WIP docs update
This commit is contained in:
Simon Forman
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"However, this creates (and destroys) an intermediate list, which is a waste."
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Variations\n",
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"\n",
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"Let's review the operation of the hylomorphism:\n",
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"\n",
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" [P] c [G] [Q] H₁ == [P] [pop c] [G] [dip Q] genrec\n",
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"\n",
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" ... a G [H₁] dip Q\n",
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" ... a′ b [H₁] dip Q\n",
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" ... a′ H₁ b Q\n",
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" <predicate omitted>\n",
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" ... a′ G [H₁] dip Q b Q\n",
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" ... a″ b′ [H₁] dip Q b Q\n",
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" ... a″ H₁ b′ Q b Q\n",
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" <predicate omitted>\n",
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" ... a″ G [H₁] dip Q b′ Q b Q\n",
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" ... a‴ b″ [H₁] dip Q b′ Q b Q\n",
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" ... a‴ H₁ b″ Q b′ Q b Q\n",
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" <predicate omitted>\n",
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" ... a‴ pop c b″ Q b′ Q b Q\n",
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" ... c b″ Q b′ Q b Q\n",
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" ... c′ b′ Q b Q\n",
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" ... c″ b Q\n",
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" ... c‴\n",
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"\n",
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"Notice that `b` values and the `Q` combiner function get pushed into the pending expression (\"continuation\")."
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### A tail-recursive version\n",
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"\n",
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"Now let's try a version that doesn't do that. When you can start with the identity value `c` on the stack and the combiner `Q` can operate as you go using the intermediate results immediately rather than queuing them up, we can do this:\n",
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"\n",
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" [P] c [G] [Q] H₂ == c [pop P] [popd] [[G] dip Q] tailrec\n",
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"\n",
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"An important difference is that the combiner function `Q` must accept its arguments in the reverse order and take into account the presence of the `a` value:\n",
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"\n",
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" ... a c [G] dip Q H₂\n",
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" ... a G c Q H₂\n",
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" ... a′ b c Q H₂\n",
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" ... a′ c′ H₂\n",
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" <predicate omitted>\n",
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" ... a′ c′ [G] dip Q H₂\n",
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" ... a′ G c′ Q H₂\n",
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" ... a″ b c′ Q H₂\n",
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" ... a″ c″ H₂\n",
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" <predicate omitted>\n",
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" ... a″ c″ [G] dip Q H₂\n",
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" ... a″ G c″ Q H₂\n",
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" ... a‴ b c″ Q H₂\n",
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" ... a‴ c‴ H₂\n",
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" <predicate omitted>\n",
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" ... a‴ c‴ popd\n",
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" ... c‴"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Note that the `c` values are processed by the `Q` combiner function in reverse order as compared to the original version.\n",
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"\n",
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"#### `range` with `H₂`\n",
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"\n",
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"For example, if we reimplement the `range` function in the tail-recursive style it generates the list with $0$ at the head rather than $n - 1$:\n",
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"\n",
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" range_reverse == [] [pop 0 <=] [popd] [[-- dup] dip cons] tailrec\n",
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" "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": []
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}
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],
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"source": [
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"clear "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": []
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}
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],
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"source": [
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"[range_reverse [] [pop 0 <=] [popd] [[-- dup] dip cons] tailrec] inscribe"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[4 3 2 1 0] [0 1 2 3 4]"
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]
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}
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],
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"source": [
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"5 [range] [range_reverse] cleave"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Instead of making `Q` take account of the `c` value we could do this:\n",
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"\n",
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" ... a′ b c [Q] ccons dip swap H₂\n",
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" ... a′ [b c Q] dip swap H₂\n",
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" ... b c Q a′ swap H₂\n",
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" ... c′ a′ swap H₂\n",
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" ... a′ c′ H₂\n",
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"\n",
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"This shows that any function `Q` can be converted to `[Q] ccons dip swap` to deal with that `a` value on the stack."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### the old version\n",
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"\n",
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"Now let's try a version that doesn't do that. When you can start with the identity value `c` on the stack and the combiner `Q` can operate as you go using the intermediate results immediately rather than queuing them up, we can do this:\n",
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"\n",
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" [P] c [G] [Q] H₂ == c swap [P] [pop] [G [Q] dip] tailrec\n",
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"\n",
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"An important difference is that the generator function must return its results in the reverse order, and both `Q` and `G` have to take into account the presence of the `c` value:\n",
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"\n",
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" ... c a G [Q] dip H₂\n",
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" ... c b a′ [Q] dip H₂\n",
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" ... c b Q a′ H₂\n",
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" ... c′ a′ H₂\n",
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" <predicate omitted>\n",
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" ... c′ a′ G [Q] dip H₂\n",
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" ... c′ b′ a″ [Q] dip H₂\n",
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" ... c′ b′ Q a″ H₂\n",
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" ... c″ a″ H₂\n",
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" <predicate omitted>\n",
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" ... c″ a″ G [Q] dip H₂\n",
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" ... c″ b″ a‴ [Q] dip H₂\n",
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" ... c″ b″ Q a‴ H₂\n",
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" ... c‴ a‴ H₂\n",
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" <predicate omitted>\n",
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" ... c‴ a‴ pop\n",
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" ... c‴\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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@ -2654,21 +2654,21 @@
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 2",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python2"
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 2
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython2",
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"version": "2.7.12"
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"pygments_lexer": "ipython3",
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"version": "3.7.10"
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}
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},
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"nbformat": 4,
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@ -470,7 +470,6 @@
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"source": [
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"### Traversal\n",
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" [key value] first [left right] [K] map i\n",
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" key [value] [left right] [K] map i\n",
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" key [left right] [K] map i\n",
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" key [lkey rkey ] i\n",
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" key lkey rkey"
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@ -873,21 +872,21 @@
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 2",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python2"
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 2
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython2",
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"version": "2.7.12"
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"pygments_lexer": "ipython3",
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"version": "3.7.10"
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}
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},
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"nbformat": 4,
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