%   ORIGINAL: h4/ringNorm/interp__sp__def_c2
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/canonical/interp__sp__def_c0: !vm sr c. h4/canonical/interp__sp sr vm (h4/canonical/SPconst c) = c
% Assm: h4/canonical/interp__sp__def_c1: !vm sr i. h4/canonical/interp__sp sr vm (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm: h4/canonical/interp__sp__def_c2: !vm sr p2 p1. h4/canonical/interp__sp sr vm (h4/canonical/SPplus p1 p2) = h4/semi__ring/semi__ring__SRP sr (h4/canonical/interp__sp sr vm p1) (h4/canonical/interp__sp sr vm p2)
% Assm: h4/canonical/interp__sp__def_c3: !vm sr p2 p1. h4/canonical/interp__sp sr vm (h4/canonical/SPmult p1 p2) = h4/semi__ring/semi__ring__SRM sr (h4/canonical/interp__sp sr vm p1) (h4/canonical/interp__sp sr vm p2)
% Assm: h4/ring/semi__ring__of__def: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm: h4/ringNorm/r__interp__sp__def: !r. h4/ringNorm/r__interp__sp r = h4/canonical/interp__sp (h4/ring/semi__ring__of r)
% Goal: !vm r p2 p1. h4/ringNorm/r__interp__sp r vm (h4/canonical/SPplus p1 p2) = h4/ring/ring__RP r (h4/ringNorm/r__interp__sp r vm p1) (h4/ringNorm/r__interp__sp r vm p2)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c0]: !vm sr c. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPconst c) = c
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c1]: !vm sr i. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c2]: !vm sr p2 p1. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPplus p1 p2) = happ (happ (h4/semi__ring/semi__ring__SRP sr) (happ (happ (h4/canonical/interp__sp sr) vm) p1)) (happ (happ (h4/canonical/interp__sp sr) vm) p2)
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c3]: !vm sr p2 p1. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPmult p1 p2) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (happ (happ (h4/canonical/interp__sp sr) vm) p1)) (happ (happ (h4/canonical/interp__sp sr) vm) p2)
% Assm [h4s_rings_semiu_u_ringu_u_ofu_u_def]: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm [h4s_ringNorms_ru_u_interpu_u_spu_u_def]: !r. h4/ringNorm/r__interp__sp r = h4/canonical/interp__sp (h4/ring/semi__ring__of r)
% Goal: !vm r p2 p1. happ (happ (h4/ringNorm/r__interp__sp r) vm) (h4/canonical/SPplus p1 p2) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ringNorm/r__interp__sp r) vm) p1)) (happ (happ (h4/ringNorm/r__interp__sp r) vm) p2)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q288256,TV_Q288252]: ![V_f, V_g]: (![V_x]: s(TV_Q288252,happ(s(t_fun(TV_Q288256,TV_Q288252),V_f),s(TV_Q288256,V_x))) = s(TV_Q288252,happ(s(t_fun(TV_Q288256,TV_Q288252),V_g),s(TV_Q288256,V_x))) => s(t_fun(TV_Q288256,TV_Q288252),V_f) = s(t_fun(TV_Q288256,TV_Q288252),V_g))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spconst(s(TV_u_27a,V_c))))) = s(TV_u_27a,V_c)).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spvar(s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spplus(s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2)))))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c3, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spmult(s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2)))))).
fof(ah4s_rings_semiu_u_ringu_u_ofu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_ringNorms_ru_u_interpu_u_spu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ch4s_ringNorms_interpu_u_spu_u_defu_c2, conjecture, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spplus(s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2)))))).
