:: A Tree of Execution of a Macroinstruction
:: by Artur Korni{\l}owicz
::
:: Received December 10, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

:: LocSeq

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N;
let M be Subset of NAT;
deffunc H₁( object ) -> set = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . $1;

func LocSeq (M,S) -> Sequence of NAT means :Def1: :: AMISTD_3:def 1
( dom it = card M & ( for m being set st m in card M holds
it . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) );

existence
ex b₁ being Sequence of NAT st
( dom b₁ = card M & ( for m being set st m in card M holds
b₁ . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) )

proof end;

uniqueness
for b₁, b₂ being Sequence of NAT st dom b₁ = card M & ( for m being set st m in card M holds
b₁ . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) & dom b₂ = card M & ( for m being set st m in card M holds
b₂ . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines LocSeq AMISTD_3:def 1 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N
for M being Subset of NAT
for b₄ being Sequence of NAT holds
( b₄ = LocSeq (M,S) iff ( dom b₄ = card M & ( for m being set st m in card M holds
b₄ . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) ) );

theorem :: AMISTD_3:1

for n being Nat
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N
for F being Subset of NAT st F = {n} holds
LocSeq (F,S) = 0 .--> n

proof end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N;
let M be Subset of NAT;

cluster LocSeq (M,S) -> one-to-one ;

coherence
LocSeq (M,S) is one-to-one

proof end;

end;

:: Tree of Execution

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N;
let M be non empty preProgram of S;

func ExecTree M -> DecoratedTree of NAT means :Def2: :: AMISTD_3:def 2
( it . {} = FirstLoc M & ( for t being Element of dom it holds
( succ t = { (t ^ <*k*>) where k is Nat : k in card (NIC ((M /. (it . t)),(it . t))) } & ( for m being Nat st m in card (NIC ((M /. (it . t)),(it . t))) holds
it . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (it . t)),(it . t))),S)) . m ) ) ) );

existence
ex b₁ being DecoratedTree of NAT st
( b₁ . {} = FirstLoc M & ( for t being Element of dom b₁ holds
( succ t = { (t ^ <*k*>) where k is Nat : k in card (NIC ((M /. (b₁ . t)),(b₁ . t))) } & ( for m being Nat st m in card (NIC ((M /. (b₁ . t)),(b₁ . t))) holds
b₁ . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (b₁ . t)),(b₁ . t))),S)) . m ) ) ) )

proof end;

uniqueness
for b₁, b₂ being DecoratedTree of NAT st b₁ . {} = FirstLoc M & ( for t being Element of dom b₁ holds
( succ t = { (t ^ <*k*>) where k is Nat : k in card (NIC ((M /. (b₁ . t)),(b₁ . t))) } & ( for m being Nat st m in card (NIC ((M /. (b₁ . t)),(b₁ . t))) holds
b₁ . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (b₁ . t)),(b₁ . t))),S)) . m ) ) ) & b₂ . {} = FirstLoc M & ( for t being Element of dom b₂ holds
( succ t = { (t ^ <*k*>) where k is Nat : k in card (NIC ((M /. (b₂ . t)),(b₂ . t))) } & ( for m being Nat st m in card (NIC ((M /. (b₂ . t)),(b₂ . t))) holds
b₂ . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (b₂ . t)),(b₂ . t))),S)) . m ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines ExecTree AMISTD_3:def 2 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N
for M being non empty preProgram of S
for b₄ being DecoratedTree of NAT holds
( b₄ = ExecTree M iff ( b₄ . {} = FirstLoc M & ( for t being Element of dom b₄ holds
( succ t = { (t ^ <*k*>) where k is Nat : k in card (NIC ((M /. (b₄ . t)),(b₄ . t))) } & ( for m being Nat st m in card (NIC ((M /. (b₄ . t)),(b₄ . t))) holds
b₄ . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (b₄ . t)),(b₄ . t))),S)) . m ) ) ) ) );

theorem :: AMISTD_3:2

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting standard AMI-Struct over N holds ExecTree (Stop S) = TrivialInfiniteTree --> 0

proof end;