hcju/isabelleps huggingface dataset:
wt'_wt: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e :' T" shows "P,E,h \<turnstile> e : T"
supp_def: fixes f :: "'a \<Rightarrow> 'b" shows "supp f = {x. f x \<noteq> (0::'b)}"
bin_rcat_eq_foldl: fixes n :: "nat" shows "bin_rcat n = foldl (\<lambda>u v. concat_bit n v u) 0"
unit_simps_5: fixes a :: "'b" assumes "D.obj a" shows "D.cod \<i>\<^sub>D[a] = a"
interleave_eq_size: fixes s :: "'a event list"   and t :: "'a event list"   and C :: "'a event set"   and u :: "'a event list"   and s' :: "'a event list" assumes "s setinterleaves ((t, u), C)"   and "s' setinterleaves ((t, u), C)" shows "length s = length s'"
not_reset_dur: fixes i :: "nat"   and j :: "nat"   and c :: "'c"   and xs :: "('c \<Rightarrow> real) stream" assumes "\<forall>k>i. k \<le> j \<longrightarrow> \<not> zero c ([xs !! k]\<^sub>\<R>)"   and "i \<le> j"   and "c \<in> \<X>"   and "stream_trans xs"   and "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"   and "\<forall>i. xs !! i \<in> V" shows "dur xs j - dur xs i = (xs !! j) c - (xs !! i) c"
wlsAbs: shows "wlsAbs (xs, s) (Abs xs' x X)"
\<tau>: fixes es' :: "('a, 'b, 'addr) exp list" assumes "\<tau>moves1 P h (Val v # es')" shows "\<tau>moves1 P h es'"
hseq_char: fixes \<nu> :: "'c"   and \<mu> :: "'c" shows "C.hseq \<nu> \<mu> = (C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<nu> = trg\<^sub>C \<mu>)"
simps_1: fixes stack :: "'c list"   and env :: "'d \<Rightarrow> 'c" shows "exec [] stack env = stack"
nv_4: fixes w :: "'a vec" assumes "w \<in> D" shows "normal_vector D \<bullet> w = (0::'a)"
pmaxsimpchain: fixes xs :: "'a set list" assumes "pgallery xs" shows "gallery xs"
genideal_zero: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "Idl\<^bsub>R\<^esub> {\<zero>\<^bsub>R\<^esub>} = {\<zero>\<^bsub>R\<^esub>}"
brec_inner_step_def: fixes Qi :: "('a, unit) HashMap.hashmap"   and r :: "('a, 'b) ta_rule" shows "brec_inner_step Qi r \<equiv> \<lambda>(Q, W, rcm, qwit). let c = the (hm.lookup r rcm); cond = c \<le> 1 \<and> hm.lookup (lhs r) Q = None; rcm' = hm.update r (c - 1) rcm; Q' = if cond then hm.update (lhs r) (brec_construct_witness Q r) Q else Q; W' = if cond then fifo.enqueue (lhs r) W else W; qwit' = if c \<le> 1 \<and> hs.memb (lhs r) Qi then Some (lhs r) else qwit in (Q', W', rcm', qwit')"
order_subst2: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
size_3: fixes x31 :: "fm"   and x32 :: "fm" shows "size (x31 OR x32) = size x31 + size x32 + Suc 0"
Pz_nn: fixes z :: "'d" assumes "z \<in> space P" shows "0 \<le> Pz z"
IH: shows "(\<Sum>i = a..<a + b. f (i - a)) = sum f {..<b}"
rule_sub_lem_1_a: fixes v :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v]" shows "[\<^bold>\<not>\<psi> \<^bold>\<equiv> \<^bold>\<not>\<chi> in v]"
simps_7: fixes bbs :: "bool list"   and bs :: "int list"   and a :: "Cooper.num" shows "Ifm bbs bs (Eq a) = (Inum bs a = 0)"
face_cycle_succ_cyclic: fixes a :: "'b" shows "cyclic_on face_cycle_succ (face_cycle_set a)"
interleave_intros_1: shows "[] \<in> [] \<otimes> []"
oalist_inv_raw_sort_oalist: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" shows "oalist_inv_raw x (sort_oalist x xs)"
not_eq_order_implies_strict: fixes a :: "'a \<Rightarrow>\<^sub>0 nat"   and b :: "'a \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> b"   and "b \<preceq> a" shows "b \<prec> a"
subtract_Some_all_distinct_res: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree"   and t :: "'b tree" assumes "subtract t\<^sub>1 t\<^sub>2 = Some t"   and "all_distinct t\<^sub>2" shows "all_distinct t"
bernoulli_conv_bernoulli'_if: fixes n :: "nat" assumes "n \<noteq> 1" shows "bernoulli n = (if n = 1 then - 1 / 2 else bernoulli' n)"
eq_up_to_apply: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "IFC prob"   and "IFC_def.eq_up_to \<pi> k \<pi>'"   and "j \<le> k" shows "\<pi> j = \<pi>' j"
two_add_simp: fixes x :: "one\<^sub>\<bottom>"   and y :: "one\<^sub>\<bottom>" shows "two_add\<cdot>x\<cdot>y = (if x \<sqsubseteq> none then y else if y \<sqsubseteq> none then x else many)"
simps_2: fixes xs :: "'b list" shows "shuffles xs [] = {xs}"
supset_ld_is_ld: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and A :: "'g set"   and B :: "'g set" assumes "module R M"   and "module.lin_dep R M A"   and "A \<subseteq> B" shows "module.lin_dep R M B"
axioms_2: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partial_equivalence A equiv" shows "partial_equivalence_axioms A equiv"
preserve_rtranclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. R x y \<Longrightarrow> S (f x) (f y)"   and "R\<^sup>*\<^sup>* x y" shows "S\<^sup>*\<^sup>* (f x) (f y)"
mr_in_dom: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "x \<in> dom (C a)"
of_nat_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and f :: "'d \<Rightarrow> nat"   and A :: "'d set" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. semiring_1.of_nat one plus zero (f x)) A"
IH_2: fixes ds :: "bool list" assumes "branch ds T\<^sub>2" shows "path ds T\<^sub>2"
fps_monom: fixes k :: "nat"   and c :: "'b" shows "Abs_fps (\<lambda>n. of_nat n ^ k * c ^ n) = fps_of_poly (fps_monom_poly c k) / (1 - fps_const c * fps_X) ^ (k + 1)"
attract: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubseteq> x"   and "x \<sqsubseteq> y"   and "z \<sqsubseteq> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
hom: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and g :: "'a"   and g' :: "'a" assumes "GroupHom G T"   and "g \<in> G"   and "g' \<in> G" shows "T (g + g') = T g + T g'"
chine_is_induced_map: shows "\<pi>.is_induced_map \<pi>.chine"
conga_preserves_lta: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong D E F D' E' F'"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LtA Bet Cong A' B' C' D' E' F'"
subtract_Some_dist_res: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree"   and t :: "'b tree" assumes "subtract t\<^sub>1 t\<^sub>2 = Some t" shows "set_of t\<^sub>1 \<inter> set_of t = {}"
dominates_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and g :: "'a"   and n :: "'b"   and m :: "'b" shows "graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g n m \<equiv> m \<in> set (\<alpha>n g) \<and> (\<forall>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns m \<longrightarrow> n \<in> set ns)"
res_def: fixes side_oracle :: "'b cstate \<Rightarrow> 'a \<Rightarrow> ('c option \<times> 'b cstate) spmf" shows "channel.res side_oracle \<equiv> RES (side_oracle \<oplus>\<^sub>O CIPHER.AUTH.send_oracle \<oplus>\<^sub>O CIPHER.AUTH.recv_oracle) Void"
scaleR2_rel_def: fixes A :: "('d \<times> ('e \<times> 'f) set) set" shows "\<langle>A\<rangle>scaleR2_rel = ((ereal_rel \<times>\<^sub>r ereal_rel) \<times>\<^sub>r A) O br (\<lambda>((l, u), X). scaleR2 l u X) (\<lambda>((l, u), uu_). ereal -` {l..u} \<noteq> {})"
remDupsNMT: fixes p :: "'a list" assumes "p \<noteq> []" shows "remdups p \<noteq> []"
inf_sup_aci_1: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "inf_prod x y = inf_prod y x"
per_union_def: fixes R :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" shows "per_union R a b \<equiv> R \<union> {(x, y). (x, a) \<in> R \<and> (y, b) \<in> R} \<union> {(y, x). (x, a) \<in> R \<and> (y, b) \<in> R}"
htchLoc_def: fixes lH :: "('c \<times> 'd \<times> nat) set"   and \<iota> :: "nat" shows "WHATWHERE.htchLoc lH \<iota> \<equiv> {(d, e). (d, e, \<iota>) \<in> lH}"
SUP_upper2: fixes i :: "'b"   and A :: "'b set"   and u :: "'c \<times> 'd"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "s2p A i"   and "less_eq_prod u (f i)" shows "less_eq_prod u (Sup_prod (\<P> f A))"
domComm: fixes a :: "'b \<Rightarrow> 'c option"   and b :: "'b \<Rightarrow> 'd option" shows "dom a \<inter> dom b = dom b \<inter> dom a"
cong_less_modulus_unique_nat: fixes x :: "nat"   and y :: "nat"   and m :: "nat" assumes "[x = y] (mod m)"   and "x < m"   and "y < m" shows "x = y"
Radon_partition: fixes C :: "'c set" assumes "finite C"   and "affine_dependent C" shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = C \<and> convex hull m \<inter> convex hull p \<noteq> {}"
red2_Trg_Nml: fixes t :: "'c local.term" assumes "Ide t"   and "Nml t" shows "\<lbrace>Trg t \<^bold>\<Down> t\<rbrace> = \<l>[\<lbrace>t\<rbrace>]"
e: assumes "\<forall>f\<in>set fs. irreducible G f"   and "foldr (\<otimes>) fs \<one> \<sim> a" shows "P"
inv_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree" assumes "invh l"   and "invh r"   and "bheight l = bheight r"   and "invc l"   and "invc r" shows "invh (RBT.join l r) \<and> bheight (RBT.join l r) = bheight l \<and> invc2 (RBT.join l r) \<and> (color l = Black \<and> color r = Black \<longrightarrow> invc (RBT.join l r))"
Ks'_1: shows "set (take (length Us) Ks) \<subseteq> set Ks'"
has_laplaceI: fixes s :: "complex"   and f :: "real \<Rightarrow> complex"   and L :: "complex" assumes "((\<lambda>t. exp (t *\<^sub>R - s) * f t) has_integral L) {0..}" shows "(f has_laplace L) s"
sum_list_update: fixes n :: "nat"   and xs :: "'a list"   and y :: "'a" assumes "n < length xs" shows "sum_list (xs[n := y]) = sum_list xs - xs ! n + y"
gen_keys_def: fixes kts :: "('a \<times> ('a, 'b) rbt) list"   and t :: "('a, 'b) rbt" shows "gen_keys kts t = keys t @ concat (map (\<lambda>(k, t). k # keys t) kts)"
invf: shows "invf o\<^sub>L f' x0 = id_blinfun"
pair_commute_pmf: fixes x :: "'a pmf"   and y :: "'b pmf" shows "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
HStartBallot_Inv2a_dblock: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "Inv2a_innermost s p (dblock s p)" shows "Inv2a_innermost s' p (dblock s' p)"
mult_coprime: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" assumes "multiplicative_function f"   and "coprime a b" shows "f (a * b) = f a * f b"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a" assumes "Finite_Graph c" shows "ek_analysis c"
is_regular_sequence_removeAll_zero: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list" assumes "is_regular_sequence fs" shows "is_regular_sequence (removeAll 0 fs)"
GEN_ALGO_def: fixes f :: "'d"   and \<Phi> :: "'d \<Rightarrow> 'c" shows "GEN_ALGO f \<Phi> \<equiv> \<Phi> f"
neg64: shows "SCAST(32 \<rightarrow> 64) w2 + 0x7FFFFFFF = SCAST(32 \<rightarrow> 64) w2 + 0x7FFFFFFF"
agreeDF: fixes s :: "'d \<Rightarrow> bool"   and env :: "('d, bool) mapping"   and x :: "'d" assumes "agree s env"   and "Mapping.lookup env x = Some False" shows "\<not> s x"
coextension_iff_cones_map: fixes a :: "'c"   and d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "'c"   and a' :: "'c"   and x' :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a d x"   and "\<guillemotleft>g : a' \<rightarrow>\<^sub>C a\<guillemotright>"   and "\<guillemotleft>x' : \<Delta>.map a' \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] d\<guillemotright>" shows "arrow_from_functor.is_coext (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a x a' x' g = (\<chi>.\<Phi>.Ya.Cop_S.Map x' = (\<lambda>\<chi>\<in>{\<chi>. Limit.cone (\<cdot>\<^sub>J) (\<cdot>) (\<chi>.\<Phi>.Ya.Cop_S.Map d) (C.cod g) \<chi>}. \<lambda>j. if J.arr j then \<chi> j \<cdot> g else C.null) (\<chi>.\<Phi>.Ya.Cop_S.Map x))"
case_prod_app: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'f"   and x :: "'g \<times> 'h"   and y :: "'i" shows "(case x of (x, xa) \<Rightarrow> f x xa) y = (case x of (l, r) \<Rightarrow> f l r y)"
equal: fixes equal :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
Q'eqQ: shows "Q' \<equiv>\<^sub>e Q"
double_complement: fixes A :: "'c set" shows "- (- A) = A"
RedAAssBounds: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr val" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "i <s 0 \<or> int n \<le> sint i" shows "extTA,P,t \<turnstile> \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayIndexOutOfBounds),s\<rangle>"
path_split_Cons: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge a'" and "CFG.path sourcenode targetnode valid_edge (targetnode a') as' n'"
is_path_split': fixes v :: "'V"   and p1 :: "('V \<times> 'W \<times> 'V) list"   and u :: "'V"   and w :: "'W"   and u' :: "'V"   and p2 :: "('V \<times> 'W \<times> 'V) list"   and v' :: "'V" shows "is_path v (p1 @ (u, w, u') # p2) v' = (is_path v p1 u \<and> (u, w, u') \<in> E \<and> is_path u' p2 v')"
connected_closed_set: fixes S :: "'a set" assumes "closed S" shows "connected S = (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
enat_less_enat_plusI2: fixes y :: "nat"   and z :: "enat"   and x :: "nat" assumes "enat y < z" shows "enat (x + y) < enat x + z"
is_partial_heap_except_down_def: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'b"   and l :: "int"   and m :: "int"   and r :: "int"   and i :: "int" shows "is_partial_heap_except_down cmp heap l m r i = (\<forall>x. bounded m r x \<longrightarrow> (parent l x \<noteq> i \<longrightarrow> bounded m r (parent l x) \<longrightarrow> cmp (heap (parent l x)) (heap x)) \<and> (parent l x = i \<longrightarrow> bounded m r (parent l (parent l x)) \<longrightarrow> cmp (heap (parent l (parent l x))) (heap x)))"
4: shows "True \<and> Sorted_Less.sorted (case tn of (t, n) \<Rightarrow> inorder t)"
ind_is_open_def: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and U :: "'a set" assumes "ind_topology X is_open S" shows "ind_topology.ind_is_open X is_open S U \<equiv> U \<subseteq> S \<and> (\<exists>V\<subseteq>X. is_open V \<and> U = S \<inter> V)"
divide_strict_right_mono: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and 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"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero c" shows "less (divide a c) (divide b c)"
expand_rslt_exist_eq: fixes n_ns :: "'b node \<times> 'b node set" shows "expand n_ns \<le> SPEC (expand_rslt_exist_eq n_ns)"
Int_stable_def: fixes M :: "'a set set" shows "Int_stable M = (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M)"
rtrancl_while_invariant: fixes st :: "'a list \<times> 'a set" assumes "local.rtrancl_while_invariant st"   and "local.rtrancl_while_test st" shows "local.rtrancl_while_invariant (local.rtrancl_while_step st)"
right_adjoint_to_equivalence_is_equivalence: fixes f :: "'a"   and g :: "'a" assumes "equivalence_map f"   and "adjoint_pair f g" shows "equivalence_map g"
lang_ENC_formula: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (ENC n (FOV \<phi>)) = \<Union> {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) \<phi>}"
strict_mono_def: fixes f :: "'a \<Rightarrow> 'b" shows "strict_mono f = (\<forall>x y. x < y \<longrightarrow> f x < f y)"
trace_hd: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" shows "hd (trace n f g xs x) = the (eval (Pr n f g) (x # xs))"
poly_subdegree_def: fixes p :: "'b poly" shows "poly_subdegree p = subdegree (fps_of_poly p)"
true_eqvt: fixes pi :: "('d \<times> 'd) list" shows "pi \<bullet> True = True"
chf_def: shows "chf = (\<lambda>A x. if x \<in> A then 0 else 1)"
alist_split_2: shows "priority = snd \<circ> alist"
loose_bvar1_imp_loose_bvar: fixes t :: "Core.term"   and n :: "nat" assumes "loose_bvar1 t n" shows "loose_bvar t n"
assms_4: shows "\<forall>k\<in>set vs. 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)"
power2_eq_square: fixes a :: "'a" shows "power.power (1::'a) (\<lambda>a b. b * a) a 2 = a * a"
real_le_ereal_iff: fixes y :: "ereal"   and x :: "real" shows "(real_of_ereal y \<le> x) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
remove_preserves_type_wf: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove child \<rightarrow>\<^sub>h h'" shows "type_wf h'"
cond_case_prod_eta: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<times> 'd \<Rightarrow> 'e" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
F: shows "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
AbstrLevel0: shows "AbstrLevel level0 = {sA1, sA2, sA3, sA4, sA5, sA6, sA7, sA8, sA9}"
nonces_def: shows "nonces \<equiv> (rep_msg ---> id) freenonces"
sorted_sublist_wrt_nth_le: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and i :: "nat"   and j :: "nat" assumes "sorted_sublist_wrt R xs lo hi"   and "lo \<le> hi"   and "hi < length xs"   and "lo \<le> i"   and "i < j"   and "j \<le> hi" shows "R (xs ! i) (xs ! j)"
cong_add: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "[b = c] (mod a)"   and "[d = e] (mod a)" shows "[b + d = c + e] (mod a)"
sum_list_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> set xs \<Longrightarrow> less_eq x zero" shows "less_eq (monoid_add.sum_list plus zero xs) zero"
Shuffle_def: fixes A :: "'b list set"   and B :: "'b list set" shows "A \<parallel> B = \<Union> {shuffles xs ys |xs ys. xs \<in> A \<and> ys \<in> B}"
het_base: fixes x :: "real list" assumes "pos x \<and> het x = 0" shows "gmean x = mean x"
similar_mat_four_block_0_ex: fixes A1 :: "'b mat"   and B1 :: "'b mat"   and A2 :: "'b mat"   and B2 :: "'b mat"   and A0 :: "'b mat"   and n :: "nat"   and m :: "nat" assumes "similar_mat A1 B1"   and "similar_mat A2 B2"   and "A0 \<in> carrier_mat n m"   and "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat m m" shows "\<exists>B0. B0 \<in> carrier_mat n m \<and> similar_mat (four_block_mat A1 A0 (0\<^sub>m m n) A2) (four_block_mat B1 B0 (0\<^sub>m m n) B2)"
induct: fixes P :: "('fun, 'var) extstrand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) extstrand_step list" assumes "P []"   and "\<And>x S. P S \<Longrightarrow> P (Step x # S)"   and "\<And>t S. P S \<Longrightarrow> P (Decomp t # S)" shows "P a0"
sup_id_top: fixes y :: "'a" assumes "(1::'a) \<le> y" shows "y \<cdot> \<top> = \<top>"
mset_map_split_orig: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d multiset"   and M1 :: "'c multiset"   and M2 :: "'c multiset" assumes "f `# P = M1 + M2" obtains P1 :: "'d multiset" and P2 :: "'d multiset" where "P = P1 + P2" and "f `# P1 = M1" and "f `# P2 = M2"
DSourcesA12_L1: shows "DSources level1 sA12 = {}"
set_embed_pmf: fixes f :: "'h \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "(\<Sum>\<^sup>+ x. ennreal (f x)) = 1" shows "set_pmf (embed_pmf f) = {x. f x \<noteq> 0}"
intro: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>a a' b. prod (a + a') b = prod a b + prod a' b"   and "\<And>a b b'. prod a (b + b') = prod a b + prod a b'"   and "\<And>r a b. prod (r *\<^sub>C a) b = cnj r *\<^sub>C prod a b"   and "\<And>a r b. prod a (r *\<^sub>C b) = r *\<^sub>C prod a b"   and "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" shows "bounded_sesquilinear prod"
write_state_reg_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {sreg_type WRY, sreg_type WRPSR, sreg_type WRWIM, sreg_type WRTBR}" shows "snd (write_state_reg_instr i s) = False"
wellFoundedPrecedesOrder: fixes list :: "'b list" shows "wf (precedesOrder list)"
subspace_neg: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S" shows "- x \<in> S"
set_split: fixes n :: "nat" shows "{k. k < Suc n} = {k. k < n} \<union> {n}"
cone_decomp_closed_uminus: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "cone_decomp T ps"   and "a \<in> T" shows "- a \<in> T"
unstables_def: fixes r :: "nat list \<Rightarrow> nat list \<Rightarrow> bool"   and step :: "nat \<Rightarrow> nat list \<Rightarrow> (nat \<times> nat list) list"   and \<tau>s :: "nat list list" shows "unstables r step \<tau>s = sorted_list_of_set {p. p < length \<tau>s \<and> \<not> stable r step \<tau>s p}"