt_O: ∀f.O < \len (bars f).
apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
qed.
-(*
-lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m.
-intros; rewrite > sym_plus; apply (le_S_S n (m+n)); apply (le_plus_n m n); qed.
-*)
-
-(*
-lemma all_bigger_can_concat_bigger:
- ∀l1,l2,start,b,x,n.
- (∀i.i< len l1 → nth_base l1 i < \fst b) →
- (∀i.i< len l2 → \fst b ≤ nth_base l2 i) →
- (∀i.i< len l1 → start ≤ i → x ≤ nth_base l1 i) →
- start ≤ n → n < len (l1@b::l2) → x ≤ \fst b → x ≤ nth_base (l1@b::l2) n.
-intros; cases (cmp_nat n (len l1));
-[1: unfold nth_base; rewrite > (nth_concat_lt_len ????? H6);
- apply (H2 n); assumption;
-|2: rewrite > H6; unfold nth_base; rewrite > nth_len; assumption;
-|3: unfold nth_base; rewrite > nth_concat_ge_len; [2: apply lt_to_le; assumption]
- rewrite > len_concat in H4; simplify in H4; rewrite < plus_n_Sm in H4;
- lapply linear le_S_S_to_le to H4 as K; rewrite > sym_plus in K;
- lapply linear le_plus_to_minus to K as X;
- generalize in match X; generalize in match (n - len l1); intro W; cases W; clear W X;
- [intros; assumption] intros;
- apply (q_le_trans ??? H5); apply (H1 n1); assumption;]
-qed.
-*)
-
-
-inductive value_spec (f : q_f) (i : ℚ) : ℚ → nat → CProp ≝
- value_of : ∀q,j.
- nth_height (bars f) j = q →
- nth_base (bars f) j < i →
- (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q j.
-
-
-inductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝
-| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l.
-
-lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l.
-intros 2; elim n;
-[1: elim l in H; [cases (not_le_Sn_O ? H)]
- apply (break_to ?? ? [] a l1); reflexivity;
-|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros;
- [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1;
- rewrite > len_append in H1; rewrite > plus_n_SO in H1;
- cases (not_le_Sn_n ? H1);
- |2: apply (break_to ?? ? (l1@[x]) t l3);
- [2: simplify; rewrite > associative_append; assumption;
- |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]]
-qed.
-
-definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j.
+coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝
+| value_of : ∀j,q.
+ nth_height (bars f) j = q → nth_base (bars f) j < i →
+ (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q.
+
+definition value_lemma : ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p.
intros;
letin P ≝
(λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]);
exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));]
-exists [apply (pred (find ? P (bars f) ▭))] apply value_of;
+apply (value_of ?? (pred (find ? P (bars f) ▭)));
[1: reflexivity
|2: cases (cases_find bar P (bars f) ▭);
[1: cases i1 in H H1 H2 H3; simplify; intros;
[ apply le_O_n; | assumption]]]
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 : q_f → ratio → ℚ × ℚ.
+intros; cases (value_lemma q r); apply w; qed.
-inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝
- | value_ok : ∀n,q. n ≤ (len (bars f)) →
- q = \snd (nth (bars f) ▭ n) →
- sum_bases (bars f) n ≤ ⅆ[i,start f] →
- ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉.
-
-lemma value_ok:
- ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) →
- value_ok_spec f i (\fst (value f i)).
-intros; cases (value f i); simplify;
-cases H3; simplify; clear H3; cases H4; clear 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: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros;
- constructor 1; assumption;]
-qed.
+lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i).
+intros; unfold value; cases (value_lemma f i); assumption; qed.
-definition same_values ≝
- λl1,l2:q_f.
- ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
+definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input.
-definition same_bases ≝
- λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
+definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i).
alias symbol "lt" = "Q less than".
lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
intro; cases x; intros; [2:exists [apply r] reflexivity]
cases (?:False);
-[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
+[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
qed.
notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
(* *)
(**************************************************************************)
-include "nat_ordered_set.ma".
-include "models/q_shift.ma".
+include "models/q_bars.ma".
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "models/q_bars.ma".
-
-lemma initial_shift_same_values:
- ∀l1:q_f.∀init.init < start l1 →
- same_values l1
- (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
-[apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
-intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
-cases (unpos (start l1-init) H1); intro input;
-simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
-cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
-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)]
- simplify in ⊢ (? ? ? (? ? ? %));
- 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;]
- simplify in ⊢ (? ? ? (? ? ? %));
- rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
-|3: simplify in Hi1; destruct Hi1;
-|4: cut (start l1 ≤ input → \snd (\fst (value l1 input))=\snd v1) as solution;[2:
- intro H4; cases Hi1; clear Hi1; cases H3; clear H3;
- simplify in H5 H6 H8 H9 H7:(? ? (? % %)) ⊢ (? ? ? (? ? ? %));
- generalize in match (refl_eq ? (bars l1):bars l1 = bars l1);
- generalize in ⊢ (???% → ?); intro X; cases X; clear X; intro Hb;
- [1: rewrite > (value_OQ_e ?? Hb); rewrite > Hv12; rewrite > Hb in Hv11 ⊢ %;
- simplify in Hv11 ⊢ %; cases (\fst v1) in Hv11; [intros; reflexivity]
- cases n; [intros; reflexivity] intro X; cases (not_le_Sn_O ? (le_S_S_to_le ?? X));
- |2: cases (value_ok l1 input);
- [2: rewrite > Hb; intro W; destruct W;
- |3: assumption;
- |4: apply (q_lt_le_trans ??? H7); rewrite > H2;
- rewrite > q_plus_sym; rewrite < q_plus_assoc;
- rewrite > q_plus_sym; apply q_le_inj_plus_r;
- apply q_le_minus; apply q_eq_to_le; reflexivity;
- |1: rewrite > Hv12; rewrite > Hb; clear Hv12; simplify;
- rewrite > H10; rewrite > Hb;
- cut (O < \fst v1);[2: cases (\fst v1) in H9; intros; [2: autobatch]
- cases (?:False); generalize in match H9;
- rewrite > q_d_sym; rewrite > q_d_noabs; [2,4: assumption]
- rewrite > H2; simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- repeat rewrite > q_elim_minus;
- intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
- apply (q_lt_le_incompat ?? Y); assumption;]
- cases (\fst v1) in H8 H9 Hcut; [1:intros (_ _ X); cases (not_le_Sn_O ? X)]
- intros; clear H13; simplify;
- rewrite > (sum_bases_n_m n n1 (b::l)); [1,4: reflexivity] rewrite < Hb;
- [2: simplify in H8; apply (q_le_lt_trans ??? (q_le_plus_r ??? H8));
- apply (q_le_lt_trans ???? H12); rewrite > H2;
- rewrite > q_d_sym; rewrite > q_d_noabs; [2: assumption]
- rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
- rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
- rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? (? % ?) ?);
- rewrite > (q_plus_sym ? init); do 2 rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_OQ;
- rewrite > q_d_sym; rewrite > q_d_noabs; [2:assumption]
- apply q_eq_to_le; reflexivity;
- |*: apply (q_le_lt_trans ??? H11);
- rewrite > q_d_sym; rewrite > q_d_noabs; [2:assumption;]
- generalize in match H9; rewrite > q_d_sym; rewrite > q_d_noabs; [2: assumption]
- rewrite > H2; intro X;
- lapply (q_lt_inj_plus_r ?? (Qopp (start l1-init)) X) as Y; clear X;
- rewrite < q_plus_assoc in Y; repeat rewrite < q_elim_minus in Y;
- rewrite > q_plus_minus in Y; rewrite > q_plus_OQ in Y;
- apply (q_le_lt_trans ???? Y);
- rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
- rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
- rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? ? (? % ?));
- rewrite > (q_plus_sym ? init); rewrite < (q_elim_minus init);
- rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity;]]]]]
- cases (q_cmp input (start l1));
- [2: simplify in ⊢ (? ? ? (? ? ? %)); rewrite > (value_OQ_l ?? H4);
- change with (OQ = \snd v1); rewrite > Hv12;
- cases H3; clear H3; simplify in H5; cases (\fst v1) in H5;[intros;reflexivity]
- simplify; rewrite > q_d_sym; rewrite > q_d_noabs; [2:cases Hi1; apply H5]
- rewrite > H2; do 2 rewrite > q_elim_minus;rewrite > q_plus_assoc;
- intro X; lapply (q_le_canc_plus_r ??? X) as Y; clear X;
- cases (?:False); apply (q_lt_le_incompat input (start l1)); try assumption;
- apply (q_le_S ???? Y); try assumption; apply sum_bases_ge_OQ;
- |1: apply solution; apply (q_eq_to_le ?? (sym_eq ??? H4));
- |3: apply solution; apply (q_lt_to_le ?? H4);]
-qed.
-
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "models/q_shift.ma".
-
-lemma q_cmp2:
- ∀a,b:ℚ.a < b ∨ b ≤ a.
-intros; cases (q_cmp a b);
-[1: right; apply q_eq_to_le; symmetry; assumption;
-|2: left; assumption;
-|3: right; apply q_lt_to_le; assumption;]
-qed.
-
-
-
-lemma value_skip:
- ∀f,i,b,tl. bars f = b::tl → start f + Qpos (\fst b) < i →
- \fst (value f i) = \fst (value (mk_q_f (start f + Qpos (\fst b)) tl) i).
-intros; cases (value f i); cases H2; clear H2;
-[1: cases (?:False); apply (q_lt_corefl (start f)); cases H3; clear H3 H4 H5;
- apply (q_lt_trans ???? H2);
-|2:
-simplify in ⊢ (? ? % ?);
projects / helm.git / commitdiff
