hcju/isabelleps huggingface dataset:
weak_arrow_of_homs_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" shows "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F \<equiv> (horizontal_homs C src\<^sub>C trg\<^sub>C \<and> horizontal_homs D src\<^sub>D trg\<^sub>D) \<and> functor C D F \<and> weak_arrow_of_homs_axioms C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"
H_eq_3: fixes s :: "'s"   and t :: "'s" shows "H s t = U s t * H t t"
\<pi>_def: shows "\<pi> \<equiv> \<lambda>n. card {p. prime p \<and> p \<le> n}"
scale_minus_left: fixes a :: "'a"   and x :: "'c" shows "- a *b x = - (a *b x)"
group_homomorphism_hom_relboundary: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "hom_relboundary p X S T \<in> hom (relative_homology_group p X S) (relative_homology_group (p - 1) (subtopology X S) T)"
r_into_trancl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
scc_root_exists: fixes scc :: "'v set" assumes "is_scc E scc"   and "scc \<inter> dom (discovered s') \<noteq> {}" shows "\<exists>r. scc_root s' r scc"
sorted_spvec_abs_spvec: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sorted_spvec (abs_spvec v)"
equiv_unconnected_on: fixes V :: "'b set"   and E' :: "'b uprod set" shows "equiv V (Restr (uconnected E') V)"
density_context_\<alpha>: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "density_context (set vs) (set vs') \<Gamma> (\<lambda>x. ennreal (extract_real (cexpr_sem x \<delta>)))"
uncurry_splits_1: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c" shows "P (uncurry f p) = (\<forall>x1 x2. p = (x1, x2) \<longrightarrow> P (f x1 x2))"
a: shows "\<forall>i<j. s i = (0::'a)"
wf_subtype_acc: fixes wf_mb :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog wf_mb P" shows "acc (subtype P)"
lcoset_rel_def: fixes A :: "'b set" shows "lcoset_rel A \<equiv> {(x, y). - x + y \<in> A}"
coretract_inverse: fixes y :: "'a" assumes "y \<in> carrier \<Y>\<^bsub>F\<^esub>" shows "\<pi>\<^sup>*\<^bsub>F\<^esub> (\<pi>\<^sub>*\<^bsub>F\<^esub> y) = y"
simps_94: fixes x51 :: "'a formula"   and x52 :: "'a formula" shows "atoms (x51 \<^bold>\<or> x52) = atoms x51 \<union> atoms x52"
comp_assoc_assoc'_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>\<^sup>-\<^sup>1[f, g, h] \<cdot> \<a>[f, g, h] = (f \<star> g) \<star> h"
monom_mult_one_left: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.monom_mult (1::'b) 0 p = p"
show_law_int: fixes i :: "int" shows "show_law showsp_int i"
iso_FG_constituent: fixes Us :: "('b, 'a) aezfun set list" assumes "V \<noteq> 0"   and "\<forall>U\<in>set Us. IrrFinGroupRepresentation G (*) U"   and "0 \<notin> set Us"   and "FG = (\<Oplus>U\<leftarrow>Us. U)" shows "\<exists>U\<in>set Us. isomorphic (*) U"
tree_map'_ID': fixes xs :: "'r tree" shows "tree_map'\<cdot>ID\<cdot>xs = xs"
vsubset_antisym: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "B \<subseteq>\<^sub>\<circ> A" shows "A = B"
simps_10: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and x2 :: "'f" shows "(case Skip x2 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f2 x2"
rules_def: shows "rules = Idle ## sinterleaves [mkRules NegL, mkRules NegR, mkRulePairs ConjL, mkRulePairs ConjR] (smap Ax nats)"
mor: shows "h \<in> mor \<and> g \<in> mor"
SumAssert1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and \<phi> :: "'c"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<phi> Q \<longmapsto> Rs"
looseD_hindrance: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "loose \<Gamma>" shows "\<not> hindrance \<Gamma> (\<lambda>_. 0)"
InvariantReasonClausesHoldsForPrefixElements: fixes F :: "Literal list list"   and p :: "(Literal \<times> bool) list"   and M :: "(Literal \<times> bool) list"   and literal :: "Literal" assumes "InvariantReasonClauses F p"   and "isPrefix p M"   and "literal el elements p"   and "\<not> literal el decisions M" shows "\<exists>clause. formulaEntailsClause F clause \<and> isReason clause literal (elements M)"
measure_space_measure_of_st_vec': fixes x :: "'a st_vec" shows "measure_space UNIV UNIV (measure_of_st_vec' x)"
subset_code_1: fixes xs :: "'d list"   and B :: "'d set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
is_int_rat: fixes x :: "rat" shows "is_int_rat x = (x \<in> \<int>)"
LIMSEQ_unique: fixes "open" :: "'c set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.t2_space open"   and "filterlim X (topological_space.nhds open a) sequentially"   and "filterlim X (topological_space.nhds open b) sequentially" shows "a = b"
fcompose_smult: fixes a :: "'b"   and p :: "'b poly"   and q1 :: "'b poly"   and q2 :: "'b poly" shows "fcompose (smult a p) q1 q2 = smult a (fcompose p q1 q2)"
invertible_iff_is_unit: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = (det A dvd (1::'b))"
c0: shows "coeff (smult lc r - b * d) dr = (0::'a)"
divide_fpxs_def: fixes f :: "'a fpxs"   and g :: "'a fpxs" shows "f div g = f * inverse g"
step\<^sub>2': shows "\<langle>c\<^sub>1, mds, mem\<^sub>2\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>1', mds', mem\<^sub>2'\<rangle>"
poly_rel_dvd: shows "(poly_rel ===> poly_rel ===> (=)) (dvd_poly_i ops) (dvd)"
rotate_rotate: fixes m :: "nat"   and n :: "nat"   and xs :: "'a list" shows "rotate m (rotate n xs) = rotate (m + n) xs"
VA: fixes T :: "'a set" assumes "T \<in> \<C>" shows "\<V> T \<in> U \<and> \<V> T \<notin> S \<and> \<A> T \<in> S \<and> T \<subseteq> ball (\<V> T) (setdist {\<V> T} S / 2) \<and> dist (\<V> T) (\<A> T) \<le> 2 * setdist {\<V> T} S"
go_is_no_brk_impl_def: fixes s :: "('q set \<times> 'q set) option \<times> ('q \<Rightarrow> node_state option)" shows "go_is_no_brk_impl s \<equiv> fst s = None"
GammaNegForall: fixes A :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and G :: "('a, 'b) form list" assumes "\<turnstile> Neg (A[t/0]) # G" shows "\<turnstile> Neg (Forall A) # G"
simps_2: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" shows "inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
build_rule_index_def: shows "build_rule_index \<equiv> hll_idx.idx_build"
rev_is_rev_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "(rev xs = rev ys) = (xs = ys)"
of_int_minus: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- z) = uminus (ring_1.of_int one plus zero minus z)"
PiProf: fixes n :: "nat" shows "(\<lambda>i. if h i < n then Lab else Lba) \<in> Prof"
while_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and e :: "'d"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((While e c, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
impl_of: shows "impl_of t \<in> {t. invar_trie t}"
absorb_iff1: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "((a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)) = (a \<sqinter> b = a)"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter"
bij_lens_equiv: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "bij_lens X"   and "X \<approx>\<^sub>L Y" shows "bij_lens Y"
prems: shows "is_tangle_diagram ((z * zs) \<circ> y)"
Bin1OpRed1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and bop :: "bop"   and e2 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e \<guillemotleft>bop\<guillemotright> e2,s\<rangle> -ta\<rightarrow> \<langle>e' \<guillemotleft>bop\<guillemotright> e2,s'\<rangle>"
basis_inI_aux: fixes X :: "'a set" assumes "\<E>.basis X" shows "basis_in \<E> X"
sectionI: fixes g :: "'a"   and f :: "'a" assumes "C.ide (g \<cdot> f)" shows "C.section f"
trm_rep_is_lower_subt_red: fixes S :: "'a eclause set"   and x :: "'a trm" assumes "subterm_reduction_applicable S x" shows "(trm_rep x S, x) \<in> trm_ord"
method_exit_unique: fixes n :: "'b"   and n' :: "'b" assumes "method_exit n"   and "method_exit n'"   and "get_proc n = get_proc n'" shows "n = n'"
wpt_le_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "times x y = y" shows "less_eq (mbt_algebra.wpt one times inf top y) x"
subsJustified: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "\<not> terminal subs gamma"   and "\<forall>sigma\<in>subs gamma. sequent sigma \<in> deductions CutFreePC" shows "sequent gamma \<in> deductions CutFreePC"
filter_True: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "filter P xs = xs"
seqE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.dom C g = partial_magma.cod C f"
his: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([x, y], rev qs @ h)"
prems_3: shows "n \<notin> fst (pn_atoms F)"
intros: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "is_App t"   and "chop t \<ge>\<^sub>t s" shows "gt_chop t s"
def_lfp_induct: fixes A :: "'a"   and f :: "'a \<Rightarrow> 'a"   and P :: "'a" assumes "A \<equiv> lfp f"   and "mono f"   and "f (inf A P) \<le> P" shows "A \<le> P"
app6: fixes env :: "v sem_env"   and s1 :: "'a state"   and es :: "exp0 list"   and s2 :: "'a state"   and err :: "v error_result"   and op0 :: "op0" assumes "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 (rev es) (s2, Rerr err)" shows "Big_Step_Unclocked_Single.evaluate env s1 (App op0 es) (s2, Rerr err)"
le_max_insert: fixes ffp :: "'a"   and x0 :: "'a"   and x1 :: "'b \<Rightarrow> 'a"   and x2 :: "'c \<Rightarrow> 'b"   and gs :: "'c list" shows "ffp \<le> max x0 (Max (insert ffp (x1 ` x2 ` set gs)))"
abs_eq: fixes xa :: "'e \<Rightarrow> 'c"   and x :: "'d \<Rightarrow> 'e" shows "uop xa (mk\<^sub>e x) = mk\<^sub>e (\<lambda>b. xa (x b))"
select_convs_1: fixes xc :: "int"   and yc :: "int"   and more :: "'a" shows "xc \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = xc"
space_subprob_algebra_empty_iff: fixes N :: "'a measure" shows "(space (subprob_algebra N) = {}) = (space N = {})"
tfr\<^sub>s\<^sub>t_def: fixes S :: "('fun, ('fun, 'atom) Term.term \<times> nat) strand_step list" shows "tfr\<^sub>s\<^sub>t S \<equiv> tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S) \<and> list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
separating_essential: fixes G :: "('k, 'l) graph_scheme"   and A :: "'k set"   and B :: "'k set"   and S :: "'k set" assumes "separating_gen G A B S" shows "separating_gen G A B {x \<in> S. essential G B S x}"
h_chop_middle1: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> left (ext v) \<le> right (ext u)"
zero_ennreal_def: shows "0 \<equiv> e2ennreal 0"
mset_sublist_incl: fixes xs' :: "'c list"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" assumes "mset xs' = mset xs"   and "\<And>i. i < lo \<Longrightarrow> xs' ! i = xs ! i"   and "\<And>j. \<lbrakk>hi < j; j < length xs\<rbrakk> \<Longrightarrow> xs' ! j = xs ! j"   and "lo \<le> hi"   and "hi < length xs" shows "set (WB_Sort.sublist xs' lo hi) \<subseteq> set (WB_Sort.sublist xs lo hi)"
mat_delete_index: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "A \<in> carrier_mat (Suc n) (Suc n)"   and "i < Suc n"   and "j < Suc n"   and "i' < n"   and "j' < n" shows "A $$ (insert_index i i', insert_index j j') = mat_delete A i j $$ (i', j')"
pp_inf_below_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> - z) = (- - x \<sqinter> y \<le> - z)"
infinite_greachable_gtrace: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) stream"   and v :: "nat \<times> 'b"   and u :: "nat \<times> 'b" assumes "grun A w V r v"   and "u \<in> sset (gtrace r v)" shows "infinite (greachable A w V u)"
well_def_signs1: fixes j :: "rat list" assumes "j \<in> set (get_signs (calculate_data p qs1))" shows "length j = length qs1"
map_transfer: fixes Rb :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sd :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((Rb ===> Sd) ===> list_all2 Rb ===> list_all2 Sd) map map"
sets_PiM_cong: fixes I :: "'d set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and N :: "'d \<Rightarrow> 'e measure" assumes "I = J"   and "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (Pi\<^sub>M I M) = sets (Pi\<^sub>M J N)"
rref_condition1_explicit: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f" assumes "reduced_row_echelon_form A"   and "is_zero_row i A" shows "\<forall>j>i. is_zero_row j A"
comm_ring_1_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" shows "class.comm_ring_1 times one plus zero minus uminus \<equiv> class.comm_monoid_mult times one \<and> class.comm_ring plus zero minus uminus times \<and> class.zero_neq_one one zero"
ground_fmla_aux_simps_6: fixes A :: "fmla"   and S :: "atom set" shows "ground_fmla_aux (Robinson_Arithmetic.neg A) S = ground_fmla_aux A S"
ENR_from_Un_Int_gen: fixes S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T"   and "ENR (S \<union> T)"   and "ENR (S \<inter> T)" shows "ENR S"
gcd_pos_nat: fixes m :: "nat"   and n :: "nat" shows "(0 < gcd m n) = (m \<noteq> 0 \<or> n \<noteq> 0)"
get_root_node_parent_child_rel: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and child :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "heap_is_wellformed h"   and "h \<turnstile> get_root_node child \<rightarrow>\<^sub>r root"   and "known_ptrs h"   and "type_wf h" shows "(root, child) \<in> (parent_child_rel h)\<^sup>*"
augment_edge_cf: fixes u :: "nat"   and v :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "(u, v) \<in> E \<union> E\<inverse>" shows "cf_of (augment_edge f (u, v) \<Delta>) = (cf_of f)((u, v) := cf_of f (u, v) - \<Delta>, (v, u) := cf_of f (v, u) + \<Delta>)"
L_transition_AC_strong: fixes X :: "'a"   and f :: "'effect"   and F :: "'effect fs_set"   and P :: "'state"   and \<alpha>\<^sub>L :: "('act, 'effect) L_action"   and P\<^sub>L' :: "('state, 'effect) L_state" assumes "finite (supp X)"   and "AC (f, F, P) \<rightarrow>\<^sub>L \<langle>\<alpha>\<^sub>L,P\<^sub>L'\<rangle>" shows "\<exists>\<alpha> P'. P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<and> \<langle>\<alpha>\<^sub>L,P\<^sub>L'\<rangle> = \<langle>L_action.Act \<alpha>,EF (L (\<alpha>, F, f), P')\<rangle> \<and> bn \<alpha> \<sharp>* X"
rref_imp_upper_triagular_upt: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form A" shows "upper_triangular_upt_k A k"
Inl_not_Inr: fixes a :: "'a"   and b :: "'b" shows "Inl a \<noteq> Inr b"
chop_leq2: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<longrightarrow> t \<le> r"
hom_connected_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "(\<nu> \<star> \<mu> \<noteq> null) = (local.dom \<nu> \<star> \<mu> \<noteq> null)"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'c"   and T1 :: "'a trm"   and T2 :: "'a trm" assumes "U.Struct wtFsym wtPsym arOf parOf D intF intP" shows "U.Struct.satA intF intP \<xi> (Eq T1 T2) = (U.Struct.int intF \<xi> T1 = U.Struct.int intF \<xi> T2)"
src_dom: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (B.dom \<mu>) = src \<mu>"
pos_le_divide_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (divide b c) = less_eq (times a c) b"
simps_6: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (Le (CN 0 c e)) = (0 < c \<and> numbound0 e)"
exists_next_propagate': fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc" assumes "c_work c loc \<noteq> {#}\<^sub>z" shows "\<exists>c'. next_propagate c c'"
block_comp_elem_alt_right: fixes ps :: "'a set"   and bl :: "'a set" assumes "ps \<subseteq> \<V>"   and "\<And>x. x \<in> ps \<Longrightarrow> x \<notin> bl" shows "ps \<subseteq> bl\<^sup>c"
compile_correct_halt: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and stp :: "nat"   and am :: "nat list"   and n :: "nat"   and rs :: "nat"   and off :: "nat" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (0, lm) (Suc 0, l, r) ires"   and "abc_steps_l (0, lm) ap stp = (length ap, am)"   and "n < length am"   and "abc_lm_v am n = rs"   and "off = length tp div 2" shows "\<exists>stp i j. steps0 (Suc 0, l, r) (tp @ shift (mopup n) off) stp = (0, Bk \<up> i @ Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> j)"
freeword_funlift_add: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c freeword"   and g' :: "'c freeword" shows "freeword_funlift f (g + g') = freeword_funlift f g + freeword_funlift f g'"
get_scdom_component_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "ptr' |\<in>| object_ptr_kinds h"
dist_riemann_sphere_sym: fixes M1 :: "riemann_sphere"   and M2 :: "riemann_sphere" shows "dist_riemann_sphere' M1 M2 = dist_riemann_sphere' M2 M1"
converse_tranclpE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a" assumes "r\<^sup>+\<^sup>+ x z" obtains "r x z"   | y :: "'a" where "r x y" and "r\<^sup>+\<^sup>+ y z"
tcopy_begin_def: shows "tcopy_begin \<equiv> [(W0, 0), (R, 2), (R, 3), (R, 2), (W1, 3), (L, 4), (L, 4), (L, 0)]"
CauchyD: fixes X :: "nat \<Rightarrow> 'a"   and e :: "real" assumes "Cauchy X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. \<parallel>X m - X n\<parallel> < e"
recvmsg_simps_6: fixes P :: "'d \<Rightarrow> bool"   and d :: "nat" shows "recvmsg P (deliver d) = True"
distrib_right: fixes a :: "'b hrel"   and b :: "'b hrel"   and c :: "'b hrel" shows "a \<sqinter> b ;; c = (a ;; c) \<sqinter> (b ;; c)"
HFinite_not_HInfinite: fixes x :: "'a star" assumes "x \<in> HFinite" shows "x \<notin> HInfinite"
Pi_I': fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and B :: "'h \<Rightarrow> 'i set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
IH: shows "label_list (a2 @ b) = State_Monad.ap (State_Monad.ap (State_Monad.return (@)) (label_list a2)) (label_list b)"