hcju/isabelleps huggingface dataset:
RedT_def: shows "RedT \<equiv> mthr.Trsys"
tm_append_fetch_first: fixes A :: "(action \<times> nat) list"   and s :: "nat"   and b :: "cell"   and ac :: "action"   and ns :: "nat"   and B :: "(action \<times> nat) list" assumes "fetch A s b = (ac, ns)"   and "ns \<noteq> 0" shows "fetch (A @ B) s b = (ac, ns)"
ipresSwap_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'c, 'h, 'i) Iteration.model_scheme" shows "ipresSwap h MOD \<equiv> \<forall>zs z1 z2 s X. wls s X \<longrightarrow> h (X #[z1 \<and> z2]_zs) = igSwap MOD zs z1 z2 (h X)"
absc_repc_id: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg" shows "absc (repc cfg) = cfg"
sat_thms_p: fixes p :: "'b pl"   and tt :: "'b set" assumes "{} |= p" shows "hyps p tt |- p"
total: fixes act :: "('a \<times> 'a) set" assumes "act \<in> Acts F" shows "Domain act = UNIV"
rep_real_alg_def: shows "rep_real_alg \<equiv> quot_type.rep Rep_real_alg"
cla_def: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and Z :: "('d \<Rightarrow> real) set" assumes "AlphaClosure X" shows "AlphaClosure.cla X k Z = \<Union> {R \<in> {region X I r |I r. valid_region X k I r}. R \<inter> Z \<noteq> {}}"
filter_min_aux_supset: fixes ys :: "'b list"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "set ys \<subseteq> set (General.filter_min_aux rel xs ys)"
simps_127: fixes x1 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>x1\<rangle>) = fv x1"
substEnv_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wlsEnv rho"   and "wls (asSort ys) Y" shows "wlsEnv (rho &[Y / y]_ys)"
BoxDmdBox: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<box>\<diamond>\<box>F) = (\<diamond>\<box>F)"
up'_inplace: fixes p' :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat"   and l :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "p' \<notin> grid p ds"   and "d \<in> ds" shows "snd (up' d l p \<alpha>) p' = \<alpha> p'"
hom_into_subgroup_eq: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme"   and h :: "'c \<Rightarrow> 'a"   and K :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "(h \<in> hom K (subgroup_generated G H)) = (h \<in> hom K G \<and> h ` carrier K \<subseteq> H)"
interp_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.interp S N C = \<Union> (ground_resolution_with_selection.production S N ` {D. D < C})"
pair_in_Id_conv: fixes a :: "'k"   and b :: "'k" shows "((a, b) \<in> Id) = (a = b)"
sta: shows "\<forall>(q, \<tau>)\<in>set (step p (start ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> start ! q"
snoc_5: shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> n'"
Cauchy_higher_deriv_bound: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and y :: "complex"   and B0 :: "real"   and n :: "nat" assumes "f holomorphic_on ball z r"   and "continuous_on (cball z r) f"   and "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"   and "0 < r"   and "0 < n" shows "cmod ((deriv ^^ n) f z) \<le> fact n * B0 / r ^ n"
arrI: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.arr (fst f) \<and> B.VV.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (fst (snd f))" shows "B.VVV.arr f"
colRec: shows "col (Matrix_Legacy.transpose nr m) i = row m i"
lookup_pair_sort_oalist': fixes xs :: "('a \<times> 'c) list"   and x :: "'o" assumes "distinct (map fst xs)" shows "lookup_pair x (sort_oalist x xs) = lookup_dflt xs"
i_len: shows "enat i < llength (lmap snd (ltake r E))"
distinct_113: fixes x41 :: "Syntax.var"   and x42 :: "Syntax.term"   and x13 :: "Syntax.term" shows "Rec x41 x42 \<noteq> Roll x13"
measure_spmf_spmf_of_pmf: fixes p :: "'c pmf" shows "measure_spmf (spmf_of_pmf p) = measure_pmf p"
proots_pcompose_bij_eq: fixes q :: "'a poly"   and A :: "'a set"   and B :: "'a set"   and p :: "'a poly" assumes "bij_betw (poly q) A B"   and "p \<noteq> 0"   and "degree q = 1" shows "proots_count p B = proots_count (p \<circ>\<^sub>p q) A"
bac_close: shows "(\<lambda>n. real_of_int (B * int (nth_prime n)) / real_of_int (a n) - real_of_int (c n)) \<longlonglongrightarrow> 0"
Rep_data_select: fixes D :: "'b data" shows "(DataValue D, Data.DataSpace D) \<in> data"
not_equiv_trans1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<not> x \<sim> y"   and "y \<sim> z" shows "\<not> x \<sim> z"
is_shortest_path_onD2: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list"   and V :: "nat set"   and p' :: "nat list" assumes "is_shortest_path_on G m n p V"   and "p' \<in> path_set_on G m n V" shows "path_weight G p \<le> path_weight G p'"
fps_shift_subdegree: fixes n :: "nat"   and f :: "'a fps" assumes "n \<le> subdegree f" shows "subdegree (fps_shift n f) = subdegree f - n"
outs_\<I>_mono: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('i, 'j) \<I>" assumes "\<I> \<le> \<I>'" shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
Inf_top_conv_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(top = Inf A) = (\<forall>x\<in>A. x = top)"
q: fixes x :: "'call"   and s :: "'s" assumes "consider' x" shows "interaction_bound consider (callee s x) \<le> q"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('e, 'c) resource"   and b :: "('e, 'd) resource" assumes "rel_resource' R a b" obtains xa :: "'e \<Rightarrow> ('c \<times> ('e, 'c) resource) spmf" and ya :: "'e \<Rightarrow> ('d \<times> ('e, 'd) resource) spmf" where "a = Resource xa" and "b = Resource ya" and "rel_fun (=) (rel_spmf (rel_prod R (rel_resource' R))) xa ya"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr)|\<^sub>r"
oexp_gt_0_iff: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>" shows "(0 < \<alpha> \<up> \<beta>) = (0 < \<alpha> \<or> \<beta> = 0)"
assms: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
add_vars'_def: shows "add_vars' = fold_aterms (\<lambda>x l. case x of Fv idn s \<Rightarrow> List.insert (idn, s) l | _ \<Rightarrow> l)"
cong: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
ah_leq_il_right: fixes h1 :: "'b \<Rightarrow> 'b set"   and h2 :: "'b \<Rightarrow> 'b set"   and h2' :: "'b \<Rightarrow> 'b set" assumes "h1 [*] h2"   and "h2' \<le> h2" shows "h1 [*] h2'"
pair_wf_digraph_axioms: shows "pair_wf_digraph (subdivide G e w)"
padic_a_inv_in_padic_set: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p"   and "prime p" shows "padic_a_inv p f \<in> padic_set p"
per_union_memI3: fixes x :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and y :: "'b" assumes "(x, b) \<in> R"   and "(a, y) \<in> R" shows "(x, y) \<in> per_union R a b"
is_zero_row_utp_0': fixes A :: "(('e, 'f) vec, 'd) vec" shows "\<forall>m. is_zero_row_upt_k m 0 A"
preserves_arr: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.arr (Left_a.L f)"
new_character_data_new_ptr: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr|}"
rel_refl_strong: fixes x :: "'d pmf"   and Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> Ra z z" shows "rel_pmf Ra x x"
FlexRayArch_nGet: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and i :: "nat" assumes "FlexRayArch n nReturn nC nStore nGet"   and "\<forall>i<n. msg (Suc 0) (nReturn i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "i < n" shows "msg (Suc 0) (nGet i)"
combine_altdef: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m1 :: "('b, 'c) mapping"   and m2 :: "('b, 'c) mapping" shows "combine f m1 m2 = combine_with_key (\<lambda>_. f) m1 m2"
abs_eq: fixes xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "MPoly xa * MPoly x = MPoly (xa * x)"
NullPointer_def: shows "NullPointer \<equiv> ''NullPointer''"
ereal_le_distrib: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" shows "c * (a + b) \<le> c * a + c * b"
D1: shows "D1 \<in> carrier_mat (dim_row A - 1) (dim_col A)"
sqrt: shows "\<bar>x\<bar> < sqrt * sqrt \<and> sqrt * sqrt - \<bar>x\<bar> < \<epsilon>"
path_set_on_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and V :: "nat set" shows "path_set_on G m n V = {p \<in> path_set G m n. int_pts p \<subseteq> V}"
itypeI': fixes c :: "'c" shows "c ::\<^sub>i TYPE('d)"
no_zero_divisors: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors plus zero times"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "times a b \<noteq> zero"
L: shows "c * (d * (s' + a' * b')) = d' * e' \<and> IntAgrK (S (Card n, n, run)) = Some d' \<and> ExtAgrK (S (Card n, n, run)) = Some e'"
assms_2: shows "is_imgu \<theta> (fterm_to_iterm ` ts \<times> fterm_to_iterm ` ts)"
drop_numeral_Cons_1: fixes x :: "'b"   and xs :: "['b]" shows "Data_List.drop\<cdot>1\<cdot>(x : xs) = xs"
diamond_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "diamond R = commute R R"
fused_wiring_def: shows "fused_wiring \<equiv> (parallel_wiring |\<^sub>= parallel_wiring) \<odot> parallel_wiring"
inputFreshDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> N" shows "x \<sharp> P'"
lemma1: fixes as :: "letter list"   and ws :: "letter list list"   and a :: "letter" assumes "L as ws" shows "L (a # as) ws"
elims: fixes x :: "fterm literal"   and xa :: "char list \<Rightarrow> fterm"   and y :: "fterm literal" assumes "x \<cdot>\<^sub>l xa = y" obtains p :: "char list" and ts :: "fterm list" and \<sigma> :: "char list \<Rightarrow> fterm" where "x = Pos p__ ts__" and "xa = \<sigma>__" and "y = Pos p__ (ts__ \<cdot>\<^sub>t\<^sub>s \<sigma>__)"   | p :: "char list" and ts :: "fterm list" and \<sigma> :: "char list \<Rightarrow> fterm" where "x = Neg p ts" and "xa = \<sigma>" and "y = Neg p (ts \<cdot>\<^sub>t\<^sub>s \<sigma>)"
len_div2: fixes x :: "nat" shows "len (x div 2) = len x - 1"
prop_make_vert_equiv_3: fixes n :: "nat" shows "mat (rat_poly.row_length (make_vert_equiv n)) (length (make_vert_equiv n)) (make_vert_equiv n)"
simps_1: fixes a :: "symbol list"   and b :: "symbol list" shows "LeftDerivation a [] b = (a = b)"
Un_insert_right: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A \<union> insert a B = insert a (A \<union> B)"
act_emp: fixes x :: "'a" shows "\<alpha> x (\<Squnion> {}) = \<bottom>"
expectation_gpv_const_le: assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv 0 \<I> (\<lambda>_. 1) gpv \<le> max 1 0"
chine_assoc'_def: shows "trg_trg_f.chine_assoc' \<equiv> \<langle>\<langle>trg_trg_f.Prj\<^sub>1 \<lbrakk>trg.leg0, trg.leg1\<rbrakk> trg_trg_f.Prj\<^sub>1\<^sub>0\<rangle> \<lbrakk>trg.leg0 \<cdot> trg_trg.prj\<^sub>0, f.leg1\<rbrakk> trg_trg_f.Prj\<^sub>0\<^sub>0\<rangle>"
set_map_of_compr: fixes xs :: "('c \<times> 'd) list" assumes "distinct (map fst xs)" shows "set xs = {(k, v). map_of xs k = Some v}"
simps_4: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cCurrentF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cCurrentF"
wf_extend_oprod2: fixes r :: "('d \<times> 'd) set"   and A :: "'e set" assumes "wf r" shows "wf {((x1, y), x2, y). (x1, x2) \<in> r \<and> y \<in> A}"
preserves_cones: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'c"   and a :: "'c"   and \<chi> :: "'a \<Rightarrow> 'c" assumes "Limit.cone J (\<cdot>) D a \<chi>" shows "Limit.cone J (\<cdot>) (C.map \<circ> D) (C.map a) (C.map \<circ> \<chi>)"
matrix_of_matrix_vector_mul: fixes A :: "(('b, 'c) vec, 'd) vec" shows "matrix ((*v) A) = A"
simps_1: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "R ^^ 0 = (=)"
SS': shows "{xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1} = {xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (config\<^sub>p (rTS h) init (as @ [x] @ bs @ [x] @ take i cs)) \<and> count_list (take (index (snd (config\<^sub>p (rTS h) init (as @ [x] @ bs @ [x] @ take i cs))) (cs ! i)) (snd (config\<^sub>p (rTS h) init (as @ [x] @ bs @ [x] @ take i cs)))) xa \<le> 1}"
paracompact: fixes S :: "'c set"   and \<C> :: "'c set set" assumes "S \<subseteq> \<Union> \<C>"   and "\<And>T. T \<in> \<C> \<Longrightarrow> open T" obtains \<C>' :: "'c set set" where "S \<subseteq> \<Union> \<C>'" and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>'. U \<inter> V \<noteq> {}}"
mds\<^sub>0_def: shows "mds\<^sub>0 \<equiv> \<lambda>m. case m of AsmNoReadOrWrite \<Rightarrow> {reg\<^sub>C} | _ \<Rightarrow> {}"
opp_add: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2" shows "ell_field.opp R (ell_field.add R a p\<^sub>1 p\<^sub>2) = ell_field.add R a (ell_field.opp R p\<^sub>1) (ell_field.opp R p\<^sub>2)"
index_vec_of_list: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "vec_of_list xs $ i = xs ! i"
reindex: fixes h :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'b \<Rightarrow> 'c" assumes "inj_on h A" shows "prod g (h ` A) = prod (g \<circ> h) A"
rstep_def: fixes l :: "'l"   and s :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "C.rstep l s i \<equiv> clstep l (s i) (s (Suc i))"
comp_eq_dest_lhs: fixes a :: "'d \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'd"   and c :: "'a \<Rightarrow> 'b"   and v :: "'a" assumes "a \<circ> b = c" shows "a (b v) = c v"
it_to_it_def: fixes it :: "('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list" shows "it_to_it it \<equiv> foldli (it (\<lambda>_. True) (\<lambda>x l. l @ [x]) [])"
open_negations: fixes S :: "'b set" assumes "open S" shows "open (uminus ` S)"
ND0aux3: fixes x :: "'b"   and y :: "'b"   and p :: "'c"   and b :: "('b, 'c) Combinators list" assumes "AllowPortFromTo x y p \<in> set b" shows "x \<in> set (net_list_aux b)"
options_def: fixes g :: "game" shows "options g \<equiv> zunion (left_options g) (right_options g)"
d: shows "\<eta> $$ X maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ X to NTCod \<eta> @@ X"
simps: fixes V :: "nat set"   and s :: "(nat set \<times> 'a option list set) set" shows "projectQuery V s = projectTable V ` s"
nnf_BigAnd: fixes F :: "'a formula list" shows "nnf \<^bold>\<And>F = \<^bold>\<And>map nnf F"
nth_eq_iff_index_eq: fixes xs :: "'c list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "(xs ! i = xs ! j) = (i = j)"
inject_2: fixes x2 :: "ref_ty"   and y2 :: "ref_ty" shows "(RefT x2 = RefT y2) = (x2 = y2)"
frac_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and w :: "'b"   and z :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero y"   and "less_eq x y"   and "less zero w"   and "less_eq w z" shows "less_eq (divide x z) (divide y w)"
inverse_diff_inverse: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
dvd_mult2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "a dvd b * c"
surjective_top_closed: shows "bounded_distrib_lattice_class.comp_inf.surjective top"
sup_subset_mset_def: fixes A :: "'k multiset"   and B :: "'k multiset" shows "A \<union># B = A + (B - A)"
fmpred_fmmap_keys: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmpred P (fmmap_keys f m) = fmpred (\<lambda>a b. P a (f a b)) m"
Init_def: shows "DFIG.Init = (\<lambda>(A, A'). retidS A' = {} \<and> DFIG_alphabet.max A' = 0)"
assms_3: shows "linearly_independent_over_int (\<alpha> ` {..<n})"
kmp_SPEC_altdefs_1: fixes s :: "'a list"   and t :: "'a list" shows "kmp_SPEC s t = SPEC (\<lambda>x. case x of None \<Rightarrow> \<nexists>i. sublist_at s t i | Some i \<Rightarrow> is_arg_min id (sublist_at s t) i)"
ord_isom_onto: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Order E"   and "ord_isom D E f" shows "f ` carrier D = carrier E"
simps: fixes a :: "sterm" shows "is_value a = ((\<exists>cs. a = Sabs cs) \<or> (\<exists>vs name. a = name $$ vs \<and> list_all is_value vs \<and> name |\<in>| C))"
disjnt_subset2: fixes X :: "'c set"   and Y :: "'c set"   and Z :: "'c set" assumes "disjnt X Y"   and "Z \<subseteq> Y" shows "disjnt X Z"
REP_CLS: fixes f :: "'a" assumes "B.is_left_adjoint f" shows "REP \<lbrakk>\<lbrakk>f\<rbrakk>\<rbrakk> \<cong>\<^sub>B f"
finite_RedBlack_def: fixes prb :: "('d, 'e, 'f) pre_RedBlack" shows "finite_RedBlack prb \<equiv> finite (pred (init_conf prb)) \<and> (\<forall>e\<in>pred (init_conf prb). finite (Bexp.vars e)) \<and> finite_lts (black prb)"
leof_True_rule: fixes \<Phi> :: "'b \<Rightarrow> bool"   and m :: "'b nres" assumes "\<And>x. \<Phi> x" shows "m \<le>\<^sub>n SPEC \<Phi>"
Arr_implies_Ide_Cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Ide C (monoidal_language.Cod C t)"
discounted_init: fixes r :: "real"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real" shows "\<forall>w. discounted_value r X 0 w = X 0 w"
eq_iff: fixes a :: "'a list"   and b :: "'a list" shows "(a = b) = (a \<le>lex b \<and> b \<le>lex a)"
set_disconnected_nodes_locs_def: shows "set_disconnected_nodes_locs = (\<lambda>document_ptr. all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update))"
PsiInv_alpha1: shows "\<turnstile> alpha1 \<and> $PsiInv \<longrightarrow> PsiInv$"
sum_nonneg_eq_0_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "(comm_monoid_add.sum (+) {#} f A = {#}) = (\<forall>x\<in>A. f x = {#})"
substNext_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.substNext u p g \<equiv> \<lambda>v. if v = uninst.chooseNext u p g then uninst.substitution u p g else v"
monad_state_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return bind get put" shows "monad_state (return_option return) (bind_option return bind) (get_option get) (put_option put)"
card_gt_0_iff: fixes A :: "'a set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
j_def: shows "j \<le> i \<and> vs @ xs \<in> C M2 M1 \<Omega> V m j"
simps_2: fixes uv :: "'b option" shows "Applicative.ap None uv = None"
semigroup_mult_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "semigroup_mult_hom hom \<equiv> times_hom hom"
function_mult_eval_car: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "a \<in> S"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "(f \<otimes>\<^bsub>function_ring S R\<^esub> g) a = f a \<otimes>\<^bsub>R\<^esub> g a"
Input: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> K\<lparr>N[xvec::=Tvec]\<rparr> \<prec> P[xvec::=Tvec]"
Nil_in_shuffles: fixes xs :: "'g list"   and ys :: "'g list" shows "([] \<in> shuffles xs ys) = (xs = [] \<and> ys = [])"
gen_igba_impl_rel_eext_def: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i \<Rightarrow> bool)) set" shows "\<langle>Rm, Rl\<rangle>gen_igba_impl_rel_eext = {(\<lparr>igbai_L = Li, \<dots> = mi\<rparr>, \<lparr>igba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
intersect_list'_nodes_finite: fixes AA :: "('c, 'd) dba list" assumes "list_all (finite \<circ> DBA.nodes) AA" shows "finite (DGBA.nodes (intersect_list' AA))"
Red_F_\<G>_def: fixes N :: "'f set" shows "local.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F C. D \<in> Red_F_G (\<G>_Fset N) \<or> (\<exists>E\<in>N. Prec_F_g D E C \<and> D \<in> \<G>_F E)}"
1: shows "\<exists>\<psi>. \<guillemotleft>\<psi> : F \<I>\<^sub>C \<rightarrow>\<^sub>D \<I>\<^sub>D\<guillemotright> \<and> D.iso \<psi>"