:: Cages, external approximation of Jordan's curve
:: by Czes{\l}aw Byli\'nski and Mariusz \.Zynel
::
:: Received June 22, 1999
:: Copyright (c) 1999-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, XBOOLE_0, FINSEQ_1, XXREAL_0, ARYTM_1,
ARYTM_3, PRE_TOPC, RELAT_2, CONNSP_1, TARSKI, RELAT_1, FINSEQ_5,
PARTFUN1, GOBOARD1, EUCLID, REAL_1, MATRIX_1, COMPLEX1, GOBRD13, FUNCT_1,
TOPREAL1, RFINSEQ, RLTOPSP1, GOBOARD5, TOPS_1, TREES_1, SPPOL_1, MCART_1,
CARD_1, GOBOARD9, RCOMP_1, NAT_1, JORDAN8, PSCOMP_1, NEWTON, SPRECT_2,
ORDINAL4, STRUCT_0, PCOMPS_1, METRIC_1, JORDAN9, CONVEX1;
notations TARSKI, XBOOLE_0, SUBSET_1, GOBOARD5, ORDINAL1, CARD_1, NUMBERS,
XCMPLX_0, XREAL_0, REAL_1, NAT_1, NAT_D, RELAT_1, FUNCT_1, PARTFUN1,
FUNCT_2, COMPLEX1, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSEQ_5, NEWTON,
RFINSEQ, STRUCT_0, XXREAL_0, MATRIX_0, MATRIX_1, METRIC_1, PRE_TOPC,
TOPS_1, COMPTS_1, CONNSP_1, PCOMPS_1, RLTOPSP1, EUCLID, TOPREAL1,
GOBOARD1, SPPOL_1, PSCOMP_1, SPRECT_2, GOBOARD9, JORDAN8, GOBRD13;
constructors REAL_1, FINSEQ_4, NEWTON, RFINSEQ, NAT_D, TOPS_1, CONNSP_1,
COMPTS_1, SPPOL_1, PSCOMP_1, GOBOARD9, SPRECT_2, JORDAN8, GOBRD13,
RELSET_1, FUNCSDOM, PCOMPS_1, CONVEX1, MATRIX_1;
registrations RELAT_1, FUNCT_1, ORDINAL1, XXREAL_0, XREAL_0, NAT_1, FINSEQ_1,
STRUCT_0, EUCLID, SPPOL_2, PSCOMP_1, GOBOARD9, SPRECT_1, SPRECT_2,
JORDAN8, NEWTON, FINSET_1, SPPOL_1, JORDAN1, MATRIX_0;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions TARSKI, GOBOARD5, GOBRD13, XBOOLE_0, SEQM_3;
equalities GOBOARD5, XBOOLE_0, PSCOMP_1;
expansions TARSKI, GOBRD13;
theorems NAT_1, FINSEQ_1, GOBOARD1, FINSEQ_4, EUCLID, FINSEQ_3, SPPOL_2,
TARSKI, JORDAN3, PSCOMP_1, FINSEQ_5, FINSEQ_6, GOBOARD7, TOPREAL1,
JORDAN5B, GOBOARD5, SPRECT_2, SPPOL_1, ABSVALUE, FUNCT_1, FUNCT_2,
GOBOARD9, FINSEQ_2, UNIFORM1, SUBSET_1, GOBRD11, SPRECT_3, CARD_1,
RFINSEQ, GOBOARD6, TOPREAL3, TOPMETR, TOPS_1, JORDAN8, GOBRD13, SPRECT_4,
CONNSP_1, PARTFUN2, RELSET_1, SPRECT_1, XBOOLE_0, XBOOLE_1, XREAL_0,
XCMPLX_1, XREAL_1, NEWTON, XXREAL_0, ORDINAL1, PARTFUN1, MATRIX_0, NAT_D,
RLTOPSP1, SEQ_4;
schemes NAT_1, RECDEF_1;
begin :: Generalities
reserve i,j,k,n for Nat,
D for non empty set,
f, g for FinSequence of D;
Lm1: for n st 1 <= n holds n-'1+2 = n+1
proof
let n;
assume 1 <= n;
hence n-'1+2 = n+2-'1 by NAT_D:38
.= n+1+1 - 1 by NAT_D:37
.= n+1;
end;
theorem Th1:
for T being non empty TopSpace for B,C1,C2,D being Subset of T st
B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 &
B meets C2 & B c= D holds C1 = C2
proof
let T be non empty TopSpace;
let B,C1,C2,D be Subset of T;
assume that
A1: B is connected and
A2: C1 is_a_component_of D & C2 is_a_component_of D and
A3: B meets C1 and
A4: B meets C2 & B c= D;
A5: B <> {} by A3,XBOOLE_1:65;
B c= C1 & B c= C2 by A1,A2,A3,A4,GOBOARD9:4;
hence thesis by A2,A5,GOBOARD9:1,XBOOLE_1:68;
end;
theorem Th2:
(for n holds f|n = g|n) implies f = g
proof
assume
A1: for n holds f|n = g|n;
A2: now
assume
A3: len f <> len g;
per cases by A3,XXREAL_0:1;
suppose
A4: len f < len g;
A5: g|len g = g by FINSEQ_1:58;
f|len g = f by A4,FINSEQ_1:58;
hence contradiction by A1,A4,A5;
end;
suppose
A6: len g < len f;
then f|len f = f & g|len f = g by FINSEQ_1:58;
hence contradiction by A1,A6;
end;
end;
f|len f = f & g|len g = g by FINSEQ_1:58;
hence thesis by A1,A2;
end;
theorem Th3:
n in dom f implies ex k st k in dom Rev f & n+k = len f+1 & f/.n = (Rev f)/.k
proof
assume
A1: n in dom f;
take k = len f+1-'n;
1 <= n by A1,FINSEQ_3:25;
then k+1 <= k+n by XREAL_1:6;
then
A2: k+1-'1 <= k+n -'1 by NAT_D:42;
A3: n <= len f by A1,FINSEQ_3:25;
then n+1 <= len f+1 by XREAL_1:6;
then
A4: 1 <= k by NAT_D:55;
n <= len f+1 by A3,XREAL_1:145;
then
A5: k+n = len f+1 by XREAL_1:235;
then k+n-'1 = len f by NAT_D:34;
then k <= len f by A2,NAT_D:34;
then k in dom f by A4,FINSEQ_3:25;
hence thesis by A1,A5,FINSEQ_5:57,66;
end;
theorem Th4:
n in dom Rev f implies ex k st k in dom f & n+k = len f+1 & (Rev f)/.n = f/.k
proof
assume n in dom Rev f;
then n in dom f by FINSEQ_5:57;
then consider k such that
A1: k in dom Rev f & n+k = len f+1 and
f/.n = (Rev f)/.k by Th3;
A2: dom f = dom Rev f by FINSEQ_5:57;
then (Rev f)/.n = f/.k by A1,FINSEQ_5:66;
hence thesis by A1,A2;
end;
begin :: Go-board preliminaries
reserve G for Go-board,
f, g for FinSequence of TOP-REAL 2,
p for Point of TOP-REAL 2,
r, s for Real,
x for set;
theorem Th5:
for D being non empty set for G being Matrix of D for f being
FinSequence of D holds f is_sequence_on G iff Rev f is_sequence_on G
proof
let D be non empty set;
let G be Matrix of D;
let f be FinSequence of D;
hereby
assume
A1: f is_sequence_on G;
A2: for n being Nat st n in dom Rev f & n+1 in dom Rev f
for m,k,i,j being Nat
st [m,k] in Indices G & [i,j] in Indices G & (Rev f)/.n = G*(m,k
) & (Rev f)/.(n+1) = G*(i,j) holds |.m-i.|+|.k-j.| = 1
proof
let n be Nat such that
A3: n in dom Rev f and
A4: n+1 in dom Rev f;
consider l being Nat such that
A5: l in dom f and
A6: n+l = len f+1 and
A7: (Rev f)/.n = f/.l by A3,Th4;
let m,k,i,j be Nat such that
A8: [m,k] in Indices G & [i,j] in Indices G & (Rev f)/.n = G*(m,k)
& (Rev f)/.(n+1) = G*(i,j);
A9: |.i-m.| = |.m-i.| & |.j-k.| = |.k-j.| by UNIFORM1:11;
consider l9 being Nat such that
A10: l9 in dom f and
A11: n+1+l9 = len f+1 and
A12: (Rev f)/.(n+1) = f/.l9 by A4,Th4;
n+(1+l9) = n+l by A6,A11;
hence thesis by A1,A8,A5,A7,A10,A12,A9,GOBOARD1:def 9;
end;
for n being Nat st n in dom Rev f
ex i,j being Nat st [i,j] in Indices G & (Rev f)/.n = G*(i,j)
proof
let n be Nat;
assume n in dom Rev f;
then consider k such that
A13: k in dom f and
n+k = len f+1 and
A14: (Rev f)/.n = f/.k by Th4;
consider i,j being Nat such that
A15: [i,j] in Indices G & f/.k = G*(i,j) by A1,A13,GOBOARD1:def 9;
take i,j;
thus thesis by A14,A15;
end;
hence Rev f is_sequence_on G by A2,GOBOARD1:def 9;
end;
assume
A16: Rev f is_sequence_on G;
A17: for n being Nat st n in dom f & n+1 in dom f
for m,k,i,j being Nat
st [m,k] in Indices G & [i,j] in Indices G & f/.n = G*(m,k) & f/.(n+1) = G*
(i,j) holds |.m-i.|+|.k-j.| = 1
proof
let n be Nat such that
A18: n in dom f and
A19: n+1 in dom f;
consider l being Nat such that
A20: l in dom Rev f and
A21: n+l = len f+1 and
A22: f/.n = (Rev f)/.l by A18,Th3;
let m,k,i,j be Nat such that
A23: [m,k] in Indices G & [i,j] in Indices G & f/.n = G*(m,k) & f/.(n+
1) = G*(i,j);
A24: |.i-m.| = |.m-i.| & |.j-k.| = |.k-j.| by UNIFORM1:11;
consider l9 being Nat such that
A25: l9 in dom Rev f and
A26: n+1+l9 = len f+1 and
A27: f/.(n+1) = (Rev f)/.l9 by A19,Th3;
n+(1+l9) = n+l by A21,A26;
hence thesis by A16,A23,A20,A22,A25,A27,A24,GOBOARD1:def 9;
end;
for n being Nat st n in dom f
ex i,j being Nat st [i,j] in Indices G & f/.n = G*(i,j)
proof
let n be Nat;
assume n in dom f;
then consider k such that
A28: k in dom Rev f and
n+k = len f+1 and
A29: f/.n = (Rev f)/.k by Th3;
consider i,j being Nat such that
A30: [i,j] in Indices G & (Rev f)/.k = G*(i,j) by A16,A28,GOBOARD1:def 9;
take i,j;
thus thesis by A29,A30;
end;
hence thesis by A17,GOBOARD1:def 9;
end;
theorem Th6:
for G being Matrix of TOP-REAL 2 for f being FinSequence of
TOP-REAL 2 st f is_sequence_on G
for k being Nat st 1 <= k & k <= len f holds f/.k in
Values G
proof
let G be Matrix of TOP-REAL 2;
let f be FinSequence of TOP-REAL 2;
assume that
A1: f is_sequence_on G;
let k be Nat such that
A2: 1 <= k & k <= len f;
A3: k in dom f by A2,FINSEQ_3:25;
then f/.k = f.k by PARTFUN1:def 6;
then
A4: f/.k in rng f by A3,FUNCT_1:def 3;
rng f c= Values G by A1,GOBRD13:8;
hence thesis by A4;
end;
Lm2: f is_sequence_on G & 1 <= k & k <= len f
implies ex i,j being Nat st [i,j] in Indices G & f/.k = G*(i,j)
proof
assume that
A1: f is_sequence_on G and
A2: 1 <= k & k <= len f;
k in dom f by A2,FINSEQ_3:25;
then consider i,j being Nat such that
A3: [i,j] in Indices G & f/.k = G*(i,j) by A1,GOBOARD1:def 9;
take i,j;
thus thesis by A3;
end;
theorem Th7:
n <= len f & x in L~(f/^n)
implies ex i being Nat st n+1 <= i & i+1 <= len f & x in LSeg(f,i)
proof
assume that
A1: n <= len f and
A2: x in L~(f/^n);
consider j being Nat such that
A3: 1 <= j and
A4: j+1 <= len(f/^n) and
A5: x in LSeg(f/^n,j) by A2,SPPOL_2:13;
j+1 <= len f - n by A1,A4,RFINSEQ:def 1;
then
A6: n+(j+1) <= len f by XREAL_1:19;
take n+j;
j+1 <= len f - n by A1,A4,RFINSEQ:def 1;
hence thesis by A3,A5,A6,SPPOL_2:5,XREAL_1:6;
end;
theorem Th8:
f is_sequence_on G implies
for k being Nat st 1 <= k & k+1 <= len f
holds f/.k in left_cell(f,k,G) & f/.k in right_cell(f,k,G)
proof
assume that
A1: f is_sequence_on G;
let k be Nat such that
A2: 1 <= k & k+1 <= len f;
set p = f/.k;
LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A2,TOPREAL1:def 3;
then p in LSeg(f,k) by RLTOPSP1:68;
then p in left_cell(f,k,G) /\ right_cell(f,k,G) by A1,A2,GOBRD13:29;
hence thesis by XBOOLE_0:def 4;
end;
theorem Th9:
f is_sequence_on G implies
for k being Nat st 1 <= k & k+1 <= len f
holds Int left_cell(f,k,G) <> {} & Int right_cell(f,k,G) <> {}
proof
assume
A1: f is_sequence_on G;
let k be Nat;
assume
A2: 1 <= k & k+1 <= len f;
then consider i1,j1,i2,j2 being Nat such that
A3: [i1,j1] in Indices G and
A4: f/.k = G*(i1,j1) and
A5: [i2,j2] in Indices G and
A6: f/.(k+1) = G*(i2,j2) and
A7: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
i1 = i2 & j1 = j2+1 by A1,JORDAN8:3;
A8: i2 <= len G by A5,MATRIX_0:32;
A9: i1 <= len G by A3,MATRIX_0:32;
then
A10: i1-'1 <= len G by NAT_D:44;
A11: j2 <= width G by A5,MATRIX_0:32;
then
A12: j2-'1 <= width G by NAT_D:44;
A13: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A14: j1 <= width G by A3,MATRIX_0:32;
then
A15: j1-'1 <= width G by NAT_D:44;
A16: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
per cases by A7;
suppose
i1 = i2 & j1+1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1,
j1) by A1,A3,A4,A5,A6,A13,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A14,A10,GOBOARD9:14;
end;
suppose
i1+1 = i2 & j1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1,
j1) by A1,A3,A4,A5,A6,A16,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A14,A15,GOBOARD9:14;
end;
suppose
i1 = i2+1 & j1 = j2;
then
right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-'
1) by A1,A3,A4,A5,A6,A16,GOBRD13:def 2,def 3,A2;
hence thesis by A8,A11,A12,GOBOARD9:14;
end;
suppose
i1 = i2 & j1 = j2+1;
then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1,
j2) by A1,A3,A4,A5,A6,A13,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A11,A10,GOBOARD9:14;
end;
end;
theorem Th10:
f is_sequence_on G implies
for k being Nat st 1 <= k & k+1 <= len f
holds Int left_cell(f,k,G) is convex & Int right_cell(f,k,G) is convex
proof
assume
A1: f is_sequence_on G;
let k be Nat;
assume
A2: 1 <= k & k+1 <= len f;
then consider i1,j1,i2,j2 being Nat such that
A3: [i1,j1] in Indices G and
A4: f/.k = G*(i1,j1) and
A5: [i2,j2] in Indices G and
A6: f/.(k+1) = G*(i2,j2) and
A7: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
i1 = i2 & j1 = j2+1 by A1,JORDAN8:3;
A8: i2 <= len G by A5,MATRIX_0:32;
A9: i1 <= len G by A3,MATRIX_0:32;
then
A10: i1-'1 <= len G by NAT_D:44;
A11: j2 <= width G by A5,MATRIX_0:32;
then
A12: j2-'1 <= width G by NAT_D:44;
A13: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A14: j1 <= width G by A3,MATRIX_0:32;
then
A15: j1-'1 <= width G by NAT_D:44;
A16: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
per cases by A7;
suppose
i1 = i2 & j1+1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1,
j1) by A1,A3,A4,A5,A6,A13,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A14,A10,GOBOARD9:17;
end;
suppose
i1+1 = i2 & j1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1,
j1) by A1,A3,A4,A5,A6,A16,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A14,A15,GOBOARD9:17;
end;
suppose
i1 = i2+1 & j1 = j2;
then
right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-'
1) by A1,A3,A4,A5,A6,A16,GOBRD13:def 2,def 3,A2;
hence thesis by A8,A11,A12,GOBOARD9:17;
end;
suppose
i1 = i2 & j1 = j2+1;
then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1,
j2) by A1,A3,A4,A5,A6,A13,GOBRD13:def 2,def 3,A2;
hence thesis by A9,A11,A10,GOBOARD9:17;
end;
end;
theorem Th11:
f is_sequence_on G & 1 <= k & k+1 <= len f implies Cl Int
left_cell(f,k,G) = left_cell(f,k,G) & Cl Int right_cell(f,k,G) = right_cell(f,k
,G)
proof
assume
A1: f is_sequence_on G & 1 <= k & k+1 <= len f;
then consider i1,j1,i2,j2 being Nat such that
A2: [i1,j1] in Indices G and
A3: f/.k = G*(i1,j1) and
A4: [i2,j2] in Indices G and
A5: f/.(k+1) = G*(i2,j2) and
A6: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
i1 = i2 & j1 = j2+1 by JORDAN8:3;
A7: i2 <= len G by A4,MATRIX_0:32;
A8: i1 <= len G by A2,MATRIX_0:32;
then
A9: i1-'1 <= len G by NAT_D:44;
A10: j2 <= width G by A4,MATRIX_0:32;
then
A11: j2-'1 <= width G by NAT_D:44;
A12: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A13: j1 <= width G by A2,MATRIX_0:32;
then
A14: j1-'1 <= width G by NAT_D:44;
A15: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
per cases by A6;
suppose
i1 = i2 & j1+1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1,
j1) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3;
hence thesis by A8,A13,A9,GOBRD11:35;
end;
suppose
i1+1 = i2 & j1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1,
j1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3;
hence thesis by A8,A13,A14,GOBRD11:35;
end;
suppose
i1 = i2+1 & j1 = j2;
then
right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-'
1) by A1,A2,A3,A4,A5,A15,GOBRD13:def 2,def 3;
hence thesis by A7,A10,A11,GOBRD11:35;
end;
suppose
i1 = i2 & j1 = j2+1;
then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1,
j2) by A1,A2,A3,A4,A5,A12,GOBRD13:def 2,def 3;
hence thesis by A8,A10,A9,GOBRD11:35;
end;
end;
theorem Th12:
f is_sequence_on G & LSeg(f,k) is horizontal implies ex j st 1
<= j & j <= width G & for p st p in LSeg(f,k) holds p`2 = G*(1,j)`2
proof
assume that
A1: f is_sequence_on G and
A2: LSeg(f,k) is horizontal;
per cases;
suppose
A3: 1 <= k & k+1 <= len f;
k <= k+1 by NAT_1:11;
then k <= len f by A3,XXREAL_0:2;
then consider i,j being Nat such that
A4: [i,j] in Indices G and
A5: f/.k = G*(i,j) by A1,A3,Lm2;
reconsider j as Nat;
take j;
thus
A6: 1 <= j & j <= width G by A4,MATRIX_0:32;
A7: f/.k in LSeg(f,k) by A3,TOPREAL1:21;
let p;
A8: 1 <= i & i <= len G by A4,MATRIX_0:32;
assume p in LSeg(f,k);
hence p`2 = (f/.k)`2 by A2,A7,SPPOL_1:def 2
.= G*(1,j)`2 by A5,A6,A8,GOBOARD5:1;
end;
suppose
A9: not(1 <= k & k+1 <= len f);
take 1;
width G <> 0 by MATRIX_0:def 10;
hence 1 <= 1 & 1 <= width G by NAT_1:14;
thus thesis by A9,TOPREAL1:def 3;
end;
end;
theorem Th13:
f is_sequence_on G & LSeg(f,k) is vertical implies ex i st 1 <=
i & i <= len G & for p st p in LSeg(f,k) holds p`1 = G*(i,1)`1
proof
assume that
A1: f is_sequence_on G and
A2: LSeg(f,k) is vertical;
per cases;
suppose
A3: 1 <= k & k+1 <= len f;
k <= k+1 by NAT_1:11;
then k <= len f by A3,XXREAL_0:2;
then consider i,j being Nat such that
A4: [i,j] in Indices G and
A5: f/.k = G*(i,j) by A1,A3,Lm2;
reconsider i as Nat;
take i;
thus
A6: 1 <= i & i <= len G by A4,MATRIX_0:32;
A7: f/.k in LSeg(f,k) by A3,TOPREAL1:21;
let p;
A8: 1 <= j & j <= width G by A4,MATRIX_0:32;
assume p in LSeg(f,k);
hence p`1 = (f/.k)`1 by A2,A7,SPPOL_1:def 3
.= G*(i,1)`1 by A5,A6,A8,GOBOARD5:2;
end;
suppose
A9: not(1 <= k & k+1 <= len f);
take 1;
0 <> len G by MATRIX_0:def 10;
hence 1 <= 1 & 1 <= len G by NAT_1:14;
thus thesis by A9,TOPREAL1:def 3;
end;
end;
theorem Th14:
f is_sequence_on G & f is special
implies for i,j being Nat st i <= len G & j <= width G
holds Int cell(G,i,j) misses L~f
proof
assume that
A1: f is_sequence_on G and
A2: f is special;
let i,j be Nat such that
A3: i <= len G and
A4: j <= width G;
A5: Int cell(G,i,j) = Int v_strip(G,i) /\ Int h_strip(G,j) by TOPS_1:17;
assume Int cell(G,i,j) meets L~f;
then consider x being object such that
A6: x in Int cell(G,i,j) and
A7: x in L~f by XBOOLE_0:3;
L~f = union { LSeg(f,k) where k is Nat:
1 <= k & k+1 <= len f } by TOPREAL1:def 4;
then consider X being set such that
A8: x in X and
A9: X in { LSeg(f,k) where k is Nat
: 1 <= k & k+1 <= len f } by A7,TARSKI:def 4;
consider k being Nat such that
A10: X = LSeg(f,k) and
1 <= k and
k+1 <= len f by A9;
reconsider p = x as Point of TOP-REAL 2 by A8,A10;
per cases by A2,SPPOL_1:19;
suppose
LSeg(f,k) is horizontal;
then consider j0 being Nat such that
A11: 1 <= j0 and
A12: j0 <= width G and
A13: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`2 = G*(
1,j0)`2 by A1,Th12;
now
A14: j0 > j implies j0 >= j+1 by NAT_1:13;
assume
A15: p in Int h_strip(G,j);
per cases by A14,XXREAL_0:1;
suppose
A16: j0 < j;
0 <> len G by MATRIX_0:def 10;
then 1 <= len G by NAT_1:14;
then
A17: G*(1,j)`2 > G*(1,j0)`2 by A4,A11,A16,GOBOARD5:4;
j >= 1 by A11,A16,XXREAL_0:2;
then p`2 > G*(1,j)`2 by A4,A15,GOBOARD6:27;
hence contradiction by A8,A10,A13,A17;
end;
suppose
j0 = j;
then p`2 > G*(1,j0)`2 by A11,A12,A15,GOBOARD6:27;
hence contradiction by A8,A10,A13;
end;
suppose
A18: j0 > j+1;
then j+1 <= width G by A12,XXREAL_0:2;
then j < width G by NAT_1:13;
then
A19: p`2 < G*(1,j+1)`2 by A15,GOBOARD6:28;
0 <> len G by MATRIX_0:def 10;
then
A20: 1 <= len G by NAT_1:14;
j+1 >= 1 by NAT_1:14;
then G*(1,j+1)`2 < G*(1,j0)`2 by A12,A18,A20,GOBOARD5:4;
hence contradiction by A8,A10,A13,A19;
end;
suppose
A21: j0 = j+1;
then j < width G by A12,NAT_1:13;
then p`2 < G*(1,j0)`2 by A15,A21,GOBOARD6:28;
hence contradiction by A8,A10,A13;
end;
end;
hence contradiction by A6,A5,XBOOLE_0:def 4;
end;
suppose
LSeg(f,k) is vertical;
then consider i0 being Nat such that
A22: 1 <= i0 and
A23: i0 <= len G and
A24: for p being Point of TOP-REAL 2 st p in LSeg(f,k) holds p`1 = G*(
i0,1)`1 by A1,Th13;
now
A25: i0 > i implies i0 >= i+1 by NAT_1:13;
assume
A26: p in Int v_strip(G,i);
per cases by A25,XXREAL_0:1;
suppose
A27: i0 < i;
0 <> width G by MATRIX_0:def 10;
then 1 <= width G by NAT_1:14;
then
A28: G*(i,1)`1 > G*(i0,1)`1 by A3,A22,A27,GOBOARD5:3;
i >= 1 by A22,A27,XXREAL_0:2;
then p`1 > G*(i,1)`1 by A3,A26,GOBOARD6:29;
hence contradiction by A8,A10,A24,A28;
end;
suppose
i0 = i;
then p`1 > G*(i0,1)`1 by A22,A23,A26,GOBOARD6:29;
hence contradiction by A8,A10,A24;
end;
suppose
A29: i0 > i+1;
then i+1 <= len G by A23,XXREAL_0:2;
then i < len G by NAT_1:13;
then
A30: p`1 < G*(i+1,1)`1 by A26,GOBOARD6:30;
0 <> width G by MATRIX_0:def 10;
then
A31: 1 <= width G by NAT_1:14;
i+1 >= 1 by NAT_1:14;
then G*(i+1,1)`1 < G*(i0,1)`1 by A23,A29,A31,GOBOARD5:3;
hence contradiction by A8,A10,A24,A30;
end;
suppose
A32: i0 = i+1;
then i < len G by A23,NAT_1:13;
then p`1 < G*(i0,1)`1 by A26,A32,GOBOARD6:30;
hence contradiction by A8,A10,A24;
end;
end;
hence contradiction by A6,A5,XBOOLE_0:def 4;
end;
end;
theorem Th15:
f is_sequence_on G & f is special implies
for k being Nat st 1 <= k & k+1 <= len f
holds Int left_cell(f,k,G) misses L~f & Int right_cell(f,k,G) misses L~f
proof
assume that
A1: f is_sequence_on G and
A2: f is special;
let k be Nat such that
A3: 1 <= k & k+1 <= len f;
consider i1,j1,i2,j2 being Nat such that
A4: [i1,j1] in Indices G and
A5: f/.k = G*(i1,j1) and
A6: [i2,j2] in Indices G and
A7: f/.(k+1) = G*(i2,j2) and
A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
i1 = i2 & j1 = j2+1 by A1,A3,JORDAN8:3;
A9: i2 <= len G by A6,MATRIX_0:32;
A10: i1 <= len G by A4,MATRIX_0:32;
then
A11: i1-'1 <= len G by NAT_D:44;
A12: j2 <= width G by A6,MATRIX_0:32;
then
A13: j2-'1 <= width G by NAT_D:44;
A14: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A15: j1 <= width G by A4,MATRIX_0:32;
then
A16: j1-'1 <= width G by NAT_D:44;
A17: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
per cases by A8;
suppose
i1 = i2 & j1+1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1) & left_cell(f,k,G) = cell(G,i1-'1,
j1) by A1,A3,A4,A5,A6,A7,A14,GOBRD13:def 2,def 3;
hence thesis by A1,A2,A10,A15,A11,Th14;
end;
suppose
i1+1 = i2 & j1 = j2;
then right_cell(f,k,G) = cell(G,i1,j1-'1) & left_cell(f,k,G) = cell(G,i1,
j1) by A1,A3,A4,A5,A6,A7,A17,GOBRD13:def 2,def 3;
hence thesis by A1,A2,A10,A15,A16,Th14;
end;
suppose
i1 = i2+1 & j1 = j2;
then
right_cell(f,k,G) = cell(G,i2,j2) & left_cell(f,k,G) = cell(G,i2,j2-'
1) by A1,A3,A4,A5,A6,A7,A17,GOBRD13:def 2,def 3;
hence thesis by A1,A2,A9,A12,A13,Th14;
end;
suppose
i1 = i2 & j1 = j2+1;
then right_cell(f,k,G) = cell(G,i1-'1,j2) & left_cell(f,k,G) = cell(G,i1,
j2) by A1,A3,A4,A5,A6,A7,A14,GOBRD13:def 2,def 3;
hence thesis by A1,A2,A10,A12,A11,Th14;
end;
end;
theorem Th16:
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j)
`1 = G*(i,j+1)`1 & G*(i,j)`2 = G*(i+1,j)`2 & G*(i+1,j+1)`1 = G*(i+1,j)`1 & G*(i
+1,j+1)`2 = G*(i,j+1)`2
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: j < width G by A4,NAT_1:13;
A6: 1 <= j+1 by NAT_1:11;
A7: i < len G by A2,NAT_1:13;
hence G*(i,j)`1 = G*(i,1)`1 by A1,A3,A5,GOBOARD5:2
.= G*(i,j+1)`1 by A1,A4,A7,A6,GOBOARD5:2;
A8: 1 <= i+1 by NAT_1:11;
thus G*(i,j)`2 = G*(1,j)`2 by A1,A3,A7,A5,GOBOARD5:1
.= G*(i+1,j)`2 by A2,A3,A5,A8,GOBOARD5:1;
thus G*(i+1,j+1)`1 = G*(i+1,1)`1 by A2,A4,A8,A6,GOBOARD5:2
.= G*(i+1,j)`1 by A2,A3,A5,A8,GOBOARD5:2;
thus G*(i+1,j+1)`2 = G*(1,j+1)`2 by A2,A4,A8,A6,GOBOARD5:1
.= G*(i,j+1)`2 by A1,A4,A7,A6,GOBOARD5:1;
end;
theorem Th17:
for i,j being Nat st 1 <= i & i+1 <= len G & 1 <= j &
j+1 <= width G holds p in cell(G,i,j) iff G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1
& G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2
proof
let i,j be Nat such that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: i < len G & j < width G by A2,A4,NAT_1:13;
then
A6: h_strip(G,j) = { |[r,s]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } by A1,A3,
GOBOARD5:5;
A7: v_strip(G,i) = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 } by A1,A3,A5,
GOBOARD5:8;
hereby
assume
A8: p in cell(G,i,j);
then p in v_strip(G,i) by XBOOLE_0:def 4;
then ex r,s st |[r,s]| = p & G*(i,j)`1 <= r & r <= G*(i+1,j)`1 by A7;
hence G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 by EUCLID:52;
p in h_strip(G,j) by A8,XBOOLE_0:def 4;
then ex r,s st |[r,s]| = p & G*(i,j)`2 <= s & s <= G*(i,j+1)`2 by A6;
hence G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 by EUCLID:52;
end;
assume that
A9: G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 and
A10: G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2;
A11: p = |[p`1,p`2]| by EUCLID:53;
then
A12: p in h_strip(G,j) by A6,A10;
p in v_strip(G,i) by A7,A9,A11;
hence thesis by A12,XBOOLE_0:def 4;
end;
theorem
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies cell(G,i,j) =
{ |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+1)
`2 }
proof
set A = { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s
<= G*(i,j+1)`2 };
assume
A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G;
now
let p be object;
assume
A2: p in cell(G,i,j);
then reconsider q=p as Point of TOP-REAL 2;
A3: G*(i,j)`2 <= q`2 & q`2 <= G*(i,j+1)`2 by A1,A2,Th17;
A4: p = |[q`1,q`2]| by EUCLID:53;
G*(i,j)`1 <= q`1 & q`1 <= G*(i+1,j)`1 by A1,A2,Th17;
hence p in A by A4,A3;
end;
hence cell(G,i,j) c= A;
now
let p be object;
assume p in A;
then consider r,s such that
A5: |[r,s]| = p and
A6: G*(i,j)`1 <= r & r <= G*(i+1,j)`1 & G*(i,j)`2 <= s & s <= G*(i,j+ 1)`2;
reconsider q=p as Point of TOP-REAL 2 by A5;
r = q`1 & s = q`2 by A5,EUCLID:52;
hence p in cell(G,i,j) by A1,A6,Th17;
end;
hence thesis;
end;
theorem Th19:
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G
& p in cell(G,i,j) implies p = G*(i,j) or p = G*(i,j+1) or p = G*(i+1,j+1) or p
= G*(i+1,j)
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G and
A5: p in Values G and
A6: p in cell(G,i,j);
A7: Values G = { G*(k,l) where k,l is Nat: [k,l] in Indices G }
by MATRIX_0:39;
A8: i < len G by A2,NAT_1:13;
A9: j < width G by A4,NAT_1:13;
consider k,l being Nat such that
A10: p = G*(k,l) and
A11: [k,l] in Indices G by A5,A7;
A12: 1 <= k by A11,MATRIX_0:32;
A13: l <= width G by A11,MATRIX_0:32;
A14: 1 <= l by A11,MATRIX_0:32;
A15: k <= len G by A11,MATRIX_0:32;
A16: 1 <= j+1 by NAT_1:11;
A17: now
assume
A18: l <> j & l <> j+1;
per cases by A18,NAT_1:9;
suppose
l < j;
then G*(k,l)`2 < G*(k,j)`2 by A9,A12,A15,A14,GOBOARD5:4;
then G*(k,l)`2 < G*(1,j)`2 by A3,A9,A12,A15,GOBOARD5:1;
then G*(k,l)`2 < G*(i,j)`2 by A1,A3,A8,A9,GOBOARD5:1;
hence contradiction by A1,A2,A3,A4,A6,A10,Th17;
end;
suppose
j+1 < l;
then G*(k,j+1)`2 < G*(k,l)`2 by A16,A12,A15,A13,GOBOARD5:4;
then G*(1,j+1)`2 < G*(k,l)`2 by A4,A16,A12,A15,GOBOARD5:1;
then G*(i,j+1)`2 < G*(k,l)`2 by A1,A4,A8,A16,GOBOARD5:1;
hence contradiction by A1,A2,A3,A4,A6,A10,Th17;
end;
end;
A19: 1 <= i+1 by NAT_1:11;
now
assume
A20: k <> i & k <> i+1;
per cases by A20,NAT_1:9;
suppose
k < i;
then G*(k,l)`1 < G*(i,l)`1 by A8,A12,A14,A13,GOBOARD5:3;
then G*(k,l)`1 < G*(i,1)`1 by A1,A8,A14,A13,GOBOARD5:2;
then G*(k,l)`1 < G*(i,j)`1 by A1,A3,A8,A9,GOBOARD5:2;
hence contradiction by A1,A2,A3,A4,A6,A10,Th17;
end;
suppose
i+1 < k;
then G*(i+1,l)`1 < G*(k,l)`1 by A19,A15,A14,A13,GOBOARD5:3;
then G*(i+1,1)`1 < G*(k,l)`1 by A2,A19,A14,A13,GOBOARD5:2;
then G*(i+1,j)`1 < G*(k,l)`1 by A2,A3,A9,A19,GOBOARD5:2;
hence contradiction by A1,A2,A3,A4,A6,A10,Th17;
end;
end;
hence thesis by A10,A17;
end;
theorem Th20:
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j)
in cell(G,i,j) & G*(i,j+1) in cell(G,i,j) & G*(i+1,j+1) in cell(G,i,j) & G*(i+1
,j) in cell(G,i,j)
proof
assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: i < i+1 & j < width G by A4,NAT_1:13;
then
A6: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,GOBOARD5:3;
A7: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3;
A8: j < j+1 & i < len G by A2,NAT_1:13;
then
A9: G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,GOBOARD5:4;
A10: G*(i+1,j+1)`1 = G*(i+1,j)`1 by A1,A2,A3,A4,Th16;
then
A11: G*(i,j)`1 <= G*(i+1,j+1)`1 by A1,A2,A3,A5,GOBOARD5:3;
G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
hence G*(i,j) in cell(G,i,j) by A1,A2,A3,A4,A6,Th17;
A12: G*(i,j)`1 = G*(i,j+1)`1 by A1,A2,A3,A4,Th16;
then G*(i,j+1)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3;
hence G*(i,j+1) in cell(G,i,j) by A1,A2,A3,A4,A12,A9,Th17;
A13: G*(i+1,j+1)`2 = G*(i,j+1)`2 by A1,A2,A3,A4,Th16;
then G*(i,j)`2 <= G*(i+1,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
hence G*(i+1,j+1) in cell(G,i,j) by A1,A2,A3,A4,A10,A11,A13,Th17;
A14: G*(i,j)`2 = G*(i+1,j)`2 by A1,A2,A3,A4,Th16;
then G*(i+1,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
hence thesis by A1,A2,A3,A4,A7,A14,Th17;
end;
theorem Th21:
1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & p in Values G
& p in cell(G,i,j) implies p is_extremal_in cell(G,i,j)
proof
assume that
A1: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G and
A2: p in Values G and
A3: p in cell(G,i,j);
for a,b being Point of TOP-REAL 2 st p in LSeg(a,b) & LSeg(a,b) c= cell(
G,i,j) holds p = a or p = b
proof
let a,b be Point of TOP-REAL 2 such that
A4: p in LSeg(a,b) and
A5: LSeg(a,b) c= cell(G,i,j);
A6: a in LSeg(a,b) by RLTOPSP1:68;
A7: b in LSeg(a,b) by RLTOPSP1:68;
assume that
A8: a <> p and
A9: b <> p;
per cases by A1,A2,A3,Th19;
suppose
A10: p = G*(i,j);
then
A11: p`2 <= b`2 by A1,A5,A7,Th17;
A12: p`1 <= a`1 by A1,A5,A6,A10,Th17;
A13: p`1 <= b`1 by A1,A5,A7,A10,Th17;
A14: p`2 <= a`2 by A1,A5,A6,A10,Th17;
now
per cases;
suppose
A15: a`1 <= b`1 & a`2 <= b`2;
then a`2 <= p`2 by A4,TOPREAL1:4;
then
A16: a`2 = p`2 by A14,XXREAL_0:1;
a`1 <= p`1 by A4,A15,TOPREAL1:3;
then a`1 = p`1 by A12,XXREAL_0:1;
hence contradiction by A8,A16,TOPREAL3:6;
end;
suppose
A17: a`1 <= b`1 & b`2 < a`2;
then b`2 <= p`2 by A4,TOPREAL1:4;
then
A18: b`2 = p`2 by A11,XXREAL_0:1;
A19: a`1 <= p`1 by A4,A17,TOPREAL1:3;
then
A20: a`1 = p`1 by A12,XXREAL_0:1;
then a`2 <> p`2 by A8,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A6,A12,A19,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A9,A20,A18,TOPREAL3:6;
end;
suppose
A21: b`1 < a`1 & a`2 <= b`2;
then a`2 <= p`2 by A4,TOPREAL1:4;
then
A22: a`2 = p`2 by A14,XXREAL_0:1;
A23: b`1 <= p`1 by A4,A21,TOPREAL1:3;
then
A24: b`1 = p`1 by A13,XXREAL_0:1;
then b`2 <> p`2 by A9,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A7,A13,A23,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A8,A24,A22,TOPREAL3:6;
end;
suppose
A25: b`1 < a`1 & b`2 < a`2;
then b`2 <= p`2 by A4,TOPREAL1:4;
then
A26: b`2 = p`2 by A11,XXREAL_0:1;
b`1 <= p`1 by A4,A25,TOPREAL1:3;
then b`1 = p`1 by A13,XXREAL_0:1;
hence contradiction by A9,A26,TOPREAL3:6;
end;
end;
hence contradiction;
end;
suppose
A27: p = G*(i,j+1);
then
A28: b`2 <= p`2 by A1,A5,A7,Th17;
A29: p`1 = G*(i,j)`1 by A1,A27,Th16;
then
A30: p`1 <= a`1 by A1,A5,A6,Th17;
A31: p`1 <= b`1 by A1,A5,A7,A29,Th17;
A32: a`2 <= p`2 by A1,A5,A6,A27,Th17;
now
per cases;
suppose
A33: a`1 <= b`1 & a`2 <= b`2;
then p`2 <= b`2 by A4,TOPREAL1:4;
then
A34: b`2 = p`2 by A28,XXREAL_0:1;
A35: a`1 <= p`1 by A4,A33,TOPREAL1:3;
then
A36: a`1 = p`1 by A30,XXREAL_0:1;
then a`2 <> p`2 by A8,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A6,A30,A35,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A9,A36,A34,TOPREAL3:6;
end;
suppose
A37: a`1 <= b`1 & b`2 < a`2;
then p`2 <= a`2 by A4,TOPREAL1:4;
then
A38: a`2 = p`2 by A32,XXREAL_0:1;
a`1 <= p`1 by A4,A37,TOPREAL1:3;
then a`1 = p`1 by A30,XXREAL_0:1;
hence contradiction by A8,A38,TOPREAL3:6;
end;
suppose
A39: b`1 < a`1 & a`2 <= b`2;
then p`2 <= b`2 by A4,TOPREAL1:4;
then
A40: b`2 = p`2 by A28,XXREAL_0:1;
b`1 <= p`1 by A4,A39,TOPREAL1:3;
then b`1 = p`1 by A31,XXREAL_0:1;
hence contradiction by A9,A40,TOPREAL3:6;
end;
suppose
A41: b`1 < a`1 & b`2 < a`2;
then p`2 <= a`2 by A4,TOPREAL1:4;
then
A42: a`2 = p`2 by A32,XXREAL_0:1;
A43: b`1 <= p`1 by A4,A41,TOPREAL1:3;
then
A44: b`1 = p`1 by A31,XXREAL_0:1;
then b`2 <> p`2 by A9,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A7,A31,A43,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A8,A44,A42,TOPREAL3:6;
end;
end;
hence contradiction;
end;
suppose
A45: p = G*(i+1,j+1);
then
A46: p`1 = G*(i+1,j)`1 by A1,Th16;
then
A47: a`1 <= p`1 by A1,A5,A6,Th17;
A48: p`2 = G*(i,j+1)`2 by A1,A45,Th16;
then
A49: b`2 <= p`2 by A1,A5,A7,Th17;
A50: b`1 <= p`1 by A1,A5,A7,A46,Th17;
A51: a`2 <= p`2 by A1,A5,A6,A48,Th17;
now
per cases;
suppose
A52: a`1 <= b`1 & a`2 <= b`2;
then p`2 <= b`2 by A4,TOPREAL1:4;
then
A53: b`2 = p`2 by A49,XXREAL_0:1;
p`1 <= b`1 by A4,A52,TOPREAL1:3;
then b`1 = p`1 by A50,XXREAL_0:1;
hence contradiction by A9,A53,TOPREAL3:6;
end;
suppose
A54: a`1 <= b`1 & b`2 < a`2;
then p`2 <= a`2 by A4,TOPREAL1:4;
then
A55: a`2 = p`2 by A51,XXREAL_0:1;
A56: p`1 <= b`1 by A4,A54,TOPREAL1:3;
then
A57: b`1 = p`1 by A50,XXREAL_0:1;
then b`2 <> p`2 by A9,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A7,A50,A56,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A8,A57,A55,TOPREAL3:6;
end;
suppose
A58: b`1 < a`1 & a`2 <= b`2;
then p`2 <= b`2 by A4,TOPREAL1:4;
then
A59: b`2 = p`2 by A49,XXREAL_0:1;
A60: p`1 <= a`1 by A4,A58,TOPREAL1:3;
then
A61: a`1 = p`1 by A47,XXREAL_0:1;
then a`2 <> p`2 by A8,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A6,A47,A60,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A9,A61,A59,TOPREAL3:6;
end;
suppose
A62: b`1 < a`1 & b`2 < a`2;
then p`2 <= a`2 by A4,TOPREAL1:4;
then
A63: a`2 = p`2 by A51,XXREAL_0:1;
p`1 <= a`1 by A4,A62,TOPREAL1:3;
then a`1 = p`1 by A47,XXREAL_0:1;
hence contradiction by A8,A63,TOPREAL3:6;
end;
end;
hence contradiction;
end;
suppose
A64: p = G*(i+1,j);
then
A65: p`2 = G*(i,j)`2 by A1,Th16;
then
A66: p`2 <= b`2 by A1,A5,A7,Th17;
A67: a`1 <= p`1 by A1,A5,A6,A64,Th17;
A68: b`1 <= p`1 by A1,A5,A7,A64,Th17;
A69: p`2 <= a`2 by A1,A5,A6,A65,Th17;
now
per cases;
suppose
A70: a`1 <= b`1 & a`2 <= b`2;
then a`2 <= p`2 by A4,TOPREAL1:4;
then
A71: a`2 = p`2 by A69,XXREAL_0:1;
A72: p`1 <= b`1 by A4,A70,TOPREAL1:3;
then
A73: b`1 = p`1 by A68,XXREAL_0:1;
then b`2 <> p`2 by A9,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A7,A68,A72,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A8,A73,A71,TOPREAL3:6;
end;
suppose
A74: a`1 <= b`1 & b`2 < a`2;
then b`2 <= p`2 by A4,TOPREAL1:4;
then
A75: b`2 = p`2 by A66,XXREAL_0:1;
p`1 <= b`1 by A4,A74,TOPREAL1:3;
then b`1 = p`1 by A68,XXREAL_0:1;
hence contradiction by A9,A75,TOPREAL3:6;
end;
suppose
A76: b`1 < a`1 & a`2 <= b`2;
then a`2 <= p`2 by A4,TOPREAL1:4;
then
A77: a`2 = p`2 by A69,XXREAL_0:1;
p`1 <= a`1 by A4,A76,TOPREAL1:3;
then a`1 = p`1 by A67,XXREAL_0:1;
hence contradiction by A8,A77,TOPREAL3:6;
end;
suppose
A78: b`1 < a`1 & b`2 < a`2;
then b`2 <= p`2 by A4,TOPREAL1:4;
then
A79: b`2 = p`2 by A66,XXREAL_0:1;
A80: p`1 <= a`1 by A4,A78,TOPREAL1:3;
then
A81: a`1 = p`1 by A67,XXREAL_0:1;
then a`2 <> p`2 by A8,TOPREAL3:6;
then LSeg(a,b) is vertical by A4,A6,A67,A80,SPPOL_1:18,XXREAL_0:1;
then a`1 = b`1 by SPPOL_1:16;
hence contradiction by A9,A81,A79,TOPREAL3:6;
end;
end;
hence contradiction;
end;
end;
hence thesis by A3,SPPOL_1:def 1;
end;
theorem Th22:
2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <= len f
implies
ex i,j being Nat st 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G & LSeg(
f,k) c= cell(G,i,j)
proof
assume that
A1: 2 <= len G and
A2: 2 <= width G and
A3: f is_sequence_on G and
A4: 1 <= k & k+1 <= len f;
consider i1,j1,i2,j2 being Nat such that
A5: [i1,j1] in Indices G and
A6: f/.k = G*(i1,j1) and
A7: [i2,j2] in Indices G and
A8: f/.(k+1) = G*(i2,j2) and
A9: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
or i1 = i2 & j1 = j2+1 by A3,A4,JORDAN8:3;
A10: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A4,TOPREAL1:def 3;
A11: 1 <= i2 by A7,MATRIX_0:32;
A12: 1 <= i1 by A5,MATRIX_0:32;
A13: 1 <= j2 by A7,MATRIX_0:32;
A14: 1 <= j1 by A5,MATRIX_0:32;
A15: i2 <= len G by A7,MATRIX_0:32;
A16: i1 <= len G by A5,MATRIX_0:32;
A17: j2 <= width G by A7,MATRIX_0:32;
A18: j1 <= width G by A5,MATRIX_0:32;
per cases by A9;
suppose
A19: i1 = i2 & j1+1 = j2;
then
A20: j1 < width G by A17,XREAL_1:145;
now
per cases by A16,XXREAL_0:1;
suppose
A21: i1 < len G;
take i1,j1;
A22: i1+1 <= len G by A21,NAT_1:13;
LSeg(f,k) c= cell(G,i1,j1) by A10,A6,A8,A12,A16,A14,A17,A19,GOBOARD5:19
,XREAL_1:145;
hence thesis by A12,A14,A17,A19,A22;
end;
suppose
A23: i1 = len G;
reconsider i19=i1-'1,j1 as Nat;
take i19,j1;
2-1 <= 2-'1 & 2-'1 <= i19 by A1,A23,NAT_D:42,XREAL_0:def 2;
then
A24: 1 <= i19 by XXREAL_0:2;
A25: i19+1 = i1 by A12,XREAL_1:235;
then i19 < len G by A16,NAT_1:13;
then LSeg(f,k) c= cell(G,i19,j1) by A10,A6,A8,A14,A19,A20,A25,
GOBOARD5:18;
hence thesis by A16,A14,A17,A19,A24,A25;
end;
end;
hence thesis;
end;
suppose
A26: i1+1 = i2 & j1 = j2;
then
A27: i1 < len G by A15,XREAL_1:145;
now
per cases by A18,XXREAL_0:1;
suppose
A28: j1 < width G;
take i1,j1;
A29: j1+1 <= width G by A28,NAT_1:13;
LSeg(f,k) c= cell(G,i1,j1) by A10,A6,A8,A12,A14,A18,A15,A26,GOBOARD5:22
,XREAL_1:145;
hence thesis by A12,A14,A15,A26,A29;
end;
suppose
A30: j1 = width G;
reconsider i1,j19=j1-'1 as Nat;
take i1,j19;
2-1 <= 2-'1 & 2-'1 <= j19 by A2,A30,NAT_D:42,XREAL_0:def 2;
then
A31: 1 <= j19 by XXREAL_0:2;
A32: j19+1=j1 by A14,XREAL_1:235;
then j19 < width G by A30,NAT_1:13;
then LSeg(f,k) c= cell(G,i1,j19) by A10,A6,A8,A12,A26,A27,A32,
GOBOARD5:21;
hence thesis by A12,A18,A15,A26,A31,A32;
end;
end;
hence thesis;
end;
suppose
A33: i1 = i2+1 & j1 = j2;
then
A34: i2 < len G by A16,XREAL_1:145;
now
per cases by A18,XXREAL_0:1;
suppose
A35: j1 < width G;
take i2,j1;
A36: j1+1 <= width G by A35,NAT_1:13;
LSeg(f,k) c= cell(G,i2,j1) by A10,A6,A8,A16,A11,A13,A17,A33,GOBOARD5:22
,XREAL_1:145;
hence thesis by A16,A14,A11,A33,A36;
end;
suppose
A37: j1 = width G;
reconsider i2,j19=j1-'1 as Nat;
take i2,j19;
2-1 <= 2-'1 & 2-'1 <= j19 by A2,A37,NAT_D:42,XREAL_0:def 2;
then
A38: 1 <= j19 by XXREAL_0:2;
A39: j19+1=j1 by A14,XREAL_1:235;
then j19 < width G by A37,NAT_1:13;
then LSeg(f,k) c= cell(G,i2,j19) by A10,A6,A8,A11,A33,A34,A39,
GOBOARD5:21;
hence thesis by A16,A18,A11,A33,A38,A39;
end;
end;
hence thesis;
end;
suppose
A40: i1 = i2 & j1 = j2+1;
then
A41: j2 < width G by A18,XREAL_1:145;
now
per cases by A16,XXREAL_0:1;
suppose
A42: i1 < len G;
take i1,j2;
A43: i1+1 <= len G by A42,NAT_1:13;
LSeg(f,k) c= cell(G,i1,j2) by A10,A6,A8,A18,A11,A15,A13,A40,GOBOARD5:19
,XREAL_1:145;
hence thesis by A12,A18,A13,A40,A43;
end;
suppose
A44: i1 = len G;
reconsider i19=i1-'1,j2 as Nat;
take i19,j2;
2-1 <= 2-'1 & 2-'1 <= i19 by A1,A44,NAT_D:42,XREAL_0:def 2;
then
A45: 1 <= i19 by XXREAL_0:2;
A46: i19+1 = i1 by A12,XREAL_1:235;
then i19 < len G by A16,NAT_1:13;
then LSeg(f,k) c= cell(G,i19,j2) by A10,A6,A8,A13,A40,A41,A46,
GOBOARD5:18;
hence thesis by A16,A18,A13,A40,A45,A46;
end;
end;
hence thesis;
end;
end;
theorem Th23:
2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k+1 <=
len f & p in Values G & p in LSeg(f,k) implies p = f/.k or p = f/.(k+1)
proof
assume that
A1: 2 <= len G & 2 <= width G & f is_sequence_on G and
A2: 1 <= k & k+1 <= len f and
A3: p in Values G and
A4: p in LSeg(f,k);
A5: LSeg(f,k) = LSeg(f/.k, f/.(k+1)) by A2,TOPREAL1:def 3;
consider i,j being Nat such that
A6: 1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G and
A7: LSeg(f,k) c= cell(G,i,j) by A1,A2,Th22;
p is_extremal_in cell(G,i,j) by A3,A4,A6,A7,Th21;
hence thesis by A4,A7,A5,SPPOL_1:def 1;
end;
theorem
[i,j] in Indices G & 1 <= k & k <= width G implies G*(i,j)`1 <= G* (
len G,k)`1
proof
assume that
A1: [i,j] in Indices G and
A2: 1 <= k & k <= width G;
A3: 1 <= i by A1,MATRIX_0:32;
A4: i <= len G by A1,MATRIX_0:32;
then
A5: i < len G or i = len G by XXREAL_0:1;
1 <= j & j <= width G by A1,MATRIX_0:32;
then G*(i,j)`1 = G*(i,1)`1 by A3,A4,GOBOARD5:2
.= G*(i,k)`1 by A2,A3,A4,GOBOARD5:2;
hence thesis by A2,A3,A5,GOBOARD5:3;
end;
theorem
[i,j] in Indices G & 1 <= k & k <= len G implies G*(i,j)`2 <= G* (k,
width G)`2
proof
assume that
A1: [i,j] in Indices G and
A2: 1 <= k & k <= len G;
A3: 1 <= j by A1,MATRIX_0:32;
A4: j <= width G by A1,MATRIX_0:32;
then
A5: j < width G or j = width G by XXREAL_0:1;
1 <= i & i <= len G by A1,MATRIX_0:32;
then G*(i,j)`2 = G*(1,j)`2 by A3,A4,GOBOARD5:1
.= G*(k,j)`2 by A2,A3,A4,GOBOARD5:1;
hence thesis by A2,A3,A5,GOBOARD5:4;
end;
theorem Th26:
f is_sequence_on G & f is special & L~g c= L~f & 1 <= k & k+1 <=
len f implies for A being Subset of TOP-REAL 2 st A = right_cell(f,k,G)\L~g or
A = left_cell(f,k,G)\L~g holds A is connected
proof
assume that
A1: f is_sequence_on G and
A2: f is special and
A3: L~g c= L~f and
A4: 1 <= k & k+1 <= len f;
let A be Subset of TOP-REAL 2 such that
A5: A = right_cell(f,k,G)\L~g or A = left_cell(f,k,G)\L~g;
per cases by A5;
suppose
A6: A = right_cell(f,k,G)\L~g;
Int right_cell(f,k,G) misses L~f by A1,A2,A4,Th15;
then Int right_cell(f,k,G) misses L~g by A3,XBOOLE_1:63;
then
A7: Int right_cell(f,k,G) c= (L~g)` by SUBSET_1:23;
A c= right_cell(f,k,G) by A6,XBOOLE_1:36;
then
A8: A c= Cl Int right_cell(f,k,G) by A1,A4,Th11;
A9: A = right_cell(f,k,G) /\ (L~g)` by A6,SUBSET_1:13;
Int right_cell(f,k,G) is convex & Int right_cell(f,k,G) c=
right_cell(f,k,G) by A1,A4,Th10,TOPS_1:16;
hence thesis by A9,A7,A8,CONNSP_1:18,XBOOLE_1:19;
end;
suppose
A10: A = left_cell(f,k,G)\L~g;
Int left_cell(f,k,G) misses L~f by A1,A2,A4,Th15;
then Int left_cell(f,k,G) misses L~g by A3,XBOOLE_1:63;
then
A11: Int left_cell(f,k,G) c= (L~g)` by SUBSET_1:23;
A c= left_cell(f,k,G) by A10,XBOOLE_1:36;
then
A12: A c= Cl Int left_cell(f,k,G) by A1,A4,Th11;
A13: A = left_cell(f,k,G) /\ (L~g)` by A10,SUBSET_1:13;
Int left_cell(f,k,G) is convex & Int left_cell(f,k,G) c= left_cell
(f,k,G) by A1,A4,Th10,TOPS_1:16;
hence thesis by A13,A11,A12,CONNSP_1:18,XBOOLE_1:19;
end;
end;
theorem Th27:
for f being non constant standard special_circular_sequence st f
is_sequence_on G for k st 1 <= k & k+1 <= len f holds right_cell(f,k,G)\L~f c=
RightComp f & left_cell(f,k,G)\L~f c= LeftComp f
proof
let f be non constant standard special_circular_sequence such that
A1: f is_sequence_on G;
let k such that
A2: 1 <= k & k+1 <= len f;
A3: Int right_cell(f,k,G) <> {} by A1,A2,Th9;
set rc = right_cell(f,k,G)\L~f;
rc \/ L~f = right_cell(f,k,G) \/ L~f by XBOOLE_1:39;
then
Int right_cell(f,k,G) c= right_cell(f,k,G) & right_cell(f,k,G) c= rc \/
L~f by TOPS_1:16,XBOOLE_1:7;
then
A4: Int right_cell(f,k,G) c= rc \/ L~f;
set lc = left_cell(f,k,G)\L~f;
rc = right_cell(f,k,G) /\ (L~f)` by SUBSET_1:13;
then
A5: RightComp f is_a_component_of (L~f)` & rc c= (L~f)` by GOBOARD9:def 2
,XBOOLE_1:17;
rc c= right_cell(f,k,G) & right_cell(f,k,G) c= right_cell(f,k) by A1,A2,
GOBRD13:33,XBOOLE_1:36;
then rc c= right_cell(f,k);
then
A6: Int rc c= Int right_cell(f,k) by TOPS_1:19;
Int right_cell(f,k) c= RightComp f by A2,GOBOARD9:25;
then
A7: Int rc c= RightComp f by A6;
Int right_cell(f,k,G) misses L~f by A1,A2,Th15;
then
A8: Int Int right_cell(f,k,G) c= Int rc by A4,TOPS_1:19,XBOOLE_1:73;
Int right_cell(f,k,G) c= rc by A1,A2,A4,Th15,XBOOLE_1:73;
then
A9: rc meets Int rc by A3,A8,XBOOLE_1:68;
rc is connected by A1,A2,Th26;
hence right_cell(f,k,G)\L~f c= RightComp f by A7,A9,A5,GOBOARD9:4;
lc = left_cell(f,k,G) /\ (L~f)` by SUBSET_1:13;
then
A10: LeftComp f is_a_component_of (L~f)` & lc c= (L~f)` by GOBOARD9:def 1
,XBOOLE_1:17;
lc \/ L~f = left_cell(f,k,G) \/ L~f by XBOOLE_1:39;
then
Int left_cell(f,k,G) c= left_cell(f,k,G) & left_cell(f,k,G) c= lc \/ L~
f by TOPS_1:16,XBOOLE_1:7;
then
A11: Int left_cell(f,k,G) c= lc \/ L~f;
lc c= left_cell(f,k,G) & left_cell(f,k,G) c= left_cell(f,k) by A1,A2,
GOBRD13:33,XBOOLE_1:36;
then lc c= left_cell(f,k);
then
A12: Int lc c= Int left_cell(f,k) by TOPS_1:19;
Int left_cell(f,k) c= LeftComp f by A2,GOBOARD9:21;
then
A13: Int lc c= LeftComp f by A12;
A14: Int left_cell(f,k,G) <> {} by A1,A2,Th9;
Int left_cell(f,k,G) misses L~f by A1,A2,Th15;
then
A15: Int Int left_cell(f,k,G) c= Int lc by A11,TOPS_1:19,XBOOLE_1:73;
Int left_cell(f,k,G) c= lc by A1,A2,A11,Th15,XBOOLE_1:73;
then
A16: lc meets Int lc by A14,A15,XBOOLE_1:68;
lc is connected by A1,A2,Th26;
hence thesis by A13,A16,A10,GOBOARD9:4;
end;
begin :: Cages
reserve C for compact non vertical non horizontal non empty Subset of TOP-REAL
2,
l, m, i1, i2, j1, j2 for Nat;
theorem Th28:
for n being Nat ex i st 1 <= i & i+1 <= len Gauge(C,n) & N-min C
in cell(Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width
Gauge(C,n)-'1)
proof
let n be Nat;
set G = Gauge(C,n);
defpred P[Nat] means 1 <= $1 & $1 < len G & G*($1,(width G)-'1)`1 < (N-min C
)`1;
A1: for k be Nat st P[k] holds k <= len G;
A2: len G = width G by JORDAN8:def 1;
(NW-corner C)`1 <= (N-min C)`1 by PSCOMP_1:38;
then
A3: W-bound C <= (N-min C)`1 by EUCLID:52;
A4: len G >= 4 by JORDAN8:10;
then
A5: (len G)-'1 <= len G & 2 <= len G by NAT_D:35,XXREAL_0:2;
A6: 1 < len G by A4,XXREAL_0:2;
then
A7: 1 <= (len G)-'1 by NAT_D:49;
A8: n in NAT & len G = width G by JORDAN8:def 1,ORDINAL1:def 12;
then G*(2,(width G)-'1)`1 = W-bound C by A7,JORDAN8:11,NAT_D:35;
then G*(1,(width G)-'1)`1 < W-bound C by A2,A7,A5,GOBOARD5:3;
then G*(1,(width G)-'1)`1 < (N-min C)`1 by A3,XXREAL_0:2;
then
A9: ex k be Nat st P[k] by A6;
ex i being Nat st P[i] & for n be Nat st P[n] holds n <= i from NAT_1:
sch 6(A1,A9);
then consider i being Nat such that
A10: 1 <= i and
A11: i < len G and
A12: G*(i,(width G)-'1)`1 < (N-min C)`1 and
A13: for n be Nat st P[n] holds n <= i;
reconsider i as Nat;
A14: 1 <= i+1 & i < i+1 by NAT_1:12,13;
A15: (N-min C)`2 = N-bound C by EUCLID:52;
A16: i+1 <= len G by A11,NAT_1:13;
then
A17: (N-min C)`2 = G* (i+1,(width G)-'1)`2 by A8,A15,JORDAN8:14,NAT_1:12;
now
assume i+1 = len G;
then len G-'1 = i by NAT_D:34;
then
A18: G*(i,(width G)-'1)`1 = E-bound C by A8,A7,JORDAN8:12,NAT_D:35;
(NE-corner C)`1 >= (N-min C)`1 by PSCOMP_1:38;
hence contradiction by A12,A18,EUCLID:52;
end;
then i+1 < len G by A16,XXREAL_0:1;
then
A19: (N-min C)`1 <= G*(i+1,(width G)-'1)`1 by A13,A14;
G*(i,(width G)-'1)`2 = (N-min C)`2 by A8,A10,A11,A15,JORDAN8:14;
then
A20: N-min C in LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) by A12,A17,A19,
GOBOARD7:8;
take i;
thus 1 <= i & i+1 <= len G by A10,A11,NAT_1:13;
LSeg(G*(i,(width G)-'1),G*(i+1,(width G)-'1)) c= cell(G,i,(width G)-'1)
by A2,A7,A10,A11,GOBOARD5:22,NAT_D:35;
hence N-min C in cell(G,i,(width G)-'1) by A20;
thus thesis by A12;
end;
theorem Th29:
for n, i1, i2 being Nat holds 1 <= i1 & i1+1 <= len Gauge(C,n) &
N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i1,
width Gauge(C,n)-'1) & 1 <= i2 & i2+1 <= len Gauge(C,n) & N-min C in cell(Gauge
(C,n),i2,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i2,width Gauge(C,n)-'1)
implies i1 = i2
proof
let n, i1, i2 be Nat;
set G = Gauge(C,n), j = width G-'1;
A1: 2|^n >= n+1 by NEWTON:85;
A2: 1+ (n+3) > 1+0 by XREAL_1:6;
A3: len G = width G by JORDAN8:def 1;
A4: len G = 2|^n+3 by JORDAN8:def 1;
then
A5: len G >= n+1+3 by A1,XREAL_1:6;
then len G > 1 by A2,XXREAL_0:2;
then
A6: len G >= 1+1 by NAT_1:13;
then
A7: 1 <= j by A3,JORDAN5B:2;
A8: j+1 = len G by A3,A5,A2,XREAL_1:235,XXREAL_0:2;
then
A9: j < len G by NAT_1:13;
assume that
A10: 1 <= i1 and
A11: i1+1 <= len G and
A12: N-min C in cell(G,i1,j) and
A13: N-min C <> G*(i1,j) and
A14: 1 <= i2 and
A15: i2+1 <= len G and
A16: N-min C in cell(G,i2,j) and
A17: N-min C <> G*(i2,j) and
A18: i1 <> i2;
A19: cell(G,i1,j) meets cell(G,i2,j) by A12,A16,XBOOLE_0:3;
A20: i1 < len G by A11,NAT_1:13;
A21: i2 < len G by A15,NAT_1:13;
per cases by A18,XXREAL_0:1;
suppose
A22: i1 < i2;
then
A23: i2-'i1+i1 = i2 by XREAL_1:235;
then i2-'i1 <= 1 by A21,A3,A19,A7,A9,JORDAN8:7;
then i2-'i1 < 1 or i2-'i1 = 1 by XXREAL_0:1;
then i2-'i1 = 0 or i2-'i1 = 1 by NAT_1:14;
then cell(G,i1,j) /\ cell(G,i2,j) = LSeg(G*(i2,j),G*(i2,j+1)) by A20,A3
,A6,A9,A22,A23,GOBOARD5:25,JORDAN5B:2;
then
A24: N-min C in LSeg(G*(i2,j),G*(i2,j+1)) by A12,A16,XBOOLE_0:def 4;
1 <= j+1 by NAT_1:12;
then
A25: [i2,j+1] in Indices G by A14,A21,A3,A8,MATRIX_0:30;
set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1);
set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2);
set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i2-2);
j = (2|^n+2+1)-'1 by A4,JORDAN8:def 1
.= (2|^n+2) by NAT_D:34;
then
A26: (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A1,
XCMPLX_1:87;
[i2,j] in Indices G by A14,A21,A3,A7,A9,MATRIX_0:30;
then
A27: G*(i2,j) = |[x,y1]| by JORDAN8:def 1;
then
A28: G*(i2,j)`1 = x by EUCLID:52;
j+1-(1+1) = j-1;
then G*(i2,j+1) = |[x,y2]| by A25,JORDAN8:def 1;
then G*(i2,j+1)`1 = x by EUCLID:52;
then LSeg(G*(i2,j),G*(i2,j+1)) is vertical by A28,SPPOL_1:16;
then (N-min C)`1 = G*(i2,j)`1 by A24,SPPOL_1:41;
hence contradiction by A17,A27,A28,A26,EUCLID:52;
end;
suppose
A29: i2 < i1;
then
A30: i1-'i2+i2 = i1 by XREAL_1:235;
then i1-'i2 <= 1 by A20,A3,A19,A7,A9,JORDAN8:7;
then i1-'i2 < 1 or i1-'i2 = 1 by XXREAL_0:1;
then i1-'i2 = 0 or i1-'i2 = 1 by NAT_1:14;
then cell(G,i2,j) /\ cell(G,i1,j) = LSeg(G*(i1,j),G*(i1,j+1)) by A21,A3
,A6,A9,A29,A30,GOBOARD5:25,JORDAN5B:2;
then
A31: N-min C in LSeg(G*(i1,j),G*(i1,j+1)) by A12,A16,XBOOLE_0:def 4;
1 <= j+1 by NAT_1:12;
then
A32: [i1,j+1] in Indices G by A10,A20,A3,A8,MATRIX_0:30;
set y2 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-1);
set y1 = (S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2);
set x = (W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i1-2);
j = (2|^n+2+1)-'1 by A4,JORDAN8:def 1
.= (2|^n+2) by NAT_D:34;
then
A33: (((N-bound C)-(S-bound C))/(2|^n))*(j-2) = (N-bound C)-(S-bound C) by A1,
XCMPLX_1:87;
[i1,j] in Indices G by A10,A20,A3,A7,A9,MATRIX_0:30;
then
A34: G*(i1,j) = |[x,y1]| by JORDAN8:def 1;
then
A35: G*(i1,j)`1 = x by EUCLID:52;
j+1-(1+1) = j-1;
then G*(i1,j+1) = |[x,y2]| by A32,JORDAN8:def 1;
then G*(i1,j+1)`1 = x by EUCLID:52;
then LSeg(G*(i1,j),G*(i1,j+1)) is vertical by A35,SPPOL_1:16;
then (N-min C)`1 = G*(i1,j)`1 by A31,SPPOL_1:41;
hence contradiction by A13,A34,A35,A33,EUCLID:52;
end;
end;
theorem Th30:
for n being Nat for f being standard non constant
special_circular_sequence st f is_sequence_on Gauge(C,n) &
(for k being Nat st 1 <= k & k
+1 <= len f holds left_cell(f,k,Gauge(C,n)) misses C & right_cell(f,k,Gauge(C,n
)) meets C) &
(ex i being Nat st 1 <= i & i+1 <= len Gauge(C,n) & f/.1 = Gauge(C,n)*(i,
width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) & N-min C in cell(
Gauge(C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'
1)) holds N-min L~f = f/.1
proof
let n be Nat;
set G = Gauge(C,n);
let f be standard non constant special_circular_sequence such that
A1: f is_sequence_on G and
A2: for k being Nat st 1 <= k & k+1 <= len f holds left_cell(f,k,G)
misses C & right_cell(f,k,G) meets C;
N-min L~f in rng f by SPRECT_2:39;
then consider m being Nat such that
A3: m in dom f and
A4: f.m = N-min L~f by FINSEQ_2:10;
reconsider m as Nat;
consider i,j being Nat such that
A5: [i,j] in Indices G and
A6: f/.m = G*(i,j) by A1,A3,GOBOARD1:def 9;
A7: f/.m = f.m by A3,PARTFUN1:def 6;
A8: (N-min L~f)`2 = N-bound L~f by EUCLID:52;
set W = W-bound C, S = S-bound C, E = E-bound C, N = N-bound C;
given i9 being Nat such that
A9: 1 <= i9 and
A10: i9+1 <= len G and
A11: f/.1 = G*(i9,width G) and
A12: f/.2 = G*(i9+1,width G) and
A13: N-min C in cell(G,i9,width G-'1) and
A14: N-min C <> G*(i9,width G-'1);
A15: G*(i9,len G-'1) = |[G*(i9,len G-'1)`1,G*(i9,len G-'1)`2]| & (N-min C)`2
= N by EUCLID:52,53;
G*(i,j) = |[W+((E-W)/(2|^n))*(i-2), S+((N-S)/(2|^n))*(j-2)]| by A5,
JORDAN8:def 1;
then
A16: S+((N-S)/(2|^n))*(j-2) = N-bound L~f by A4,A7,A8,A6,EUCLID:52;
N > S by JORDAN8:9;
then 2|^n > 0 & N-S > 0 by NEWTON:83,XREAL_1:50;
then
A17: (N-S)/(2|^n) > 0 by XREAL_1:139;
A18: (NW-corner L~f)`1 = W-bound L~f & (NE-corner L~f)`1 = E-bound L~f by
EUCLID:52;
A19: 1 <= i by A5,MATRIX_0:32;
A20: (NW-corner L~f)`2 = N-bound L~f & (NE-corner L~f)`2 = N-bound L~f by
EUCLID:52;
A21: m <= len f by A3,FINSEQ_3:25;
A22: 1 <= j by A5,MATRIX_0:32;
len G = 2|^n+3 by JORDAN8:def 1;
then
A23: len G >= 3 by NAT_1:12;
then
A24: 1 < len G by XXREAL_0:2;
then
A25: 1 <= len G-'1 by NAT_D:49;
then
A26: len G-'1 < len G by NAT_D:51;
A27: i <= len G by A5,MATRIX_0:32;
A28: j <= width G by A5,MATRIX_0:32;
then
A29: G*(i,j)`2 = G* (1,j)`2 by A19,A27,A22,GOBOARD5:1;
A30: len f > 4 by GOBOARD7:34;
1 in dom f by FINSEQ_5:6;
then
A31: f/.1 in L~f by A30,GOBOARD1:1,XXREAL_0:2;
then
A32: N-bound L~f >= (f/.1)`2 by PSCOMP_1:24;
A33: len G = width G by JORDAN8:def 1;
A34: i9 < len G by A10,NAT_1:13;
then G*(i9,j)`2 = G*(1,j)`2 by A9,A22,A28,GOBOARD5:1;
then G*(i,j)`2 <= G*(i9,len G)`2 by A9,A34,A33,A22,A28,A29,SPRECT_3:12;
then
A35: N-bound L~f = (f/.1)`2 by A11,A33,A4,A7,A8,A6,A32,XXREAL_0:1;
[i9,len G] in Indices G by A9,A34,A33,A24,MATRIX_0:30;
then G*(i9,len G)=|[W+((E-W)/(2|^n))*(i9-2),S+((N-S)/(2|^n))*(len G-2)]| by
JORDAN8:def 1;
then S+((N-S)/(2|^n))*(len G-2) = N-bound L~f by A11,A33,A35,EUCLID:52;
then
A36: len G-2 = j-2 by A17,A16,XCMPLX_1:5;
then
A37: G*(i9,len G)`1 = G*(i9,1)`1 by A9,A34,A33,A22,GOBOARD5:2;
W-bound L~f <= (f/.1)`1 & (f/.1)`1 <= E-bound L~f by A31,PSCOMP_1:24;
then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) by A35,A18,A20,GOBOARD7:8;
then f/.1 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A31,XBOOLE_0:def 4;
then
A38: (N-min L~f)`1 <= (f/.1)`1 by PSCOMP_1:39;
then
A39: i <= i9 by A9,A11,A33,A4,A7,A6,A27,A22,A36,GOBOARD5:3;
then
A40: i < len G by A34,XXREAL_0:2;
then
A41: i+1 <= len G by NAT_1:13;
A42: len G-'1+1 = len G by A23,XREAL_1:235,XXREAL_0:2;
then
N-min C in { |[r9,s9]| where r9,s9 is Real:
G*(i9,1)`1 <= r9 & r9 <= G*
(i9+1,1)`1 & G*(1,len G-'1)`2 <= s9 & s9 <= G*(1,len G)`2 } by A9,A13,A34,A33
,A25,A26,GOBRD11:32;
then
ex r9,s9 being Real
st N-min C = |[r9,s9]| & G*(i9,1)`1 <= r9 & r9 <= G*
(i9+1,1)`1 & G*(1,len G-'1)`2 <= s9 & s9 <= G*(1,len G)`2;
then
A43: (f/.1)`1 <= (N-min C)`1 by A11,A33,A37,EUCLID:52;
then
A44: (N-min L~f)`1 <= (N-min C)`1 by A38,XXREAL_0:2;
A45: 1 <= m by A3,FINSEQ_3:25;
A46: G*(i9,len G-'1)`2 = N by A9,A34,JORDAN8:14;
A47: N-min C = |[(N-min C)`1,(N-min C)`2]| by EUCLID:53;
A48: (NW-corner C)`2 = N & (NE-corner C)`2 = N by EUCLID:52;
A49: (NW-corner C)`1 = W & (NE-corner C)`1 = E by EUCLID:52;
A50: len G = width G by JORDAN8:def 1;
G*(i9,len G-'1)`1 = G*(i9,1)`1 by A9,A34,A33,A25,A26,GOBOARD5:2;
then
A51: G*(i9,len G-'1)`1 < (N-min C)`1 by A11,A14,A33,A37,A43,A47,A15,A46,
XXREAL_0:1;
A52: G*(i,len G)`1 = G*(i,1)`1 by A19,A27,A22,A28,A36,GOBOARD5:2;
per cases by A21,XXREAL_0:1;
suppose
m = len f;
hence thesis by A4,A7,FINSEQ_6:def 1;
end;
suppose
m < len f;
then
A53: m+1 <= len f by NAT_1:13;
then consider i1,j1,i2,j2 being Nat such that
A54: [i1,j1] in Indices G & f/.m = G*(i1,j1) and
A55: [i2,j2] in Indices G and
A56: f/.(m+1) = G*(i2,j2) and
A57: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
or i1 = i2 & j1 = j2+1 by A1,A45,JORDAN8:3;
A58: right_cell(f,m,G) meets C by A2,A45,A53;
then consider p being object such that
A59: p in right_cell(f,m,G) and
A60: p in C by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A59;
A61: W <= p`1 & p`1 <= E by A60,PSCOMP_1:24;
A62: (N-min C)`2 = N by EUCLID:52;
then
A63: p`2 <= (N-min C)`2 by A60,PSCOMP_1:24;
A64: G*(1,len G-'1)`2 < G*(1,len G)`2 by A50,A24,A25,A26,GOBOARD5:4;
A65: G*(1,len G-'1)`2 = N by A24,JORDAN8:14;
A66: j2 <= len G by A50,A55,MATRIX_0:32;
now
per cases by A5,A6,A36,A54,A57,GOBOARD1:5;
suppose
i = i2 & len G+1 = j2;
hence thesis by A66,NAT_1:13;
end;
suppose
A67: i+1 = i2 & len G = j2;
A68: cell(G,i,len G-'1) = {|[r,s]| : G*(i,1)`1 <= r & r <= G*(i+1,1)`1
& G*(1,len G-'1)`2 <= s & s <= G*(1,len G-'1+1)`2 } by A50,A19,A25,A26,A40,
GOBRD11:32;
right_cell(f,m,G) = cell(G,i,len G-'1) by A1,A45,A5,A6,A36,A53,A55,A56
,A67,GOBRD13:24;
then consider r,s such that
A69: p = |[r,s]| and
G*(i,1)`1 <= r and
A70: r <= G*(i+1,1)`1 and
A71: G*(1,len G-'1)`2 <= s and
s <= G*(1,len G-'1+1)`2 by A59,A68;
p`2 = s by A69,EUCLID:52;
then p`2 = N by A62,A63,A65,A71,XXREAL_0:1;
then p in LSeg(NW-corner C, NE-corner C) by A49,A48,A61,GOBOARD7:8;
then p in LSeg(NW-corner C, NE-corner C) /\ C by A60,XBOOLE_0:def 4;
then
A72: (N-min C)`1 <= p`1 by PSCOMP_1:39;
p`1 = r by A69,EUCLID:52;
then (N-min C)`1 <= G*(i+1,1)`1 by A70,A72,XXREAL_0:2;
then
A73: N-min C in cell(G,i,width G-'1) by A33,A4,A7,A6,A36,A52,A42,A44,A47,A62
,A65,A64,A68;
N-min C <> G*(i,len G-'1) by A34,A33,A19,A25,A26,A51,A39,SPRECT_3:13;
hence thesis by A9,A10,A11,A13,A14,A33,A4,A7,A6,A19,A36,A41,A73,Th29;
end;
suppose
i = i2+1 & len G = j2;
then right_cell(f,m,G) = cell(G,i2,len G) & i2 < len G by A1,A45,A5,A6
,A27,A36,A53,A55,A56,GOBRD13:26,NAT_1:13;
hence thesis by A2,A45,A53,JORDAN8:15;
end;
suppose
A74: i = i2 & len G = j2+1;
then
A75: j2 = len G-'1 by NAT_D:34;
then
A76: right_cell(f,m,G) = cell(G,i-'1,len G-'1) by A1,A45,A5,A6,A36,A53,A55
,A56,A74,GOBRD13:28;
m-'1 <= m by NAT_D:35;
then
A77: m-'1 <= len f by A21,XXREAL_0:2;
now
1 <= i9+1 by A9,NAT_1:13;
then
A78: G*(i9+1,len G)`2 = G* (1,len G)`2 by A10,A33,A24,GOBOARD5:1;
assume
A79: m = 1;
G*(i9,len G)`2 = G*(1,len G)`2 by A9,A34,A33,A24,GOBOARD5:1;
hence contradiction by A11,A12,A33,A6,A19,A27,A36,A25,A26,A56,A74,A75
,A79,A78,GOBOARD5:4;
end;
then m > 1 by A45,XXREAL_0:1;
then
A80: m-'1 >= 1 by NAT_D:49;
A81: m-'1+1 = m by A45,XREAL_1:235;
then consider i19,j19,i29,j29 being Nat such that
A82: [i19,j19] in Indices G and
A83: f/.(m-'1) = G*(i19,j19) and
A84: [i29,j29] in Indices G & f/.m = G*(i29,j29) &( i19 = i29 &
j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 = i29+1 & j19 = j29 or i19 = i29
& j19 = j29+1) by A1,A21,A80,JORDAN8:3;
A85: 1 <= i19 by A82,MATRIX_0:32;
A86: i19 <= len G by A82,MATRIX_0:32;
now
per cases by A5,A6,A36,A84,GOBOARD1:5;
suppose
A87: i19 = i & j19+1 = len G;
then j19 = len G-'1 by NAT_D:34;
then left_cell(f,m-'1,G) = cell(G,i-'1,len G-'1) by A1,A21,A5,A6
,A36,A80,A81,A82,A83,A87,GOBRD13:21;
hence contradiction by A2,A21,A58,A76,A80,A81;
end;
suppose
A88: i19+1 = i & j19 = len G;
A89: G*(i19,j)`2 = G*(1,j)`2 & G*(i,j)`2 = G*(1,j)`2 by A19,A27,A22,A28
,A85,A86,GOBOARD5:1;
m-'1 in dom f by A80,A77,FINSEQ_3:25;
then
A90: f/.(m-'1) in L~f by A30,GOBOARD1:1,XXREAL_0:2;
then W-bound L~f <= (f/.(m-'1))`1 & (f/.(m-'1))`1 <= E-bound L~f
by PSCOMP_1:24;
then f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) by A4,A7,A8,A6
,A36,A18,A20,A83,A88,A89,GOBOARD7:8;
then
A91: f/.(m-'1) in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A90,
XBOOLE_0:def 4;
i19 < i by A88,NAT_1:13;
then (f/.(m-'1))`1 < (f/.m)`1 by A6,A27,A22,A28,A36,A83,A85,A88,
GOBOARD5:3;
hence contradiction by A4,A7,A91,PSCOMP_1:39;
end;
suppose
i19 = i+1 & j19 = len G;
then right_cell(f,m-'1,G) = cell(G,i,len G) by A1,A21,A5,A6,A36,A80
,A81,A82,A83,GOBRD13:26;
hence contradiction by A2,A21,A27,A80,A81,JORDAN8:15;
end;
suppose
i19 = i & j19 = len G+1;
then len G+1 <= len G+0 by A50,A82,MATRIX_0:32;
hence contradiction by XREAL_1:6;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
definition
let C be compact non vertical non horizontal non empty Subset of TOP-REAL 2;
let n be Nat;
assume
A1: C is connected;
func Cage(C,n) -> clockwise_oriented standard non constant
special_circular_sequence means
:Def1:
it is_sequence_on Gauge(C,n) &
(ex i being Nat st 1 <= i & i+1 <= len Gauge(C,n) &
it/.1 = Gauge(C,n)*(i,width Gauge(C,n)) &
it/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) &
N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) &
N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1)) &
for k being Nat st 1 <=
k & k+2 <= len it holds (front_left_cell(it,k,Gauge(C,n)) misses C &
front_right_cell(it,k,Gauge(C,n)) misses C implies it turns_right k,Gauge(C,n))
& (front_left_cell(it,k,Gauge(C,n)) misses C & front_right_cell(it,k,Gauge(C,n)
) meets C implies it goes_straight k,Gauge(C,n)) & (front_left_cell(it,k,Gauge(
C,n)) meets C implies it turns_left k,Gauge(C,n));
existence
proof
set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C;
set G = Gauge(C,n);
defpred P[Nat,set,set] means ($1 = 0 implies ex i st 1 <= i & i
+1 <= len G & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) &
$3 = <*G*(i,width G)*>) & ($1 = 1 implies ex i st 1 <= i & i+1 <= len G & N-min
C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) & $3 = <*G*(i,width G),
G*(i+1,width G)*>) & ($1 > 1 & $2 is FinSequence of TOP-REAL 2 implies for f
being FinSequence of TOP-REAL 2 st $2 = f holds (len f = $1 implies (f
is_sequence_on G & right_cell(f,len f-'1,G) meets C implies (front_left_cell(f,
(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G) misses C implies ex i
,j st f^<*G*(i,j)*> turns_right (len f)-'1,G & $3 = f^<*G*(i,j)*>) & (
front_left_cell(f,(len f)-'1,G) misses C & front_right_cell(f,(len f)-'1,G)
meets C implies ex i,j st f^<*G*(i,j)*> goes_straight (len f)-'1,G & $3 = f^<*G
*(i,j)*>) & (front_left_cell(f,(len f)-'1,G) meets C implies ex i,j st f^<*G*(i
,j)*> turns_left (len f)-'1,G & $3 = f^<*G*(i,j)*>)) & (not f is_sequence_on G
or right_cell(f,len f-'1,G) misses C implies $3 = f^<*G*(1,1)*>)) & (len f <>
$1 implies $3 = {})) & ($1 > 1 & $2 is not FinSequence of TOP-REAL 2 implies $3
= {});
A2: len G = width G by JORDAN8:def 1;
A3: for k being Nat, x being set ex y being set st P[k,x,y]
proof
let k be Nat, x be set;
consider m being Nat such that
A4: 1 <= m & m+1 <= len G & N-min C in cell(G,m,width G-'1) & N-min
C <> G*(m,width G-'1) by Th28;
per cases by NAT_1:25;
suppose
A5: k=0;
take <*G*(m,width G)*>;
thus thesis by A4,A5;
end;
suppose
A6: k = 1;
take <*G*(m,width G),G*(m+1,width G)*>;
thus thesis by A4,A6;
end;
suppose that
A7: k > 1 and
A8: x is FinSequence of TOP-REAL 2;
reconsider f = x as FinSequence of TOP-REAL 2 by A8;
per cases;
suppose
A9: len f = k;
per cases;
suppose
A10: f is_sequence_on G & right_cell(f,len f-'1,G) meets C;
A11: (len f) -'1 +1 = len f by A7,A9,XREAL_1:235;
then
A12: (len f)-'1+(1+1) = (len f)+1;
A13: (len f)-'1+1 in dom f by A7,A9,A11,FINSEQ_3:25;
A14: 1 <= (len f)-'1 by A7,A9,NAT_D:49;
then consider i1,j1,i2,j2 being Nat such that
A15: [i1,j1] in Indices G and
A16: f/.((len f) -'1) = G*(i1,j1) and
A17: [i2,j2] in Indices G and
A18: f/.((len f) -'1+1) = G*(i2,j2) and
A19: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2
+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A10,A11,JORDAN8:3;
A20: i1 <= len G by A15,MATRIX_0:32;
A21: 1 <= j2+1 by NAT_1:12;
A22: 1 <= i2 by A17,MATRIX_0:32;
A23: j1 <= width G by A15,MATRIX_0:32;
A24: 1 <= i2+1 by NAT_1:12;
A25: 1 <= j2 by A17,MATRIX_0:32;
(len f)-'1 <= len f by NAT_D:35;
then
A26: (len f)-'1 in dom f by A14,FINSEQ_3:25;
A27: j2 <= width G by A17,MATRIX_0:32;
then
A28: j2-'1 <= width G by NAT_D:44;
A29: i2 <= len G by A17,MATRIX_0:32;
then
A30: i2-'1 <= len G by NAT_D:44;
per cases;
suppose
A31: front_left_cell(f,(len f)-'1,G) misses C &
front_right_cell(f,(len f)-'1,G) misses C;
per cases by A19;
suppose
A32: i1 = i2 & j1+1 = j2;
take f1 = f^<*G*(i2+1,j2)*>;
now take i=i2+1,j=j2;
thus f1 turns_right (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A33: [i19,j19] in Indices G and
A34: [i29,j29] in Indices G and
A35: f1/.((len f)-'1) = G*(i19,j19) and
A36: f1/.((len f)-'1+1) = G*(i29,j29);
A37: f/.((len f)-'1) = G*(i19,j19) by A26,A35,
FINSEQ_4:68;
then
A38: i1 = i19 by A15,A16,A33,GOBOARD1:5;
A39: j1 = j19 by A15,A16,A33,A37,GOBOARD1:5;
A40: f/.((len f)-'1+1) = G*(i29,j29) by A13,A36,
FINSEQ_4:68;
then
A41: i2 = i29 by A17,A18,A34,GOBOARD1:5;
A42: j2 = j29 by A17,A18,A34,A40,GOBOARD1:5;
per cases by A15,A16,A19,A33,A37,A41,A42,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
now
assume i2+1 > len G;
then
A43: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A29,XREAL_1:6;
then i2+1 = (len G)+1 by A43,XXREAL_0:1;
then cell(G,len G,j1) meets C by A10,A14,A11
,A15,A16,A17,A18,A32,GOBRD13:22;
hence contradiction by A2,A23,JORDAN8:16;
end;
hence
[i29+1,j29] in Indices G by A25,A27,A24,A41,A42,
MATRIX_0:30;
thus thesis by A12,A41,A42,FINSEQ_4:67;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A32,A38,A41;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A32,A38,A41;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A32,A39,A42;
end;
end;
end;
hence thesis by A7,A9,A10,A31;
end;
suppose
A44: i1+1 = i2 & j1 = j2;
take f1 = f^<*G*(i2,j2-'1)*>;
now
take i=i2,j=j2-'1;
thus f1 turns_right (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A45: [i19,j19] in Indices G and
A46: [i29,j29] in Indices G and
A47: f1/.((len f)-'1) = G*(i19,j19) and
A48: f1/.((len f)-'1+1) = G*(i29,j29);
A49: f/.((len f)-'1+1) = G*(i29,j29) by A13,A48,
FINSEQ_4:68;
then
A50: i2 = i29 by A17,A18,A46,GOBOARD1:5;
A51: f/.((len f)-'1) = G*(i19,j19) by A26,A47,
FINSEQ_4:68;
then
A52: i1 = i19 by A15,A16,A45,GOBOARD1:5;
A53: j2 = j29 by A17,A18,A46,A49,GOBOARD1:5;
per cases by A15,A16,A19,A45,A51,A50,A53,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
hence thesis by A44,A52,A50;
end;
case
i19+1 = i29 & j19 = j29;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A25,XXREAL_0:1;
then cell(G,i1,1-'1) meets C by A10,A14,A11,A15
,A16,A17,A18,A44,GOBRD13:24;
then cell(G,i1,0) meets C by XREAL_1:232;
hence contradiction by A20,JORDAN8:17;
end;
hence [i29,j29-'1] in Indices G by A22,A29,A28
,A50,A53,MATRIX_0:30;
thus thesis by A12,A50,A53,FINSEQ_4:67;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A44,A52,A50;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A44,A52,A50;
end;
end;
end;
hence thesis by A7,A9,A10,A31;
end;
suppose
A54: i1 = i2+1 & j1 = j2;
take f1 = f^<*G*(i2,j2+1)*>;
now
take i=i2,j=j2+1;
thus f1 turns_right (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A55: [i19,j19] in Indices G and
A56: [i29,j29] in Indices G and
A57: f1/.((len f)-'1) = G*(i19,j19) and
A58: f1/.((len f)-'1+1) = G*(i29,j29);
A59: f/.((len f)-'1+1) = G*(i29,j29) by A13,A58,
FINSEQ_4:68;
then
A60: i2 = i29 by A17,A18,A56,GOBOARD1:5;
A61: f/.((len f)-'1) = G*(i19,j19) by A26,A57,
FINSEQ_4:68;
then
A62: i1 = i19 by A15,A16,A55,GOBOARD1:5;
A63: j2 = j29 by A17,A18,A56,A59,GOBOARD1:5;
per cases by A15,A16,A19,A55,A61,A60,A63,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
hence thesis by A54,A62,A60;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A54,A62,A60;
end;
case
i19 = i29+1 & j19 = j29;
now
assume j2+1 > len G;
then
A64: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A27,XREAL_1:6;
then j2+1 = (len G)+1 by A64,XXREAL_0:1;
then cell(G,i2,len G) meets C by A10,A14,A11
,A15,A16,A17,A18,A54,GOBRD13:26;
hence contradiction by A29,JORDAN8:15;
end;
hence
[i29,j29+1] in Indices G by A2,A22,A29,A21,A60
,A63,MATRIX_0:30;
thus thesis by A12,A60,A63,FINSEQ_4:67;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A54,A62,A60;
end;
end;
end;
hence thesis by A7,A9,A10,A31;
end;
suppose
A65: i1 = i2 & j1 = j2+1;
take f1 = f^<*G*(i2-'1,j2)*>;
now
reconsider i=i2-'1 ,j=j2 as Nat;
take i,j;
thus f1 turns_right (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A66: [i19,j19] in Indices G and
A67: [i29,j29] in Indices G and
A68: f1/.((len f)-'1) = G*(i19,j19) and
A69: f1/.((len f)-'1+1) = G*(i29,j29);
A70: f/.((len f)-'1) = G*(i19,j19) by A26,A68,
FINSEQ_4:68;
then
A71: i1 = i19 by A15,A16,A66,GOBOARD1:5;
A72: j1 = j19 by A15,A16,A66,A70,GOBOARD1:5;
A73: f/.((len f)-'1+1) = G*(i29,j29) by A13,A69,
FINSEQ_4:68;
then
A74: i2 = i29 by A17,A18,A67,GOBOARD1:5;
A75: j2 = j29 by A17,A18,A67,A73,GOBOARD1:5;
per cases by A15,A16,A19,A66,A70,A74,A75,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
hence thesis by A65,A72,A75;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A65,A71,A74;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A65,A71,A74;
end;
case
i19 = i29 & j19 = j29+1;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A22,XXREAL_0:1;
then cell(G,1-'1,j2) meets C by A10,A14,A11,A15
,A16,A17,A18,A65,GOBRD13:28;
then cell(G,0,j2) meets C by XREAL_1:232;
hence contradiction by A2,A27,JORDAN8:18;
end;
hence [i29-'1,j29] in Indices G by A25,A27,A30
,A74,A75,MATRIX_0:30;
thus thesis by A12,A74,A75,FINSEQ_4:67;
end;
end;
end;
hence thesis by A7,A9,A10,A31;
end;
end;
suppose
A76: front_left_cell(f,(len f)-'1,G) misses C &
front_right_cell(f,(len f)-'1,G) meets C;
per cases by A19;
suppose
A77: i1 = i2 & j1+1 = j2;
take f1 = f^<*G*(i2,j2+1)*>;
now
take i=i2,j=j2+1;
thus f1 goes_straight (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A78: [i19,j19] in Indices G and
A79: [i29,j29] in Indices G and
A80: f1/.((len f)-'1) = G*(i19,j19) and
A81: f1/.((len f)-'1+1) = G*(i29,j29);
A82: f/.((len f)-'1) = G*(i19,j19) by A26,A80,
FINSEQ_4:68;
then
A83: i1 = i19 by A15,A16,A78,GOBOARD1:5;
A84: j1 = j19 by A15,A16,A78,A82,GOBOARD1:5;
A85: f/.((len f)-'1+1) = G*(i29,j29) by A13,A81,
FINSEQ_4:68;
then
A86: i2 = i29 by A17,A18,A79,GOBOARD1:5;
A87: j2 = j29 by A17,A18,A79,A85,GOBOARD1:5;
per cases by A15,A16,A19,A78,A82,A86,A87,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
now
assume j2+1 > len G;
then
A88: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A27,XREAL_1:6;
then j2+1 = (len G)+1 by A88,XXREAL_0:1;
then cell(G,i1,len G) meets C by A10,A14,A11
,A15,A16,A17,A18,A76,A77,GOBRD13:35;
hence contradiction by A20,JORDAN8:15;
end;
hence
[i29,j29+1] in Indices G by A2,A22,A29,A21,A86
,A87,MATRIX_0:30;
thus thesis by A12,A86,A87,FINSEQ_4:67;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A77,A83,A86;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A77,A83,A86;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A77,A84,A87;
end;
end;
end;
hence thesis by A7,A9,A10,A76;
end;
suppose
A89: i1+1 = i2 & j1 = j2;
take f1 = f^<*G*(i2+1,j2)*>;
now
take i=i2+1 ,j=j2;
thus f1 goes_straight (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A90: [i19,j19] in Indices G and
A91: [i29,j29] in Indices G and
A92: f1/.((len f)-'1) = G*(i19,j19) and
A93: f1/.((len f)-'1+1) = G*(i29,j29);
A94: f/.((len f)-'1+1) = G*(i29,j29) by A13,A93,
FINSEQ_4:68;
then
A95: i2 = i29 by A17,A18,A91,GOBOARD1:5;
A96: f/.((len f)-'1) = G*(i19,j19) by A26,A92,
FINSEQ_4:68;
then
A97: i1 = i19 by A15,A16,A90,GOBOARD1:5;
A98: j2 = j29 by A17,A18,A91,A94,GOBOARD1:5;
per cases by A15,A16,A19,A90,A96,A95,A98,GOBOARD1:5
;
case
i19 = i29 & j19+1 = j29;
hence thesis by A89,A97,A95;
end;
case
i19+1 = i29 & j19 = j29;
now
assume i2+1 > len G;
then
A99: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A29,XREAL_1:6;
then i2+1 = (len G)+1 by A99,XXREAL_0:1;
then cell(G,len G,j1-'1) meets C by A10,A14,A11
,A15,A16,A17,A18,A76,A89,GOBRD13:37;
hence contradiction by A2,A23,JORDAN8:16
,NAT_D:44;
end;
hence [i29+1,j29] in Indices G by A25,A27,A24,A95
,A98,MATRIX_0:30;
thus thesis by A12,A95,A98,FINSEQ_4:67;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A89,A97,A95;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A89,A97,A95;
end;
end;
end;
hence thesis by A7,A9,A10,A76;
end;
suppose
A100: i1 = i2+1 & j1 = j2;
take f1 = f^<*G*(i2-'1,j2)*>;
now
take i=i2-'1 ,j=j2;
thus f1 goes_straight (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A101: [i19,j19] in Indices G and
A102: [i29,j29] in Indices G and
A103: f1/.((len f)-'1) = G*(i19,j19) and
A104: f1/.((len f)-'1+1) = G*(i29,j29);
A105: f/.((len f)-'1+1) = G*(i29,j29) by A13,A104,
FINSEQ_4:68;
then
A106: i2 = i29 by A17,A18,A102,GOBOARD1:5;
A107: f/.((len f)-'1) = G*(i19,j19) by A26,A103,
FINSEQ_4:68;
then
A108: i1 = i19 by A15,A16,A101,GOBOARD1:5;
A109: j2 = j29 by A17,A18,A102,A105,GOBOARD1:5;
per cases by A15,A16,A19,A101,A107,A106,A109,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
hence thesis by A100,A108,A106;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A100,A108,A106;
end;
case
i19 = i29+1 & j19 = j29;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A22,XXREAL_0:1;
then cell(G,1-'1,j1) meets C by A10,A14,A11,A15
,A16,A17,A18,A76,A100,GOBRD13:39;
then cell(G,0,j1) meets C by XREAL_1:232;
hence contradiction by A2,A23,JORDAN8:18;
end;
hence [i29-'1,j29] in Indices G by A25,A27,A30
,A106,A109,MATRIX_0:30;
thus thesis by A12,A106,A109,FINSEQ_4:67;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A100,A108,A106;
end;
end;
end;
hence thesis by A7,A9,A10,A76;
end;
suppose
A110: i1 = i2 & j1 = j2+1;
take f1 = f^<*G*(i2,j2-'1)*>;
now
take i=i2,j=j2-'1;
thus f1 goes_straight (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A111: [i19,j19] in Indices G and
A112: [i29,j29] in Indices G and
A113: f1/.((len f)-'1) = G*(i19,j19) and
A114: f1/.((len f)-'1+1) = G*(i29,j29);
A115: f/.((len f)-'1) = G*(i19,j19) by A26,A113,
FINSEQ_4:68;
then
A116: i1 = i19 by A15,A16,A111,GOBOARD1:5;
A117: j1 = j19 by A15,A16,A111,A115,GOBOARD1:5;
A118: f/.((len f)-'1+1) = G*(i29,j29) by A13,A114,
FINSEQ_4:68;
then
A119: i2 = i29 by A17,A18,A112,GOBOARD1:5;
A120: j2 = j29 by A17,A18,A112,A118,GOBOARD1:5;
per cases by A15,A16,A19,A111,A115,A119,A120,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
hence thesis by A110,A117,A120;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A110,A116,A119;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A110,A116,A119;
end;
case
i19 = i29 & j19 = j29+1;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A25,XXREAL_0:1;
then cell(G,i1-'1,1-'1) meets C by A10,A14,A11
,A15,A16,A17,A18,A76,A110,GOBRD13:41;
then cell(G,i1-'1,0) meets C by XREAL_1:232;
hence contradiction by A20,JORDAN8:17,NAT_D:44;
end;
hence [i29,j29-'1] in Indices G by A22,A29,A28
,A119,A120,MATRIX_0:30;
thus thesis by A12,A119,A120,FINSEQ_4:67;
end;
end;
end;
hence thesis by A7,A9,A10,A76;
end;
end;
suppose
A121: front_left_cell(f,(len f)-'1,G) meets C;
per cases by A19;
suppose
A122: i1 = i2 & j1+1 = j2;
take f1 = f^<*G*(i2-'1,j2)*>;
now
take i=i2-'1 ,j=j2;
thus f1 turns_left (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A123: [i19,j19] in Indices G and
A124: [i29,j29] in Indices G and
A125: f1/.((len f)-'1) = G*(i19,j19) and
A126: f1/.((len f)-'1+1) = G*(i29,j29);
A127: f/.((len f)-'1) = G*(i19,j19) by A26,A125,
FINSEQ_4:68;
then
A128: i1 = i19 by A15,A16,A123,GOBOARD1:5;
A129: j1 = j19 by A15,A16,A123,A127,GOBOARD1:5;
A130: f/.((len f)-'1+1) = G*(i29,j29) by A13,A126,
FINSEQ_4:68;
then
A131: i2 = i29 by A17,A18,A124,GOBOARD1:5;
A132: j2 = j29 by A17,A18,A124,A130,GOBOARD1:5;
per cases by A15,A16,A19,A123,A127,A131,A132,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A22,XXREAL_0:1;
then cell(G,1-'1,j2) meets C by A10,A14,A11,A15
,A16,A17,A18,A121,A122,GOBRD13:34;
then cell(G,0,j2) meets C by XREAL_1:232;
hence contradiction by A2,A27,JORDAN8:18;
end;
hence [i29-'1,j29] in Indices G by A25,A27,A30
,A131,A132,MATRIX_0:30;
thus thesis by A12,A131,A132,FINSEQ_4:67;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A122,A128,A131;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A122,A128,A131;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A122,A129,A132;
end;
end;
end;
hence thesis by A7,A9,A10,A121;
end;
suppose
A133: i1+1 = i2 & j1 = j2;
take f1 = f^<*G*(i2,j2+1)*>;
now
take i=i2,j=j2+1;
thus f1 turns_left (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A134: [i19,j19] in Indices G and
A135: [i29,j29] in Indices G and
A136: f1/.((len f)-'1) = G*(i19,j19) and
A137: f1/.((len f)-'1+1) = G*(i29,j29);
A138: f/.((len f)-'1+1) = G*(i29,j29) by A13,A137,
FINSEQ_4:68;
then
A139: i2 = i29 by A17,A18,A135,GOBOARD1:5;
A140: f/.((len f)-'1) = G*(i19,j19) by A26,A136,
FINSEQ_4:68;
then
A141: i1 = i19 by A15,A16,A134,GOBOARD1:5;
A142: j2 = j29 by A17,A18,A135,A138,GOBOARD1:5;
per cases by A15,A16,A19,A134,A140,A139,A142,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
hence thesis by A133,A141,A139;
end;
case
i19+1 = i29 & j19 = j29;
now
assume j2+1 > len G;
then
A143: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A27,XREAL_1:6;
then j2+1 = (len G)+1 by A143,XXREAL_0:1;
then cell(G,i2,len G) meets C by A10,A14,A11
,A15,A16,A17,A18,A121,A133,GOBRD13:36;
hence contradiction by A29,JORDAN8:15;
end;
hence
[i29,j29+1] in Indices G by A2,A22,A29,A21,A139
,A142,MATRIX_0:30;
thus thesis by A12,A139,A142,FINSEQ_4:67;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A133,A141,A139;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A133,A141,A139;
end;
end;
end;
hence thesis by A7,A9,A10,A121;
end;
suppose
A144: i1 = i2+1 & j1 = j2;
take f1 = f^<*G*(i2,j2-'1)*>;
now
take i=i2,j=j2-'1;
thus f1 turns_left (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A145: [i19,j19] in Indices G and
A146: [i29,j29] in Indices G and
A147: f1/.((len f)-'1) = G*(i19,j19) and
A148: f1/.((len f)-'1+1) = G*(i29,j29);
A149: f/.((len f)-'1+1) = G*(i29,j29) by A13,A148,
FINSEQ_4:68;
then
A150: i2 = i29 by A17,A18,A146,GOBOARD1:5;
A151: f/.((len f)-'1) = G*(i19,j19) by A26,A147,
FINSEQ_4:68;
then
A152: i1 = i19 by A15,A16,A145,GOBOARD1:5;
A153: j2 = j29 by A17,A18,A146,A149,GOBOARD1:5;
per cases by A15,A16,A19,A145,A151,A150,A153,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
hence thesis by A144,A152,A150;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A144,A152,A150;
end;
case
i19 = i29+1 & j19 = j29;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A25,XXREAL_0:1;
then cell(G,i2-'1,1-'1) meets C by A10,A14,A11
,A15,A16,A17,A18,A121,A144,GOBRD13:38;
then cell(G,i2-'1,0) meets C by XREAL_1:232;
hence contradiction by A29,JORDAN8:17,NAT_D:44;
end;
hence [i29,j29-'1] in Indices G by A22,A29,A28
,A150,A153,MATRIX_0:30;
thus thesis by A12,A150,A153,FINSEQ_4:67;
end;
case
i19 = i29 & j19 = j29+1;
hence thesis by A144,A152,A150;
end;
end;
end;
hence thesis by A7,A9,A10,A121;
end;
suppose
A154: i1 = i2 & j1 = j2+1;
take f1 = f^<*G*(i2+1,j2)*>;
now
take i=i2+1 ,j=j2;
thus f1 turns_left (len f)-'1,G
proof
let i19,j19,i29,j29 be Nat;
assume that
A155: [i19,j19] in Indices G and
A156: [i29,j29] in Indices G and
A157: f1/.((len f)-'1) = G*(i19,j19) and
A158: f1/.((len f)-'1+1) = G*(i29,j29);
A159: f/.((len f)-'1) = G*(i19,j19) by A26,A157,
FINSEQ_4:68;
then
A160: i1 = i19 by A15,A16,A155,GOBOARD1:5;
A161: j1 = j19 by A15,A16,A155,A159,GOBOARD1:5;
A162: f/.((len f)-'1+1) = G*(i29,j29) by A13,A158,
FINSEQ_4:68;
then
A163: i2 = i29 by A17,A18,A156,GOBOARD1:5;
A164: j2 = j29 by A17,A18,A156,A162,GOBOARD1:5;
per cases by A15,A16,A19,A155,A159,A163,A164,
GOBOARD1:5;
case
i19 = i29 & j19+1 = j29;
hence thesis by A154,A161,A164;
end;
case
i19+1 = i29 & j19 = j29;
hence thesis by A154,A160,A163;
end;
case
i19 = i29+1 & j19 = j29;
hence thesis by A154,A160,A163;
end;
case
i19 = i29 & j19 = j29+1;
now
assume i2+1 > len G;
then
A165: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A29,XREAL_1:6;
then i2+1 = (len G)+1 by A165,XXREAL_0:1;
then cell(G,len G,j2-'1) meets C by A10,A14,A11
,A15,A16,A17,A18,A121,A154,GOBRD13:40;
hence contradiction by A2,A27,JORDAN8:16
,NAT_D:44;
end;
hence [i29+1,j29] in Indices G by A25,A27,A24
,A163,A164,MATRIX_0:30;
thus thesis by A12,A163,A164,FINSEQ_4:67;
end;
end;
end;
hence thesis by A7,A9,A10,A121;
end;
end;
end;
suppose
A166: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C;
take f^<*G*(1,1)*>;
thus thesis by A7,A9,A166;
end;
end;
suppose
A167: len f <> k;
take {};
thus thesis by A7,A167;
end;
end;
suppose
A168: k > 1 & x is not FinSequence of TOP-REAL 2;
take {};
thus thesis by A168;
end;
end;
consider F being Function such that
A169: dom F = NAT and
A170: F.0 = {} and
A171: for k being Nat holds P[k,F.k,F.(k+1)] from RECDEF_1:
sch 1 (A3);
defpred P[Nat] means F.$1 is FinSequence of TOP-REAL 2;
A172: {} = <*>(the carrier of TOP-REAL 2);
A173: for k st P[k] holds P[k+1]
proof
let k such that
A174: F.k is FinSequence of TOP-REAL 2;
reconsider k as Element of NAT by ORDINAL1:def 12;
A175: P[k,F.k,F.(k+1)] by A171;
per cases by NAT_1:25;
suppose
k = 0;
hence thesis by A175;
end;
suppose
k = 1;
hence thesis by A175;
end;
suppose
A176: k > 1;
reconsider f = F.k as FinSequence of TOP-REAL 2 by A174;
per cases;
suppose
A177: len f = k;
per cases;
suppose
A178: f is_sequence_on G & right_cell(f,len f-'1,G) meets C;
then
A179: front_left_cell(f,(len f)-'1,G) meets C implies ex i,j
st f^<*G*(i,j) *> turns_left (len f)-'1,G & F.(k+1) = f^<*G*(i,j)*> by A171
,A176,A177;
A180: front_left_cell(f,(len f)-'1,G) misses C &
front_right_cell(f,(len f) -'1,G) meets C implies ex i,j st f^<*G*(i,j)*>
goes_straight (len f)-'1,G & F.( k+1) = f^<*G*(i,j)*> by A171,A176,A177,A178;
front_left_cell(f,(len f)-'1,G) misses C &
front_right_cell(f,(len f) -'1,G) misses C implies ex i,j st f^<*G*(i,j)*>
turns_right (len f)-'1,G & F.(k +1) = f^<*G*(i,j)*> by A171,A176,A177,A178;
hence thesis by A180,A179;
end;
suppose
A181: not f is_sequence_on G or right_cell(f,len f-'1,G) misses C;
f^<*G*(1,1)*> is FinSequence of TOP-REAL 2;
hence thesis by A171,A176,A177,A181;
end;
end;
suppose
len f <> k;
hence thesis by A171,A172,A176;
end;
end;
end;
A182: P[0] by A170,A172;
A183: for k holds P[k] from NAT_1:sch 2(A182,A173);
rng F c= (the carrier of TOP-REAL 2)*
proof
let y be object;
assume y in rng F;
then ex x being object st x in dom F & F.x = y by FUNCT_1:def 3;
then y is FinSequence of TOP-REAL 2 by A169,A183;
hence thesis by FINSEQ_1:def 11;
end;
then reconsider F as sequence of (the carrier of TOP-REAL 2)* by A169,
FUNCT_2:def 1,RELSET_1:4;
defpred P[Nat] means len(F.$1) = $1;
A184: for k st P[k] holds P[k+1]
proof
let k such that
A185: len(F.k) = k;
A186: P[k,F.k,F.(k+1)] by A171;
per cases by NAT_1:25;
suppose
k = 0;
hence thesis by A186,FINSEQ_1:39;
end;
suppose
k = 1;
hence thesis by A186,FINSEQ_1:44;
end;
suppose
A187: k > 1;
thus thesis
proof
per cases;
suppose
A188: F.k is_sequence_on G & right_cell(F.k,len(F.k)-'1,G) meets C;
then
A189: front_left_cell(F.k,(len(F.k))-'1,G) meets C implies ex i,j
st (F.k)^ <*G*(i,j)*> turns_left (len(F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*>
by A171,A185,A187;
A190: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k, (len(F.k))-'1,G) meets C implies ex i,j st (F.k)^<*G*(i,j
)*> goes_straight (len (F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A185
,A187,A188;
front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k, (len(F.k))-'1,G) misses C implies ex i,j st (F.k)^<*G*(i,
j)*> turns_right (len( F.k))-'1,G & F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A185
,A187,A188;
hence thesis by A185,A190,A189,FINSEQ_2:16;
end;
suppose
not F.k is_sequence_on G or right_cell(F.k,len(F.k)-'1,G
) misses C;
then F.(k+1) = (F.k)^<*G*(1,1)*> by A171,A185,A187;
hence thesis by A185,FINSEQ_2:16;
end;
end;
end;
end;
A191: P[0] by A170,CARD_1:27;
A192: for k holds P[k] from NAT_1:sch 2(A191,A184);
A193: now
let k such that
A194: F.k is_sequence_on G and
A195: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G)
misses C & right_cell(F.k,m,G) meets C and
A196: k > 1;
len(F.k) = k by A192;
then
A197: 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A196,NAT_D:49
,XREAL_1:235;
then
A198: right_cell(F.k,(len(F.k))-'1,G) meets C by A195;
let i1,j1,i2,j2 be Nat such that
A199: [i1,j1] in Indices G and
A200: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and
A201: [i2,j2] in Indices G and
A202: (F.k)/.len(F.k) = G*(i2,j2);
A203: i2 <= len G by A201,MATRIX_0:32;
A204: 1 <= i2+1 by NAT_1:12;
A205: 1 <= j2 by A201,MATRIX_0:32;
A206: j2 <= width G by A201,MATRIX_0:32;
A207: j1 <= width G by A199,MATRIX_0:32;
hereby
assume
A208: i1 = i2 & j1+1 = j2;
now
assume i2+1 > len G;
then
A209: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A203,XREAL_1:6;
then i2+1 = (len G)+1 by A209,XXREAL_0:1;
then cell(G,len G,j1) meets C by A194,A199,A200,A201,A202,A197,A198
,A208,GOBRD13:22;
hence contradiction by A2,A207,JORDAN8:16;
end;
hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_0:30;
end;
A210: i1 <= len G by A199,MATRIX_0:32;
A211: 1 <= i2 by A201,MATRIX_0:32;
A212: j2-'1 <= width G by A206,NAT_D:44;
hereby
assume
A213: i1+1 = i2 & j1 = j2;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A205,XXREAL_0:1;
then cell(G,i1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A198
,A213,GOBRD13:24;
then cell(G,i1,0) meets C by XREAL_1:232;
hence contradiction by A210,JORDAN8:17;
end;
hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_0:30;
end;
A214: 1 <= j2+1 by NAT_1:12;
hereby
assume
A215: i1 = i2+1 & j1 = j2;
now
assume j2+1 > len G;
then
A216: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A206,XREAL_1:6;
then j2+1 = (len G)+1 by A216,XXREAL_0:1;
then cell(G,i2,len G) meets C by A194,A199,A200,A201,A202,A197,A198
,A215,GOBRD13:26;
hence contradiction by A203,JORDAN8:15;
end;
hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_0:30;
end;
A217: i2-'1 <= len G by A203,NAT_D:44;
hereby
assume
A218: i1 = i2 & j1 = j2+1;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A211,XXREAL_0:1;
then cell(G,1-'1,j2) meets C by A194,A199,A200,A201,A202,A197,A198
,A218,GOBRD13:28;
then cell(G,0,j2) meets C by XREAL_1:232;
hence contradiction by A2,A206,JORDAN8:18;
end;
hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_0:30;
end;
hereby
assume
A219: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 +1 = j2;
now
assume j2+1 > len G;
then
A220: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A206,XREAL_1:6;
then j2+1 = (len G)+1 by A220,XXREAL_0:1;
then cell(G,i1,len G) meets C by A194,A199,A200,A201,A202,A197,A219,
GOBRD13:35;
hence contradiction by A210,JORDAN8:15;
end;
hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_0:30;
end;
hereby
assume
A221: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2;
now
assume i2+1 > len G;
then
A222: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A203,XREAL_1:6;
then i2+1 = (len G)+1 by A222,XXREAL_0:1;
then cell(G,len G,j1-'1) meets C by A194,A199,A200,A201,A202,A197
,A221,GOBRD13:37;
hence contradiction by A2,A207,JORDAN8:16,NAT_D:44;
end;
hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_0:30;
end;
hereby
assume
A223: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A211,XXREAL_0:1;
then cell(G,1-'1,j1) meets C by A194,A199,A200,A201,A202,A197,A223,
GOBRD13:39;
then cell(G,0,j1) meets C by XREAL_1:232;
hence contradiction by A2,A207,JORDAN8:18;
end;
hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_0:30;
end;
hereby
assume
A224: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A205,XXREAL_0:1;
then cell(G,i1-'1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A224
,GOBRD13:41;
then cell(G,i1-'1,0) meets C by XREAL_1:232;
hence contradiction by A210,JORDAN8:17,NAT_D:44;
end;
hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_0:30;
end;
hereby
assume
A225: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+ 1 = j2;
now
assume i2-'1 < 1;
then i2 <= 1 by NAT_1:14,NAT_D:36;
then i2 = 1 by A211,XXREAL_0:1;
then cell(G,1-'1,j2) meets C by A194,A199,A200,A201,A202,A197,A225,
GOBRD13:34;
then cell(G,0,j2) meets C by XREAL_1:232;
hence contradiction by A2,A206,JORDAN8:18;
end;
hence [i2-'1,j2] in Indices G by A205,A206,A217,MATRIX_0:30;
end;
hereby
assume
A226: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 = j2;
now
assume j2+1 > len G;
then
A227: (len G)+1 <= j2+1 by NAT_1:13;
j2+1 <= (len G)+1 by A2,A206,XREAL_1:6;
then j2+1 = (len G)+1 by A227,XXREAL_0:1;
then cell(G,i2,len G) meets C by A194,A199,A200,A201,A202,A197,A226,
GOBRD13:36;
hence contradiction by A203,JORDAN8:15;
end;
hence [i2,j2+1] in Indices G by A2,A211,A203,A214,MATRIX_0:30;
end;
hereby
assume
A228: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 = j2;
now
assume j2-'1 < 1;
then j2 <= 1 by NAT_1:14,NAT_D:36;
then j2 = 1 by A205,XXREAL_0:1;
then cell(G,i2-'1,1-'1) meets C by A194,A199,A200,A201,A202,A197,A228
,GOBRD13:38;
then cell(G,i2-'1,0) meets C by XREAL_1:232;
hence contradiction by A203,JORDAN8:17,NAT_D:44;
end;
hence [i2,j2-'1] in Indices G by A211,A203,A212,MATRIX_0:30;
end;
hereby
assume
A229: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2+1;
now
assume i2+1 > len G;
then
A230: (len G)+1 <= i2+1 by NAT_1:13;
i2+1 <= (len G)+1 by A203,XREAL_1:6;
then i2+1 = (len G)+1 by A230,XXREAL_0:1;
then cell(G,len G,j2-'1) meets C by A194,A199,A200,A201,A202,A197
,A229,GOBRD13:40;
hence contradiction by A2,A206,JORDAN8:16,NAT_D:44;
end;
hence [i2+1,j2] in Indices G by A205,A206,A204,MATRIX_0:30;
end;
end;
defpred P[Nat] means F.$1 is_sequence_on G & for m st 1 <= m &
m+1 <= len(F.$1) holds left_cell(F.$1,m,G) misses C & right_cell(F.$1,m,G)
meets C;
A231: len G = 2|^n+3 by JORDAN8:def 1;
A232: for k st P[k] holds P[k+1]
proof
A233: 2|^n > 0 by NEWTON:83;
A234: 1 <= len G by A231,NAT_1:12;
let k such that
A235: F.k is_sequence_on G and
A236: for m st 1 <= m & m+1 <= len(F.k) holds left_cell(F.k,m,G)
misses C & right_cell(F.k,m,G) meets C;
A237: len(F.k) = k by A192;
A238: len(F.(k+1)) = k+1 by A192;
per cases by NAT_1:25;
suppose
A239: k = 0;
then consider i such that
A240: 1 <= i and
A241: i+1 <= len G and
N-min C in cell(G,i,width G-'1) and
N-min C <> G*(i,width G-'1) and
A242: F.(k+1) = <*G*(i,width G)*> by A171;
i < len G by A241,NAT_1:13;
then
A243: [i,len G] in Indices G by A2,A234,A240,MATRIX_0:30;
A244: now
let l;
assume l in dom(F.(k+1));
then 1 <= l & l <= 1 by A238,A239,FINSEQ_3:25;
then l = 1 by XXREAL_0:1;
hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G* (i,j) by A2
,A242,A243,FINSEQ_4:16;
end;
now
let l;
assume that
A245: l in dom(F.(k+1)) and
A246: l+1 in dom(F.(k+1));
1 <= l & l <= 1 by A238,A239,A245,FINSEQ_3:25;
then l = 1 by XXREAL_0:1;
hence for i1,j1,i2,j2 st [i1,j1] in Indices G & [i2,j2] in Indices G
& (F.(k+1))/.l = G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2)
holds |.i1-i2.|+|.j1-j2.| = 1 by A238,A239,A246,FINSEQ_3:25;
end;
hence F.(k+1) is_sequence_on G by A244,GOBOARD1:def 9;
let m;
assume that
A247: 1 <= m and
A248: m+1 <= len(F.(k+1));
1 <= m+1 by NAT_1:12;
then m+1 = 0+1 by A238,A239,A248,XXREAL_0:1;
hence thesis by A247;
end;
suppose
A249: k = 1;
then consider i such that
A250: 1 <= i and
A251: i+1 <= len G and
A252: N-min C in cell(G,i,width G-'1) and
N-min C <> G*(i,width G-'1) and
A253: F.(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A171;
A254: i < len G by A251,NAT_1:13;
then
A255: [i,len G] in Indices G by A2,A234,A250,MATRIX_0:30;
1 <= i+1 by A250,NAT_1:13;
then
A256: [i+1,len G] in Indices G by A2,A234,A251,MATRIX_0:30;
A257: (F.(k+1))/.1 = G*(i,width G) & (F.(k+1))/.2 = G*(i+1,width G) by A253,
FINSEQ_4:17;
A258: now
let l;
assume that
A259: l in dom(F.(k+1)) and
A260: l+1 in dom(F.(k+1));
l <= 2 by A238,A249,A259,FINSEQ_3:25;
then
A261: l = 0 or ... or l = 2;
let i1,j1,i2,j2 such that
A262: [i1,j1] in Indices G & [i2,j2] in Indices G & (F.(k+1))/.l
= G*(i1,j1) & (F.(k+1))/.(l+1) = G*(i2,j2);
j1 = len G & j2 = len G by A2,A238,A249,A257,A255,A256,A259,A260,A261
,A262,FINSEQ_3:25,GOBOARD1:5;
then
A263: |.j1-j2.| = 0 by ABSVALUE:def 1;
i1 = i & i2 = i+1 by A2,A238,A249,A257,A255,A256,A259,A260,A261,A262,
FINSEQ_3:25,GOBOARD1:5;
then |.i2-i1.| = 1 by ABSVALUE:def 1;
hence |.i1-i2.|+|.j1-j2.| = 1 by A263,UNIFORM1:11;
end;
now
let l;
assume
A264: l in dom(F.(k+1));
then l <= 2 by A238,A249,FINSEQ_3:25;
then l = 0 or ... or l = 2;
hence ex i,j st [i,j] in Indices G & (F.(k+1))/.l = G*(i,j) by A2
,A257,A255,A256,A264,FINSEQ_3:25;
end;
hence
A265: F.(k+1) is_sequence_on G by A258,GOBOARD1:def 9;
A266: i < i+1 & i+1 < (i+1)+1 by NAT_1:13;
let m;
assume that
A267: 1 <= m and
A268: m+1 <= len(F.(k+1));
1+1 <= m+1 by A267,XREAL_1:6;
then
A269: m+1 = 1+1 by A238,A249,A268,XXREAL_0:1;
then
A270: left_cell(F.(k+1),m,G) = cell(G,i,len G) by A2,A257,A255,A256,A265,A268
,A266,GOBRD13:def 3;
now
N > S by JORDAN8:9;
then N-S > S-S by XREAL_1:9;
then (N-S)/(2|^n) > 0 by A233,XREAL_1:139;
then
A271: N+0 < N+(N-S)/(2|^n) by XREAL_1:6;
[1,len G] in Indices G by A2,A234,MATRIX_0:30;
then G* (1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2|^n))*((len
G)-2)]| by JORDAN8:def 1;
then
A272: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by EUCLID:52;
A273: cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)`1
& G*(1,len G)`2 <= s } by A2,A250,A254,GOBRD11:31;
assume left_cell(F.(k+1),m,G) meets C;
then consider p being object such that
A274: p in cell(G,i,len G) and
A275: p in C by A270,XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A274;
reconsider p as Element of TOP-REAL 2;
A276: p`2 <= N by A275,PSCOMP_1:24;
consider r,s such that
A277: p = |[r,s]| and
G*(i,1)`1 <= r and
r <= G*(i+1,1)`1 and
A278: G*(1,len G)`2 <= s by A274,A273;
((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2|^
n))*1 by A231
.= (N-S)+(N-S)/(2|^n) by A233,XCMPLX_1:87;
then N < s by A278,A272,A271,XXREAL_0:2;
hence contradiction by A277,A276,EUCLID:52;
end;
hence left_cell(F.(k+1),m,G) misses C;
N-min C in C & N-min C in right_cell(F.(k+1),m,G) by A2,A252,A257,A255
,A256,A265,A268,A269,A266,GOBRD13:def 2,SPRECT_1:11;
hence thesis by XBOOLE_0:3;
end;
suppose
A279: k > 1;
then
A280: len(F.k) in dom(F.k) by A237,FINSEQ_3:25;
A281: (len(F.k)) -'1 +1 = len(F.k) by A237,A279,XREAL_1:235;
A282: 1 <= (len(F.k))-'1 by A237,A279,NAT_D:49;
then consider i1,j1,i2,j2 being Nat such that
A283: [i1,j1] in Indices G and
A284: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and
A285: [i2,j2] in Indices G and
A286: (F.k)/.len(F.k) = G*(i2,j2) and
i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
or i1 = i2 & j1 = j2+1 by A235,A281,JORDAN8:3;
A287: i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G by A193,A235,A236
,A279,A283,A284,A285,A286;
A288: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 =
j2 implies [i2,j2-'1] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A289: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 =
j2 implies [i2,j2+1] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A290: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 =
j2 implies [i2-'1,j2] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A291: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 =
j2+1 implies [i2,j2-'1] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286
;
A292: i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G by A193,A235,A236
,A279,A283,A284,A285,A286;
A293: i1 +1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G by A193,A235,A236
,A279,A283,A284,A285,A286;
A294: 1 <= j2 by A285,MATRIX_0:32;
A295: right_cell(F.k,(len(F.k))-'1,G) meets C by A236,A282,A281;
A296: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2+1 & j1 =
j2 implies [i2-'1,j2] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A297: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1+1 = i2 & j1 =
j2 implies [i2+1,j2] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A298: front_right_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1+1 =
j2 implies [i2,j2+1] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
A299: i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G by A193,A235,A236
,A279,A283,A284,A285,A286;
A300: front_left_cell(F.k,(len(F.k))-'1,G) meets C & i1 = i2 & j1 = j2
+1 implies [i2+1,j2] in Indices G by A193,A235,A236,A279,A283,A284,A285,A286;
(len(F.k))-'1 <= len(F.k) by NAT_D:35;
then
A301: (len(F.k))-'1 in dom(F.k) by A282,FINSEQ_3:25;
A302: 1 <= i2 by A285,MATRIX_0:32;
thus
A303: F.(k+1) is_sequence_on G
proof
per cases;
suppose
front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) misses C;
then consider i,j such that
A304: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and
A305: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
thus thesis
proof
set f = (F.k)^<*G*(i,j)*>;
A306: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67;
A307: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A284
,A286,A301,A280,FINSEQ_4:68;
per cases by A281,A283,A285,A304,A307;
suppose that
A308: i1 = i2 & j1+1 = j2 and
A309: f/.(len(F.k)+1) = G*(i2+1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A310: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G*(i29,j29);
then j2 = j19 & j2 = j29 by A285,A286,A287,A308,GOBOARD1:5;
then
A311: |.j29-j19.| = 0 by ABSVALUE:def 1;
i2 = i19 & i2+1 = i29 by A285,A286,A287,A308,A310,GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A311,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A287,A305,A306,A308,A309,
CARD_1:27,JORDAN8:6;
end;
suppose that
A312: i1+1 = i2 & j1 = j2 and
A313: f/.(len(F.k)+1) = G*(i2,j2-'1);
now
let i19,j19,i29,j29 be Nat;
assume
A314: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G* (i29,j29);
then j2 = j19 & j2-'1 = j29 by A285,A286,A293,A312,GOBOARD1:5
;
then j19-j29 = j2-(j2-1) by A294,XREAL_1:233;
then
A315: |.j19-j29.| = 1 by ABSVALUE:def 1;
i2 = i19 & i2 = i29 by A285,A286,A293,A312,A314,GOBOARD1:5;
then |.i29-i19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A315,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A293,A305,A306,A312,A313,
CARD_1:27,JORDAN8:6;
end;
suppose that
A316: i1 = i2+1 & j1 = j2 and
A317: f/.(len(F.k)+1) = G*(i2,j2+1);
now
let i19,j19,i29,j29 be Nat;
assume
A318: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G*(i29,j29);
then i2 = i19 & i2 = i29 by A285,A286,A292,A316,GOBOARD1:5;
then
A319: |.i29-i19.| = 0 by ABSVALUE:def 1;
j2 = j19 & j2+1 = j29 by A285,A286,A292,A316,A318,GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A319,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A292,A305,A306,A316,A317,
CARD_1:27,JORDAN8:6;
end;
suppose that
A320: i1 = i2 & j1 = j2+1 and
A321: f/.(len(F.k)+1) = G*(i2-'1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A322: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G* (i29,j29);
then i2 = i19 & i2-'1 = i29 by A285,A286,A299,A320,GOBOARD1:5
;
then i19-i29 = i2-(i2-1) by A302,XREAL_1:233;
then
A323: |.i19-i29.| = 1 by ABSVALUE:def 1;
j2 = j19 & j2 = j29 by A285,A286,A299,A320,A322,GOBOARD1:5;
then |.j29-j19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A323,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A299,A305,A306,A320,A321,
CARD_1:27,JORDAN8:6;
end;
end;
end;
suppose
A324: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C;
then consider i,j such that
A325: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and
A326: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
thus thesis
proof
set f = (F.k)^<*G*(i,j)*>;
A327: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67;
A328: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A284
,A286,A301,A280,FINSEQ_4:68;
per cases by A281,A283,A285,A325,A328;
suppose that
A329: i1 = i2 & j1+1 = j2 and
A330: f/.(len(F.k)+1) = G*(i2,j2+1);
now
let i19,j19,i29,j29 be Nat;
assume
A331: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G*(i29,j29);
then i2 = i19 & i2 = i29 by A285,A286,A298,A324,A329,
GOBOARD1:5;
then
A332: |.i29-i19.| = 0 by ABSVALUE:def 1;
j2 = j19 & j2+1 = j29 by A285,A286,A298,A324,A329,A331,
GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A332,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A298,A324,A326,A327,A329,A330,
CARD_1:27,JORDAN8:6;
end;
suppose that
A333: i1+1 = i2 & j1 = j2 and
A334: f/.(len(F.k)+1) = G*(i2+1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A335: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G* (i29,j29);
then j2 = j19 & j2 = j29 by A285,A286,A297,A324,A333,
GOBOARD1:5;
then
A336: |.j29-j19.| = 0 by ABSVALUE:def 1;
i2 = i19 & i2+1 = i29 by A285,A286,A297,A324,A333,A335,
GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A336,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A297,A324,A326,A327,A333,A334,
CARD_1:27,JORDAN8:6;
end;
suppose that
A337: i1 = i2+1 & j1 = j2 and
A338: f/.(len(F.k)+1) = G*(i2-'1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A339: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G*(i29,j29);
then i2 = i19 & i2-'1 = i29 by A285,A286,A296,A324,A337,
GOBOARD1:5;
then i19-i29 = i2-(i2-1) by A302,XREAL_1:233;
then
A340: |.i19-i29.| = 1 by ABSVALUE:def 1;
j2 = j19 & j2 = j29 by A285,A286,A296,A324,A337,A339,
GOBOARD1:5;
then |.j29-j19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A340,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A296,A324,A326,A327,A337,A338,
CARD_1:27,JORDAN8:6;
end;
suppose that
A341: i1 = i2 & j1 = j2+1 and
A342: f/.(len(F.k)+1) = G*(i2,j2-'1);
now
let i19,j19,i29,j29 be Nat;
assume
A343: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G* (i29,j29);
then j2 = j19 & j2-'1 = j29 by A285,A286,A291,A324,A341,
GOBOARD1:5;
then j19-j29 = j2-(j2-1) by A294,XREAL_1:233;
then
A344: |.j19-j29.| = 1 by ABSVALUE:def 1;
i2 = i19 & i2 = i29 by A285,A286,A291,A324,A341,A343,
GOBOARD1:5;
then |.i29-i19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A344,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A291,A324,A326,A327,A341,A342,
CARD_1:27,JORDAN8:6;
end;
end;
end;
suppose
A345: front_left_cell(F.k,(len(F.k))-'1,G) meets C;
then consider i,j such that
A346: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and
A347: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
thus thesis
proof
set f = (F.k)^<*G*(i,j)*>;
A348: f/.(len(F.k)+1) = G*(i,j) by FINSEQ_4:67;
A349: f/.(len(F.k)-'1) = G*(i1,j1) & f/.len(F.k) = G*(i2,j2) by A284
,A286,A301,A280,FINSEQ_4:68;
per cases by A281,A283,A285,A346,A349;
suppose that
A350: i1 = i2 & j1+1 = j2 and
A351: f/.(len(F.k)+1) = G*(i2-'1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A352: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2-'1,j2) = G*(i29,j29);
then i2 = i19 & i2-'1 = i29 by A285,A286,A290,A345,A350,
GOBOARD1:5;
then i19-i29 = i2-(i2-1) by A302,XREAL_1:233;
then
A353: |.i19-i29.| = 1 by ABSVALUE:def 1;
j2 = j19 & j2 = j29 by A285,A286,A290,A345,A350,A352,
GOBOARD1:5;
then |.j29-j19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A353,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A290,A345,A347,A348,A350,A351,
CARD_1:27,JORDAN8:6;
end;
suppose that
A354: i1+1 = i2 & j1 = j2 and
A355: f/.(len(F.k)+1) = G*(i2,j2+1);
now
let i19,j19,i29,j29 be Nat;
assume
A356: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2+1) = G* (i29,j29);
then i2 = i19 & i2 = i29 by A285,A286,A289,A345,A354,
GOBOARD1:5;
then
A357: |.i29-i19.| = 0 by ABSVALUE:def 1;
j2 = j19 & j2+1 = j29 by A285,A286,A289,A345,A354,A356,
GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A357,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A289,A345,A347,A348,A354,A355,
CARD_1:27,JORDAN8:6;
end;
suppose that
A358: i1 = i2+1 & j1 = j2 and
A359: f/.(len(F.k)+1) = G*(i2,j2-'1);
now
let i19,j19,i29,j29 be Nat;
assume
A360: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2,j2-'1) = G*(i29,j29);
then j2 = j19 & j2-'1 = j29 by A285,A286,A288,A345,A358,
GOBOARD1:5;
then j19-j29 = j2-(j2-1) by A294,XREAL_1:233;
then
A361: |.j19-j29.| = 1 by ABSVALUE:def 1;
i2 = i19 & i2 = i29 by A285,A286,A288,A345,A358,A360,
GOBOARD1:5;
then |.i29-i19.| = 0 by ABSVALUE:def 1;
hence |.i29-i19.|+|.j29-j19.| = 1 by A361,UNIFORM1:11;
end;
hence thesis by A235,A237,A279,A288,A345,A347,A348,A358,A359,
CARD_1:27,JORDAN8:6;
end;
suppose that
A362: i1 = i2 & j1 = j2+1 and
A363: f/.(len(F.k)+1) = G*(i2+1,j2);
now
let i19,j19,i29,j29 be Nat;
assume
A364: [i19,j19] in Indices G & [i29,j29] in Indices G &
(F.k)/.len(F.k) = G* (i19,j19) & G*(i2+1,j2) = G* (i29,j29);
then j2 = j19 & j2 = j29 by A285,A286,A300,A345,A362,
GOBOARD1:5;
then
A365: |.j29-j19.| = 0 by ABSVALUE:def 1;
i2 = i19 & i2+1 = i29 by A285,A286,A300,A345,A362,A364,
GOBOARD1:5;
hence |.i29-i19.|+|.j29-j19.| = 1 by A365,ABSVALUE:def 1;
end;
hence thesis by A235,A237,A279,A300,A345,A347,A348,A362,A363,
CARD_1:27,JORDAN8:6;
end;
end;
end;
end;
let m such that
A366: 1 <= m and
A367: m+1 <= len(F.(k+1));
A368: left_cell(F.k,(len(F.k))-'1,G) misses C by A236,A282,A281;
now
per cases;
suppose
A369: m+1 = len(F.(k+1));
A370: (j2-'1)+1 = j2 by A294,XREAL_1:235;
A371: (i2-'1)+1 = i2 by A302,XREAL_1:235;
thus thesis
proof
per cases;
suppose
A372: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) misses C;
then
A373: ex i,j st (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G &
F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
then
A374: (F.(k+1))/.len(F.k) = G* (i2,j2) by A286,A280,FINSEQ_4:68;
A375: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A284,A301,A373,
FINSEQ_4:68;
now
per cases by A281,A283,A285,A373,A375,A374;
suppose that
A376: i1 = i2 & j1+1 = j2 and
A377: (F.(k+1))/.((len(F.k))+1) = G*(i2+1,j2);
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2
) by A235,A282,A281,A283,A284,A285,A286,A376,GOBRD13:35;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A287,A303,A366,A369,A372,A374,A376,A377,GOBRD13:23;
j2-'1 = j1 & cell(G,i1,j1) meets C by A235,A282,A281,A283
,A284,A285,A286,A295,A376,GOBRD13:22,NAT_D:34;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A287,A303,A366,A369,A374,A376,A377,GOBRD13:24;
end;
suppose that
A378: i1+1 = i2 & j1 = j2 and
A379: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2-'1);
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2
-'1) by A235,A282,A281,A283,A284,A285,A286,A378,GOBRD13:37;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A293,A303,A366,A369,A370,A372,A374,A378,A379,GOBRD13:27;
i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A235,A282,A281
,A283,A284,A285,A286,A295,A378,GOBRD13:24,NAT_D:34;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A293,A303,A366,A369,A370,A374,A378,A379,GOBRD13:28;
end;
suppose that
A380: i1 = i2+1 & j1 = j2 and
A381: (F.(k+1))/.((len(F.k))+1) = G*(i2,j2+1);
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1
,j2) by A235,A282,A281,A283,A284,A285,A286,A380,GOBRD13:39;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A292,A303,A366,A369,A372,A374,A380,A381,GOBRD13:21;
cell(G,i2,j2) meets C by A235,A282,A281,A283,A284,A285,A286
,A295,A380,GOBRD13:26;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A292,A303,A366,A369,A374,A380,A381,GOBRD13:22;
end;
suppose that
A382: i1 = i2 & j1 = j2+1 and
A383: (F.(k+1))/.((len(F.k))+1) = G*(i2-'1,j2);
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1
,j2-'1 ) by A235,A282,A281,A283,A284,A285,A286,A382,GOBRD13:41;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A299,A303,A366,A369,A371,A372,A374,A382,A383,GOBRD13:25;
cell(G,i2-'1,j2) meets C by A235,A282,A281,A283,A284,A285
,A286,A295,A382,GOBRD13:28;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A299,A303,A366,A369,A371,A374,A382,A383,GOBRD13:26;
end;
end;
hence thesis;
end;
suppose
A384: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C;
then
A385: ex i,j st (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G
& F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
then
A386: (F.(k+1))/.len(F.k) = G*(i2,j2) by A286,A280,FINSEQ_4:68;
A387: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A284,A301,A385,
FINSEQ_4:68;
now
per cases by A281,A283,A285,A385,A387,A386;
suppose that
A388: i1 = i2 & j1+1 = j2 and
A389: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1);
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1,
j2) by A235,A282,A281,A283,A284,A285,A286,A388,GOBRD13:34;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A298,A303,A366,A369,A384,A386,A388,A389,GOBRD13:21;
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i1,j2
) by A235,A282,A281,A283,A284,A285,A286,A388,GOBRD13:35;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A298,A303,A366,A369,A384,A386,A388,A389,GOBRD13:22;
end;
suppose that
A390: i1+1 = i2 & j1 = j2 and
A391: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2);
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2)
by A235,A282,A281,A283,A284,A285,A286,A390,GOBRD13:36;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A297,A303,A366,A369,A384,A386,A390,A391,GOBRD13:23;
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2
-'1) by A235,A282,A281,A283,A284,A285,A286,A390,GOBRD13:37;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A297,A303,A366,A369,A384,A386,A390,A391,GOBRD13:24;
end;
suppose that
A392: i1 = i2+1 & j1 = j2 and
A393: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2);
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,
j2-'1) by A235,A282,A281,A283,A284,A285,A286,A392,GOBRD13:38;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A296,A303,A366,A369,A371,A384,A386,A392,A393,GOBRD13:25;
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1
,j2) by A235,A282,A281,A283,A284,A285,A286,A392,GOBRD13:39;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A296,A303,A366,A369,A371,A384,A386,A392,A393,GOBRD13:26;
end;
suppose that
A394: i1 = i2 & j1 = j2+1 and
A395: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1);
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2
-'1) by A235,A282,A281,A283,A284,A285,A286,A394,GOBRD13:40;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A291,A303,A366,A369,A370,A384,A386,A394,A395,GOBRD13:27;
front_right_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1
,j2-'1 ) by A235,A282,A281,A283,A284,A285,A286,A394,GOBRD13:41;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A291,A303,A366,A369,A370,A384,A386,A394,A395,GOBRD13:28;
end;
end;
hence thesis;
end;
suppose
A396: front_left_cell(F.k,(len(F.k))-'1,G) meets C;
then
A397: ex i,j st (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G &
F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
then
A398: (F.(k+1))/.len(F.k) = G* (i2,j2) by A286,A280,FINSEQ_4:68;
A399: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A284,A301,A397,
FINSEQ_4:68;
now
per cases by A281,A283,A285,A397,A399,A398;
suppose that
A400: i1 = i2 & j1+1 = j2 and
A401: (F.(k+1))/.(len(F.k)+1) = G*(i2-'1,j2);
j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A235,A282,A281
,A283,A284,A285,A286,A368,A400,GOBRD13:21,NAT_D:34;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A290,A303,A366,A369,A371,A396,A398,A400,A401,GOBRD13:25;
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i1-'1,
j2) by A235,A282,A281,A283,A284,A285,A286,A400,GOBRD13:34;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A290,A303,A366,A369,A371,A396,A398,A400,A401,GOBRD13:26;
end;
suppose that
A402: i1+1 = i2 & j1 = j2 and
A403: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2+1);
i2-'1 = i1 & cell(G,i1,j1) misses C by A235,A282,A281,A283
,A284,A285,A286,A368,A402,GOBRD13:23,NAT_D:34;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A289,A303,A366,A369,A396,A398,A402,A403,GOBRD13:21;
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2)
by A235,A282,A281,A283,A284,A285,A286,A402,GOBRD13:36;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A289,A303,A366,A369,A396,A398,A402,A403,GOBRD13:22;
end;
suppose that
A404: i1 = i2+1 & j1 = j2 and
A405: (F.(k+1))/.(len(F.k)+1) = G*(i2,j2-'1);
cell(G,i2,j2-'1) misses C by A235,A282,A281,A283,A284,A285
,A286,A368,A404,GOBRD13:25;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A288,A303,A366,A369,A370,A396,A398,A404,A405,GOBRD13:27;
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2-'1,
j2-'1) by A235,A282,A281,A283,A284,A285,A286,A404,GOBRD13:38;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A288,A303,A366,A369,A370,A396,A398,A404,A405,GOBRD13:28;
end;
suppose that
A406: i1 = i2 & j1 = j2+1 and
A407: (F.(k+1))/.(len(F.k)+1) = G*(i2+1,j2);
cell(G,i2,j2) misses C by A235,A282,A281,A283,A284,A285
,A286,A368,A406,GOBRD13:27;
hence left_cell(F.(k+1),m,G) misses C by A237,A238,A285
,A300,A303,A366,A369,A396,A398,A406,A407,GOBRD13:23;
front_left_cell(F.k,(len(F.k))-'1,G) = cell(G,i2,j2
-'1) by A235,A282,A281,A283,A284,A285,A286,A406,GOBRD13:40;
hence right_cell(F.(k+1),m,G) meets C by A237,A238,A285
,A300,A303,A366,A369,A396,A398,A406,A407,GOBRD13:24;
end;
end;
hence thesis;
end;
end;
end;
suppose
m+1 <> len(F.(k+1));
then m+1 < len(F.(k+1)) by A367,XXREAL_0:1;
then
A408: m+1 <= len(F.k)by A237,A238,NAT_1:13;
then consider i1,j1,i2,j2 being Nat such that
A409: [i1,j1] in Indices G and
A410: (F.k)/.m = G*(i1,j1) and
A411: [i2,j2] in Indices G and
A412: (F.k)/.(m+1) = G*(i2,j2) and
A413: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1
& j1 = j2 or i1 = i2 & j1 = j2+1 by A235,A366,JORDAN8:3;
A414: left_cell(F.k,m,G) misses C & right_cell(F.k,m,G) meets C by A236
,A366,A408;
A415: now
per cases;
suppose
front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) misses C;
then consider i,j such that
(F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and
A416: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
take i,j;
thus F.(k+1) = (F.k)^<*G*(i,j)*> by A416;
end;
suppose
front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C;
then consider i,j such that
(F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and
A417: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
take i,j;
thus F.(k+1) = (F.k)^<*G*(i,j)*> by A417;
end;
suppose
front_left_cell(F.k,(len(F.k))-'1,G) meets C;
then consider i,j such that
(F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and
A418: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A235,A237,A279,A295;
take i,j;
thus F.(k+1) = (F.k)^<*G*(i,j)*> by A418;
end;
end;
1 <= m+1 by NAT_1:12;
then m+1 in dom(F.k) by A408,FINSEQ_3:25;
then
A419: (F.(k+1))/.(m+1) = G*(i2,j2) by A412,A415,FINSEQ_4:68;
m <= len(F.k) by A408,NAT_1:13;
then m in dom(F.k) by A366,FINSEQ_3:25;
then
A420: (F.(k+1))/.m = G*(i1,j1) by A410,A415,FINSEQ_4:68;
now
per cases by A413;
suppose
A421: i1 = i2 & j1+1 = j2;
then left_cell(F.k,m,G) = cell(G,i1-'1,j1) & right_cell(F.k,m
,G) = cell(G,i1,j1) by A235,A366,A408,A409,A410,A411,A412,GOBRD13:21,22;
hence thesis by A303,A366,A367,A409,A411,A414,A420,A419,A421,
GOBRD13:21,22;
end;
suppose
A422: i1+1 = i2 & j1 = j2;
then left_cell(F.k,m,G) = cell(G,i1,j1) & right_cell(F.k,m,G)
= cell(G,i1,j1-'1) by A235,A366,A408,A409,A410,A411,A412,GOBRD13:23,24;
hence thesis by A303,A366,A367,A409,A411,A414,A420,A419,A422,
GOBRD13:23,24;
end;
suppose
A423: i1 = i2+1 & j1 = j2;
then left_cell(F.k,m,G) = cell(G,i2,j2-'1) & right_cell(F.k,m
,G) = cell(G,i2,j2) by A235,A366,A408,A409,A410,A411,A412,GOBRD13:25,26;
hence thesis by A303,A366,A367,A409,A411,A414,A420,A419,A423,
GOBRD13:25,26;
end;
suppose
A424: i1 = i2 & j1 = j2+1;
then left_cell(F.k,m,G) = cell(G,i2,j2) & right_cell(F.k,m,G)
= cell(G,i1-'1,j2) by A235,A366,A408,A409,A410,A411,A412,GOBRD13:27,28;
hence thesis by A303,A366,A367,A409,A411,A414,A420,A419,A424,
GOBRD13:27,28;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
defpred Q[Nat] means ex w being Nat st w = $1 & $1 >= 1 & ex m
st m in dom(F.w) & m <> len(F.w) & (F.w)/.m = (F.w)/.len(F.w);
A425: P[0]
proof
( for n st n in dom(F.0) ex i,j st [i,j] in Indices G & (F.0)/.n =
G*(i,j))& for n st n in dom(F.0) & n+1 in dom(F.0) holds for m,k,i,j st [m,k]
in Indices G & [i,j] in Indices G & (F.0)/.n = G*(m,k) & (F.0)/.(n+1) = G*(i,j)
holds |.m-i.|+|.k-j.| = 1 by A170;
hence F.0 is_sequence_on G by GOBOARD1:def 9;
let m;
assume that
1 <= m and
A426: m+1 <= len(F.0);
thus thesis by A170,A426,CARD_1:27;
end;
A427: for k holds P[k] from NAT_1:sch 2(A425,A232);
A428: for k,i1,i2,j1,j2 st k > 1 & [i1,j1] in Indices G & (F.k)/.((len(F.k
)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) holds
(front_left_cell(F.k,(len(F.k))-'1,G) misses C & front_right_cell(F.k,(len(F.k)
)-'1,G) misses C implies F.(k+1) turns_right (len(F.k))-'1,G & (i1 = i2 & j1+1
= j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1+1
= i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)
*>) & (i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G
*(i2,j2+1)*>)& (i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) =
(F.k)^<*G*(i2-'1,j2)*>)) & (front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C implies F.(k+1) goes_straight (
len(F.k))-'1,G & (i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1)
= (F.k)^<*G*(i2,j2+1)*>) & (i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G
& F.(k+1) = (F.k)^<*G*(i2+1,j2)*>) & (i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in
Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)*>)& (i1 = i2 & j1 = j2+1 implies [i2
,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)*>)) & (front_left_cell(F.
k,(len(F.k))-'1,G) meets C implies F.(k+1) turns_left (len(F.k))-'1,G & (i1 =
i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)
*>) & (i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G
*(i2,j2+1)*>) & (i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1)
= (F.k)^<*G*(i2,j2-'1)*>)& (i1 = i2 & j1 = j2+1 implies [i2+1,j2] in Indices G
& F.(k+1) = (F.k)^<*G*(i2+1,j2)*>))
proof
let k,i1,i2,j1,j2 such that
A429: k > 1 and
A430: [i1,j1] in Indices G and
A431: (F.k)/.((len(F.k)) -'1) = G*(i1,j1) and
A432: [i2,j2] in Indices G and
A433: (F.k)/.len(F.k) = G*(i2,j2);
A434: len(F.k) = k by A192;
then
A435: (len(F.k)) -'1 +1 = len(F.k) by A429,XREAL_1:235;
A436: 1 <= (len(F.k))-'1 by A429,A434,NAT_D:49;
then
A437: right_cell(F.k,(len(F.k))-'1,G) meets C by A427,A435;
(len(F.k))-'1 <= len(F.k) by NAT_D:35;
then
A438: (len(F.k))-'1 in dom(F.k) by A436,FINSEQ_3:25;
A439: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A440: F.k is_sequence_on G by A427;
A441: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
A442: len(F.k) in dom(F.k) by A429,A434,FINSEQ_3:25;
hereby
assume front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F. k))-'1,G) misses C;
then consider i,j such that
A443: (F.k)^<*G*(i,j)*> turns_right (len(F.k))-'1,G and
A444: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A429,A440,A434,A437;
thus F.(k+1) turns_right (len(F.k))-'1,G by A443,A444;
A445: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A444,FINSEQ_4:67;
A446: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G*
(i2,j2) by A431,A433,A438,A442,A444,FINSEQ_4:68;
hence
i1 = i2 & j1+1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F
.k)^<*G*(i2+1,j2)*> by A430,A432,A435,A439,A443,A444,A445;
thus i1+1 = i2 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (
F.k)^<*G*(i2,j2-'1)*> by A430,A432,A435,A441,A443,A444,A446,A445;
thus i1 = i2+1 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F
.k)^<*G*(i2,j2+1)*> by A430,A432,A435,A441,A443,A444,A446,A445;
thus i1 = i2 & j1 = j2+1 implies [i2-'1,j2] in Indices G & F.(k+1) = (
F.k)^<*G*(i2-'1,j2)*> by A430,A432,A435,A439,A443,A444,A446,A445;
end;
hereby
assume front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F. k))-'1,G) meets C;
then consider i,j such that
A447: (F.k)^<*G*(i,j)*> goes_straight (len(F.k))-'1,G and
A448: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A429,A440,A434,A437;
thus F.(k+1) goes_straight (len(F.k))-'1,G by A447,A448;
A449: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A448,FINSEQ_4:67;
A450: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) & (F.(k+1))/.len(F.k) = G*
(i2,j2) by A431,A433,A438,A442,A448,FINSEQ_4:68;
hence
i1 = i2 & j1+1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F
.k)^<*G*(i2,j2+1)*> by A430,A432,A435,A439,A447,A448,A449;
thus i1+1 = i2 & j1 = j2 implies [i2+1,j2] in Indices G & F.(k+1) = (F
.k)^<*G*(i2+1,j2)*> by A430,A432,A435,A441,A447,A448,A450,A449;
thus i1 = i2+1 & j1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (
F.k)^<*G*(i2-'1,j2)*> by A430,A432,A435,A441,A447,A448,A450,A449;
thus i1 = i2 & j1 = j2+1 implies [i2,j2-'1] in Indices G & F.(k+1) = (
F.k)^<*G*(i2,j2-'1)*> by A430,A432,A435,A439,A447,A448,A450,A449;
end;
assume front_left_cell(F.k,(len(F.k))-'1,G) meets C;
then consider i,j such that
A451: (F.k)^<*G*(i,j)*> turns_left (len(F.k))-'1,G and
A452: F.(k+1) = (F.k)^<*G*(i,j)*> by A171,A429,A440,A434,A437;
A453: (F.(k+1))/.len(F.k) = G* (i2,j2) by A433,A442,A452,FINSEQ_4:68;
thus F.(k+1) turns_left (len(F.k))-'1,G by A451,A452;
A454: (F.(k+1))/.(len(F.k)+1) = G*(i,j) by A452,FINSEQ_4:67;
A455: (F.(k+1))/.(len(F.k)-'1) = G*(i1,j1) by A431,A438,A452,FINSEQ_4:68;
hence
i1 = i2 & j1+1 = j2 implies [i2-'1,j2] in Indices G & F.(k+1) = (F.
k)^<*G*(i2-'1,j2)*> by A430,A432,A435,A439,A451,A452,A453,A454;
thus i1+1 = i2 & j1 = j2 implies [i2,j2+1] in Indices G & F.(k+1) = (F.k
)^<*G*(i2,j2+1)*> by A430,A432,A435,A441,A451,A452,A455,A453,A454;
thus i1 = i2+1 & j1 = j2 implies [i2,j2-'1] in Indices G & F.(k+1) = (F.
k)^<*G*(i2,j2-'1)*> by A430,A432,A435,A441,A451,A452,A455,A453,A454;
thus thesis by A430,A432,A435,A439,A451,A452,A455,A453,A454;
end;
A456: for k st k > 1 holds (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C
& front_right_cell(F.k,k-'1,Gauge(C,n)) misses C implies F.(k+1) turns_right k
-'1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) misses C &
front_right_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1) goes_straight k-'
1,Gauge(C,n)) & (front_left_cell(F.k,k-'1,Gauge(C,n)) meets C implies F.(k+1)
turns_left k-'1,Gauge(C,n))
proof
let k such that
A457: k > 1;
A458: F.k is_sequence_on G by A427;
A459: len(F.k) = k by A192;
then 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A457,NAT_D:49
,XREAL_1:235;
then ex i1,j1,i2,j2 being Nat st [i1,j1] in Indices G & (F.k
)/.((len(F.k)) -'1) = G*(i1,j1) & [i2,j2] in Indices G & (F.k)/.len(F.k) = G*(
i2,j2) & (i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
i1 = i2 & j1 = j2+1) by A458,JORDAN8:3;
hence thesis by A428,A457,A459;
end;
defpred P[Nat] means for m st m <= $1 holds (F.$1)|m = F.m;
A460: P[0]
proof
let m;
assume
A461: m <= 0;
then (F.0)|0 = (F.0)|m;
hence thesis by A170,A461;
end;
A462: for k holds ex i,j st [i,j] in Indices G & F.(k+1) = (F.k)^<*G*(i,j) *>
proof
let k;
A463: F.k is_sequence_on G by A427;
A464: len(F.k) = k by A192;
len G >= 4 by JORDAN8:10;
then
A465: len G = width G & 1 < len G by JORDAN8:def 1,XXREAL_0:2;
per cases by XXREAL_0:1;
suppose
A466: k < 1;
consider i such that
A467: 1 <= i and
A468: i+1 <= len G and
N-min C in cell(G,i,width G-'1) and
N-min C <> G*(i,width G-'1) and
A469: F.(0+1) = <*G*(i,width G)*> by A171;
take i, j = width G;
i < len G by A468,NAT_1:13;
hence [i,j] in Indices G by A465,A467,MATRIX_0:30;
k = 0 by A466,NAT_1:14;
hence thesis by A170,A469,FINSEQ_1:34;
end;
suppose
A470: k = 1;
consider i such that
A471: 1 <= i and
A472: i+1 <= len G and
A473: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1 ) and
A474: F.(0+1) = <*G*(i,width G)*> by A171;
take i+1,j = width G;
1 <= i+1 by A471,NAT_1:13;
hence [i+1,j] in Indices G by A465,A472,MATRIX_0:30;
consider i9 being Nat such that
A475: 1 <= i9 & i9+1 <= len G & N-min C in cell(G,i9,width G-'1) &
N-min C <> G*(i9,width G-'1) and
A476: F.(1+1) = <*G*(i9,width G),G*(i9+1,width G)*> by A171;
i = i9 by A471,A472,A473,A475,Th29;
hence thesis by A470,A474,A476,FINSEQ_1:def 9;
end;
suppose
A477: k > 1;
then 1 <= (len(F.k))-'1 & (len(F.k)) -'1 +1 = len(F.k) by A464,NAT_D:49
,XREAL_1:235;
then consider i1,j1,i2,j2 being Nat such that
A478: [i1,j1] in Indices G & (F.k)/.((len(F.k)) -'1) = G*(i1,j1) &
[i2,j2] in Indices G & (F.k)/.len(F.k) = G*(i2,j2) and
A479: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1
= j2 or i1 = i2 & j1 = j2+1 by A463,JORDAN8:3;
now
per cases;
suppose
A480: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) misses C;
now
per cases by A479;
suppose
i1 = i2 & j1+1 = j2;
then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>
by A428,A477,A478,A480;
hence thesis;
end;
suppose
i1+1 = i2 & j1 = j2;
then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)
*> by A428,A477,A478,A480;
hence thesis;
end;
suppose
i1 = i2+1 & j1 = j2;
then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>
by A428,A477,A478,A480;
hence thesis;
end;
suppose
i1 = i2 & j1 = j2+1;
then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)
*> by A428,A477,A478,A480;
hence thesis;
end;
end;
hence thesis;
end;
suppose
A481: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C;
now
per cases by A479;
suppose
i1 = i2 & j1+1 = j2;
then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>
by A428,A477,A478,A481;
hence thesis;
end;
suppose
i1+1 = i2 & j1 = j2;
then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>
by A428,A477,A478,A481;
hence thesis;
end;
suppose
i1 = i2+1 & j1 = j2;
then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)
*> by A428,A477,A478,A481;
hence thesis;
end;
suppose
i1 = i2 & j1 = j2+1;
then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)
*> by A428,A477,A478,A481;
hence thesis;
end;
end;
hence thesis;
end;
suppose
A482: front_left_cell(F.k,(len(F.k))-'1,G) meets C;
now
per cases by A479;
suppose
i1 = i2 & j1+1 = j2;
then [i2-'1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2-'1,j2)
*> by A428,A477,A478,A482;
hence thesis;
end;
suppose
i1+1 = i2 & j1 = j2;
then [i2,j2+1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2+1)*>
by A428,A477,A478,A482;
hence thesis;
end;
suppose
i1 = i2+1 & j1 = j2;
then [i2,j2-'1] in Indices G & F.(k+1) = (F.k)^<*G*(i2,j2-'1)
*> by A428,A477,A478,A482;
hence thesis;
end;
suppose
i1 = i2 & j1 = j2+1;
then [i2+1,j2] in Indices G & F.(k+1) = (F.k)^<*G*(i2+1,j2)*>
by A428,A477,A478,A482;
hence thesis;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
A483: for k st P[k] holds P[k+1]
proof
let k such that
A484: for m st m <= k holds (F.k)|m = F.m;
let m such that
A485: m <= k+1;
per cases by A485,XXREAL_0:1;
suppose
m < k+1;
then
A486: m <= k by NAT_1:13;
len(F.k) = k & ex i,j st [i,j] in Indices G & F.(k+1) = F.k^<*G*
(i,j)*> by A192,A462;
then (F.(k+1))|m = (F.k)|m by A486,FINSEQ_5:22;
hence thesis by A484,A486;
end;
suppose
A487: m = k+1;
len(F.(k+1)) = k+1 by A192;
hence thesis by A487,FINSEQ_1:58;
end;
end;
A488: for k holds P[k] from NAT_1:sch 2(A460,A483);
defpred P[Nat] means F.$1 is unfolded;
A489: for k st P[k] holds P[k+1]
proof
let k such that
A490: F.k is unfolded;
A491: F.k is_sequence_on G by A427;
per cases;
suppose
k <= 1;
then k+1 <= 1+1 by XREAL_1:6;
then len(F.(k+1)) <= 2 by A192;
hence thesis by SPPOL_2:26;
end;
suppose
A492: k > 1;
set m = k-'1;
A493: m+1 = k by A492,XREAL_1:235;
A494: len(F.k) = k by A192;
A495: 1 <= m by A492,NAT_D:49;
then consider i1,j1,i2,j2 being Nat such that
A496: [i1,j1] in Indices G and
A497: (F.k)/.m = G*(i1,j1) and
A498: [i2,j2] in Indices G and
A499: (F.k)/.len(F.k) = G*(i2,j2) and
A500: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1
= j2 or i1 = i2 & j1 = j2+1 by A491,A493,A494,JORDAN8:3;
A501: LSeg(F.k,m) = LSeg(G*(i1,j1),G*(i2,j2)) by A495,A493,A494,A497,A499,
TOPREAL1:def 3;
A502: 1 <= j2 by A498,MATRIX_0:32;
then
A503: (j2-'1)+1 = j2 by XREAL_1:235;
A504: 1 <= j1 by A496,MATRIX_0:32;
A505: 1 <= i2 by A498,MATRIX_0:32;
then
A506: (i2-'1)+1 = i2 by XREAL_1:235;
A507: i1 <= len G by A496,MATRIX_0:32;
A508: j2 <= width G by A498,MATRIX_0:32;
A509: 1 <= i1 by A496,MATRIX_0:32;
A510: j1 <= width G by A496,MATRIX_0:32;
A511: i2 <= len G by A498,MATRIX_0:32;
now
per cases;
suppose
A512: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) misses C;
now
per cases by A500;
suppose
A513: i1 = i2 & j1+1 = j2;
then [i2+1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A512;
then i2+1 <= len G by MATRIX_0:32;
then
A514: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.
k)/.len(F.k)} by A499,A509,A504,A508,A501,A513,GOBOARD7:15;
F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A512,A513;
hence thesis by A490,A493,A494,A514,SPPOL_2:30;
end;
suppose
A515: i1+1 = i2 & j1 = j2;
then [i2,j2-'1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A512;
then 1 <= j2-'1 by MATRIX_0:32;
then
A516: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.
k)/.len(F.k)} by A499,A509,A510,A511,A503,A501,A515,GOBOARD7:16;
F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A428,A492,A494,A496,A497
,A498,A499,A512,A515;
hence thesis by A490,A493,A494,A516,SPPOL_2:30;
end;
suppose
A517: i1 = i2+1 & j1 = j2;
then [i2,j2+1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A512;
then j2+1 <= width G by MATRIX_0:32;
then
A518: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.
k)/.len(F.k)} by A499,A507,A504,A505,A501,A517,GOBOARD7:17;
F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A428,A492,A494,A496,A497
,A498,A499,A512,A517;
hence thesis by A490,A493,A494,A518,SPPOL_2:30;
end;
suppose
A519: i1 = i2 & j1 = j2+1;
then [i2-'1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A512;
then 1 <= i2-'1 by MATRIX_0:32;
then
A520: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.
k)/.len(F.k)} by A499,A507,A510,A502,A506,A501,A519,GOBOARD7:18;
F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A512,A519;
hence thesis by A490,A493,A494,A520,SPPOL_2:30;
end;
end;
hence thesis;
end;
suppose
A521: front_left_cell(F.k,(len(F.k))-'1,G) misses C &
front_right_cell(F.k,(len(F.k))-'1,G) meets C;
now
per cases by A500;
suppose
A522: i1 = i2 & j1+1 = j2;
then [i2,j2+1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A521;
then
A523: j2+1 <= width G by MATRIX_0:32;
j2+1 = j1+(1+1) by A522;
then
A524: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.
k)/.len(F.k)} by A499,A509,A507,A504,A501,A522,A523,GOBOARD7:13;
F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A428,A492,A494,A496,A497
,A498,A499,A521,A522;
hence thesis by A490,A493,A494,A524,SPPOL_2:30;
end;
suppose
A525: i1+1 = i2 & j1 = j2;
then [i2+1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A521;
then
A526: i2+1 <= len G by MATRIX_0:32;
i2+1 = i1+(1+1) by A525;
then
A527: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.
k)/.len(F.k)} by A499,A509,A504,A510,A501,A525,A526,GOBOARD7:14;
F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A521,A525;
hence thesis by A490,A493,A494,A527,SPPOL_2:30;
end;
suppose
A528: i1 = i2+1 & j1 = j2;
then [i2-'1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A521;
then
A529: 1 <= i2-'1 by MATRIX_0:32;
i2-'1+1+1 = i2-'1+(1+1);
then
A530: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.
k)/.len(F.k)} by A499,A507,A504,A510,A506,A501,A528,A529,GOBOARD7:14;
F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A521,A528;
hence thesis by A490,A493,A494,A530,SPPOL_2:30;
end;
suppose
A531: i1 = i2 & j1 = j2+1;
then [i2,j2-'1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A521;
then
A532: 1 <= j2-'1 by MATRIX_0:32;
j2-'1+1+1 = j2-'1+(1+1);
then
A533: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.
k)/.len(F.k)} by A499,A509,A507,A510,A503,A501,A531,A532,GOBOARD7:13;
F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A428,A492,A494,A496,A497
,A498,A499,A521,A531;
hence thesis by A490,A493,A494,A533,SPPOL_2:30;
end;
end;
hence thesis;
end;
suppose
A534: front_left_cell(F.k,(len(F.k))-'1,G) meets C;
now
per cases by A500;
suppose
A535: i1 = i2 & j1+1 = j2;
then [i2-'1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A534;
then 1 <= i2-'1 by MATRIX_0:32;
then
A536: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2-'1,j2)) = {(F.
k)/.len(F.k)} by A499,A507,A504,A508,A506,A501,A535,GOBOARD7:16;
F.(k+1) = (F.k)^ <*G*(i2-'1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A534,A535;
hence thesis by A490,A493,A494,A536,SPPOL_2:30;
end;
suppose
A537: i1+1 = i2 & j1 = j2;
then [i2,j2+1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A534;
then j2+1 <= width G by MATRIX_0:32;
then
A538: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2,j2+1)) = {(F.
k)/.len(F.k)} by A499,A509,A504,A511,A501,A537,GOBOARD7:18;
F.(k+1) = (F.k)^ <*G*(i2,j2+1)*> by A428,A492,A494,A496,A497
,A498,A499,A534,A537;
hence thesis by A490,A493,A494,A538,SPPOL_2:30;
end;
suppose
A539: i1 = i2+1 & j1 = j2;
then [i2,j2-'1] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A534;
then 1 <= j2-'1 by MATRIX_0:32;
then
A540: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G*(i2,j2-'1)) = {(F.
k)/.len(F.k)} by A499,A507,A510,A505,A503,A501,A539,GOBOARD7:15;
F.(k+1) = (F.k)^ <*G*(i2,j2-'1)*> by A428,A492,A494,A496,A497
,A498,A499,A534,A539;
hence thesis by A490,A493,A494,A540,SPPOL_2:30;
end;
suppose
A541: i1 = i2 & j1 = j2+1;
then [i2+1,j2] in Indices G by A428,A492,A494,A496,A497,A498
,A499,A534;
then i2+1 <= len G by MATRIX_0:32;
then
A542: LSeg(F.k,m) /\ LSeg((F.k)/.len(F.k),G* (i2+1,j2)) = {(F.
k)/.len(F.k)} by A499,A509,A510,A502,A501,A541,GOBOARD7:17;
F.(k+1) = (F.k)^ <*G*(i2+1,j2)*> by A428,A492,A494,A496,A497
,A498,A499,A534,A541;
hence thesis by A490,A493,A494,A542,SPPOL_2:30;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
now
defpred P[Nat] means F.$1 is one-to-one;
assume
A543: for k st k >= 1 holds for m st m in dom(F.k) & m <> len(F.k)
holds (F.k)/.m <> (F.k)/.len(F.k);
A544: for k st P[k] holds P[k+1]
proof
let k;
assume
A545: F.k is one-to-one;
now
let n,m be Element of NAT such that
A546: n in dom(F.(k+1)) and
A547: m in dom(F.(k+1)) and
A548: (F.(k+1))/.n = (F.(k+1))/.m;
A549: n <= len(F.(k+1)) & m <= len(F.(k+1)) by A546,A547,FINSEQ_3:25;
A550: 1 <= m by A547,FINSEQ_3:25;
A551: 1 <= n by A546,FINSEQ_3:25;
A552: ex i,j st [i,j] in Indices G & F.(k+1) = (F.k)^<*G*(i,j) *> by A462;
A553: len(F.k) = k by A192;
A554: len(F.(k+1)) = k+1 by A192;
per cases by A549,A554,NAT_1:8;
suppose
A555: n <= k & m <= k;
then
A556: m in dom(F.k) by A550,A553,FINSEQ_3:25;
then
A557: (F.(k+1))/.m = (F.k)/.m by A552,FINSEQ_4:68;
A558: n in dom(F.k) by A551,A553,A555,FINSEQ_3:25;
then (F.(k+1))/.n = (F.k)/.n by A552,FINSEQ_4:68;
hence n = m by A545,A548,A558,A556,A557,PARTFUN2:10;
end;
suppose
n = k+1 & m <= k;
hence n = m by A543,A547,A548,A554,NAT_1:12;
end;
suppose
n <= k & m = k+1;
hence n = m by A543,A546,A548,A554,NAT_1:12;
end;
suppose
n = k+1 & m = k+1;
hence n = m;
end;
end;
hence thesis by PARTFUN2:9;
end;
A559: P[0] by A170;
A560: for k holds P[k] from NAT_1:sch 2(A559,A544);
A561: for k holds card rng(F.k) = k
proof
let k;
F.k is one-to-one by A560;
hence card rng(F.k) = len(F.k) by FINSEQ_4:62
.= k by A192;
end;
set k = (len G)*(width G)+1;
F.k is_sequence_on G by A427;
then card Values G <= (len G)*(width G) & card rng(F.k) <= card Values
G by GOBRD13:8,MATRIX_0:40,NAT_1:43;
then k > (len G)*(width G) & card rng(F.k) <= (len G)*(width G) by
NAT_1:13,XXREAL_0:2;
hence contradiction by A561;
end;
then
A562: ex k be Nat st Q[k];
consider k be Nat such that
A563: Q[k] and
A564: for l be Nat st Q[l] holds k <= l from NAT_1:sch 5(A562);
reconsider k as Nat;
consider m such that
A565: m in dom(F.k) and
A566: m <> len(F.k) and
A567: (F.k)/.m = (F.k)/.len(F.k) by A563;
A568: 1 <= m by A565,FINSEQ_3:25;
reconsider f = F.k as non empty FinSequence of TOP-REAL 2 by A563;
A569: f is_sequence_on G by A427;
A570: m <= len f by A565,FINSEQ_3:25;
then
A571: m < len f by A566,XXREAL_0:1;
then 1 < len f by A568,XXREAL_0:2;
then
A572: len f >= 1+1 by NAT_1:13;
A573: P[0] by A170,CARD_1:27,SPPOL_2:26;
for k holds P[k] from NAT_1:sch 2(A573,A489);
then reconsider
f as non constant special unfolded non empty FinSequence of
TOP-REAL 2 by A569,A572,JORDAN8:4,5;
A574: m+1 <= len f by A571,NAT_1:13;
set g = f/^(m-'1);
m-'1 <= m by NAT_D:44;
then m-'1 < m+1 by NAT_1:13;
then
A575: m-'1 < len f by A574,XXREAL_0:2;
then
A576: len g = len f - (m-'1) by RFINSEQ:def 1;
then (m-'1)-(m-'1) < len g by A575,XREAL_1:9;
then reconsider g as non empty FinSequence of TOP-REAL 2 by CARD_1:27;
len g in dom g by FINSEQ_5:6;
then
A577: g/.len g = f/.(m-'1+len g) by FINSEQ_5:27
.= f/.len f by A576;
A578: len(F.k) = k by A192;
A579: for i st 1 <= i & i < len g & 1 <= j & j < len g & g/.i = g/.j
holds i = j
proof
let i such that
A580: 1 <= i and
A581: i < len g and
A582: 1 <= j and
A583: j < len g and
A584: g/.i = g/.j and
A585: i <> j;
A586: i in dom g by A580,A581,FINSEQ_3:25;
then
A587: g/.i = f/.(m-'1+i) by FINSEQ_5:27;
A588: j in dom g by A582,A583,FINSEQ_3:25;
then
A589: g/.j = f/.(m-'1+j) by FINSEQ_5:27;
per cases by A585,XXREAL_0:1;
suppose
A590: i < j;
set l = m-'1+j, m9= m-'1+i;
A591: m9 < l by A590,XREAL_1:6;
A592: len(F.l) = l by A192;
A593: l < k by A578,A576,A583,XREAL_1:20;
then
A594: f|l = F.l by A488;
0+j <= l by XREAL_1:6;
then
A595: 1 <= l by A582,XXREAL_0:2;
then l in dom(F.l) by A592,FINSEQ_3:25;
then
A596: (F.l)/.l = f/.l by A594,FINSEQ_4:70;
0+i <= m9 by XREAL_1:6;
then 1 <= m9 by A580,XXREAL_0:2;
then
A597: m9 in dom(F.l) by A591,A592,FINSEQ_3:25;
then (F.l)/.m9 = f/.m9 by A594,FINSEQ_4:70;
hence contradiction by A564,A584,A587,A588,A591,A593,A595,A592,A597
,A596,FINSEQ_5:27;
end;
suppose
A598: j < i;
set l = m-'1+i, m9= m-'1+j;
A599: m9 < l by A598,XREAL_1:6;
A600: len(F.l) = l by A192;
A601: l < k by A578,A576,A581,XREAL_1:20;
then
A602: f|l = F.l by A488;
0+i <= l by XREAL_1:6;
then
A603: 1 <= l by A580,XXREAL_0:2;
then l in dom(F.l) by A600,FINSEQ_3:25;
then
A604: (F.l)/.l = f/.l by A602,FINSEQ_4:70;
0+j <= m9 by XREAL_1:6;
then 1 <= m9 by A582,XXREAL_0:2;
then
A605: m9 in dom(F.l) by A599,A600,FINSEQ_3:25;
then (F.l)/.m9 = f/.m9 by A602,FINSEQ_4:70;
hence contradiction by A564,A584,A586,A589,A599,A601,A603,A600,A605
,A604,FINSEQ_5:27;
end;
end;
A606: now
consider i such that
A607: 1 <= i & i+1 <= len G and
A608: N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1) and
A609: F.(1+1) = <*G*(i,width G),G*(i+1,width G)*> by A171;
take i;
thus 1 <= i & i+1 <= len G by A607;
A610: f|2 = F.2 by A488,A578,A572;
A611: len(f|2) = 2 by A572,FINSEQ_1:59;
then 1 in dom(f|2) by FINSEQ_3:25;
hence f/.1 = (f|2)/.1 by FINSEQ_4:70
.= G*(i,width G) by A609,A610,FINSEQ_4:17;
2 in dom(f|2) by A611,FINSEQ_3:25;
hence f/.2 = (f|2)/.2 by FINSEQ_4:70
.= G*(i+1,width G) by A609,A610,FINSEQ_4:17;
thus N-min C in cell(G,i,width G-'1) & N-min C <> G* (i,width G-'1) by
A608;
end;
1 in dom g by FINSEQ_5:6;
then
A612: g/.1 = f/.(m-'1+1) by FINSEQ_5:27
.= f/.m by A568,XREAL_1:235;
A613: for i st 1 < i & i < j & j <= len g holds g/.i <> g/.j
proof
let i such that
A614: 1 < i and
A615: i < j and
A616: j <= len g and
A617: g/.i = g/.j;
A618: 1 < j by A614,A615,XXREAL_0:2;
A619: i < len g by A615,A616,XXREAL_0:2;
then
A620: 1 < len g by A614,XXREAL_0:2;
per cases;
suppose
j <> len g;
then j < len g by A616,XXREAL_0:1;
hence contradiction by A579,A614,A615,A617,A618,A619;
end;
suppose
j = len g;
hence contradiction by A567,A612,A577,A579,A614,A615,A617,A620;
end;
end;
m+1-(m-'1) <= len g by A574,A576,XREAL_1:9;
then
A621: m+1-(m-1) <= len g by A568,XREAL_1:233;
then
A622: 1+m-m+1 <= len g;
A623: g is_sequence_on G by A427,JORDAN8:2;
then
A624: g is standard by JORDAN8:4;
A625: g is non constant
proof
take 1,2;
thus
A626: 1 in dom g by FINSEQ_5:6;
thus
A627: 2 in dom g by A621,FINSEQ_3:25;
then g/.1 <> g/.(1+1) by A624,FINSEQ_5:6,GOBOARD7:29;
then g.1 <> g/.(1+1) by A626,PARTFUN1:def 6;
hence thesis by A627,PARTFUN1:def 6;
end;
A628: for i st 1 <= i & i < j & j < len g holds g/.i <> g/.j
proof
let i such that
A629: 1 <= i & i < j & j < len g and
A630: g/.i = g/.j;
1 < j & i < len g by A629,XXREAL_0:2;
hence contradiction by A579,A629,A630;
end;
g is s.c.c.
proof
let i,j such that
A631: i+1 < j and
A632: i > 1 & j < len g or j+1 < len g;
A633: 1 < j by A631,NAT_1:12;
A634: 1 <= i+1 by NAT_1:12;
A635: j <= j+1 by NAT_1:12;
then
A636: i+1 < j+1 by A631,XXREAL_0:2;
i < j by A631,NAT_1:13;
then
A637: i < j+1 by A635,XXREAL_0:2;
per cases by A632,NAT_1:14;
suppose
A638: i > 1 & j < len g;
then
A639: j+1 <= len g by NAT_1:13;
then
A640: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A633,TOPREAL1:def 3;
consider i19,j19,i29,j29 being Nat such that
A641: [i19,j19] in Indices G and
A642: g/.j = G*(i19,j19) and
A643: [i29,j29] in Indices G and
A644: g/.(j+1) = G*(i29,j29) and
A645: i19 = i29 & j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 =
i29+1 & j19 = j29 or i19 = i29 & j19 = j29+1 by A623,A633,A639,JORDAN8:3;
A646: 1 <= i19 by A641,MATRIX_0:32;
A647: j29 <= width G by A643,MATRIX_0:32;
A648: 1 <= i29 by A643,MATRIX_0:32;
A649: i19 <= len G by A641,MATRIX_0:32;
A650: 1 <= j29 by A643,MATRIX_0:32;
A651: j19 <= width G by A641,MATRIX_0:32;
A652: i29 <= len G by A643,MATRIX_0:32;
A653: 1 <= j19 by A641,MATRIX_0:32;
A654: i+1 < len g by A631,A638,XXREAL_0:2;
then
A655: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A638,TOPREAL1:def 3;
A656: i < len g by A654,NAT_1:13;
consider i1,j1,i2,j2 such that
A657: [i1,j1] in Indices G and
A658: g/.i = G*(i1,j1) and
A659: [i2,j2] in Indices G and
A660: g/.(i+1) = G*(i2,j2) and
A661: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 &
j1 = j2 or i1 = i2 & j1 = j2+1 by A623,A638,A654,JORDAN8:3;
A662: 1 <= i1 by A657,MATRIX_0:32;
A663: j2 <= width G by A659,MATRIX_0:32;
A664: j1 <= width G by A657,MATRIX_0:32;
A665: 1 <= j2 by A659,MATRIX_0:32;
A666: 1 <= j1 by A657,MATRIX_0:32;
A667: i2 <= len G by A659,MATRIX_0:32;
A668: i1 <= len G by A657,MATRIX_0:32;
A669: 1 < i+1 by A638,NAT_1:13;
assume
A670: LSeg(g,i) meets LSeg(g,j);
A671: 1 <= i2 by A659,MATRIX_0:32;
now
per cases by A661,A645;
suppose
A672: i1 = i2 & j1+1 = j2 & i19 = i29 & j19+1 = j29;
then
A673: j1 = j19 or j1 = j19+1 or j1+1 = j19 by A655,A658,A660,A662,A668
,A666,A663,A640,A642,A644,A646,A649,A653,A647,A670,GOBOARD7:22;
i1 = i19 by A655,A658,A660,A662,A668,A666,A663,A640,A642,A644,A646
,A649,A653,A647,A670,A672,GOBOARD7:19;
hence contradiction by A579,A613,A631,A635,A634,A633,A637,A638,A654
,A656,A658,A660,A639,A642,A644,A672,A673;
end;
suppose
A674: i1 = i2 & j1+1 = j2 & i19+1 = i29 & j19 = j29;
then i1 = i19 & j1 = j19 or i1 = i19 & j1+1 = j19 or i1 = i19+1
& j1 = j19 or i1 = i19+1 & j1+1 = j19 by A655,A658,A660,A662,A668,A666,A663
,A640,A642,A644,A646,A653,A651,A652,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A674;
end;
suppose
A675: i1 = i2 & j1+1 = j2 & i19 = i29+1 & j19 = j29;
then i1 = i29 & j19 = j1 or i1 = i29 & j1+1 = j19 or i1 = i29+1
& j19 = j1 or i1 = i29+1 & j1+1 = j19 by A655,A658,A660,A662,A668,A666,A663
,A640,A642,A644,A649,A653,A651,A648,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A675;
end;
suppose
A676: i1 = i2 & j1+1 = j2 & i19 = i29 & j19 = j29+1;
then
A677: j1 = j29 or j1 = j29+1 or j1+1 = j29 by A655,A658,A660,A662,A668
,A666,A663,A640,A642,A644,A646,A649,A651,A650,A670,GOBOARD7:22;
i1 = i19 by A655,A658,A660,A662,A668,A666,A663,A640,A642,A644,A646
,A649,A651,A650,A670,A676,GOBOARD7:19;
hence contradiction by A579,A613,A631,A636,A633,A638,A669,A656,A658
,A660,A639,A642,A644,A676,A677;
end;
suppose
A678: i1+1 = i2 & j1 = j2 & i19 = i29 & j19+1 = j29;
then i19 = i1 & j1 = j19 or i19 = i1 & j19+1 = j1 or i19 = i1+1
& j1 = j19 or i19 = i1+1 & j19+1 = j1 by A655,A658,A660,A662,A666,A664,A667
,A640,A642,A644,A646,A649,A653,A647,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A678;
end;
suppose
A679: i1+1 = i2 & j1 = j2 & i19+1 = i29 & j19 = j29;
then
A680: i1 = i19 or i1 = i19+1 or i1+1 = i19 by A655,A658,A660,A662,A666
,A664,A667,A640,A642,A644,A646,A653,A651,A652,A670,GOBOARD7:23;
j1 = j19 by A655,A658,A660,A662,A666,A664,A667,A640,A642,A644,A646
,A653,A651,A652,A670,A679,GOBOARD7:20;
hence contradiction by A579,A613,A631,A635,A634,A633,A637,A638,A654
,A656,A658,A660,A639,A642,A644,A679,A680;
end;
suppose
A681: i1+1 = i2 & j1 = j2 & i19 = i29+1 & j19 = j29;
then
A682: i1 = i29 or i1 = i29+1 or i1+1 = i29 by A655,A658,A660,A662,A666
,A664,A667,A640,A642,A644,A649,A653,A651,A648,A670,GOBOARD7:23;
j1 = j19 by A655,A658,A660,A662,A666,A664,A667,A640,A642,A644,A649
,A653,A651,A648,A670,A681,GOBOARD7:20;
hence contradiction by A579,A613,A636,A633,A637,A638,A669,A656,A658
,A660,A639,A642,A644,A681,A682;
end;
suppose
A683: i1+1 = i2 & j1 = j2 & i19 = i29 & j19 = j29+1;
then i19 = i1 & j1 = j29 or i19 = i1 & j29+1 = j1 or i19 = i1+1
& j1 = j29 or i19 = i1+1 & j29+1 = j1 by A655,A658,A660,A662,A666,A664,A667
,A640,A642,A644,A646,A649,A651,A650,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A683;
end;
suppose
A684: i1 = i2+1 & j1 = j2 & i19 = i29 & j19+1 = j29;
then i19 = i2 & j19 = j1 or i19 = i2 & j19+1 = j1 or i19 = i2+1
& j19 = j1 or i19 = i2+1 & j19+1 = j1 by A655,A658,A660,A668,A666,A664,A671
,A640,A642,A644,A646,A649,A653,A647,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A684;
end;
suppose
A685: i1 = i2+1 & j1 = j2 & i19+1 = i29 & j19 = j29;
then
A686: i2 = i19 or i2 = i19+1 or i2+1 = i19 by A655,A658,A660,A668,A666
,A664,A671,A640,A642,A644,A646,A653,A651,A652,A670,GOBOARD7:23;
j1 = j19 by A655,A658,A660,A668,A666,A664,A671,A640,A642,A644,A646
,A653,A651,A652,A670,A685,GOBOARD7:20;
hence contradiction by A579,A613,A631,A636,A633,A638,A669,A656,A658
,A660,A639,A642,A644,A685,A686;
end;
suppose
A687: i1 = i2+1 & j1 = j2 & i19 = i29+1 & j19 = j29;
then
A688: i2 = i29 or i2 = i29+1 or i2+1 = i29 by A655,A658,A660,A668,A666
,A664,A671,A640,A642,A644,A649,A653,A651,A648,A670,GOBOARD7:23;
j1 = j19 by A655,A658,A660,A668,A666,A664,A671,A640,A642,A644,A649
,A653,A651,A648,A670,A687,GOBOARD7:20;
hence contradiction by A613,A631,A636,A637,A638,A669,A658,A660,A639
,A642,A644,A687,A688;
end;
suppose
A689: i1 = i2+1 & j1 = j2 & i19 = i29 & j19 = j29+1;
then i19 = i2 & j29 = j1 or i19 = i2 & j29+1 = j1 or i19 = i2+1
& j29 = j1 or i19 = i2+1 & j29+1 = j1 by A655,A658,A660,A668,A666,A664,A671
,A640,A642,A644,A646,A649,A651,A650,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A689;
end;
suppose
A690: i1 = i2 & j1 = j2+1 & i19 = i29 & j19+1 = j29;
then
A691: j2 = j19 or j2 = j19+1 or j2+1 = j19 by A655,A658,A660,A662,A668
,A664,A665,A640,A642,A644,A646,A649,A653,A647,A670,GOBOARD7:22;
i1 = i19 by A655,A658,A660,A662,A668,A664,A665,A640,A642,A644,A646
,A649,A653,A647,A670,A690,GOBOARD7:19;
hence contradiction by A579,A613,A631,A636,A633,A638,A669,A656,A658
,A660,A639,A642,A644,A690,A691;
end;
suppose
A692: i1 = i2 & j1 = j2+1 & i19+1 = i29 & j19 = j29;
then i1 = i19 & j2 = j19 or i1 = i19 & j2+1 = j19 or i1 = i19+1
& j2 = j19 or i1 = i19+1 & j2+1 = j19 by A655,A658,A660,A662,A668,A664,A665
,A640,A642,A644,A646,A653,A651,A652,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A692;
end;
suppose
A693: i1 = i2 & j1 = j2+1 & i19 = i29+1 & j19 = j29;
then i1 = i29 & j2 = j19 or i1 = i29 & j2+1 = j19 or i1 = i29+1
& j2 = j19 or i1 = i29+1 & j2+1 = j19 by A655,A658,A660,A662,A668,A664,A665
,A640,A642,A644,A649,A653,A651,A648,A670,GOBOARD7:21;
hence contradiction by A579,A613,A631,A636,A633,A637,A638,A669,A656
,A658,A660,A639,A642,A644,A693;
end;
suppose
A694: i1 = i2 & j1 = j2+1 & i19 = i29 & j19 = j29+1;
then
A695: j2 = j29 or j2 = j29+1 or j2+1 = j29 by A655,A658,A660,A662,A668
,A664,A665,A640,A642,A644,A646,A649,A651,A650,A670,GOBOARD7:22;
i1 = i19 by A655,A658,A660,A662,A668,A664,A665,A640,A642,A644,A646
,A649,A651,A650,A670,A694,GOBOARD7:19;
hence contradiction by A613,A631,A636,A637,A638,A669,A658,A660,A639
,A642,A644,A694,A695;
end;
end;
hence contradiction;
end;
suppose
i = 0 & j+1 < len g;
then LSeg(g,i) = {} by TOPREAL1:def 3;
hence thesis by XBOOLE_1:65;
end;
suppose
A696: 1 <= i & j+1 < len g;
then
A697: i+1 < len g by A636,XXREAL_0:2;
then
A698: LSeg(g,i) = LSeg(g/.i,g/.(i+1)) by A696,TOPREAL1:def 3;
A699: i < len g by A697,NAT_1:13;
consider i1,j1,i2,j2 such that
A700: [i1,j1] in Indices G and
A701: g/.i = G*(i1,j1) and
A702: [i2,j2] in Indices G and
A703: g/.(i+1) = G*(i2,j2) and
A704: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 &
j1 = j2 or i1 = i2 & j1 = j2+1 by A623,A696,A697,JORDAN8:3;
A705: 1 <= i1 by A700,MATRIX_0:32;
A706: j2 <= width G by A702,MATRIX_0:32;
A707: j1 <= width G by A700,MATRIX_0:32;
A708: 1 <= j2 by A702,MATRIX_0:32;
A709: 1 <= j1 by A700,MATRIX_0:32;
A710: i2 <= len G by A702,MATRIX_0:32;
A711: i1 <= len G by A700,MATRIX_0:32;
A712: 1 < i+1 by A696,NAT_1:13;
assume
A713: LSeg(g,i) meets LSeg(g,j);
consider i19,j19,i29,j29 being Nat such that
A714: [i19,j19] in Indices G and
A715: g/.j = G*(i19,j19) and
A716: [i29,j29] in Indices G and
A717: g/.(j+1) = G*(i29,j29) and
A718: i19 = i29 & j19+1 = j29 or i19+1 = i29 & j19 = j29 or i19 =
i29+1 & j19 = j29 or i19 = i29 & j19 = j29+1 by A623,A633,A696,JORDAN8:3;
A719: 1 <= i19 by A714,MATRIX_0:32;
A720: j29 <= width G by A716,MATRIX_0:32;
A721: 1 <= i29 by A716,MATRIX_0:32;
A722: i19 <= len G by A714,MATRIX_0:32;
A723: 1 <= j29 by A716,MATRIX_0:32;
A724: j19 <= width G by A714,MATRIX_0:32;
A725: i29 <= len G by A716,MATRIX_0:32;
A726: 1 <= j19 by A714,MATRIX_0:32;
A727: j < len g by A696,NAT_1:12;
A728: LSeg(g,j) = LSeg(g/.j,g/.(j+1)) by A633,A696,TOPREAL1:def 3;
A729: 1 <= i2 by A702,MATRIX_0:32;
now
per cases by A704,A718;
suppose
A730: i1 = i2 & j1+1 = j2 & i19 = i29 & j19+1 = j29;
then
A731: j1 = j19 or j1 = j19+1 or j1+1 = j19 by A698,A701,A703,A705,A711
,A709,A706,A728,A715,A717,A719,A722,A726,A720,A713,GOBOARD7:22;
i1 = i19 by A698,A701,A703,A705,A711,A709,A706,A728,A715,A717,A719
,A722,A726,A720,A713,A730,GOBOARD7:19;
hence contradiction by A628,A631,A636,A634,A637,A696,A727,A701,A703
,A715,A717,A730,A731;
end;
suppose
A732: i1 = i2 & j1+1 = j2 & i19+1 = i29 & j19 = j29;
then i1 = i19 & j1 = j19 or i1 = i19 & j1+1 = j19 or i1 = i19+1
& j1 = j19 or i1 = i19+1 & j1+1 = j19 by A698,A701,A703,A705,A711,A709,A706
,A728,A715,A717,A719,A726,A724,A725,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A732;
end;
suppose
A733: i1 = i2 & j1+1 = j2 & i19 = i29+1 & j19 = j29;
then i1 = i29 & j19 = j1 or i1 = i29 & j1+1 = j19 or i1 = i29+1
& j19 = j1 or i1 = i29+1 & j1+1 = j19 by A698,A701,A703,A705,A711,A709,A706
,A728,A715,A717,A722,A726,A724,A721,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A733;
end;
suppose
A734: i1 = i2 & j1+1 = j2 & i19 = i29 & j19 = j29+1;
then
A735: j1 = j29 or j1 = j29+1 or j1+1 = j29 by A698,A701,A703,A705,A711
,A709,A706,A728,A715,A717,A719,A722,A724,A723,A713,GOBOARD7:22;
i1 = i19 by A698,A701,A703,A705,A711,A709,A706,A728,A715,A717,A719
,A722,A724,A723,A713,A734,GOBOARD7:19;
hence contradiction by A579,A613,A631,A636,A633,A696,A712,A699,A727
,A701,A703,A715,A717,A734,A735;
end;
suppose
A736: i1+1 = i2 & j1 = j2 & i19 = i29 & j19+1 = j29;
then i19 = i1 & j1 = j19 or i19 = i1 & j19+1 = j1 or i19 = i1+1
& j1 = j19 or i19 = i1+1 & j19+1 = j1 by A698,A701,A703,A705,A709,A707,A710
,A728,A715,A717,A719,A722,A726,A720,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A634,A633,A637,A696,A699
,A727,A701,A703,A715,A717,A736;
end;
suppose
A737: i1+1 = i2 & j1 = j2 & i19+1 = i29 & j19 = j29;
then
A738: i1 = i19 or i1 = i19+1 or i1+1 = i19 by A698,A701,A703,A705,A709
,A707,A710,A728,A715,A717,A719,A726,A724,A725,A713,GOBOARD7:23;
j1 = j19 by A698,A701,A703,A705,A709,A707,A710,A728,A715,A717,A719
,A726,A724,A725,A713,A737,GOBOARD7:20;
hence contradiction by A628,A631,A636,A634,A637,A696,A727,A701,A703
,A715,A717,A737,A738;
end;
suppose
A739: i1+1 = i2 & j1 = j2 & i19 = i29+1 & j19 = j29;
then
A740: i1 = i29 or i1 = i29+1 or i1+1 = i29 by A698,A701,A703,A705,A709
,A707,A710,A728,A715,A717,A722,A726,A724,A721,A713,GOBOARD7:23;
j1 = j19 by A698,A701,A703,A705,A709,A707,A710,A728,A715,A717,A722
,A726,A724,A721,A713,A739,GOBOARD7:20;
hence contradiction by A579,A628,A636,A633,A637,A696,A712,A699,A727
,A701,A703,A715,A717,A739,A740;
end;
suppose
A741: i1+1 = i2 & j1 = j2 & i19 = i29 & j19 = j29+1;
then i19 = i1 & j1 = j29 or i19 = i1 & j29+1 = j1 or i19 = i1+1
& j1 = j29 or i19 = i1+1 & j29+1 = j1 by A698,A701,A703,A705,A709,A707,A710
,A728,A715,A717,A719,A722,A724,A723,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A741;
end;
suppose
A742: i1 = i2+1 & j1 = j2 & i19 = i29 & j19+1 = j29;
then i19 = i2 & j19 = j1 or i19 = i2 & j19+1 = j1 or i19 = i2+1
& j19 = j1 or i19 = i2+1 & j19+1 = j1 by A698,A701,A703,A711,A709,A707,A729
,A728,A715,A717,A719,A722,A726,A720,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A742;
end;
suppose
A743: i1 = i2+1 & j1 = j2 & i19+1 = i29 & j19 = j29;
then
A744: i2 = i19 or i2 = i19+1 or i2+1 = i19 by A698,A701,A703,A711,A709
,A707,A729,A728,A715,A717,A719,A726,A724,A725,A713,GOBOARD7:23;
j1 = j19 by A698,A701,A703,A711,A709,A707,A729,A728,A715,A717,A719
,A726,A724,A725,A713,A743,GOBOARD7:20;
hence contradiction by A579,A613,A631,A636,A633,A696,A712,A699,A727
,A701,A703,A715,A717,A743,A744;
end;
suppose
A745: i1 = i2+1 & j1 = j2 & i19 = i29+1 & j19 = j29;
then
A746: i2 = i29 or i2 = i29+1 or i2+1 = i29 by A698,A701,A703,A711,A709
,A707,A729,A728,A715,A717,A722,A726,A724,A721,A713,GOBOARD7:23;
j1 = j19 by A698,A701,A703,A711,A709,A707,A729,A728,A715,A717,A722
,A726,A724,A721,A713,A745,GOBOARD7:20;
hence contradiction by A628,A631,A636,A637,A696,A712,A727,A701,A703
,A715,A717,A745,A746;
end;
suppose
A747: i1 = i2+1 & j1 = j2 & i19 = i29 & j19 = j29+1;
then i19 = i2 & j29 = j1 or i19 = i2 & j29+1 = j1 or i19 = i2+1
& j29 = j1 or i19 = i2+1 & j29+1 = j1 by A698,A701,A703,A711,A709,A707,A729
,A728,A715,A717,A719,A722,A724,A723,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A747;
end;
suppose
A748: i1 = i2 & j1 = j2+1 & i19 = i29 & j19+1 = j29;
then
A749: j2 = j19 or j2 = j19+1 or j2+1 = j19 by A698,A701,A703,A705,A711
,A707,A708,A728,A715,A717,A719,A722,A726,A720,A713,GOBOARD7:22;
i1 = i19 by A698,A701,A703,A705,A711,A707,A708,A728,A715,A717,A719
,A722,A726,A720,A713,A748,GOBOARD7:19;
hence contradiction by A579,A613,A631,A636,A633,A696,A712,A699,A727
,A701,A703,A715,A717,A748,A749;
end;
suppose
A750: i1 = i2 & j1 = j2+1 & i19+1 = i29 & j19 = j29;
then i1 = i19 & j2 = j19 or i1 = i19 & j2+1 = j19 or i1 = i19+1
& j2 = j19 or i1 = i19+1 & j2+1 = j19 by A698,A701,A703,A705,A711,A707,A708
,A728,A715,A717,A719,A726,A724,A725,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A750;
end;
suppose
A751: i1 = i2 & j1 = j2+1 & i19 = i29+1 & j19 = j29;
then i1 = i29 & j2 = j19 or i1 = i29 & j2+1 = j19 or i1 = i29+1
& j2 = j19 or i1 = i29+1 & j2+1 = j19 by A698,A701,A703,A705,A711,A707,A708
,A728,A715,A717,A722,A726,A724,A721,A713,GOBOARD7:21;
hence contradiction by A579,A628,A631,A636,A633,A637,A696,A712,A699
,A727,A701,A703,A715,A717,A751;
end;
suppose
A752: i1 = i2 & j1 = j2+1 & i19 = i29 & j19 = j29+1;
then
A753: j2 = j29 or j2 = j29+1 or j2+1 = j29 by A698,A701,A703,A705,A711
,A707,A708,A728,A715,A717,A719,A722,A724,A723,A713,GOBOARD7:22;
i1 = i19 by A698,A701,A703,A705,A711,A707,A708,A728,A715,A717,A719
,A722,A724,A723,A713,A752,GOBOARD7:19;
hence contradiction by A628,A631,A636,A637,A696,A712,A727,A701,A703
,A715,A717,A752,A753;
end;
end;
hence contradiction;
end;
end;
then reconsider
g as standard non constant special_circular_sequence by A567,A612,A577,A623
,A625,FINSEQ_6:def 1,JORDAN8:4;
A754: for i st 1 <= i & i+1 <= len f holds right_cell(f,i,G) = Cl Int
right_cell(f,i,G)
proof
let i such that
A755: 1 <= i & i+1 <= len f;
consider i1,j1,i2,j2 such that
A756: [i1,j1] in Indices G and
A757: f/.i = G*(i1,j1) and
A758: [i2,j2] in Indices G and
A759: f/.(i+1) = G*(i2,j2) and
A760: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1
= j2 or i1 = i2 & j1 = j2+1 by A569,A755,JORDAN8:3;
A761: i1 <= len G by A756,MATRIX_0:32;
A762: j2 <= width G by A758,MATRIX_0:32;
A763: j1 <= width G by A756,MATRIX_0:32;
A764: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A765: i2 <= len G by A758,MATRIX_0:32;
A766: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
per cases by A760;
suppose
i1 = i2 & j1+1 = j2;
then right_cell(f,i,G) = cell(G,i1,j1) by A569,A755,A756,A757,A758,A759
,A764,GOBRD13:def 2;
hence thesis by A761,A763,GOBRD11:35;
end;
suppose
A767: i1+1 = i2 & j1 = j2;
A768: j1-'1 <= width G by A763,NAT_D:44;
right_cell(f,i,G) = cell(G,i1,j1-'1) by A569,A755,A756,A757,A758,A759
,A766,A767,GOBRD13:def 2;
hence thesis by A761,A768,GOBRD11:35;
end;
suppose
i1 = i2+1 & j1 = j2;
then right_cell(f,i,G) = cell(G,i2,j2) by A569,A755,A756,A757,A758,A759
,A766,GOBRD13:def 2;
hence thesis by A765,A762,GOBRD11:35;
end;
suppose
A769: i1 = i2 & j1 = j2+1;
A770: i1-'1 <= len G by A761,NAT_D:44;
right_cell(f,i,G) = cell(G,i1-'1,j2) by A569,A755,A756,A757,A758,A759
,A764,A769,GOBRD13:def 2;
hence thesis by A762,A770,GOBRD11:35;
end;
end;
now
A771: for h being standard non constant special_circular_sequence st L~
h c= L~f for Comp being Subset of TOP-REAL 2 st Comp is_a_component_of (L~h)`
for n st 1 <= n & n+1 <= len f & f/.n in Comp & not f/.n in L~h holds C meets
Comp
proof
let h be standard non constant special_circular_sequence such that
A772: L~h c= L~f;
let Comp be Subset of TOP-REAL 2 such that
A773: Comp is_a_component_of (L~h)`;
let n such that
A774: 1 <= n & n+1 <= len f and
A775: f/.n in Comp and
A776: not f/.n in L~h;
reconsider rc = right_cell(f,n,G)\L~h as Subset of TOP-REAL 2;
f/.n in right_cell(f,n,G) by A569,A774,Th8;
then f/.n in rc by A776,XBOOLE_0:def 5;
then
A777: rc meets Comp by A775,XBOOLE_0:3;
A778: rc meets C
proof
right_cell(f,n,G) meets C by A427,A774;
then consider p being object such that
A779: p in right_cell(f,n,G) and
A780: p in C by XBOOLE_0:3;
reconsider p as Element of TOP-REAL 2 by A779;
now
take a = p;
now
assume p in L~h;
then consider j such that
A781: 1 <= j & j+1 <= len f and
A782: p in LSeg(f,j) by A772,SPPOL_2:13;
p in left_cell(f,j,G) /\ right_cell(f,j,G) by A427,A781,A782,
GOBRD13:29;
then
A783: p in left_cell(f,j,G) by XBOOLE_0:def 4;
left_cell(f,j,G) misses C by A427,A781;
hence contradiction by A780,A783,XBOOLE_0:3;
end;
hence a in rc by A779,XBOOLE_0:def 5;
thus a in C by A780;
end;
hence thesis by XBOOLE_0:3;
end;
Int right_cell(f,n,G) misses L~f by A569,A774,Th15;
then Int right_cell(f,n,G) misses L~h by A772,XBOOLE_1:63;
then
A784: Int right_cell(f,n,G) c= (L~h)` by SUBSET_1:23;
A785: rc = right_cell(f,n,G) /\ (L~h)` by SUBSET_1:13;
then
A786: rc c= (L~h)` by XBOOLE_1:17;
rc c= right_cell(f,n,G) by XBOOLE_1:36;
then
A787: Int right_cell(f,n,G) c= right_cell(f,n,G) & rc c= Cl Int
right_cell(f,n,G) by A569,A774,Th11,TOPS_1:16;
Int right_cell(f,n,G) is convex by A569,A774,Th10;
then rc is connected by A785,A784,A787,CONNSP_1:18,XBOOLE_1:19;
then rc c= Comp by A773,A777,A786,GOBOARD9:4;
hence thesis by A778,XBOOLE_1:63;
end;
(L~g)` is open by TOPS_1:3;
then
A788: (L~g)` = Int (L~g)` by TOPS_1:23;
A789: L~g c= L~f by JORDAN3:40;
A790: for j,k st 1 <= j & j <= k holds (F.k)/.j = (F.j)/.j
proof
let j,k;
assume that
A791: 1 <= j and
A792: j <= k;
j <= len(F.k) by A192,A792;
then len(F.k|j) = j by FINSEQ_1:59;
then
A793: j in dom((F.k)|j) by A791,FINSEQ_3:25;
(F.k)|j = F.j by A488,A792;
hence thesis by A793,FINSEQ_4:70;
end;
assume m <> 1;
then
A794: 1 < m by A568,XXREAL_0:1;
A795: for n st 1 <= n & n <= m-'1 holds not f/.n in L~g
proof
A796: 2 <= len G by A231,NAT_1:12;
let n such that
A797: 1 <= n and
A798: n <= m-'1;
set p = f/.n;
A799: n <= len f by A575,A798,XXREAL_0:2;
then
A800: p in Values G by A427,A797,Th6;
assume p in L~g;
then consider j such that
A801: m-'1+1 <= j and
A802: j+1 <= len f and
A803: p in LSeg(f,j) by A575,Th7;
A804: j < k by A578,A802,NAT_1:13;
A805: n < m-'1+1 by A798,NAT_1:13;
then
A806: n < j by A801,XXREAL_0:2;
A807: m-'1+1=m by A568,XREAL_1:235;
then
A808: 1 < j by A794,A801,XXREAL_0:2;
per cases by A2,A427,A802,A803,A808,A796,A800,Th23;
suppose
A809: p = f/.j;
A810: n <> len(F.j) by A192,A801,A805;
n <= len(F.j) by A192,A806;
then
A811: n in dom(F.j) by A797,FINSEQ_3:25;
(F.j)/.n = (F.n)/.n by A790,A797,A806
.= p by A578,A790,A797,A799
.= (F.j)/.j by A790,A808,A804,A809
.= (F.j)/.len(F.j) by A192;
hence contradiction by A564,A808,A804,A811,A810;
end;
suppose
A812: p = f/.(j+1);
now
per cases by A578,A802,XXREAL_0:1;
suppose
A813: j+1 = k;
A814: n <> len(F.m) by A192,A805,A807;
n <= len(F.m) by A192,A805,A807;
then
A815: n in dom(F.m) by A797,FINSEQ_3:25;
(F.m)/.n = (F.n)/.n by A790,A797,A805,A807
.= (F.k)/.k by A578,A790,A797,A799,A812,A813
.= (F.m)/.m by A567,A578,A568,A570,A790
.= (F.m)/.len(F.m) by A192;
hence contradiction by A564,A578,A568,A571,A815,A814;
end;
suppose
A816: j+1 < k;
set l = j+1;
A817: 1 <= l by NAT_1:11;
A818: n < n+1 & n+1 < l by A806,XREAL_1:6,29;
then
A819: n <> len(F.l) by A192;
A820: n < l by A818,XXREAL_0:2;
then n <= len(F.l) by A192;
then
A821: n in dom(F.l) by A797,FINSEQ_3:25;
(F.l)/.n = (F.n)/.n by A790,A797,A820
.= p by A578,A790,A797,A799
.= (F.l)/.l by A790,A812,A816,A817
.= (F.l)/.len(F.l) by A192;
hence contradiction by A564,A816,A821,A819,NAT_1:11;
end;
end;
hence contradiction;
end;
end;
C meets RightComp Rev g
proof
1 <= len g by A622,XREAL_1:145;
then
A822: len g-'1+2 = len g+1 by Lm1;
A823: 1 - 1 < m - 1 by A794,XREAL_1:9;
A824: m-'1+2 = m+1 by A568,Lm1;
set l = (m-'1)+(len g-'1);
set a = f/.(m-'1);
set rg=Rev g;
set p = rg/.1, q = rg/.2;
A825: 1+1 - 1 <= len g - 1 by A621,XREAL_1:9;
1+1-'1 <= len g-'1 by A621,NAT_D:42;
then
A826: 1 <= len g-'1 by NAT_D:34;
then (m-'1)+1 <= l by XREAL_1:6;
then m-'1 < l by NAT_1:13;
then
A827: m-'1 <> len(F.l) by A192;
A828: 1+1 <= len rg by A621,FINSEQ_5:def 3;
then 1+1-'1 <= len rg-'1 by NAT_D:42;
then
A829: 1 <= len rg -'1 by NAT_D:34;
1 < len rg by A828,NAT_1:13;
then
A830: len rg -'1+1 = len rg by XREAL_1:235;
A831: rg is_sequence_on G by A623,Th5;
then consider p1,p2,q1,q2 being Nat such that
A832: [p1,p2] in Indices G and
A833: p = G*(p1,p2) and
A834: [q1,q2] in Indices G and
A835: q = G*(q1,q2) and
A836: p1 = q1 & p2+1 = q2 or p1+1 = q1 & p2 = q2 or p1 = q1+1 &
p2 = q2 or p1 = q1 & p2 = q2+1 by A828,JORDAN8:3;
A837: 1 <= p1 by A832,MATRIX_0:32;
A838: p2 <= width G by A832,MATRIX_0:32;
A839: p1 <= len G by A832,MATRIX_0:32;
A840: 1 <= p2 by A832,MATRIX_0:32;
A841: p = f/.m by A567,A577,FINSEQ_5:65;
len g-'1 <= len g by NAT_D:44;
then
A842: len g-'1 in dom g by A826,FINSEQ_3:25;
then
A843: q = g/.(len g-'1) by A822,FINSEQ_5:66
.= f/.l by A842,FINSEQ_5:27;
l = m+(len g -'1)-'1 by A568,NAT_D:38
.= (len g -'1)+m - 1 by A826,NAT_D:37
.= (len g - 1)+m - 1 by A825,XREAL_0:def 2
.= ((k - (m - 1)) - 1)+m - 1 by A578,A576,A823,XREAL_0:def 2
.= k - 1;
then
A844: k = l+1;
then
A845: l < k by XREAL_1:29;
len g-'1 <= l by NAT_1:11;
then
A846: 1 <= l by A826,XXREAL_0:2;
then
A847: right_cell(f,l,G) meets C by A427,A578,A844;
A848: m-'1+1 = m by A568,XREAL_1:235;
then
A849: 1 <= m-'1 by A794,NAT_1:13;
then
A850: right_cell(f,m-'1,G) meets C by A427,A570,A848;
m-'1 <= l by NAT_1:11;
then m-'1 <= len(F.l) by A192;
then
A851: m-'1 in dom(F.l) by A849,FINSEQ_3:25;
not a in L~g by A795,A849;
then
A852: not a in L~rg by SPPOL_2:22;
per cases by A836;
suppose
A853: p1 = q1 & p2+1 = q2;
consider a1,a2,p91,p92 being Nat such that
A854: [a1,a2] in Indices G and
A855: a = G*(a1,a2) and
A856: [p91,p92] in Indices G & p = G*(p91,p92) and
A857: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 =
p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A569,A570,A841,A848,A849,JORDAN8:3
;
A858: 1 <= a2 by A854,MATRIX_0:32;
thus thesis
proof
per cases by A857;
suppose
A859: a1 = p91 & a2+1 = p92;
A860: m-'1+1 <= len (F.m) & f/.(m-'1+1) = (F.m)/.m by A192,A578,A568
,A570,A790,A848;
A861: F.k|(m+1)=F.(m+1) by A488,A578,A574;
A862: a1 = p1 by A832,A833,A856,A859,GOBOARD1:5;
A863: m-'1 <= m by A848,NAT_1:11;
A864: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A578,A575,A790,A849
.= (F.m)/.(m-'1) by A790,A849,A863;
A865: 2 in dom g by A621,FINSEQ_3:25;
A866: a2+1 = p2 by A832,A833,A856,A859,GOBOARD1:5;
then
A867: p2-'1 = a2 by NAT_D:34;
right_cell(f,l,G) = cell(G,p1-'1,p2) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A853,GOBRD13:28
.= front_left_cell(F.m,m-'1,G) by A427,A841,A848,A849,A832,A833
,A854,A855,A862,A866,A864,A860,GOBRD13:34;
then F.(m+1) turns_left m-'1,G by A456,A794,A847;
then
A868: a2+1 > a2 & f turns_left m-'1,G by A849,A824,A861,GOBRD13:44
,NAT_1:13;
len rg-'1+2 = len g +1 by A822,FINSEQ_5:def 3;
then
A869: rg/.(len rg-'1) = g/.2 by A865,FINSEQ_5:66
.= f/.(m+1) by A824,A865,FINSEQ_5:27;
A870: p = g/.1 by A567,A612,A577,FINSEQ_5:65
.= rg/.len g by FINSEQ_5:65
.= rg/.len rg by FINSEQ_5:def 3;
set rc = right_cell(rg,len rg-'1,G)\L~rg;
A871: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2;
A872: p1-'1+1 = p1 by A837,XREAL_1:235;
A873: p2+1 > a2+1 by A866,NAT_1:13;
then
A874: [p1-'1,p2] in Indices G by A841,A848,A832,A833,A854,A855,A868;
then
A875: 1 <= p1-'1 by MATRIX_0:32;
f/.(m+1) = G*(p1-'1,p2) by A841,A848,A832,A833,A854,A855
,A873,A868;
then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A831,A829
,A830,A832,A833,A874,A867,A872,A869,A870,GOBRD13:24;
then a in right_cell(rg,len rg-'1,G) by A839,A838,A855,A858,A862
,A866,A875,A872,Th20;
then
A876: a in rc by A852,XBOOLE_0:def 5;
rc c= RightComp rg & L~rg c= L~f by A789,A831,A829,A830,Th27,
SPPOL_2:22;
hence thesis by A570,A771,A848,A849,A852,A876,A871;
end;
suppose
A877: a1+1 = p91 & a2 = p92;
then a1+1 = p1 by A832,A833,A856,GOBOARD1:5;
then
A878: q1-'1 = a1 by A853,NAT_D:34;
a2 = p2 by A832,A833,A856,A877,GOBOARD1:5;
then right_cell(f,l,G) = cell(G,a1,a2) by A427,A567,A578,A841
,A844,A846,A843,A832,A833,A834,A835,A853,A878,GOBRD13:28
.= left_cell(f,m-'1,G) by A427,A570,A841,A848,A849,A854,A855
,A856,A877,GOBRD13:23;
hence thesis by A427,A570,A848,A849,A847;
end;
suppose
a1 = p91+1 & a2 = p92;
then a1 = p1+1 & a2 = p2 by A832,A833,A856,GOBOARD1:5;
then right_cell(f,m-'1,G) = cell(G,p1,p2) by A427,A570,A841,A848
,A849,A832,A833,A854,A855,GOBRD13:26
.= left_cell(f,l,G) by A427,A567,A578,A841,A844,A846,A843,A832
,A833,A834,A835,A853,GOBRD13:27;
hence thesis by A427,A578,A844,A846,A850;
end;
suppose
a1 = p91 & a2 = p92+1;
then
A879: a1 = q1 & a2 = q2 by A832,A833,A853,A856,GOBOARD1:5;
(F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A790,A849,NAT_1:11
.= q by A578,A575,A790,A849,A835,A855,A879
.= (F.l)/.l by A790,A845,A846,A843
.= (F.l)/.len(F.l) by A192;
hence thesis by A564,A845,A846,A851,A827;
end;
end;
end;
suppose
A880: p1+1 = q1 & p2 = q2;
consider a1,a2,p91,p92 being Nat such that
A881: [a1,a2] in Indices G and
A882: a = G*(a1,a2) and
A883: [p91,p92] in Indices G & p = G*(p91,p92) and
A884: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 =
p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A569,A570,A841,A848,A849,JORDAN8:3
;
A885: 1 <= a1 by A881,MATRIX_0:32;
thus thesis
proof
per cases by A884;
suppose
A886: a1 = p91 & a2+1 = p92;
then a2+1 = p2 by A832,A833,A883,GOBOARD1:5;
then
A887: q2-'1 = a2 by A880,NAT_D:34;
A888: a1 = p1 by A832,A833,A883,A886,GOBOARD1:5;
right_cell(f,m-'1,G) = cell(G,a1,a2) by A427,A570,A841,A848,A849
,A881,A882,A883,A886,GOBRD13:22
.= left_cell(f,l,G) by A427,A567,A578,A841,A844,A846,A843,A832
,A833,A834,A835,A880,A888,A887,GOBRD13:25;
hence thesis by A427,A578,A844,A846,A850;
end;
suppose
A889: a1+1 = p91 & a2 = p92;
A890: m-'1 <= m by A848,NAT_1:11;
A891: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A578,A575,A790,A849
.= (F.m)/.(m-'1) by A790,A849,A890;
A892: 2 in dom g by A621,FINSEQ_3:25;
len rg-'1+2 = len g +1 by A822,FINSEQ_5:def 3;
then
A893: rg/.(len rg-'1) = g/.2 by A892,FINSEQ_5:66
.= f/.(m+1) by A824,A892,FINSEQ_5:27;
A894: p = g/.1 by A567,A612,A577,FINSEQ_5:65
.= rg/.len g by FINSEQ_5:65
.= rg/.len rg by FINSEQ_5:def 3;
A895: a2 = p2 by A832,A833,A883,A889,GOBOARD1:5;
A896: m-'1+1 <= len (F.m) & f/.(m-'1+1) = (F.m)/.m by A192,A578,A568
,A570,A790,A848;
A897: F.k|(m+1)=F.(m+1) by A488,A578,A574;
set rc = right_cell(rg,len rg-'1,G)\L~rg;
A898: a1 < a1+1 & p1 < p1+1 by XREAL_1:29;
A899: a1+1 = p1 by A832,A833,A883,A889,GOBOARD1:5;
then
A900: a1 = p1-'1 by NAT_D:34;
right_cell(f,l,G) = cell(G,p1,p2) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A880,GOBRD13:26
.= front_left_cell(F.m,m-'1,G) by A427,A841,A848,A849,A832,A833
,A881,A882,A899,A895,A891,A896,GOBRD13:36;
then F.(m+1) turns_left m-'1,G by A456,A794,A847;
then
A901: f turns_left m-'1,G by A849,A824,A897,GOBRD13:44;
then
A902: [p1,p2+1] in Indices G by A841,A848,A832,A833,A881,A882,A899,A898
;
then
A903: p2+1 <= width G by MATRIX_0:32;
f/.(m+1) = G*(p1,p2+1) by A841,A848,A832,A833,A881,A882,A899
,A898,A901;
then right_cell(rg,len rg-'1,G) = cell(G,p1-'1,a2) by A831,A829
,A830,A832,A833,A895,A902,A893,A894,GOBRD13:28;
then a in right_cell(rg,len rg-'1,G) by A839,A840,A882,A885,A899
,A895,A903,A900,Th20;
then
A904: a in rc by A852,XBOOLE_0:def 5;
A905: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2;
rc c= RightComp rg & L~rg c= L~f by A789,A831,A829,A830,Th27,
SPPOL_2:22;
hence thesis by A570,A771,A848,A849,A852,A904,A905;
end;
suppose
a1 = p91+1 & a2 = p92;
then
A906: a1 = q1 & a2 = q2 by A832,A833,A880,A883,GOBOARD1:5;
(F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A790,A849,NAT_1:11
.= q by A578,A575,A790,A849,A835,A882,A906
.= (F.l)/.l by A790,A845,A846,A843
.= (F.l)/.len(F.l) by A192;
hence thesis by A564,A845,A846,A851,A827;
end;
suppose
a1 = p91 & a2 = p92+1;
then
A907: a1 = p1 & a2 = p2+1 by A832,A833,A883,GOBOARD1:5;
right_cell(f,l,G) = cell(G,p1,p2) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A880,GOBRD13:26
.= left_cell(f,m-'1,G) by A427,A570,A841,A848,A849,A832,A833
,A881,A882,A907,GOBRD13:27;
hence thesis by A427,A570,A848,A849,A847;
end;
end;
end;
suppose
A908: p1 = q1+1 & p2 = q2;
consider a1,a2,p91,p92 being Nat such that
A909: [a1,a2] in Indices G and
A910: a = G*(a1,a2) and
A911: [p91,p92] in Indices G & p = G*(p91,p92) and
A912: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 =
p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A569,A570,A841,A848,A849,JORDAN8:3
;
A913: a1 <= len G by A909,MATRIX_0:32;
thus thesis
proof
per cases by A912;
suppose
A914: a1 = p91 & a2+1 = p92;
then a2+1 = p2 by A832,A833,A911,GOBOARD1:5;
then
A915: q2-'1 = a2 by A908,NAT_D:34;
a1 = p1 by A832,A833,A911,A914,GOBOARD1:5;
then
A916: q1 = a1-'1 by A908,NAT_D:34;
right_cell(f,l,G) = cell(G,q1,q2-'1) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A908,GOBRD13:24
.= left_cell(f,m-'1,G) by A427,A570,A841,A848,A849,A909,A910
,A911,A914,A916,A915,GOBRD13:21;
hence thesis by A427,A570,A848,A849,A847;
end;
suppose
a1+1 = p91 & a2 = p92;
then
A917: a1+1 = p1 & a2 = p2 by A832,A833,A911,GOBOARD1:5;
(F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A790,A849,NAT_1:11
.= q by A578,A575,A790,A849,A835,A908,A910,A917
.= (F.l)/.l by A790,A845,A846,A843
.= (F.l)/.len(F.l) by A192;
hence thesis by A564,A845,A846,A851,A827;
end;
suppose
A918: a1 = p91+1 & a2 = p92;
A919: p = g/.1 by A567,A612,A577,FINSEQ_5:65
.= rg/.len g by FINSEQ_5:65
.= rg/.len rg by FINSEQ_5:def 3;
A920: a1 = p1+1 by A832,A833,A911,A918,GOBOARD1:5;
p1+1>p1 by XREAL_1:29;
then
A921: a1+1 > p1 by A920,NAT_1:13;
A922: m-'1 <= m by A848,NAT_1:11;
A923: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A578,A575,A790,A849
.= (F.m)/.(m-'1) by A790,A849,A922;
A924: 2 in dom g by A621,FINSEQ_3:25;
len rg-'1+2 = len g +1 by A822,FINSEQ_5:def 3;
then
A925: rg/.(len rg-'1) = g/.2 by A924,FINSEQ_5:66
.= f/.(m+1) by A824,A924,FINSEQ_5:27;
A926: F.k|(m+1)=F.(m+1) by A488,A578,A574;
set rc = right_cell(rg,len rg-'1,G)\L~rg;
A927: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2;
A928: p2-'1+1 = p2 by A840,XREAL_1:235;
A929: f/.(m-'1+1) = (F.m)/.m by A578,A568,A570,A790,A848;
A930: p1-'1 = q1 & m-'1+1 <= len (F.m) by A192,A848,A908,NAT_D:34;
A931: a2 = p2 by A832,A833,A911,A918,GOBOARD1:5;
right_cell(f,l,G) = cell(G,q1,q2-'1) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A908,GOBRD13:24
.= front_left_cell(F.m,m-'1,G) by A427,A841,A848,A849,A832,A833
,A908,A909,A910,A920,A931,A930,A923,A929,GOBRD13:38;
then F.(m+1) turns_left m-'1,G by A456,A794,A847;
then
A932: f turns_left m-'1,G by A849,A824,A926,GOBRD13:44;
A933: a2+1 > p2 by A931,NAT_1:13;
then
A934: [p1,p2-'1] in Indices G by A841,A848,A832,A833,A909,A910,A931
,A921,A932;
then
A935: 1 <= p2-'1 by MATRIX_0:32;
f/.(m+1) = G*(p1,p2-'1) by A841,A848,A832,A833,A909,A910
,A931,A933,A921,A932;
then right_cell(rg,len rg-'1,G) = cell(G,p1,p2-'1) by A831,A829
,A830,A832,A833,A934,A928,A925,A919,GOBRD13:22;
then a in right_cell(rg,len rg-'1,G) by A837,A838,A910,A913,A920
,A931,A935,A928,Th20;
then
A936: a in rc by A852,XBOOLE_0:def 5;
rc c= RightComp rg & L~rg c= L~f by A789,A831,A829,A830,Th27,
SPPOL_2:22;
hence thesis by A570,A771,A848,A849,A852,A936,A927;
end;
suppose
A937: a1 = p91 & a2 = p92+1;
then a1 = p1 by A832,A833,A911,GOBOARD1:5;
then
A938: q1 = a1-'1 by A908,NAT_D:34;
a2 = p2+1 by A832,A833,A911,A937,GOBOARD1:5;
then right_cell(f,m-'1,G) = cell(G,q1,q2) by A427,A570,A841,A848
,A849,A908,A909,A910,A911,A937,A938,GOBRD13:28
.= left_cell(f,l,G) by A427,A567,A578,A841,A844,A846,A843,A832
,A833,A834,A835,A908,GOBRD13:23;
hence thesis by A427,A578,A844,A846,A850;
end;
end;
end;
suppose
A939: p1 = q1 & p2 = q2+1;
consider a1,a2,p91,p92 being Nat such that
A940: [a1,a2] in Indices G and
A941: a = G*(a1,a2) and
A942: [p91,p92] in Indices G & p = G*(p91,p92) and
A943: a1 = p91 & a2+1 = p92 or a1+1 = p91 & a2 = p92 or a1 =
p91+1 & a2 = p92 or a1 = p91 & a2 = p92+1 by A569,A570,A841,A848,A849,JORDAN8:3
;
A944: a2 <= width G by A940,MATRIX_0:32;
thus thesis
proof
per cases by A943;
suppose
a1 = p91 & a2+1 = p92;
then
A945: a1 = p1 & a2+1 = p2 by A832,A833,A942,GOBOARD1:5;
(F.l)/.(m-'1) = (F.(m-'1))/.(m-'1) by A790,A849,NAT_1:11
.= q by A578,A575,A790,A849,A835,A939,A941,A945
.= (F.l)/.l by A790,A845,A846,A843
.= (F.l)/.len(F.l) by A192;
hence thesis by A564,A845,A846,A851,A827;
end;
suppose
A946: a1+1 = p91 & a2 = p92;
then a2 = p2 by A832,A833,A942,GOBOARD1:5;
then
A947: a2-'1 = q2 by A939,NAT_D:34;
a1+1 = p1 by A832,A833,A942,A946,GOBOARD1:5;
then
A948: a1 = q1-'1 by A939,NAT_D:34;
right_cell(f,m-'1,G) = cell(G,a1,a2-'1) by A427,A570,A841,A848
,A849,A940,A941,A942,A946,GOBRD13:24
.= left_cell(f,l,G) by A427,A567,A578,A841,A844,A846,A843,A832
,A833,A834,A835,A939,A948,A947,GOBRD13:21;
hence thesis by A427,A578,A844,A846,A850;
end;
suppose
A949: a1 = p91+1 & a2 = p92;
then a2 = p2 by A832,A833,A942,GOBOARD1:5;
then
A950: a2-'1 = q2 by A939,NAT_D:34;
A951: a1 = p1+1 by A832,A833,A942,A949,GOBOARD1:5;
right_cell(f,l,G) = cell(G,q1,q2) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A939,GOBRD13:22
.= left_cell(f,m-'1,G) by A427,A570,A841,A848,A849,A939,A940
,A941,A942,A949,A951,A950,GOBRD13:25;
hence thesis by A427,A570,A848,A849,A847;
end;
suppose
A952: a1 = p91 & a2 = p92+1;
set rc = right_cell(rg,len rg-'1,G)\L~rg;
A953: RightComp rg is_a_component_of (L~rg)` by GOBOARD9:def 2;
A954: 2 in dom g by A621,FINSEQ_3:25;
len rg-'1+2 = len g +1 by A822,FINSEQ_5:def 3;
then
A955: rg/.(len rg-'1) = g/.2 by A954,FINSEQ_5:66
.= f/.(m+1) by A824,A954,FINSEQ_5:27;
A956: p = g/.1 by A567,A612,A577,FINSEQ_5:65
.= rg/.len g by FINSEQ_5:65
.= rg/.len rg by FINSEQ_5:def 3;
A957: a1 = p1 by A832,A833,A942,A952,GOBOARD1:5;
A958: m-'1 <= m by A848,NAT_1:11;
A959: f/.(m-'1) = (F.(m-'1))/.(m-'1) by A578,A575,A790,A849
.= (F.m)/.(m-'1) by A790,A849,A958;
A960: p2-'1 = q2 & m-'1+1 <= len (F.m) by A192,A848,A939,NAT_D:34;
A961: f/.(m-'1+1) = (F.m)/.m by A578,A568,A570,A790,A848;
A962: F.k|(m+1)=F.(m+1) by A488,A578,A574;
A963: a2 = p2+1 by A832,A833,A942,A952,GOBOARD1:5;
right_cell(f,l,G) = cell(G,q1,q2) by A427,A567,A578,A841,A844
,A846,A843,A832,A833,A834,A835,A939,GOBRD13:22
.= front_left_cell(F.m,m-'1,G) by A427,A841,A848,A849,A832,A833
,A939,A940,A941,A957,A963,A960,A959,A961,GOBRD13:40;
then F.(m+1) turns_left m-'1,G by A456,A794,A847;
then
A964: p2+1>p2 & f turns_left m-'1,G by A849,A824,A962,GOBRD13:44
,NAT_1:13;
A965: a2+1>p2+1 by A963,NAT_1:13;
then
A966: [p1+1,p2] in Indices G by A841,A848,A832,A833,A940,A941,A964;
then
A967: p1+1 <= len G by MATRIX_0:32;
f/.(m+1) = G*(p1+1,p2) by A841,A848,A832,A833,A940,A941,A965
,A964;
then right_cell(rg,len rg-'1,G) = cell(G,p1,p2) by A831,A829,A830
,A832,A833,A966,A955,A956,GOBRD13:26;
then a in right_cell(rg,len rg-'1,G) by A837,A840,A941,A944,A957
,A963,A967,Th20;
then
A968: a in rc by A852,XBOOLE_0:def 5;
rc c= RightComp rg & L~rg c= L~f by A789,A831,A829,A830,Th27,
SPPOL_2:22;
hence thesis by A570,A771,A848,A849,A852,A968,A953;
end;
end;
end;
end;
then
A969: LeftComp g is_a_component_of (L~g)` & C meets LeftComp g by GOBOARD9:24
,def 1;
reconsider Lg9 = (L~g)` as Subset of TOP-REAL 2;
A970: RightComp g is_a_component_of (L~g)` by GOBOARD9:def 2;
A971: C c= Lg9
proof
let c be object;
assume that
A972: c in C and
A973: not c in Lg9;
reconsider c as Point of TOP-REAL 2 by A972;
consider i such that
A974: 1 <= i and
A975: i+1 <= len g and
A976: c in LSeg(g/.i,g/.(i+1)) by A973,SPPOL_2:14,SUBSET_1:29;
A977: 1 <= i+(m-'1) by A974,NAT_1:12;
i+1 in dom g by A974,A975,SEQ_4:134;
then
A978: g/.(i+1) = f/.(i+1+(m-'1)) by FINSEQ_5:27;
i+1+(m-'1) = i+(m-'1)+1;
then
A979: i+(m-'1)+1 <= (len g)+(m-'1) by A975,XREAL_1:6;
i in dom g by A974,A975,SEQ_4:134;
then g/.i = f/.(i+(m-'1)) by FINSEQ_5:27;
then c in LSeg(f,i+(m-'1)) by A576,A976,A978,A977,A979,TOPREAL1:def 3;
then c in left_cell(f,i+(m-'1),G) /\ right_cell(f,i+(m-'1),G) by A427
,A576,A977,A979,GOBRD13:29;
then c in left_cell(f,i+(m-'1),G) by XBOOLE_0:def 4;
then left_cell(f,i+(m-'1),G) meets C by A972,XBOOLE_0:3;
hence contradiction by A427,A576,A977,A979;
end;
A980: the TopStruct of TOP-REAL 2 = TopSpaceMetr (Euclid 2) by EUCLID:def 8;
C meets RightComp g
proof
right_cell(f,m,G) meets C by A427,A568,A574;
then consider p being object such that
A981: p in right_cell(f,m,G) and
A982: p in C by XBOOLE_0:3;
reconsider p as Element of TOP-REAL 2 by A981;
now
reconsider u = p as Element of Euclid 2 by TOPREAL3:8;
take a = p;
thus a in C by A982;
consider r being Real such that
A983: r > 0 and
A984: Ball(u,r) c= (L~g)` by A971,A788,A982,GOBOARD6:5;
reconsider r as Real;
A985: p in Ball(u,r) by A983,GOBOARD6:1;
reconsider B = Ball(u,r) as non empty Subset of TOP-REAL 2 by A980
,A983,GOBOARD6:1,TOPMETR:12;
A986: p in B by A983,GOBOARD6:1;
right_cell(f,m,G) = Cl Int right_cell(f,m,G) & B is open by A568,A574
,A754,GOBOARD6:3;
then
A987: Int right_cell(f,m,G) meets B by A981,A985,TOPS_1:12;
Int right_cell(g,1,G) c= Int right_cell(g,1) & Int right_cell
(g,1) c= RightComp g by A623,A622,GOBOARD9:25,GOBRD13:33,TOPS_1:19;
then Int right_cell(g,1,G) c= RightComp g;
then Int right_cell(f,m-'1+1,G) c= RightComp g by A569,A575,A622,
GOBRD13:32;
then B is connected & Int right_cell(f,m,G) c= RightComp g by A568,
SPRECT_3:7,XREAL_1:235;
then B c= RightComp g by A970,A984,A987,GOBOARD9:4;
hence a in RightComp g by A986;
end;
hence thesis by XBOOLE_0:3;
end;
hence contradiction by A1,A970,A971,A969,Th1,SPRECT_4:6;
end;
then g = f/^0 by XREAL_1:232
.= f by FINSEQ_5:28;
then reconsider f as standard non constant special_circular_sequence;
f is clockwise_oriented
proof
f/.2 in LSeg(f/.1,f/.(1+1)) by RLTOPSP1:68;
then
A988: f/.2 in L~f by A572,SPPOL_2:15;
(NW-corner L~f)`1 = W-bound L~f by EUCLID:52;
then
A989: (NW-corner L~f)`1 <= (f/.2)`1 by A988,PSCOMP_1:24;
len G >= 3 by A231,NAT_1:12;
then
A990: 1 < len G by XXREAL_0:2;
(NE-corner L~f)`1 = E-bound L~f by EUCLID:52;
then
A991: (f/.2)`1 <= (NE-corner L~f)`1 by A988,PSCOMP_1:24;
for k st 1 <= k & k+1 <= len f holds left_cell(f,k,G) misses C &
right_cell(f,k,G) meets C by A427;
then
A992: N-min L~f = f/.1 by A569,A606,Th30;
consider i such that
A993: 1 <= i and
A994: i+1 <= len G and
A995: f/.1 = G*(i,width G) & f/.2 = G*(i+1,width G) and
N-min C in cell(G,i,width G-'1) and
N-min C <> G*(i,width G-'1) by A606;
i < len G by A994,NAT_1:13;
then
A996: (N-min L~f)`2 = N-bound L~f & G*(i,width G)`2 = G*(1,width G)`2
by A2,A990,A993,EUCLID:52,GOBOARD5:1;
1 <= i+1 by NAT_1:12;
then
A997: G*(i+1,width G)`2 = G*(1,width G)`2 by A2,A990,A994,GOBOARD5:1;
(NW-corner L~f)`2 = (NE-corner L~f)`2 & (NE-corner L~f)`2 =
N-bound L~f by EUCLID:52,PSCOMP_1:27;
then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) by A992,A995,A996,A997
,A989,A991,GOBOARD7:8;
then f/.2 in LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by A988,
XBOOLE_0:def 4;
hence thesis by A992,SPRECT_2:30;
end;
then reconsider f as clockwise_oriented standard non constant
special_circular_sequence;
take f;
thus f is_sequence_on G by A427;
thus ex i st 1 <= i & i+1 <= len G & f/.1 = G*(i,width G) & f/.2 = G*(i+1,
width G) & N-min C in cell(G,i,width G-'1) & N-min C <> G*(i,width G-'1)
by A606;
let m such that
A998: 1 <= m and
A999: m+2 <= len f;
A1000: F.(m+1+1) = f|(m+1+1) by A488,A578,A999;
A1001: m+1 < m+2 by XREAL_1:6;
then
A1002: f|(m+1) = F.(m+1) by A488,A578,A999,XXREAL_0:2;
A1003: m+1 <= len f by A999,A1001,XXREAL_0:2;
then
A1004: front_left_cell(F.(m+1),m,G) = front_left_cell(f,m,G) by A569,A998
,A1002,GOBRD13:42;
A1005: m = m+1-'1 & m+1 > 1 by A998,NAT_1:13,NAT_D:34;
A1006: front_right_cell(F.(m+1),m,G) = front_right_cell(f,m,G) by A569,A998
,A1003,A1002,GOBRD13:42;
hereby
assume
front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) misses C;
then F.(m+1+1) turns_right m,G by A456,A1005,A1004,A1006;
hence f turns_right m,G by A998,A1000,GOBRD13:43;
end;
hereby
assume front_left_cell(f,m,G) misses C & front_right_cell(f,m,G) meets C;
then F.(m+1+1) goes_straight m,G by A456,A1005,A1004,A1006;
hence f goes_straight m,G by A998,A1000,GOBRD13:45;
end;
assume front_left_cell(f,m,G) meets C;
then F.(m+1+1) turns_left m,G by A456,A1005,A1004;
hence thesis by A998,A1000,GOBRD13:44;
end;
uniqueness
proof
let f1,f2 be clockwise_oriented standard non constant
special_circular_sequence such that
A1007: f1 is_sequence_on Gauge(C,n);
defpred P[Nat] means f1|$1 = f2|$1;
given i1 such that
A1008: 1 <= i1 & i1+1 <= len Gauge(C,n) and
A1009: f1/.1 = Gauge(C,n)*(i1,width Gauge(C,n)) and
A1010: f1/.2 = Gauge(C,n)*(i1+1,width Gauge(C,n)) and
A1011: N-min C in cell(Gauge(C,n),i1,width Gauge(C,n)-'1) & N-min C <>
Gauge(C,n)*( i1,width Gauge(C,n)-'1);
assume that
A1012: for k st 1 <= k & k+2 <= len f1 holds (front_left_cell(f1,k,
Gauge(C,n)) misses C & front_right_cell(f1,k,Gauge(C,n)) misses C implies f1
turns_right k,Gauge(C,n)) & (front_left_cell(f1,k,Gauge(C,n)) misses C &
front_right_cell(f1,k,Gauge(C,n)) meets C implies f1 goes_straight k,Gauge(C,n)
) & (front_left_cell(f1,k,Gauge(C,n)) meets C implies f1 turns_left k,Gauge(C,n
)) and
A1013: f2 is_sequence_on Gauge(C,n);
given i2 such that
A1014: 1 <= i2 & i2+1 <= len Gauge(C,n) and
A1015: f2/.1 = Gauge(C,n)*(i2,width Gauge(C,n)) and
A1016: f2/.2 = Gauge(C,n)*(i2+1,width Gauge(C,n)) and
A1017: N-min C in cell(Gauge(C,n),i2,width Gauge(C,n)-'1) & N-min C <>
Gauge(C,n)*( i2,width Gauge(C,n)-'1);
assume
A1018: for k st 1 <= k & k+2 <= len f2 holds (front_left_cell(f2,k,
Gauge(C,n)) misses C & front_right_cell(f2,k,Gauge(C,n)) misses C implies f2
turns_right k,Gauge(C,n)) & (front_left_cell(f2,k,Gauge(C,n)) misses C &
front_right_cell(f2,k,Gauge(C,n)) meets C implies f2 goes_straight k,Gauge(C,n)
) & (front_left_cell(f2,k,Gauge(C,n)) meets C implies f2 turns_left k,Gauge(C,n
));
A1019: for k st P[k] holds P[k+1]
proof
A1020: len f1 > 4 by GOBOARD7:34;
A1021: f1|1 = <*f1/.1*> & f2|1 = <*f2/.1*> by FINSEQ_5:20;
A1022: i1 = i2 & len f2 > 4 by A1008,A1011,A1014,A1017,Th29,GOBOARD7:34;
let k such that
A1023: f1|k = f2|k;
per cases by NAT_1:25;
suppose
k = 0;
hence thesis by A1008,A1009,A1011,A1014,A1015,A1017,A1021,Th29;
end;
suppose
A1024: k = 1;
f1|2 = <*f1/.1,f1/.2*> by A1020,FINSEQ_5:81,XXREAL_0:2;
hence thesis by A1009,A1010,A1015,A1016,A1022,A1024,FINSEQ_5:81
,XXREAL_0:2;
end;
suppose
A1025: k > 1;
A1026: f1/.1 = f1/.len f1 & f2/.1 = f2/.len f2 by FINSEQ_6:def 1;
now
per cases;
suppose
A1027: len f1 > k;
set m = k-'1;
A1028: 1 <= m by A1025,NAT_D:49;
then
A1029: m+1 = k by NAT_D:43;
then
A1030: front_left_cell (f1,m,Gauge(C,n))= front_left_cell(f1|k,m,
Gauge(C,n)) by A1007,A1027,A1028,GOBRD13:42;
A1031: m+(1+1) = k+1 by A1029;
A1032: k+1 <= len f1 by A1027,NAT_1:13;
A1033: now
A1034: 1 < len f2 by GOBOARD7:34,XXREAL_0:2;
assume
A1035: len f2 <= k;
then
A1036: f2 = f2|k by FINSEQ_1:58;
then len f2 in dom(f2|k) by FINSEQ_5:6;
then
A1037: (f1|k)/.len f2 = f1/.len f2 by A1023,FINSEQ_4:70;
1 in dom(f2|k) & len f2 <= len f1 by A1023,A1036,FINSEQ_5:6,16;
hence
contradiction by A1023,A1026,A1027,A1035,A1036,A1037,A1034,
FINSEQ_4:70,GOBOARD7:38;
end;
then
A1038: k+1 <= len f2 by NAT_1:13;
A1039: front_right_cell(f2,m,Gauge(C,n)) = front_right_cell(f2|k,m
,Gauge(C,n)) by A1013,A1028,A1029,A1033,GOBRD13:42;
A1040: front_left_cell(f2,m,Gauge(C,n)) = front_left_cell(f2|k,m,
Gauge(C,n)) by A1013,A1028,A1029,A1033,GOBRD13:42;
A1041: front_right_cell(f1,m,Gauge(C,n)) = front_right_cell(f1|k,m
,Gauge(C,n)) by A1007,A1027,A1028,A1029,GOBRD13:42;
now
per cases;
suppose
front_left_cell(f1,m,Gauge(C,n)) misses C &
front_right_cell(f1,m,Gauge(C,n)) misses C;
then f1 turns_right m,Gauge(C,n) & f2 turns_right m,Gauge(C,
n) by A1012,A1018,A1023,A1028,A1038,A1032,A1030,A1041,A1040,A1039,A1031;
hence thesis by A1013,A1023,A1025,A1038,A1032,GOBRD13:46;
end;
suppose
front_left_cell(f1,m,Gauge(C,n)) misses C &
front_right_cell(f1,m,Gauge(C,n)) meets C;
then f1 goes_straight m,Gauge(C,n) & f2 goes_straight m,
Gauge(C,n) by A1012,A1018,A1023,A1028,A1038,A1032,A1030,A1041,A1040,A1039,A1031
;
hence thesis by A1013,A1023,A1025,A1038,A1032,GOBRD13:48;
end;
suppose
front_left_cell(f1,m,Gauge(C,n)) meets C;
then f1 turns_left m,Gauge(C,n) & f2 turns_left m,Gauge(C,n)
by A1012,A1018,A1023,A1028,A1038,A1032,A1030,A1040,A1031;
hence thesis by A1013,A1023,A1025,A1038,A1032,GOBRD13:47;
end;
end;
hence thesis;
end;
suppose
A1042: k >= len f1;
A1043: 1 < len f1 by GOBOARD7:34,XXREAL_0:2;
A1044: f1 = f1|k by A1042,FINSEQ_1:58;
then len f1 in dom(f1|k) by FINSEQ_5:6;
then
A1045: (f2|k)/.len f1 = f2/.len f1 by A1023,FINSEQ_4:70;
1 in dom(f1|k) & len f1 <= len f2 by A1023,A1044,FINSEQ_5:6,16;
then
A1046: len f2 = len f1 by A1023,A1026,A1044,A1045,A1043,FINSEQ_4:70
,GOBOARD7:38;
A1047: k+1 > len f1 by A1042,NAT_1:13;
hence f1|(k+1) = f1 by FINSEQ_1:58
.= f2 by A1023,A1042,A1044,A1046,FINSEQ_1:58
.= f2|(k+1) by A1046,A1047,FINSEQ_1:58;
end;
end;
hence thesis;
end;
end;
A1048: P[0];
for k holds P[k] from NAT_1:sch 2(A1048,A1019);
hence thesis by Th2;
end;
end;
theorem Th31:
C is connected implies
for n,k being Nat st 1 <= k & k+1 <= len Cage(C,n)
holds left_cell
(Cage(C,n),k,Gauge(C,n)) misses C & right_cell(Cage(C,n),k,Gauge(C,n)) meets C
proof
assume
A1: C is connected;
let n be Nat;
set G = Gauge(C,n), f = Cage(C,n);
set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C;
A2: f is_sequence_on G by A1,Def1;
defpred P[Nat] means
for m being Nat st 1 <= m & m+1 <= len(f|$1) holds
left_cell(f|$1,m,G) misses C & right_cell(f|$1,m,G) meets C;
A3: len G = width G by JORDAN8:def 1;
A4: len G = 2|^n+3 by JORDAN8:def 1;
A5: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A6: for m being Nat st 1 <= m & m+1 <= len(f|k) holds left_cell(f|k,m,G)
misses C & right_cell(f|k,m,G) meets C;
per cases;
suppose
k >= len f;
then f|k = f & f|(k+1) = f by FINSEQ_1:58,NAT_1:12;
hence thesis by A6;
end;
suppose
A7: k < len f;
then
A8: len(f|k) = k by FINSEQ_1:59;
A9: 1 <= len G by A4,NAT_1:12;
A10: f|k is_sequence_on G by A2,GOBOARD1:22;
A11: f|(k+1) is_sequence_on G by A2,GOBOARD1:22;
consider i being Nat such that
A12: 1 <= i and
A13: i+1 <= len G and
A14: f/.1 = G*(i,width G) & f/.2 = G*(i+1,width G) and
A15: N-min C in cell(G,i,width G-'1) and
N-min C <> G*(i,width G-'1) by A1,Def1;
let m be Nat such that
A16: 1 <= m and
A17: m+1 <= len(f|(k+1));
A18: k+1 <= len f by A7,NAT_1:13;
then
A19: len(f|(k+1)) = k+1 by FINSEQ_1:59;
A20: 2|^n >= n+1 by NEWTON:85;
now
per cases by NAT_1:25;
suppose
A21: k = 0;
1 <= m+1 by NAT_1:12;
then m+1 = 0+1 by A19,A17,A21,XXREAL_0:1;
hence thesis by A16;
end;
suppose
A22: k = 1;
1+1 <= m+1 by A16,XREAL_1:6;
then
A23: m+1 = 1+1 by A19,A17,A22,XXREAL_0:1;
f|(k+1) = <*G*(i,width G),G*(i+1,width G)*> by A18,A14,A22,
FINSEQ_5:81;
then
A24: (f|(k+1))/.1 = G*(i,width G) & (f|(k+1))/.2 = G*(i+1, width G)
by FINSEQ_4:17;
1 <= i+1 by A12,NAT_1:13;
then
A25: [i+1,len G] in Indices G by A3,A13,A9,MATRIX_0:30;
A26: i < len G by A13,NAT_1:13;
then
A27: [i,len G] in Indices G by A3,A12,A9,MATRIX_0:30;
A28: i < i+1 & i+1 < (i+1)+1 by NAT_1:13;
then
A29: left_cell(f|(k+1),m,G) = cell(G,i,len G) by A3,A11,A17,A24,A27,A25
,A23,GOBRD13:def 3;
now
N > S by JORDAN8:9;
then N-S > S-S by XREAL_1:9;
then (N-S)/(2|^n) > 0 by A20,XREAL_1:139;
then
A30: N+0 < N+(N-S)/(2|^n) by XREAL_1:6;
assume left_cell(f|(k+1),m,G) meets C;
then consider p being object such that
A31: p in cell(G,i,len G) and
A32: p in C by A29,XBOOLE_0:3;
reconsider p as Element of TOP-REAL 2 by A31;
A33: p`2 <= N by A32,PSCOMP_1:24;
[1,len G] in Indices G by A3,A9,MATRIX_0:30;
then G*(1,len G) = |[W+((E-W)/(2|^n))*(1-2),S+((N-S)/(2 |^n))*((
len G)- 2)]| by JORDAN8:def 1;
then
A34: G*(1,len G)`2 = S+((N-S)/(2|^n))*((len G)-2) by EUCLID:52;
cell(G,i,len G) = { |[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)
`1 & G*(1,len G)`2 <= s } by A3,A12,A26,GOBRD11:31;
then consider r,s such that
A35: p = |[r,s]| and
G*(i,1)`1 <= r and
r <= G*(i+1,1)`1 and
A36: G*(1,len G)`2 <= s by A31;
((N-S)/(2|^n))*((len G)-2) = ((N-S)/(2|^n))*(2|^n)+((N-S)/(2
|^n))*1 by A4
.= (N-S)+(N-S)/(2|^n) by A20,XCMPLX_1:87;
then N < s by A36,A34,A30,XXREAL_0:2;
hence contradiction by A35,A33,EUCLID:52;
end;
hence left_cell(f|(k+1),m,G) misses C;
N-min C in C & N-min C in right_cell(f|(k+1),m,G) by A3,A11,A15,A17
,A24,A27,A25,A23,A28,GOBRD13:def 2,SPRECT_1:11;
hence right_cell(f|(k+1),m,G) meets C by XBOOLE_0:3;
end;
suppose
A37: k > 1;
then
A38: (len(f|k)) -'1 +1 = len(f|k) by A8,XREAL_1:235;
A39: 1 <= (len(f|k))-'1 by A8,A37,NAT_D:49;
now
per cases;
suppose
A40: m+1 = len(f|(k+1));
A41: len(f|k) <= len f by FINSEQ_5:16;
now
left_cell(f|k,(len(f|k))-'1,G) misses C by A6,A39,A38;
then
A42: left_cell(f,(len(f|k))-'1,G) misses C by A2,A8,A39,A38,A41,
GOBRD13:31;
A43: (len(f|k))-'1+(1+1) = (len(f|k))+1 by A38;
right_cell(f|k,(len(f|k))-'1,G) meets C by A6,A39,A38;
then
A44: right_cell(f,(len(f|k))-'1,G) meets C by A2,A8,A39,A38,A41,
GOBRD13:31;
consider i1,j1,i2,j2 being Nat such that
A45: [i1,j1] in Indices G & f/.((len(f|k)) -'1) = G*(i1,j1 ) and
A46: [i2,j2] in Indices G and
A47: f/.len(f|k) = G*(i2,j2) and
i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 &
j1 = j2 or i1 = i2 & j1 = j2+1 by A2,A7,A8,A39,A38,JORDAN8:3;
1 <= i2 by A46,MATRIX_0:32;
then
A48: (i2-'1)+1 = i2 by XREAL_1:235;
1 <= j2 by A46,MATRIX_0:32;
then
A49: (j2-'1)+1 = j2 by XREAL_1:235;
per cases;
suppose
A50: front_left_cell(f,(len(f|k))-'1,G) misses C &
front_right_cell(f,(len(f|k))-'1,G) misses C;
then
A51: f turns_right (len(f|k))-'1,G by A1,A18,A8,A39,A43,Def1;
now
per cases by A38,A45,A46,A47,A51;
suppose that
A52: i1 = i2 & j1+1 = j2 and
A53: [i2+1,j2] in Indices G & f/.((len(f|k))+1) = G*
(i2+1,j2);
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2)
by A2,A39,A38,A41,A45,A46,A47,A52,GOBRD13:35;
then
left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40,A46
,A47,A50,A52,A53,GOBRD13:23;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
j2-'1 = j1 & cell(G,i1,j1) meets C by A2,A39,A38,A41,A45
,A46,A47,A44,A52,GOBRD13:22,NAT_D:34;
then
right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40,A46
,A47,A52,A53,GOBRD13:24;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A54: i1+1 = i2 & j1 = j2 and
A55: [i2,j2-'1] in Indices G & f/.((len(f|k))+1) = G
*(i2,j2-'1);
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2
-'1) by A2,A39,A38,A41,A45,A46,A47,A54,GOBRD13:37;
then
left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40,A46
,A47,A49,A50,A55,GOBRD13:27;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
i2-'1 = i1 & cell(G,i1,j1-'1) meets C by A2,A39,A38,A41
,A45,A46,A47,A44,A54,GOBRD13:24,NAT_D:34;
then
right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40,A46
,A47,A49,A54,A55,GOBRD13:28;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A56: i1 = i2+1 & j1 = j2 and
A57: [i2,j2+1] in Indices G & f/.((len(f|k))+1) = G*
(i2,j2+1);
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,
j2) by A2,A39,A38,A41,A45,A46,A47,A56,GOBRD13:39;
then
left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40,A46
,A47,A50,A57,GOBRD13:21;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
cell(G,i2,j2) meets C by A2,A39,A38,A41,A45,A46,A47,A44
,A56,GOBRD13:26;
then
right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40,A46
,A47,A57,GOBRD13:22;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A58: i1 = i2 & j1 = j2+1 and
A59: [i2-'1,j2] in Indices G & f/.((len(f|k))+1) = G
*(i2-'1,j2);
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,
j2-'1 ) by A2,A39,A38,A41,A45,A46,A47,A58,GOBRD13:41;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A50,A59,GOBRD13:25;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
cell(G,i2-'1,j2) meets C by A2,A39,A38,A41,A45,A46,A47
,A44,A58,GOBRD13:28;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A59,GOBRD13:26;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
end;
hence thesis;
end;
suppose
A60: front_left_cell(f,(len(f|k))-'1,G) misses C &
front_right_cell(f,(len(f|k))-'1,G) meets C;
then
A61: f goes_straight (len(f|k))-'1,G by A1,A18,A8,A39,A43,Def1;
now
per cases by A38,A45,A46,A47,A61;
suppose that
A62: i1 = i2 & j1+1 = j2 and
A63: [i2,j2+1] in Indices G & f/.(len(f|k)+1) = G*( i2,j2+1);
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1,
j2) by A2,A39,A38,A41,A45,A46,A47,A62,GOBRD13:34;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A60,A62,A63,GOBRD13:21;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i1,j2
) by A2,A39,A38,A41,A45,A46,A47,A62,GOBRD13:35;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A60,A62,A63,GOBRD13:22;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A64: i1+1 = i2 & j1 = j2 and
A65: [i2+1,j2] in Indices G & f/.(len(f|k)+1) = G*( i2+1,j2);
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2)
by A2,A39,A38,A41,A45,A46,A47,A64,GOBRD13:36;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A60,A65,GOBRD13:23;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2
-'1) by A2,A39,A38,A41,A45,A46,A47,A64,GOBRD13:37;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A60,A65,GOBRD13:24;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A66: i1 = i2+1 & j1 = j2 and
A67: [i2-'1,j2] in Indices G & f/.(len(f|k)+1) = G*
(i2-'1,j2);
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,
j2-'1) by A2,A39,A38,A41,A45,A46,A47,A66,GOBRD13:38;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A60,A67,GOBRD13:25;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1
,j2) by A2,A39,A38,A41,A45,A46,A47,A66,GOBRD13:39;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A60,A67,GOBRD13:26;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A68: i1 = i2 & j1 = j2+1 and
A69: [i2,j2-'1] in Indices G & f/.(len(f|k)+1) = G*
(i2,j2-'1);
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2
-'1) by A2,A39,A38,A41,A45,A46,A47,A68,GOBRD13:40;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A49,A60,A69,GOBRD13:27;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_right_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1
,j2-'1 ) by A2,A39,A38,A41,A45,A46,A47,A68,GOBRD13:41;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A49,A60,A69,GOBRD13:28;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
end;
hence thesis;
end;
suppose
A70: front_left_cell(f,(len(f|k))-'1,G) meets C;
then
A71: f turns_left (len(f|k))-'1,G by A1,A18,A8,A39,A43,Def1;
now
per cases by A38,A45,A46,A47,A71;
suppose that
A72: i1 = i2 & j1+1 = j2 and
A73: [i2-'1,j2] in Indices G & f/.(len(f|k)+1) = G*
(i2-'1,j2);
j2-'1 = j1 & cell(G,i1-'1,j1) misses C by A2,A39,A38,A41
,A45,A46,A47,A42,A72,GOBRD13:21,NAT_D:34;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A72,A73,GOBRD13:25;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i1-'1,
j2) by A2,A39,A38,A41,A45,A46,A47,A72,GOBRD13:34;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A48,A70,A72,A73,GOBRD13:26;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A74: i1+1 = i2 & j1 = j2 and
A75: [i2,j2+1] in Indices G & f/.(len(f|k)+1) = G*( i2,j2+1);
i2-'1 = i1 & cell(G,i1,j1) misses C by A2,A39,A38,A41,A45
,A46,A47,A42,A74,GOBRD13:23,NAT_D:34;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A74,A75,GOBRD13:21;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2)
by A2,A39,A38,A41,A45,A46,A47,A74,GOBRD13:36;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A70,A75,GOBRD13:22;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A76: i1 = i2+1 & j1 = j2 and
A77: [i2,j2-'1] in Indices G & f/.(len(f|k)+1) = G*
(i2,j2-'1);
cell(G,i2,j2-'1) misses C by A2,A39,A38,A41,A45,A46,A47
,A42,A76,GOBRD13:25;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A49,A77,GOBRD13:27;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2-'1,
j2-'1) by A2,A39,A38,A41,A45,A46,A47,A76,GOBRD13:38;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A49,A70,A77,GOBRD13:28;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
suppose that
A78: i1 = i2 & j1 = j2+1 and
A79: [i2+1,j2] in Indices G & f/.(len(f|k)+1) = G*( i2+1,j2);
cell(G,i2,j2) misses C by A2,A39,A38,A41,A45,A46,A47,A42
,A78,GOBRD13:27;
then left_cell(f,m,G) misses C by A2,A18,A8,A19,A16,A40
,A46,A47,A79,GOBRD13:23;
hence left_cell(f|(k+1),m,G) misses C by A2,A18,A19,A16
,A40,GOBRD13:31;
front_left_cell(f,(len(f|k))-'1,G) = cell(G,i2,j2
-'1) by A2,A39,A38,A41,A45,A46,A47,A78,GOBRD13:40;
then right_cell(f,m,G) meets C by A2,A18,A8,A19,A16,A40
,A46,A47,A70,A79,GOBRD13:24;
hence right_cell(f|(k+1),m,G) meets C by A2,A18,A19,A16
,A40,GOBRD13:31;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
suppose
m+1 <> len(f|(k+1));
then m+1 < len(f|(k+1)) by A17,XXREAL_0:1;
then
A80: m+1 <= len(f|k)by A8,A19,NAT_1:13;
then consider i1,j1,i2,j2 being Nat such that
A81: [i1,j1] in Indices G and
A82: (f|k)/.m = G*(i1,j1) and
A83: [i2,j2] in Indices G and
A84: (f|k)/.(m+1) = G*(i2,j2) and
A85: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+
1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A10,A16,JORDAN8:3;
A86: left_cell(f|k,m,G) misses C & right_cell(f|k,m,G) meets C
by A6,A16,A80;
A87: f|(k+1) = (f|k)^<*f/.(k+1)*> by A18,FINSEQ_5:82;
1 <= m+1 by NAT_1:12;
then m+1 in dom(f|k) by A80,FINSEQ_3:25;
then
A88: (f|(k+1))/.(m+1) = G*(i2,j2) by A84,A87,FINSEQ_4:68;
m <= len(f|k) by A80,NAT_1:13;
then m in dom(f|k) by A16,FINSEQ_3:25;
then
A89: (f|(k+1))/.m = G*(i1,j1) by A82,A87,FINSEQ_4:68;
now
per cases by A85;
suppose
A90: i1 = i2 & j1+1 = j2;
then left_cell(f|k,m,G) = cell(G,i1-'1,j1) & right_cell(f|k
,m,G) = cell(G,i1,j1) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:21,22;
hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A90,
GOBRD13:21,22;
end;
suppose
A91: i1+1 = i2 & j1 = j2;
then left_cell(f|k,m,G) = cell(G,i1,j1) & right_cell(f|k,m,
G) = cell(G,i1,j1-'1) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:23,24;
hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A91,
GOBRD13:23,24;
end;
suppose
A92: i1 = i2+1 & j1 = j2;
then left_cell(f|k,m,G) = cell(G,i2,j2-'1) & right_cell(f|k
,m,G) = cell(G,i2,j2) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:25,26;
hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A92,
GOBRD13:25,26;
end;
suppose
A93: i1 = i2 & j1 = j2+1;
then left_cell(f|k,m,G) = cell(G,i2,j2) & right_cell(f|k,m,
G) = cell(G,i1-'1,j2) by A10,A16,A80,A81,A82,A83,A84,GOBRD13:27,28;
hence thesis by A11,A16,A17,A81,A83,A86,A89,A88,A93,
GOBRD13:27,28;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
A94: f|len f = f by FINSEQ_1:58;
A95: P[0] by CARD_1:27;
for k being Nat holds P[k] from NAT_1:sch 2(A95,A5);
hence thesis by A94;
end;
theorem
C is connected implies
for n being Nat holds N-min L~Cage(C,n) = (Cage(C,n))/.1
proof
assume
A1: C is connected;
let n be Nat;
set f = Cage(C,n);
A2: for k being Nat st 1 <= k & k+1 <= len f
holds left_cell(f,k,Gauge(C,n)) misses C
& right_cell(f,k,Gauge(C,n)) meets C by A1,Th31;
f is_sequence_on Gauge(C,n) &
ex i being Nat st 1 <= i & i+1 <= len Gauge(C,n ) &
f/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n
)) & N-min C in cell(Gauge( C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)
*(i, width Gauge(C,n)-'1) by A1,Def1;
hence thesis by A2,Th30;
end;