TPTP Problem File: DAT372^1.p
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% File : DAT372^1 : TPTP v9.2.1. Released v9.2.0.
% Domain : Data Structures
% Problem : Associativity of list append polymorphic, induction step 5a typed
% Version : Especial.
% English :
% Refs : [RRB23] Rothgang et al. (2023), Theorem Proving in Dependently
% : [Rot25] Rothgang (2025), Email to Geoff Sutcliffe
% : [RK+25] Ranalter et al. (2025), The Dependently Typed Higher-O
% Source : [Rot25]
% Names : ListAppAssoc/list-app-assoc-indstep5-typeInstantiated_alt.p [Rot25]
% Status : Theorem
% Rating : ? v9.2.0
% Syntax : Number of formulae : 17 ( 5 unt; 8 typ; 0 def)
% Number of atoms : 14 ( 14 equ; 0 cnn)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 893 ( 0 ~; 0 |; 0 &; 888 @)
% ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 7 avg)
% Number of types : 1 ( 1 usr)
% Number of type decls : 8 ( 3 !>P; 2 !>D)
% Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% Number of symbols : 8 ( 7 usr; 1 con; 0-5 aty)
% Number of variables : 55 ( 0 ^; 49 !; 0 ?; 55 :)
% ( 6 !>; 0 ?*; 0 @-; 0 @+)
% SPC : DH1_THM_EQU_NAR
% Comments :
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thf(nat_type,type,
nat: $tType ).
thf(zero_type,type,
zero: nat ).
thf(suc_type,type,
suc: nat > nat ).
thf(plus_type,type,
plus: nat > nat > nat ).
thf(list_type,type,
list: $tType > nat > $tType ).
thf(nil_type,type,
nil:
!>[A: $tType] : ( list @ A @ zero ) ).
thf(cons_type,type,
cons:
!>[A: $tType,N: nat] : ( A > ( list @ A @ N ) > ( list @ A @ ( suc @ N ) ) ) ).
thf(app_type,type,
app:
!>[A: $tType,N: nat,M: nat] : ( ( list @ A @ N ) > ( list @ A @ M ) > ( list @ A @ ( plus @ N @ M ) ) ) ).
thf(ax2,axiom,
! [N: nat,M: nat] :
( ( plus @ ( suc @ N ) @ M )
= ( suc @ ( plus @ N @ M ) ) ) ).
thf(ax4,axiom,
! [A: $tType,N: nat,M: nat,X: A,Y: list @ A @ N,Z: list @ A @ M] :
( ( app @ A @ ( suc @ N ) @ M @ ( cons @ A @ N @ X @ Y ) @ Z )
= ( cons @ A @ ( plus @ N @ M ) @ X @ ( app @ A @ N @ M @ Y @ Z ) ) ) ).
thf(plus_assoc,axiom,
! [M1: nat,M2: nat,M3: nat] :
( ( plus @ M1 @ ( plus @ M2 @ M3 ) )
= ( plus @ ( plus @ M1 @ M2 ) @ M3 ) ) ).
thf(plus_s_a,axiom,
! [N: nat,M: nat,K: nat] :
( ( suc @ ( plus @ ( plus @ N @ M ) @ K ) )
= ( plus @ ( suc @ N ) @ ( plus @ M @ K ) ) ) ).
thf(list_app_assoc_indstep1,axiom,
! [M2: nat,L2: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2,M3: nat,L3: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M3,M: nat,X: list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ),L: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M] :
( ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ ( plus @ M @ M2 ) ) @ M3 @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) ) @ L3 ) )
=> ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ ( suc @ M ) @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ M2 @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ L2 ) @ L3 ) ) ) ).
thf(list_app_assoc_indstep2,axiom,
! [M2: nat,L2: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2,M3: nat,L3: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M3,M: nat,X: list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ),L: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M] :
( ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ ( plus @ M @ M2 ) @ M3 ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) @ L3 ) ) )
=> ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ ( plus @ M @ M2 ) ) @ M3 @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) ) @ L3 ) ) ) ).
thf(list_app_assoc_indstep3,axiom,
! [M2: nat,L2: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2,M3: nat,L3: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M3,M: nat,X: list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ),L: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M] :
( ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ ( plus @ M2 @ M3 ) @ L @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) @ L3 ) )
=> ( ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ ( plus @ M2 @ M3 ) ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ ( plus @ M2 @ M3 ) @ L @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) ) ) )
=> ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ ( plus @ M @ M2 ) @ M3 ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) @ L3 ) ) ) ) ) ).
thf(list_app_assoc_indstep4,axiom,
! [M2: nat,L2: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2,M3: nat,L3: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M3,M: nat,X: list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ),L: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M] :
( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ ( plus @ M2 @ M3 ) ) @ X @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ ( plus @ M2 @ M3 ) @ L @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) ) ) ) ).
thf(list_app_assoc_indstep,conjecture,
! [M2: nat,L2: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2,M3: nat,L3: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M3,M: nat,X: list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ),L: list @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M] :
( ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ ( plus @ M2 @ M3 ) @ L @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ M @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ M2 @ L @ L2 ) @ L3 ) )
=> ( ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ ( plus @ M2 @ M3 ) @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M2 @ M3 @ L2 @ L3 ) )
= ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( plus @ ( suc @ M ) @ M2 ) @ M3 @ ( app @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ ( suc @ M ) @ M2 @ ( cons @ ( list @ ( list @ nat @ ( suc @ zero ) ) @ ( suc @ zero ) ) @ M @ X @ L ) @ L2 ) @ L3 ) ) ) ).
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