ND0263

BetaPrefixEqual(b,c,d,e,l)

Every decoded source entry at i<l also decodes in the target. Actual beta totality and functionality make this extensional prefix equality, not equality of the two code parameters.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Definition in prerequisite notation

∀ mdr_i_pfp_lowertier. ∀ mdr_a_pfp_lowertier. Lt(mdr_i_pfp_lowertier,l) → BetaAt(b,c,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier) → BetaAt(d,e,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition 
forall mdr_i_pfp_lowertier mdr_a_pfp_lowertier. (exists mdr_gap_pfp_lowertierb. mdr_gap_pfp_lowertierb + S (mdr_i_pfp_lowertier) = ((l))) -> (((exists ff_h_mdr_pfp_lowertiero. ff_h_mdr_pfp_lowertiero + S (mdr_a_pfp_lowertier) = S ((S (mdr_i_pfp_lowertier)) * (c))) /\ exists ff_q_mdr_pfp_lowertiero. (b) = ff_q_mdr_pfp_lowertiero * S ((S (mdr_i_pfp_lowertier)) * (c)) + (mdr_a_pfp_lowertier))) -> (((exists ff_h_mdr_pfp_lowertiern. ff_h_mdr_pfp_lowertiern + S (mdr_a_pfp_lowertier) = S ((S (mdr_i_pfp_lowertier)) * (e))) /\ exists ff_q_mdr_pfp_lowertiern. (d) = ff_q_mdr_pfp_lowertiern * S ((S (mdr_i_pfp_lowertier)) * (e)) + (mdr_a_pfp_lowertier)))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Lt(a,b)BetaAt(b,c,i,x)

Definitions depending on this notation

none

Checked theorems using this definition

PC0004 · polynomial_zero_extended_entry_transportPC0008 · polynomial_diagonal_term_transportPC000E · polynomial_diagonal_prefix_recodingPC000F · polynomial_diagonal_prefix_from_pointwisePC0010 · polynomial_diagonal_prefix_functionalPC0012 · polynomial_diagonal_prefix_input_transportPC0016 · prime_field_convolution_coefficient_transportPC001C · prime_field_convolution_prefix_recodingPC001D · prime_field_convolution_prefix_from_pointwisePC001E · prime_field_convolution_prefix_functionalPC0021 · prime_field_convolution_prefix_input_transportPC0026 · prime_field_polynomial_convolution_transportPC0027 · prime_field_polynomial_convolution_functionalPC0029 · prime_field_polynomial_convolution_exists_uniquePC0030 · prime_field_polynomial_represented_degree_transport