Article QC_LANG4, MML version 4.99.1005
:: QC_LANG4:th 4
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like FinSequence-like set holds
ex b3 being Relation-like Function-like FinSequence-like set st
b3 = b2 | Seg b1 & b3 c= b2;
:: QC_LANG4:th 6
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Relation-like Function-like FinSequence-like set
for b5 being Element of NAT
st b5 + 1 <= len b2 & b3 = b2 | Seg (b5 + 1) & b4 = b2 | Seg b5
holds ex b6 being Element of b1 st
b3 = b4 ^ <*b6*>;
:: QC_LANG4:th 7
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Relation-like Function-like FinSequence-like set
st 1 <= len b2 & b3 = b2 | Seg 1
holds ex b4 being Element of b1 st
b3 = <*b4*>;
:: QC_LANG4:exreg 1
registration
let a1 be non empty set;
cluster Relation-like Function-like finite DecoratedTree-like ParametrizedSubset of a1;
end;
:: QC_LANG4:th 8
theorem
for b1 being Relation-like Function-like DecoratedTree-like set
for b2 being FinSequence of NAT holds
b1 . b2 = (b1 | b2) . {};
:: QC_LANG4:th 9
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Element of proj1 b1 holds
succ(b1,b2) = b2 succ * b1;
:: QC_LANG4:th 10
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Element of proj1 b1 holds
proj1 (b2 succ * b1) = dom (b2 succ);
:: QC_LANG4:th 11
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Element of proj1 b1 holds
dom succ(b1,b2) = dom (b2 succ);
:: QC_LANG4:th 12
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Element of proj1 b1
for b3 being Element of NAT holds
b2 ^ <*b3*> in proj1 b1
iff
b3 + 1 in dom (b2 succ);
:: QC_LANG4:th 13
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being Element of NAT
st b2 ^ <*b3*> in proj1 b1
holds b1 . (b2 ^ <*b3*>) = (succ(b1,b2)) . (b3 + 1);
:: QC_LANG4:th 14
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2, b3 being Element of proj1 b1
st b2 in succ b3
holds b1 . b2 in proj2 succ(b1,b3);
:: QC_LANG4:th 15
theorem
for b1 being Relation-like Function-like DecoratedTree-like finite-branching set
for b2 being Element of proj1 b1
for b3 being set
st b3 in proj2 succ(b1,b2)
holds ex b4 being Element of proj1 b1 st
b3 = b1 . b4 & b4 in succ b2;
:: QC_LANG4:sch 1
scheme QC_LANG4:sch 1
{F1 -> non empty set,
F2 -> Element of F1(),
F3 -> FinSequence of F1()}:
ex b1 being Function-like DecoratedTree-like finite-branching ParametrizedSubset of F1() st
b1 . {} = F2() &
(for b2 being Element of proj1 b1
for b3 being Element of F1()
st b3 = b1 . b2
holds succ(b1,b2) = F3(b3))
:: QC_LANG4:th 16
theorem
for b1 being non empty Tree-like set
for b2 being Element of b1 holds
ProperPrefixes b2 is finite Chain of b1;
:: QC_LANG4:th 17
theorem
for b1 being non empty Tree-like set holds
b1 -level 0 = {{}};
:: QC_LANG4:th 18
theorem
for b1 being Element of NAT
for b2 being non empty Tree-like set holds
b2 -level (b1 + 1) = union {succ b3 where b3 is Element of b2: len b3 = b1};
:: QC_LANG4:th 19
theorem
for b1 being non empty Tree-like finite-branching set
for b2 being Element of NAT holds
b1 -level b2 is finite;
:: QC_LANG4:th 20
theorem
for b1 being non empty Tree-like finite-branching set holds
b1 is finite
iff
ex b2 being Element of NAT st
b1 -level b2 = {};
:: QC_LANG4:th 21
theorem
for b1 being non empty Tree-like finite-branching set
st b1 is infinite
holds ex b2 being Chain of b1 st
b2 is infinite;
:: QC_LANG4:th 22
theorem
for b1 being non empty Tree-like finite-branching set
st b1 is infinite
holds ex b2 being Branch-like Chain of b1 st
b2 is infinite;
:: QC_LANG4:th 23
theorem
for b1 being non empty Tree-like set
for b2 being Chain of b1
for b3 being Element of b1
st b3 in b2 & b2 is infinite
holds ex b4 being Element of b1 st
b4 in b2 & b3 c< b4;
:: QC_LANG4:th 24
theorem
for b1 being non empty Tree-like set
for b2 being Branch-like Chain of b1
for b3 being Element of b1
st b3 in b2 & b2 is infinite
holds ex b4 being Element of b1 st
b4 in b2 & b4 in succ b3;
:: QC_LANG4:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,NAT
st for b2 being Element of NAT holds
b1 . (b2 + 1) <= b1 . b2
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds b1 . b3 = b1 . b2;
:: QC_LANG4:sch 2
scheme QC_LANG4:sch 2
{F1 -> non empty set,
F2 -> Function-like DecoratedTree-like finite-branching ParametrizedSubset of F1(),
F3 -> Element of NAT}:
F2() is finite
provided
for b1, b2 being Element of proj1 F2()
for b3 being Element of F1()
st b2 in succ b1 & b3 = F2() . b2
holds F3(b3) < F3(F2() . b1);
:: QC_LANG4:th 26
theorem
for b1 being non empty set
for b2 being Function-like DecoratedTree-like ParametrizedSubset of b1
for b3 being set
st b3 in proj2 b2
holds b3 is Element of b1;
:: QC_LANG4:th 27
theorem
for b1 being non empty set
for b2 being Function-like DecoratedTree-like ParametrizedSubset of b1
for b3 being set
st b3 in proj1 b2
holds b2 . b3 is Element of b1;
:: QC_LANG4:th 28
theorem
for b1, b2 being Element of QC-WFF
st b1 is_subformula_of b2
holds len @ b1 <= len @ b2;
:: QC_LANG4:th 29
theorem
for b1, b2 being Element of QC-WFF
st b1 is_subformula_of b2 & len @ b1 = len @ b2
holds b1 = b2;
:: QC_LANG4:funcnot 1 => QC_LANG4:func 1
definition
let a1 be Element of QC-WFF;
func list_of_immediate_constituents A1 -> FinSequence of QC-WFF equals
<*> QC-WFF
if (a1 = VERUM or a1 is atomic),
<*the_argument_of a1*>
if a1 is negative,
<*the_left_argument_of a1,the_right_argument_of a1*>
if a1 is conjunctive
otherwise <*the_scope_of a1*>;
end;
:: QC_LANG4:def 1
theorem
for b1 being Element of QC-WFF holds
(b1 <> VERUM & b1 is not atomic or list_of_immediate_constituents b1 = <*> QC-WFF) &
(b1 is negative implies list_of_immediate_constituents b1 = <*the_argument_of b1*>) &
(b1 is conjunctive implies list_of_immediate_constituents b1 = <*the_left_argument_of b1,the_right_argument_of b1*>) &
(b1 <> VERUM & b1 is not atomic & b1 is not negative & b1 is not conjunctive implies list_of_immediate_constituents b1 = <*the_scope_of b1*>);
:: QC_LANG4:th 30
theorem
for b1 being Element of NAT
for b2, b3 being Element of QC-WFF
st b1 in dom list_of_immediate_constituents b2 & b3 = (list_of_immediate_constituents b2) . b1
holds b3 is_immediate_constituent_of b2;
:: QC_LANG4:th 31
theorem
for b1 being Element of QC-WFF holds
proj2 list_of_immediate_constituents b1 = {b2 where b2 is Element of QC-WFF: b2 is_immediate_constituent_of b1};
:: QC_LANG4:funcnot 2 => QC_LANG4:func 2
definition
let a1 be Element of QC-WFF;
func tree_of_subformulae A1 -> Function-like finite DecoratedTree-like ParametrizedSubset of QC-WFF means
it . {} = a1 &
(for b1 being Element of proj1 it holds
succ(it,b1) = list_of_immediate_constituents (it . b1));
end;
:: QC_LANG4:def 2
theorem
for b1 being Element of QC-WFF
for b2 being Function-like finite DecoratedTree-like ParametrizedSubset of QC-WFF holds
b2 = tree_of_subformulae b1
iff
b2 . {} = b1 &
(for b3 being Element of proj1 b2 holds
succ(b2,b3) = list_of_immediate_constituents (b2 . b3));
:: QC_LANG4:th 34
theorem
for b1 being Element of QC-WFF holds
b1 in proj2 tree_of_subformulae b1;
:: QC_LANG4:th 35
theorem
for b1 being Element of NAT
for b2 being Element of QC-WFF
for b3 being Element of proj1 tree_of_subformulae b2
st b3 ^ <*b1*> in proj1 tree_of_subformulae b2
holds ex b4 being Element of QC-WFF st
b4 = (tree_of_subformulae b2) . (b3 ^ <*b1*>) &
b4 is_immediate_constituent_of (tree_of_subformulae b2) . b3;
:: QC_LANG4:th 36
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of proj1 tree_of_subformulae b2 holds
b1 is_immediate_constituent_of (tree_of_subformulae b2) . b3
iff
ex b4 being Element of NAT st
b3 ^ <*b4*> in proj1 tree_of_subformulae b2 &
b1 = (tree_of_subformulae b2) . (b3 ^ <*b4*>);
:: QC_LANG4:th 37
theorem
for b1, b2, b3 being Element of QC-WFF
st b1 in proj2 tree_of_subformulae b2 & b3 is_immediate_constituent_of b1
holds b3 in proj2 tree_of_subformulae b2;
:: QC_LANG4:th 38
theorem
for b1, b2, b3 being Element of QC-WFF
st b1 in proj2 tree_of_subformulae b2 & b3 is_subformula_of b1
holds b3 in proj2 tree_of_subformulae b2;
:: QC_LANG4:th 39
theorem
for b1, b2 being Element of QC-WFF holds
b1 in proj2 tree_of_subformulae b2
iff
b1 is_subformula_of b2;
:: QC_LANG4:th 40
theorem
for b1 being Element of QC-WFF holds
proj2 tree_of_subformulae b1 = Subformulae b1;
:: QC_LANG4:th 41
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 in succ b3
holds (tree_of_subformulae b1) . b2 is_immediate_constituent_of (tree_of_subformulae b1) . b3;
:: QC_LANG4:th 42
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 c= b3
holds (tree_of_subformulae b1) . b3 is_subformula_of (tree_of_subformulae b1) . b2;
:: QC_LANG4:th 43
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 c< b3
holds len @ ((tree_of_subformulae b1) . b3) < len @ ((tree_of_subformulae b1) . b2);
:: QC_LANG4:th 44
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 c< b3
holds (tree_of_subformulae b1) . b3 <> (tree_of_subformulae b1) . b2;
:: QC_LANG4:th 45
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 c< b3
holds (tree_of_subformulae b1) . b3 is_proper_subformula_of (tree_of_subformulae b1) . b2;
:: QC_LANG4:th 46
theorem
for b1 being Element of QC-WFF
for b2 being Element of proj1 tree_of_subformulae b1 holds
(tree_of_subformulae b1) . b2 = b1
iff
b2 = {};
:: QC_LANG4:th 47
theorem
for b1 being Element of QC-WFF
for b2, b3 being Element of proj1 tree_of_subformulae b1
st b2 <> b3 &
(tree_of_subformulae b1) . b2 = (tree_of_subformulae b1) . b3
holds not b2,b3 are_c=-comparable;
:: QC_LANG4:funcnot 3 => QC_LANG4:func 3
definition
let a1, a2 be Element of QC-WFF;
func A1 -entry_points_in_subformula_tree_of A2 -> AntiChain_of_Prefixes of proj1 tree_of_subformulae a1 means
for b1 being Element of proj1 tree_of_subformulae a1 holds
b1 in it
iff
(tree_of_subformulae a1) . b1 = a2;
end;
:: QC_LANG4:def 3
theorem
for b1, b2 being Element of QC-WFF
for b3 being AntiChain_of_Prefixes of proj1 tree_of_subformulae b1 holds
b3 = b1 -entry_points_in_subformula_tree_of b2
iff
for b4 being Element of proj1 tree_of_subformulae b1 holds
b4 in b3
iff
(tree_of_subformulae b1) . b4 = b2;
:: QC_LANG4:th 49
theorem
for b1, b2 being Element of QC-WFF holds
b1 -entry_points_in_subformula_tree_of b2 = {b3 where b3 is Element of proj1 tree_of_subformulae b1: (tree_of_subformulae b1) . b3 = b2};
:: QC_LANG4:th 50
theorem
for b1, b2 being Element of QC-WFF holds
b1 is_subformula_of b2
iff
b2 -entry_points_in_subformula_tree_of b1 <> {};
:: QC_LANG4:th 51
theorem
for b1 being Element of NAT
for b2 being Element of QC-WFF
for b3, b4 being Element of proj1 tree_of_subformulae b2
st b3 = b4 ^ <*b1*> & (tree_of_subformulae b2) . b4 is negative
holds (tree_of_subformulae b2) . b3 = the_argument_of ((tree_of_subformulae b2) . b4) &
b1 = 0;
:: QC_LANG4:th 52
theorem
for b1 being Element of NAT
for b2 being Element of QC-WFF
for b3, b4 being Element of proj1 tree_of_subformulae b2
st b3 = b4 ^ <*b1*> &
(tree_of_subformulae b2) . b4 is conjunctive &
((tree_of_subformulae b2) . b3 = the_left_argument_of ((tree_of_subformulae b2) . b4) implies b1 <> 0)
holds (tree_of_subformulae b2) . b3 = the_right_argument_of ((tree_of_subformulae b2) . b4) &
b1 = 1;
:: QC_LANG4:th 53
theorem
for b1 being Element of NAT
for b2 being Element of QC-WFF
for b3, b4 being Element of proj1 tree_of_subformulae b2
st b3 = b4 ^ <*b1*> & (tree_of_subformulae b2) . b4 is universal
holds (tree_of_subformulae b2) . b3 = the_scope_of ((tree_of_subformulae b2) . b4) &
b1 = 0;
:: QC_LANG4:th 54
theorem
for b1 being Element of QC-WFF
for b2 being Element of proj1 tree_of_subformulae b1
st (tree_of_subformulae b1) . b2 is negative
holds b2 ^ <*0*> in proj1 tree_of_subformulae b1 &
(tree_of_subformulae b1) . (b2 ^ <*0*>) = the_argument_of ((tree_of_subformulae b1) . b2);
:: QC_LANG4:th 55
theorem
for b1 being Element of QC-WFF
for b2 being Element of proj1 tree_of_subformulae b1
st (tree_of_subformulae b1) . b2 is conjunctive
holds b2 ^ <*0*> in proj1 tree_of_subformulae b1 &
(tree_of_subformulae b1) . (b2 ^ <*0*>) = the_left_argument_of ((tree_of_subformulae b1) . b2) &
b2 ^ <*1*> in proj1 tree_of_subformulae b1 &
(tree_of_subformulae b1) . (b2 ^ <*1*>) = the_right_argument_of ((tree_of_subformulae b1) . b2);
:: QC_LANG4:th 56
theorem
for b1 being Element of QC-WFF
for b2 being Element of proj1 tree_of_subformulae b1
st (tree_of_subformulae b1) . b2 is universal
holds b2 ^ <*0*> in proj1 tree_of_subformulae b1 &
(tree_of_subformulae b1) . (b2 ^ <*0*>) = the_scope_of ((tree_of_subformulae b1) . b2);
:: QC_LANG4:th 57
theorem
for b1, b2, b3 being Element of QC-WFF
for b4 being Element of proj1 tree_of_subformulae b1
for b5 being Element of proj1 tree_of_subformulae b2
st b4 in b1 -entry_points_in_subformula_tree_of b2 & b5 in b2 -entry_points_in_subformula_tree_of b3
holds b4 ^ b5 in b1 -entry_points_in_subformula_tree_of b3;
:: QC_LANG4:th 58
theorem
for b1, b2, b3 being Element of QC-WFF
for b4 being Element of proj1 tree_of_subformulae b1
for b5 being Relation-like Function-like FinSequence-like set
st b4 in b1 -entry_points_in_subformula_tree_of b2 & b4 ^ b5 in b1 -entry_points_in_subformula_tree_of b3
holds b5 in b2 -entry_points_in_subformula_tree_of b3;
:: QC_LANG4:th 59
theorem
for b1, b2, b3 being Element of QC-WFF holds
{b4 ^ b5 where b4 is Element of proj1 tree_of_subformulae b1, b5 is Element of proj1 tree_of_subformulae b2: b4 in b1 -entry_points_in_subformula_tree_of b2 & b5 in b2 -entry_points_in_subformula_tree_of b3} c= b1 -entry_points_in_subformula_tree_of b3;
:: QC_LANG4:th 60
theorem
for b1 being Element of QC-WFF
for b2 being Element of proj1 tree_of_subformulae b1 holds
(tree_of_subformulae b1) | b2 = tree_of_subformulae ((tree_of_subformulae b1) . b2);
:: QC_LANG4:th 61
theorem
for b1, b2 being Element of QC-WFF
for b3 being Element of proj1 tree_of_subformulae b1 holds
b3 in b1 -entry_points_in_subformula_tree_of b2
iff
(tree_of_subformulae b1) | b3 = tree_of_subformulae b2;
:: QC_LANG4:th 62
theorem
for b1, b2 being Element of QC-WFF holds
b1 -entry_points_in_subformula_tree_of b2 = {b3 where b3 is Element of proj1 tree_of_subformulae b1: (tree_of_subformulae b1) | b3 = tree_of_subformulae b2};
:: QC_LANG4:th 63
theorem
for b1, b2, b3 being Element of QC-WFF
for b4 being Chain of proj1 tree_of_subformulae b1
st b2 in {(tree_of_subformulae b1) . b5 where b5 is Element of proj1 tree_of_subformulae b1: b5 in b4} &
b3 in {(tree_of_subformulae b1) . b5 where b5 is Element of proj1 tree_of_subformulae b1: b5 in b4} &
not b2 is_subformula_of b3
holds b3 is_subformula_of b2;
:: QC_LANG4:modenot 1 => QC_LANG4:mode 1
definition
let a1 be Element of QC-WFF;
mode Subformula of A1 -> Element of QC-WFF means
it is_subformula_of a1;
end;
:: QC_LANG4:dfs 4
definiens
let a1, a2 be Element of QC-WFF;
To prove
a2 is Subformula of a1
it is sufficient to prove
thus a2 is_subformula_of a1;
:: QC_LANG4:def 4
theorem
for b1, b2 being Element of QC-WFF holds
b2 is Subformula of b1
iff
b2 is_subformula_of b1;
:: QC_LANG4:modenot 2 => QC_LANG4:mode 2
definition
let a1 be Element of QC-WFF;
let a2 be Subformula of a1;
mode Entry_Point_in_Subformula_Tree of A2 -> Element of proj1 tree_of_subformulae a1 means
(tree_of_subformulae a1) . it = a2;
end;
:: QC_LANG4:dfs 5
definiens
let a1 be Element of QC-WFF;
let a2 be Subformula of a1;
let a3 be Element of proj1 tree_of_subformulae a1;
To prove
a3 is Entry_Point_in_Subformula_Tree of a2
it is sufficient to prove
thus (tree_of_subformulae a1) . a3 = a2;
:: QC_LANG4:def 5
theorem
for b1 being Element of QC-WFF
for b2 being Subformula of b1
for b3 being Element of proj1 tree_of_subformulae b1 holds
b3 is Entry_Point_in_Subformula_Tree of b2
iff
(tree_of_subformulae b1) . b3 = b2;
:: QC_LANG4:th 65
theorem
for b1 being Element of QC-WFF
for b2 being Subformula of b1
for b3, b4 being Entry_Point_in_Subformula_Tree of b2
st b3 <> b4
holds not b3,b4 are_c=-comparable;
:: QC_LANG4:funcnot 4 => QC_LANG4:func 4
definition
let a1 be Element of QC-WFF;
let a2 be Subformula of a1;
func entry_points_in_subformula_tree A2 -> non empty AntiChain_of_Prefixes of proj1 tree_of_subformulae a1 equals
a1 -entry_points_in_subformula_tree_of a2;
end;
:: QC_LANG4:def 6
theorem
for b1 being Element of QC-WFF
for b2 being Subformula of b1 holds
entry_points_in_subformula_tree b2 = b1 -entry_points_in_subformula_tree_of b2;
:: QC_LANG4:th 67
theorem
for b1 being Element of QC-WFF
for b2 being Subformula of b1
for b3 being Entry_Point_in_Subformula_Tree of b2 holds
b3 in entry_points_in_subformula_tree b2;
:: QC_LANG4:th 68
theorem
for b1 being Element of QC-WFF
for b2 being Subformula of b1 holds
entry_points_in_subformula_tree b2 = {b3 where b3 is Entry_Point_in_Subformula_Tree of b2: b3 = b3};
:: QC_LANG4:th 69
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1
for b4 being Entry_Point_in_Subformula_Tree of b2
for b5 being Element of proj1 tree_of_subformulae b2
st b5 in b2 -entry_points_in_subformula_tree_of b3
holds b4 ^ b5 is Entry_Point_in_Subformula_Tree of b3;
:: QC_LANG4:th 70
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1
for b4 being Entry_Point_in_Subformula_Tree of b3
for b5 being Relation-like Function-like FinSequence-like set
st b4 ^ b5 is Entry_Point_in_Subformula_Tree of b2
holds b5 in b3 -entry_points_in_subformula_tree_of b2;
:: QC_LANG4:th 71
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1 holds
{b4 ^ b5 where b4 is Entry_Point_in_Subformula_Tree of b2, b5 is Element of proj1 tree_of_subformulae b2: b5 in b2 -entry_points_in_subformula_tree_of b3} = {b4 ^ b5 where b4 is Element of proj1 tree_of_subformulae b1, b5 is Element of proj1 tree_of_subformulae b2: b4 in b1 -entry_points_in_subformula_tree_of b2 & b5 in b2 -entry_points_in_subformula_tree_of b3};
:: QC_LANG4:th 72
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1 holds
{b4 ^ b5 where b4 is Entry_Point_in_Subformula_Tree of b2, b5 is Element of proj1 tree_of_subformulae b2: b5 in b2 -entry_points_in_subformula_tree_of b3} c= entry_points_in_subformula_tree b3;
:: QC_LANG4:th 73
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1
st ex b4 being Entry_Point_in_Subformula_Tree of b2 st
ex b5 being Entry_Point_in_Subformula_Tree of b3 st
b4 c= b5
holds b3 is_subformula_of b2;
:: QC_LANG4:th 74
theorem
for b1 being Element of QC-WFF
for b2, b3 being Subformula of b1
st b2 is_subformula_of b3
for b4 being Entry_Point_in_Subformula_Tree of b3 holds
ex b5 being Entry_Point_in_Subformula_Tree of b2 st
b4 c= b5;