TPTP Problem File: GEO436^1.p
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%------------------------------------------------------------------------------
% File : GEO436^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Geometric quantities)
% Problem : The University of Tokyo, 1991, Humanities Course, Problem 4
% Version : [Mat16] axioms : Especial.
% English : Consider the sphere that has its center on the base and is in
% contact with all the sides of the regular quadrangular pyramid
% V. Let a be the side length of the base, then find the following
% quantities: (1) The height of V (2) The volume of the intersection
% of the sphere and the pyramid V Here, regular quadrangular pyramid
% is a solid enclosed be a square base and four congruent isosceles
% triangles whose base is a side of the square base.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : Univ-Tokyo-1991-Bun-4.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3486 ( 728 unt;1200 typ; 0 def)
% Number of atoms : 6848 (2214 equ; 0 cnn)
% Maximal formula atoms : 31 ( 2 avg)
% Number of connectives : 39680 ( 104 ~; 233 |;1189 &;36028 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4462 ( 371 atm;1203 fun; 951 num;1937 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1219 (1176 usr; 73 con; 0-9 aty)
% Number of variables : 8063 ( 406 ^;7085 !; 436 ?;8063 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-04-15
% : Answer
% ^ [V_h_dot_0: $real] :
% ( ( $less @ 0.0 @ 'a/0' )
% & ( V_h_dot_0
% = ( $product @ ( $quotient @ ( 'sqrt/1' @ 2.0 ) @ 2.0 ) @ 'a/0' ) ) ) )
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include('Axioms/MAT001^0.ax').
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thf('a/0_type',type,
'a/0': $real ).
thf(p1_qustion,conjecture,
( 'find/1' @ $real
@ ^ [V_h: $real] :
? [V_A: '3d.Point',V_B: '3d.Point',V_C: '3d.Point',V_D: '3d.Point',V_E: '3d.Point',V_F: '3d.Shape',V_V: '3d.Shape'] :
( ( '3d.square-pyramid-type/1' @ V_V )
& ( '3d.is-regular-square/1' @ ( '3d.base-of/1' @ V_V ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_A ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_B ) ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_B ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_C ) ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_C ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_D ) ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_D ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_E @ V_A ) ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_A @ V_B ) )
= 'a/0' )
& ( '3d.sphere-type/1' @ V_F )
& ( '3d.on/2' @ ( '3d.center-of/1' @ V_F ) @ ( '3d.inner-part-of/1' @ ( '3d.base-of/1' @ V_V ) ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_A @ V_B ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_B @ V_C ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_C @ V_D ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_D @ V_A ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_E @ V_A ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_E @ V_B ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_E @ V_C ) )
& ( '3d.tangent/2' @ V_F @ ( '3d.seg/2' @ V_E @ V_D ) )
& ( V_h
= ( '3d.height-of/1' @ V_V ) ) ) ) ).
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