[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 1 from proof map-putrev (conclusion) (back to tree)

r ∈ rs ⊦ (r ==>(o0, a0) * hR)[rs, f(r; o1), g(r; a1)] ↔ (o0 = o1 ∧ a0 = a1) ∧ (r ==>_ * hR)[rs, f, g]

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof map-putrev (previous step) (back to tree)

¬((r ==>(o0, a0) * hR)[rs, f(r; o1), g(r; a1)] ↔ (o0 = o1 ∧ a0 = a1) ∧ (r ==>_ * hR)[rs, f, g]), r ∈ rs ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: maplet-def: ⊦ r ==>(o, a) = r __=>(o ×× a); modifysep-def: ⊦ hP[rs, f, g] ↔ hP[rs, λ r. f(r) ×× g(r)]__; size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic[refset] ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic[refset]

Proof step 3 from proof map-putrev (previous step) (back to tree)

# rs ≠ 0, ¬((r __=>(o0 ×× a0) * hR)[rs, λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__), rs ≠ ∅, r ∈ rs ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: heap-fun[heap-aux-inst], name:map-putrev, seq:r ∈ rs ⊦ (r __=>(o0) * hP)[rs, f(r; o1)]__ ↔ o0 = o1 ∧ (r ==>_ * hP)[rs, f]__, subst:[hP, r, o0, rs, o1, f] → [hR, r, o0 ×× a0, rs, o1 ×× a1, λ r. f(r) ×× g(r)]

Additional information

Used terms: hR, r, o0 ×× a0, rs, o1 ×× a1, λ r. f(r) ×× g(r)

Proof step 4 from proof map-putrev (previous step) (back to tree)

r ∈ rs ⊦ (r __=>(o0) * hP)[rs, f(r; o1)]__ ↔ o0 = o1 ∧ (r ==>_ * hP)[rs, f]__

A spec-lemma

Proof step 5 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, (λ r. f(r) ×× g(r))(r; o1 ×× a1)]__ ↔ o0 ×× a0 = o1 ×× a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, ¬((r __=>(o0 ×× a0) * hR)[rs, λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__), rs ≠ ∅, r ∈ rs ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__ ↔ o0 ×× a0 = o1 ×× a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, ¬((r __=>(o0 ×× a0) * hR)[rs, λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__), rs ≠ ∅, r ∈ rs ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: obj-aux, name:inj, seq: ⊦ obj ×× aux = obj0 ×× aux0 ↔ obj = obj0 ∧ aux = aux0, subst:[obj, aux, obj0, aux0] → [o0, a0, o1, a1], <all paths>

Additional information

Used terms: o0, a0, o1, a1

Proof step 7 from proof map-putrev (previous step) (back to tree)

⊦ obj ×× aux = obj0 ×× aux0 ↔ obj = obj0 ∧ aux = aux0

A spec-lemma

Proof step 8 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__ ↔ (o0 = o1 ∧ a0 = a1) ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, ¬((r __=>(o0 ×× a0) * hR)[rs, λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__), rs ≠ ∅, r ∈ rs ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, ¬ ( (r __=>(o0 ×× a0) * hR)[rs, λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))]__ ↔ (r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__), rs ≠ ∅, r ∈ rs ⊦

User applied rule reduce equation

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 10 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, (λ r0. (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0))) ≠ (λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))), rs ≠ ∅, r ∈ rs ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof map-putrev (previous step) (back to tree)

(r __=>(o0 ×× a0) * hR)[rs, λ r0. (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0))]__ ↔ o0 = o1 ∧ a0 = a1 ∧ (r ==>_ * hR)[rs, λ r. f(r) ×× g(r)]__, # rs ≠ 0, rs ≠ ∅, (r = r0 ⊃ o1;f(r0)) ×× (r = r0 ⊃ a1;g(r0)) ≠ (r0 = r ⊃ o1 ×× a1;f(r0) ×× g(r0)), r ∈ rs ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 4

Additional information

Proof Tree for map-putrev (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree