hcju/isabelleps huggingface dataset:
set_mode_writes: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_mode :: "shadow_root_mode"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_mode_locs shadow_root_ptr) (set_mode shadow_root_ptr shadow_root_mode) h h'"
AF_DF_Suc: fixes k :: "nat"   and i :: "nat" shows "AF k i \<lless> DF k (Suc i)"
accs_exclusive2: fixes \<delta> :: "('a, 'b) ta_rule set"   and \<delta>' :: "('a, 'b) ta_rule set"   and n :: "'b tree"   and q :: "'a" assumes "accs (\<delta> \<union> \<delta>') n q"   and "\<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}"   and "q \<in> \<delta>_states \<delta>'" shows "accs \<delta>' n q"
node_comp_def: fixes i :: "nat"   and np :: "('d, 'e seq_action) automaton"   and R\<^sub>i :: "nat set" shows "\<langle>i : np : R\<^sub>i\<rangle> \<equiv> \<lparr>init = {NodeS i s R\<^sub>i |s. s \<in> init np}, trans = node_sos (automaton.trans np)\<rparr>"
bounded_linear_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f (g x))"
ksimplex_eq_bot: fixes p :: "nat"   and n :: "nat"   and b_s :: "nat \<Rightarrow> nat"   and u_s :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and b_t :: "nat \<Rightarrow> nat"   and u_t :: "nat \<Rightarrow> nat"   and t :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" assumes "kuhn_simplex_pair p n b_s u_s s b_t u_t t"   and "a \<in> s"   and "\<And>a'. a' \<in> s \<Longrightarrow> a \<le> a'"   and "b \<in> t"   and "\<And>b'. b' \<in> t \<Longrightarrow> b \<le> b'"   and "s - {a} = t - {b}" shows "s = t"
quotient_disj: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set"   and Y :: "'b set" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r" shows "X = Y \<or> X \<inter> Y = {}"
ZFDepth_def: shows "ZFDepth = Nat2nat \<circ> ZFTFst"
dom_lookup: fixes vo :: "'d option"   and kts :: "('c \<times> ('c, 'd) trie) list" shows "dom (lookup_trie (Trie vo kts)) = (\<Union>k\<in>dom (map_of kts). (#) k ` dom (lookup_trie (the (map_of kts k)))) \<union> (if vo = None then {} else {[]})"
assms_4: shows "d \<turnstile> ev \<mapsto> d'"
approximating_semantics_iff_fun_good_ruleset: fixes rs :: "'b rule list"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and s :: "state"   and t :: "state" assumes "good_ruleset rs" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t = (approximating_bigstep_fun \<gamma> p rs s = t)"
upd_cond_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'b option"   and u :: "'a"   and v' :: "'a" shows "Prim2.upd_cond w g r Q \<pi> u v' \<equiv> (v', u) \<in> edges g \<and> v' \<noteq> r \<and> (Q v' = \<infinity> \<longrightarrow> \<pi> v' = None) \<and> enat (w {v', u}) < Q v'"
prime_in_interval_exists: fixes c :: "real" assumes "1 < c" shows "\<forall>\<^sub>F x in at_top. \<exists>p. prime p \<and> real p \<in> {x<..c * x}"
openin_Int_derived_set_of_subset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "S \<inter> X derived_set_of T \<subseteq> X derived_set_of (S \<inter> T)"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) lit" assumes "satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>__" and "xa = Pos a__" and "satA \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> satA \<xi> a"
assigns_if_good_approx': fixes G :: "prog"   and s0 :: "state"   and e :: "expr"   and b :: "val"   and s1 :: "state"   and C :: "qtname"   and L :: "lenv" assumes "G\<turnstile>s0 \<midarrow>e-\<succ>b\<rightarrow> s1"   and "normal s1"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" shows "assigns_if (the_Bool b) e \<subseteq> dom (locals (snd s1))"
meet_above_iff: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<sqsubseteq> x \<sqinter> y) = (z \<sqsubseteq> x \<and> z \<sqsubseteq> y)"
\<delta>_var_on_t: shows "\<delta> ` fv t \<subseteq> range Var"
elements_xs_range: shows "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"
ih_6: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
rank_ins2: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "rank_invar bq" shows "rank t \<le> rank (hd (ins t bq)) \<or> rank (hd (ins t bq)) = rank (hd bq) \<and> bq \<noteq> []"
elem_TT: fixes x :: "'a"   and xs :: "[:'a:]" shows "(selem\<cdot>x\<cdot>xs = TT) = (x \<in> sset xs)"
axioms_1: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2" shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"
distinct_39: fixes x2 :: "'d \<Rightarrow> 'd" shows "Basic x2 \<noteq> Throw"
Fix2: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set" assumes "Monotone \<phi>"   and "\<phi> (FIX \<phi>) (s, t, \<beta>)" shows "FIX \<phi> (s, t, \<beta>)"
retract_bc: fixes x :: "'Fbc"   and y :: "'Fb" assumes "R\<^sup>*\<^sup>* (\<pi>_Fbcb x) y" shows "\<exists>z. Ebc x z \<and> y = \<pi>_Fbcb z"
max_wp_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. maximal (wp (a (f s)))" shows "maximal (wp (Bind f a))"
outputInduct: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"   and "\<And>a b P C. F C (a{b}.P) a b P"   and "\<And>P a b P' c C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C ([c\<frown>c]P) a b P'"   and "\<And>P a b P' c d C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; \<And>C. F C P a b P'; c \<noteq> d\<rbrakk> \<Longrightarrow> F C ([c\<noteq>d]P) a b P'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b P'"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b Q'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P' \<parallel> Q)"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P \<parallel> Q')"   and "\<And>P a b P' x C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; x \<noteq> a; x \<noteq> b; x \<sharp> C; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a b (<\<nu>x>P')"   and "\<And>P a b P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; \<And>C. F C (P \<parallel> !P) a b P'\<rbrakk> \<Longrightarrow> F C (!P) a b P'" shows "F C P a b P'"
miller_rabin_def: shows "miller_rabin = primality_test (\<lambda>n a. strong_fermat_liar a n)"
that: fixes B :: "'a mat" assumes "inverts_mat A B"   and "inverts_mat B A"   and "B \<in> carrier_mat n n" shows "thesis"
quasinorm_subsetD: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm" assumes "N1 \<subseteq>\<^sub>N N2" shows "\<exists>C\<ge>0. \<forall>f. eNorm N2 f \<le> ennreal C * eNorm N1 f"
lm110: fixes Z :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "Z \<subseteq> X \<union> Y"   and "finite Z" shows "sum (X <| Y) Z = sum (X <| Y) (Z \<inter> X)"
uniformly_convergentI: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and X :: "'b set"   and l :: "'b \<Rightarrow> 'c" assumes "uniform_limit X f l sequentially" shows "uniformly_convergent_on X f"
C_eq_compileQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l))))) = Cp p"
continuous_onI_antimono: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "open (f ` A)"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<le> y\<rbrakk> \<Longrightarrow> f y \<le> f x" shows "continuous_on A f"
zero_expansion_ms_def: shows "zero_expansion = MS MSLNil (\<lambda>_. 0)"
proj_onto_sum_rw: fixes v :: "'a"   and A :: "'a set" shows "(\<Sum>x\<in>A. (x \<bullet> v / (x \<bullet> x)) *\<^sub>R x) = (\<Sum>x\<in>A. (v \<bullet> x / (x \<bullet> x)) *\<^sub>R x)"
g_bound_def: fixes fs :: "int vec list" shows "g_bound fs = (\<forall>i<m. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 \<le> rat_of_nat N)"
star_denest_var: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> (y \<cdot> x\<^sup>\<star>)\<^sup>\<star>"
eventually_at_infinity: fixes P :: "'b \<Rightarrow> bool" shows "eventually P at_infinity = (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
IH: fixes y :: "'s" assumes "(y, s) \<in> R"   and "P y"   and "y \<in> S1" shows "0 < n y"
has_integral_split: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real"   and j :: "'b" assumes "(f has_integral i) (cbox a b \<inter> {x. x \<bullet> k \<le> c})"   and "(f has_integral j) (cbox a b \<inter> {x. c \<le> x \<bullet> k})"   and "k \<in> Basis" shows "(f has_integral i + j) (cbox a b)"
no_small_separations: fixes S :: "'a set" assumes "Separation G v0 v1 S" shows "Suc (card paths) \<le> card S"
bisimE_3: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and \<Psi>' :: "'e" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(SCompose' \<Psi> \<Psi>', P, Q) \<in> env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
piecewise_C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'a"   and i :: "real set" shows "f piecewise_C1_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> f C1_differentiable_on i - S)"
Units_eq: shows "Units (subgroup_generated G A \<times>\<times> subgroup_generated G B) = carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"
dfs_eq_reachable: fixes y :: "'a"   and xs :: "'a list" assumes "is_node y"   and "list_all is_node xs" shows "memb y (dfs empt xs) = (y \<in> reachable (set xs))"
ac: fixes aa :: "nat"   and ca :: "nat"   and ba :: "nat"   and da :: "nat" assumes "aa < ca"   and "ba < b0" shows "aa * b0 + ba \<noteq> ca * b0 + da"
zip_append2: fixes xs :: "'d list"   and ys :: "'e list"   and zs :: "'e list" shows "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a = a + b)"
bv_not_bv_not: fixes w :: "bit list" shows "bv_not (bv_not w) = w"
Obs_in_L: fixes u :: "'U"   and ou :: "'Out"   and s :: "('St, 'U, 'C) state" shows "(Obs u ou \<in> L s) = (ou = out (local.getGMState s) u)"
permute_set_def: fixes p :: "perm"   and X :: "'e set" shows "p \<bullet> X = {p \<bullet> x |x. x \<in> X}"
verts_slim_in_G: shows "pverts slim \<subseteq> pverts G"
reflexive: fixes p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "p \<in> greachable A w V p"
stl_sset: fixes x :: "'b"   and a :: "'b stream" assumes "x \<in> sset (stl a)" shows "x \<in> sset a"
exhaust: fixes y :: "ephkey" obtains x1 :: "nonce_t" where "y = epublK x1"   | x2 :: "nonce_t" where "y = eprivK x2"
34: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> (y \<sqinter> z) \<squnion> u \<sqinter> z = (x \<sqinter> y \<squnion> u) \<sqinter> z"
borel_measurable_const_scaleR': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and c :: "real" assumes "f \<in> borel_measurable M" shows "c *\<^sub>R f \<in> borel_measurable M"
split_maxD: fixes t :: "('b \<times> nat) tree"   and t' :: "('b \<times> nat) tree"   and x :: "'b" assumes "split_max t = (t', x)"   and "t \<noteq> \<langle>\<rangle>"   and "invar t" shows "Tree2.inorder t' @ [x] = Tree2.inorder t"
indRelSTEQ_impl_SRel_and_TRel_weakly_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs SRel SWB"
run_foldl: fixes \<Delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "run \<Delta> q\<^sub>0 w i = foldl \<Delta> q\<^sub>0 (map w [0..<i])"
D_\<sigma>_p_1: shows "D \<in> N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)"
backward_reduction_Q: fixes D :: "'a literal multiset"   and L' :: "'a literal"   and N :: "'a literal multiset set"   and L :: "'a literal"   and \<sigma> :: "'s"   and C :: "'a literal multiset"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "D + {#L'#} \<in> N"   and "- L = L' \<cdot>l \<sigma>"   and "D \<cdot> \<sigma> \<subseteq># C" shows "(N, P, Q \<union> {C + {#L#}}) \<leadsto> (N, P \<union> {C}, Q)"
closed_except_simps_2: fixes t\<^sub>1 :: "sterm"   and t\<^sub>2 :: "sterm"   and S :: "name fset" shows "closed_except (t\<^sub>1 $\<^sub>s t\<^sub>2) S = (closed_except t\<^sub>1 S \<and> closed_except t\<^sub>2 S)"
exists_lin_indpt_subset: fixes X :: "'a vec set" assumes "X \<subseteq> carrier_vec n" shows "\<exists>Ls. lin_indpt Ls \<and> local.span Ls = local.span X \<and> Ls \<subseteq> X"
nths_vars2': shows "nths st2.dims0 (- st2.vars1') = st2.dims2"
hn_rel_def: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres" shows "hn_rel P m \<equiv> \<lambda>r. \<exists>\<^sub>Ax. P x r * \<up> (RETURN x \<le> m)"
list_all2_conv_all_nth: fixes P :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'c list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
Gen_subset_\<J>_0_0: shows "Gen (\<P> 0 0) \<subseteq> \<J> 0 0"
trm_ord_reduction_left: shows "\<forall>x1 x2 y. (x1, x2) \<in> trm_ord \<longrightarrow> (x1 \<cdot> y, x2 \<cdot> y) \<in> trm_ord"
j_in: shows "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
ide_in_hom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "partial_magma C" shows "partial_magma.ide C a = partial_magma.in_hom C a a a"
coeff_mult_Var: fixes x :: "'a mpoly"   and i :: "nat"   and p :: "nat"   and c :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (x * Var i ^ p) c = (MPoly_Type.coeff x (c - monomial p i) when p \<le> lookup c i)"
assms_2: shows "(interval.lower \<circ> X) \<longlonglongrightarrow> x"
vertex_map: fixes x :: "'a" assumes "x \<in> \<Union> X" shows "f x \<in> \<Union> Y"
ee: fixes v :: "'a"   and e :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T" shows "e * e = bot"
substEnv_psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "goodEnv rho'" shows "rho &[rho'] &[Y / y]_ys = rho &[rho' &[Y / y]_ys]"
insert_nth_remove_nth: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "insert_nth i (xs ! i) (remove_nth i xs) = xs"
strong_cases: fixes a1 :: "trm"   and a2 :: "trm"   and x :: "name"   and xa :: "name"   and xb :: "name"   and xc :: "name"   and xd :: "name"   and xe :: "name"   and xf :: "name"   and xg :: "name"   and y :: "name" assumes "a1 \<mapsto> a2" obtains s :: "trm" and s' :: "trm" and t :: "trm" where "a1 = App s__ ta__" and "a2 = App s'__ ta__" and "s__ \<mapsto> s'__"   | t :: "trm" and t' :: "trm" and s :: "trm" where "a1 = App sa__ tb__" and "a2 = App sa__ t'__" and "tb__ \<mapsto> t'__"   | s :: "trm" and t :: "trm" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = App (\<Lambda> x . tc__) sb__ \<and> a2 = tc__[x::=sb__] \<and> x \<sharp> sb__"   | t :: "trm" and t' :: "trm" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = \<Lambda> xa . td__ \<and> a2 = \<Lambda> xa . t'a__ \<and> td__ \<mapsto> t'a__"   | t :: "trm" where "\<lbrakk>xb \<sharp> a1; xb \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = \<Lambda> xb . App te__ (Var xb) \<and> a2 = te__ \<and> xb \<sharp> te__"   | s :: "trm" and s' :: "trm" and t :: "trm" where "\<lbrakk>xc \<sharp> a1; xc \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = sc__ to xc in tf__ \<and> a2 = s'a__ to xc in tf__ \<and> xc \<sharp> (sc__, s'a__) \<and> sc__ \<mapsto> s'a__"   | s :: "trm" and t :: "trm" and t' :: "trm" where "\<lbrakk>xd \<sharp> a1; xd \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = sd__ to xd in tg__ \<and> a2 = sd__ to xd in t' \<and> xd \<sharp> sd__ \<and> tg__ \<mapsto> t'"   | s :: "trm" and t :: "trm" where "\<lbrakk>xe \<sharp> a1; xe \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = [se__] to xe in th__ \<and> a2 = th__[xe::=se__] \<and> xe \<sharp> se__"   | s :: "trm" where "\<lbrakk>xf \<sharp> a1; xf \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = sf__ to xf in [Var xf] \<and> a2 = sf__ \<and> xf \<sharp> sf__"   | s :: "trm" and u :: "trm" and t :: "trm" where "\<lbrakk>xg \<noteq> y; xg \<sharp> a1; xg \<sharp> a2; y \<sharp> a1; y \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = (sg__ to xg in t) to y in u \<and> a2 = sg__ to xg in t to y in u \<and> xg \<sharp> (y, sg__, u) \<and> y \<sharp> (sg__, t)"   | s :: "trm" and s' :: "trm" where "a1 = [s]" and "a2 = [s']" and "s \<mapsto> s'"
Abs_Assertion_inverse: fixes y :: "'b" assumes "y \<in> assertion" shows "Rep_Assertion (Abs_Assertion y) = y"
map_le_def: fixes m\<^sub>1 :: "'c \<Rightarrow> 'd option"   and m\<^sub>2 :: "'c \<Rightarrow> 'd option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (m\<^sub>1 = m\<^sub>2 on dom m\<^sub>1)"
indecomposable_imp_type2: fixes \<gamma> :: "V"   and X :: "V set" assumes "indecomposable \<gamma>"   and "X \<subseteq> elts \<gamma>" shows "ordertype X VWF = \<gamma> \<or> ordertype (elts \<gamma> - X) VWF = \<gamma>"
elims: fixes x :: "'a arctic_delta"   and xa :: "'a arctic_delta"   and y :: "'a arctic_delta" assumes "x + xa = y" obtains y :: "'a arctic_delta" where "xa__ = MinInfty_delta" and "xa = yb__" and "ya__ = yb__"   | v :: "'a" where "xa__ = Num_arc_delta v" and "xa = MinInfty_delta" and "ya__ = Num_arc_delta v"   | x :: "'a" and y :: "'a" where "xa__ = Num_arc_delta x" and "xa = Num_arc_delta y" and "ya__ = Num_arc_delta (max x y)"
Factor_Node_5: fixes x :: "int poly" assumes "x \<in># factors_of_factor_tree (Factor_Node i l r)" shows "Mp x = x"
T_insert3_def: fixes a :: "'a"   and t :: "'a tree \<times> nat" shows "T_insert3 a t = time (insert3_tm a t)"
local_lipschitzE: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and t :: "'b"   and x :: "'c" assumes "local_lipschitz T X f"   and "t \<in> T"   and "x \<in> X" obtains u :: "real" and L :: "real" where "0 < u" and "\<And>s. s \<in> cball t u \<inter> T \<Longrightarrow> L-lipschitz_on (cball x u \<inter> X) (f s)"
gets_iff: fixes v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" shows "v ::= e = VC_KAD.gets v e"
set_lin_data_in': shows "set (linearize data_in') \<subseteq> set (linearize data_in)"
has_field_derivative_nat_power_aux: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < n" shows "LIM y inf (Inf (principal ` {S. open S \<and> x \<in> S})) (principal (UNIV - {x})). scaleR (inverse (norm (minus y x))) (minus (minus (nat_power n y) (nat_power n x)) (times (scaleR (ln (real n)) (nat_power n x)) (minus y x))) :> Inf (principal ` {S. open S \<and> zero \<in> S})"
Class_commutes_with_composition: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "monoid_congruence M composition unit E"   and "a \<in> M"   and "b \<in> M" shows "monoid_congruence.quotient_composition M composition E (equivalence.Class M E a) (equivalence.Class M E b) = equivalence.Class M E (composition a b)"