[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 54 from proof reduction-is-topred-in-ctxt (previous step)

context(C0), t2 = K _ v _ v0 ∧ t = v ∧ value(v) ∧ value(v0) ∨ t2 = S _ v1 _ v2 _ v3 ∧ t = v1 _ v3 _ (v2 _ v3) ∧ value(v1) ∧ value(v2) ∧ value(v3), ∀ t'. v4 => t' ↔ (∃ C, t0, t2. context(C) ∧ v4 = C[t0] ∧ t0 =>top t2 ∧ t' = C[t2]), ∀ t'. C0[t2] => t' ↔ (∃ C, t, t1. context(C) ∧ C0[t2] = C[t] ∧ t =>top t1 ∧ t' = C[t1]) ⊦ ∃ v5, v6. C0[t2] _ v4 = C0[K _ v5 _ v6] _ v4 ∧ C0[t] _ v4 = C0[v5] _ v4 ∧ value(v5) ∧ value(v6), ∃ v5, v6, v7. C0[t2] _ v4 = C0[S _ v5 _ v6 _ v7] _ v4 ∧ C0[t] _ v4 = C0[v5 _ v7 _ (v6 _ v7)] _ v4 ∧ value(v5) ∧ value(v6) ∧ value(v7)

User applied rule case distinction

And obtained the following new goals

Proof step 51:
t2 = K _ v _ v0 ∧ t = v ∧ value(v) ∧ value(v0), context(C0), ∀ t'. v4 => t' ↔ (∃ C, t0, t2. context(C) ∧ v4 = C[t0] ∧ t0 =>top t2 ∧ t' = C[t2]), ∀ t'. C0[t2] => t' ↔ (∃ C, t, t1. context(C) ∧ C0[t2] = C[t] ∧ t =>top t1 ∧ t' = C[t1]) ⊦ ∃ v5, v6. C0[t2] _ v4 = C0[K _ v5 _ v6] _ v4 ∧ C0[t] _ v4 = C0[v5] _ v4 ∧ value(v5) ∧ value(v6), ∃ v5, v6, v7. C0[t2] _ v4 = C0[S _ v5 _ v6 _ v7] _ v4 ∧ C0[t] _ v4 = C0[v5 _ v7 _ (v6 _ v7)] _ v4 ∧ value(v5) ∧ value(v6) ∧ value(v7)
Proof step 53:
t2 = S _ v1 _ v2 _ v3 ∧ t = v1 _ v3 _ (v2 _ v3) ∧ value(v1) ∧ value(v2) ∧ value(v3), context(C0), ∀ t'. v4 => t' ↔ (∃ C, t0, t2. context(C) ∧ v4 = C[t0] ∧ t0 =>top t2 ∧ t' = C[t2]), ∀ t'. C0[t2] => t' ↔ (∃ C, t, t1. context(C) ∧ C0[t2] = C[t] ∧ t =>top t1 ∧ t' = C[t1]) ⊦ ∃ v5, v6. C0[t2] _ v4 = C0[K _ v5 _ v6] _ v4 ∧ C0[t] _ v4 = C0[v5] _ v4 ∧ value(v5) ∧ value(v6), ∃ v5, v6, v7. C0[t2] _ v4 = C0[S _ v5 _ v6 _ v7] _ v4 ∧ C0[t] _ v4 = C0[v5 _ v7 _ (v6 _ v7)] _ v4 ∧ value(v5) ∧ value(v6) ∧ value(v7)

Rule argument

[CASEDARG : pos: left 2, ()]

Additional information

[[CONCRETESIMPRULES : ()]]