hcju/isabelleps huggingface dataset:
INF_cong: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e \<Rightarrow> 'f"   and D :: "'e \<Rightarrow> 'f"   and Inf :: "'f set \<Rightarrow> 'f" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
properties_distinct: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat"   and j :: "nat"   and i :: "nat" assumes "Basis.properties f x0 lambda mu"   and "j \<in> {0<..<lambda}" shows "fx0.seq' f x0 (i + j) \<noteq> fx0.seq' f x0 i"
bB0: shows "0\<^sub>v n \<notin> baseB"
e_def: shows "e = CFn\<cdot>(\<Lambda> e r. (e\<cdot>r \<down>CFn \<bottom>)\<cdot>r)"
4: fixes x :: "'a"   and y :: "'a" shows "\<not> x \<le> y \<or> x \<squnion> y = y"
Image_iff: fixes b :: "'a"   and r :: "('b \<times> 'a) set"   and A :: "'b set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
finite_UNIV: assumes "class.finite TYPE('c)" shows "finite UNIV"
map_filter_M_pure_E: fixes h :: "'l"   and f :: "'o \<Rightarrow> ('l, 'm, 'n option) prog"   and xs :: "'o list"   and ys :: "'n list"   and y :: "'n" assumes "h \<turnstile> map_filter_M f xs \<rightarrow>\<^sub>r ys"   and "y \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" obtains x :: "'o" where "h \<turnstile> f x \<rightarrow>\<^sub>r Some y" and "x \<in> set xs"
adjoint_pair_antipar_3: fixes f :: "'b"   and g :: "'b" assumes "D.adjoint_pair f g" shows "src\<^sub>D f = trg\<^sub>D g"
a: shows " \<Gamma>1 \<turnstile> t : T "
euler_witness_exists: fixes n :: "nat" assumes "odd n"   and "\<not> prime n"   and "2 < n" shows "\<exists>a. euler_witness a n \<and> coprime a (int n) \<and> 0 < a \<and> a < int n"
real2_eq: fixes u :: "(real, 2) vec"   and v :: "(real, 2) vec" assumes "u $ 1 = v $ 1"   and "u $ 2 = v $ 2" shows "u = v"
Nil_lenlex_iff2: fixes ns :: "'a list"   and r :: "('a \<times> 'a) set" shows "(ns, []) \<notin> lenlex r"
terminates_block_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and bdy :: "('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>block init bdy return c \<down> Normal s" obtains "\<Gamma>\<turnstile>bdy \<down> Normal (init s)" and "\<forall>t. \<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t \<longrightarrow> \<Gamma>\<turnstile>c s t \<down> Normal (return s t)"
unit_char_3: fixes a :: "'c" assumes "C.obj a" shows "unit a \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a)) \<cdot>\<^sub>D (unit a \<star>\<^sub>D unit a)"
openin_discrete_topology: fixes U :: "'c set"   and S :: "'c set" shows "openin (discrete_topology U) S = (S \<subseteq> U)"
owhile_def: fixes c :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'e \<Rightarrow> 'd \<Rightarrow> 'e option"   and a :: "'e" shows "owhile c b a \<equiv> \<lambda>s. option_while (\<lambda>a. c a s) (\<lambda>a. b a s) a"
assms_2: shows "(\<sigma>, s) \<in> oreachable A S U"
siso_Sbis: fixes c :: "('test, 'atom) com" assumes "siso c" shows "c \<approx>s c"
curr_val_satisfies_no_lhs_def: fixes s :: "('a, 'b) state" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<equiv> \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s \<and> \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>b \<B> s \<parallel> - lvars (\<T> s)"
52533: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<squnion> y \<sqinter> (z \<squnion> top)) \<sqinter> (x \<squnion> y) = x \<squnion> y"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c multiset" shows "set_mset (image_mset f v) = f ` set_mset v"
dma\<^sub>C_def: fixes m :: "var\<^sub>C \<Rightarrow> nat"   and x :: "var\<^sub>C" shows "dma\<^sub>C m x \<equiv> if x = buffer\<^sub>C then dma_control_var\<^sub>C (m control_var\<^sub>C) else if x = high_var\<^sub>C then High else Low"
until_until_disj: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and R :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(P \<U> Q \<U> R) \<sigma>" shows "((\<lambda>s. P s \<or> Q s) \<U> R) \<sigma>"
iso_cancel_left: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "category C"   and "category.iso C f"   and "C f g = C f g'"   and "partial_magma.arr C (C f g)" shows "g = g'"
1: shows "coreflexive (e\<^sup>T * top * e)"
assms_2: shows "(vars1 \<union> vars2) \<inter> vars3 = {}"
hyp: shows "(\<Oplus>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub>v\<in>V. a v \<odot>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub> v) = (\<Sum>v\<in>V. Polynomial.smult (a v) v)"
iso_cancel_left: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "S.iso f"   and "f \<cdot>\<^sub>S g = f \<cdot>\<^sub>S g'"   and "S.seq f g" shows "g = g'"
fold_mset_single: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d"   and x :: "'c" assumes "comp_fun_commute f" shows "fold_mset f s {#x#} = f x s"
inj_on_image_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "(f ` A = f ` B) = (A = B)"
ileq_interleave: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "w \<in> w1 \<otimes> w2" shows "w1 \<preceq> w \<and> w2 \<preceq> w"
M4: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a"   and s :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets p q"   and "meets q s"   and "meets p r"   and "meets r s" shows "q = r"
af_letter_prop_unfold_congruent: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" assumes "\<phi> \<sim>\<^sub>Q \<psi>" shows "af_letter \<phi> \<nu> \<sim>\<^sub>Q af_letter \<psi> \<nu>"
mult_mat_vec_def: fixes A :: "'a mat"   and v :: "'a Matrix.vec" shows "A *\<^sub>v v \<equiv> Matrix.vec (dim_row A) (\<lambda>i. Matrix.row A i \<bullet> v)"
diamond_imp_CR: fixes r :: "('b \<times> 'b) set" assumes "\<diamond> r" shows "CR r"
disj': shows "h\<^sub>0' \<bottom> h\<^sub>1"
state_transition_infra_def: fixes i :: "infrastructure"   and i' :: "infrastructure" shows "(i \<rightarrow>\<^sub>i i') = (i \<rightarrow>\<^sub>n i')"
simps_3: fixes key :: "'b \<Rightarrow> 'c"   and v :: "'b list"   and vb :: "'b list"   and vc :: "'b list list" shows "Mergesort_Complexity.c_merge_all key (v # vb # vc) = Mergesort_Complexity.c_merge_pairs key (v # vb # vc) + Mergesort_Complexity.c_merge_all key (merge_pairs key (v # vb # vc))"
pnet_reachable_transfer: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and n :: "net_tree"   and s :: "'c net_state"   and U :: "(nat \<Rightarrow> 'e) \<Rightarrow> (nat \<Rightarrow> 'e) \<Rightarrow> bool" assumes "openproc np onp sr"   and "wf_net_tree n"   and "s \<in> reachable (closed (pnet np n)) TT" shows "openproc.initmissing np sr (netgmap sr s) \<in> oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"
vNM_utility_implies_independence: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "independent_vnm (\<P> outcomes) \<R>"
nres_monad1: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e nres" shows "RETURN x \<bind> f = f x"
simps_3: fixes v :: "'b"   and va :: "'b" shows "first_bothNet (DenyAllFromTo v va) = bothNet (DenyAllFromTo v va)"
prefix_match_semantics_ipset_from_netmask: fixes pfx :: "'b prefix_match"   and a :: "'b word" assumes "valid_prefix pfx" shows "prefix_match_semantics pfx a = (a \<in> ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx))"
group_isomorphisms_homology_group_prod_retract: fixes S :: "'a set"   and X :: "'a topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" obtains \<H> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set" and \<K> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set" where "subgroup \<H> (homology_group p X)" and "subgroup \<K> (homology_group p X)" and "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>) (homology_group p X)" and "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)" and "hom_induced p X {} X S id \<in> Group.iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)"
exteq_sym: fixes f :: "recf"   and g :: "recf" assumes "f \<simeq> g" shows "g \<simeq> f"
emeasure_embed_measure: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and M :: "'a measure" assumes "inj f"   and "A \<in> sets (embed_measure M f)" shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> space M)"
assms_7: shows "g (- 1) $ 2 = - 1"
Lp_measurable_subset: fixes p :: "ennreal"   and M :: "'b measure" shows "space\<^sub>N (\<LL> p M) \<subseteq> borel_measurable M"
RuUSdBeqlen: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and R :: "('a list \<times> 'a list) set"   and V :: "'a list"   and V' :: "'a list" assumes "Strong_Security.d_Bisimulation_Up_To_USdB SR DA d R"   and "(V, V') \<in> R \<union> Strong_Security.USdB SR DA d" shows "length V = length V'"
Abs_Int1_mono_axioms: shows "Abs_Int1_mono num_ivl plus_ivl in_ivl filter_plus_ivl filter_less_ivl \<gamma>_ivl"
nonzero_Reals_inverse: fixes a :: "'b" assumes "a \<in> \<real>"   and "a \<noteq> (0::'b)" shows "inverse a \<in> \<real>"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "indRelRPOp a1 a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T. T \<in>T a1 \<and> T \<in>T a2) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> indRelRPOp P Q \<and> indRelRPOp Q R))"
R: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a option"   and xb :: "'a option" assumes "option_le x xa xb" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and y :: "'a option" where "x = P__" and "xa = None" and "xb = y__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xc :: "'a" and y :: "'a" where "x = P" and "xa = Some xc" and "xb = Some y" and "P xc y"
below_set_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
Cd_idem: fixes d :: "'b"   and B :: "'a set" shows "Singleton_for_d.Cd d (Singleton_for_d.Cd d B) = Singleton_for_d.Cd d B"
decr_grading_le: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat"   and t :: "'c" assumes "hom_grading d" shows "d (decr_grading d n t) \<le> n"
lookup_except: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "lookup (except p S) = (\<lambda>t. if t \<in> S then 0::'c else lookup p t)"
homeomorphism_compact: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and t :: "'d set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "\<exists>g. homeomorphism s t f g"
squareE: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'b"   and s :: "state"   and t :: "state"   and B :: "state \<times> state \<Rightarrow> bool" assumes "(s, t) \<Turnstile> [A]_v"   and "A (s, t) \<Longrightarrow> B (s, t)"   and "v t = v s \<Longrightarrow> B (s, t)" shows "B (s, t)"
map_simp: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "\<chi>.A.map f = a"
one_minus_const_fps_X_neg_power'': fixes n :: "nat"   and d :: "'a"   and c :: "'a" assumes "0 < n" shows "fps_const d / (1 - fps_const c * fps_X) ^ n = Abs_fps (\<lambda>k. poly (inverse_irred_power_poly d n) (of_nat k) * c ^ k)"
inj_on_nth: fixes xs :: "'b list"   and I :: "nat set" assumes "distinct xs"   and "\<forall>i\<in>I. i < length xs" shows "inj_on ((!) xs) I"
cases: fixes x :: "'o \<times> ('a \<times> 'd) list \<times> 'o" obtains ko :: "'o" and xs :: "('a \<times> 'd) list" and ox :: "'o" where "x = (ko, xs, ox)"
int1: shows "integrable (measure_spmf coin_spmf) (\<lambda>x. spmf (S1_14_OT (x \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), x \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), x \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), x \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) () \<bind> (\<lambda>view. A view x)) True)"
bl_xor_aux_bin: fixes n :: "nat"   and v :: "int"   and bs :: "bool list"   and w :: "int"   and cs :: "bool list" shows "map2 (\<noteq>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = bin_to_bl_aux n (v XOR w) (map2 (\<noteq>) bs cs)"
empty_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_set A {} {}"
pappus_desargues: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "is_pappus"   and "desargues_config A B C A' B' C' M N P R" shows "col M N P"
fps_zero_def: shows "0 = Abs_fps (\<lambda>n. 0::'b)"
CC_1: shows "initReachable cc \<and> <\<bottom>, outM False> \<in># msgs cc"
set_decomp_subset: fixes U :: "'b set"   and A :: "'b set"   and B :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "U = A \<union> B"   and "order.greater_eq X A"   and "order.greater_eq Y B"   and "order.greater_eq U X"   and "X \<inter> Y = {}" shows "A = X"
lt_max_keys: fixes v :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys p" shows "lex_pm v (i.lpp p)"
reciprocal_not_preserve_orientation: fixes H :: "ocircline" assumes "0\<^sub>h \<in> disc H" shows "\<not> pos_oriented (moebius_ocircline moebius_reciprocal H)"
of_int_add: fixes R :: "('b, 'c) ring_scheme"   and i :: "int"   and j :: "int" assumes "ring R" shows "\<guillemotleft>i + j\<guillemotright>\<^bsub>R\<^esub> = \<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> \<guillemotleft>j\<guillemotright>\<^bsub>R\<^esub>"
bernpoly_def: fixes n :: "nat" shows "bernpoly n = (\<lambda>x. \<Sum>k\<le>n. of_nat (n choose k) * of_real (bernoulli k) * x ^ (n - k))"
trancl_Image_in_Range: fixes R :: "('a \<times> 'a) set"   and V :: "'a set" shows "R\<^sup>+ `` V \<subseteq> Range R"
B: fixes x :: "'a" assumes "x \<in> A" shows "F x \<approx> B"
integral_0_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "(LINT x|M. \<bar>f x\<bar> = 0) = (emeasure M {x \<in> space M. f x \<noteq> 0} = 0)"
apps_preserves_beta2: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s" shows "r \<degree>\<degree> ss \<rightarrow>\<^sub>\<beta>\<^sup>* s \<degree>\<degree> ss"
vwalkI_append_r: fixes q :: "'a list"   and p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "q \<noteq> []"   and "vwalk (p @ q) G" shows "vwalk q G"
tllist_all2_TCons: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'a"   and x22 :: "('a, 'c) tllist"   and y21 :: "'b"   and y22 :: "('b, 'd) tllist" shows "tllist_all2 R1 R2 (TCons x21 x22) (TCons y21 y22) = (R1 x21 y21 \<and> tllist_all2 R1 R2 x22 y22)"
neq_Leaf_iff: fixes t :: "'d tree" shows "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
bef: shows "T\<^sub>p init (take (Suc x) qs) (take (Suc x) Strat) = T\<^sub>p init (take (Suc x) qs) (take (Suc x) (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat)) + 1"
inject_3: fixes x1 :: "name"   and x2 :: "lam"   and y1 :: "name"   and y2 :: "lam" shows "(Lam [x1].x2 = Lam [y1].y2) = ([x1].x2 = [y1].y2)"
big_d'_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "big_d' a b y i = (let l = length y; n = length b in if n < l then [] else let k = n - l in map (\<lambda>j. dij a b i (j + k) - 1) (filter (\<lambda>j. eij a b i (j + k) \<le> y ! j) [0..<length y]))"
wrm_set: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<and> sortedTree h t \<longrightarrow> setOf (wrm h t) = setOf t - {rm h t}"
Sup_multiset_in_multiset: fixes A :: "'k multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_above A" shows "(\<lambda>i. SUP X\<in>A. count X i) \<in> multiset"
all_security_requirements_fulfilled_imp_get_offending_empty: fixes M :: "'a SecurityInvariant_configured list"   and G :: "'a graph" assumes "valid_reqs M"   and "all_security_requirements_fulfilled M G" shows "get_offending_flows M G = {}"
sorted_distinct_set_unique: fixes xs :: "'a Word.word list"   and ys :: "'a Word.word list" assumes "signed.sorted xs"   and "distinct xs"   and "signed.sorted ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
simps_2: fixes v1 :: "'a"   and e :: "'a edge"   and es :: "'a edge list"   and v2 :: "'a" shows "ces v1 (e # es) v2 = (src e = v1 \<and> ces (tgt e) es v2)"
hyp_foldl: shows "foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k] = (fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]), if \<forall>m. is_zero_row_upt_k m (Suc k) (fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])) then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]))) + 1)"
irc_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "irc_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. a \<notin> f (B \<union> {a}) \<longrightarrow> f (B \<union> {a}) = f B)"
f11: shows "get_operand2 (snd instr) s1 = get_operand2 (snd instr) (snd (fst (get_curr_win () s2)))"
P: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<exists>P. eventually P F \<and> (\<forall>x. P x \<longrightarrow> x \<in> S) \<and> (\<forall>x y. P x \<longrightarrow> P y \<longrightarrow> dist x y < \<epsilon>)"
coclop_iso_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "coclop f" shows "x \<le> y \<longrightarrow> f x \<le> f y"
shift_snth_ge: fixes xs :: "'b list"   and p :: "nat"   and s :: "'b stream" assumes "length xs \<le> p" shows "(xs @- s) !! p = s !! (p - length xs)"
span_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "module.span scale S = module.subspace scale hull S"
n_add_comp_var: fixes x :: "'a" shows "t x + !x = (1::'a)"
Abs: fixes t :: "dB" assumes "NF t" shows "NF (Abs t)"
spectral_decomposition_extract_diag: fixes P :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and i :: "nat" assumes "P \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "unitary P"   and "diagonal_mat B"   and "i < n" shows "inner_prod (col P i) (P * B * adjoint P *\<^sub>v col P i) = B $$ (i, i)"
monom_mult_zero_right: fixes c :: "'c"   and t :: "'a" shows "monom_mult c t 0 = 0"
str_eq_def: fixes A :: "'b list set" shows "\<approx>A \<equiv> {(x, y). \<forall>z. (x @ z \<in> A) = (y @ z \<in> A)}"
rel_pred_neg_distr: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_pred_neg_distr_cond A B" shows "rel_pred (A OO B) \<le> rel_pred A OO rel_pred B"
add_mset_remove_trivial_eq: fixes N :: "'d multiset"   and a :: "'d" shows "(N = add_mset a (N - {#a#})) = (a \<in># N)"
stepE: fixes i :: "nat" obtains l :: "'b" where "lstep l (s i) (s (Suc i))"
map_Pair_zip_replicate_conv: fixes c :: "'b"   and l :: "'a list" shows "map (\<lambda>x. (x, c)) l = zip l (replicate (length l) c)"
norm_id_minus_trunc_ell2: fixes x :: "'c ell2"   and S :: "'c set" shows "(norm (x - trunc_ell2 S x))\<^sup>2 = (norm x)\<^sup>2 - (norm (trunc_ell2 S x))\<^sup>2"
S2: shows "A = {x. \<exists>y. pA x y = 0}"
Sup_atLeastAtMost: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<Squnion> {x..y} = y"
arrI: fixes f :: "'a \<times> 'a \<times> 'a" assumes "arr (fst f) \<and> VV.arr (snd f) \<and> I.map (fst f) = I.map (fst (snd f))" shows "VVV.arr f"
monad_stateT: assumes "monad return bind" shows "monad local.return_state local.bind_state"
l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes 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 set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<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" shows "l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel \<equiv> l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"