hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'a scene"   and xa :: "'a scene" shows "(x ##\<^sub>S xa) = (\<forall>s\<^sub>1 s\<^sub>2. Rep_scene xa (Rep_scene x s\<^sub>1 s\<^sub>2) s\<^sub>2 = Rep_scene x (Rep_scene xa s\<^sub>1 s\<^sub>2) s\<^sub>2)"
mem_val_alt_8_mem_equal_1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 1) s1 = mem_val_alt 8 ((pa AND 68719476732) + 1) s2"
subcocycle_birkhoff: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real" assumes "mpt M T"   and "integrable M f" shows "mpt.subcocycle M T (qmpt.birkhoff_sum T f)"
min_of_mono: fixes f :: "'a word \<Rightarrow> 'b"   and m :: "'a word"   and n :: "'a word" assumes "signed.mono f" shows "min (f m) (f n) = f (signed.min m n)"
substT_compose_eq_or: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and t1 :: "'b"   and t2 :: "'b"   and x1 :: "'a"   and x2 :: "'a" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "s \<in> Syntax_Arith.atrm var Var num suc pls tms"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<in> var"   and "x2 \<in> var"   and "x1 = x2 \<or> x2 \<notin> FvarsT s" shows "substT (substT s t1 x1) t2 x2 = substT s (substT t1 t2 x2) x1"
left_cancel: fixes a :: "'b perm"   and b :: "'b perm"   and c :: "'b perm" shows "(a * b = a * c) = (b = c)"
2: shows "rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) B = rat_poly_times A B - rat_poly_times (B ^ 3) B"
simps_2: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
init_segment_subset: fixes S :: "nat set"   and T :: "nat set" assumes "init_segment S T" shows "S \<subseteq> T"
B: fixes n :: "nat" shows "B n = (\<Prod>(a, i)\<in>as - UNIV \<times> {0..<n}. a)"
lnull_0_llength: fixes xs :: "'c llist" assumes "\<not> lnull xs" shows "enat 0 < llength xs"
first_step_select: fixes s' :: "nat"   and t :: "transition"   and r :: "nat \<Rightarrow>f value option"   and aa :: "String.literal"   and b :: "value list" assumes "(s', t) |\<in>| possible_steps drinks 0 r aa b" shows "s' = 1 \<and> t = select"
UP_simps_3: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr V\<^sub>B \<mu>" shows "strictified_bicategory.trg V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (strictified_bicategory.UP V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<mu>) = strictified_bicategory.UP\<^sub>0 V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (trg\<^sub>B \<mu>)"
rotate1_def: shows "rotate1 \<equiv> rec_list [] (\<lambda>x xs xsa. xs @ [x])"
degree_pderiv_mult_less: fixes C :: "'b poly"   and B :: "'b poly" assumes "pderiv C \<noteq> 0" shows "degree (pderiv C * B) < degree B + degree C"
ipurge_tr_rev_aux_single_dom: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list" shows "ipurge_tr_rev_aux I D {u} xs = ipurge_tr_rev I D u xs"
StaticDownCast: 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 e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs @ [C] @ Cs'),s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs @ [C]),s\<^sub>1\<rangle>"
Seq_append_pair: fixes s1 :: "hf"   and k1 :: "hf"   and s2 :: "hf"   and n :: "hf"   and y :: "hf" assumes "Seq s1 k1"   and "Seq s2 (succ n)"   and "\<langle>n, y\<rangle> \<^bold>\<in> s2"   and "Ord n" shows "\<langle>k1 @+ n, y\<rangle> \<^bold>\<in> seq_append k1 s1 s2"
last_append: fixes xs :: "'e list"   and ys :: "'e list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
rep_eq: fixes x :: "('d, 'e) uexpr"   and xa :: "'e \<Longrightarrow> 'c" shows "\<lbrakk>x \<oplus>\<^sub>p xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x\<rbrakk>\<^sub>e (get\<^bsub>xa\<^esub> b))"
invalid_assn_const: fixes P :: "assn"   and x :: "'a"   and y :: "'b" shows "invalid_assn (\<lambda>_ _. P) x y = \<up> (vassn_tag P) * true"
acc_boundI: fixes x :: "nat"   and q :: "'Q" assumes "x \<in> acc q" shows "x < num_acc"
g2: fixes z :: "real" assumes "1 < z * 2"   and "z \<le> 1"   and "z * 2 - 1 \<notin> s2" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1) within {0..1})"
Rep_uint64_numeral: fixes n :: "num" shows "Rep_uint64 (numeral n) = numeral n"
gseq_def: shows "gseq = {(f, g). f \<in> SEQ (lists A) \<and> g \<in> SEQ (lists A) \<and> (\<exists>i. length (g i) < length (f i) \<and> (\<forall>j<i. f j = g j))}"
hd_append: fixes xs :: "'c list"   and ys :: "'c list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
extended_Gromov_distance_continuous: fixes x :: "'a Gromov_completion" shows "continuous_on UNIV (extended_Gromov_distance x)"
OK_le_err_OK: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" shows "(OK x \<sqsubseteq>\<^bsub>Err.le r\<^esub> OK y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y)"
simps: fixes a :: "'b list"   and G :: "'b list set" shows "(a \<in> \<langle>G\<rangle>) = (a = \<epsilon> \<or> (\<exists>w1 w2. a = w1 \<cdot> w2 \<and> w1 \<in> G \<and> w2 \<in> \<langle>G\<rangle>))"
bij_inv: fixes f :: "'d \<Rightarrow> 'd" assumes "bijection f" shows "bij (inv f)"
constructive_security_obsf_refl: fixes \<eta> :: "nat"   and \<A> :: "('a + 'b, 'c + 'd) distinguisher_obsf" shows "constructive_security_obsf (middle \<eta>) (ideal \<eta>) (sim_inner \<eta>) (\<I>_middle \<eta>) (\<I>_inner \<eta>) (\<I>_common \<eta>) \<A> (advantage \<A> (obsf_resource (sim_inner \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal \<eta>)) (obsf_resource (middle \<eta>)))"
that: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, ti\<rangle> \<Rightarrow> t" shows "thesis"
a_known_ptrs_def: fixes h :: "(_) heap" shows "DocumentClass.known_ptrs h = (\<forall>ptr\<in>fset (object_ptr_kinds h). DocumentClass.known_ptr ptr)"
inv_into_inv_into_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and A' :: "'e set"   and a :: "'d" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A' (inv_into A f) a = f a"
sc_eq_box_box_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<phi>) in v]"
d_fixpoint: fixes x :: "('a \<times> 'a) set" shows "(\<exists>y. x = rdom y) = (x = rdom x)"
set_nths: fixes xs :: "'a list"   and I :: "nat set" shows "set (nths xs I) = {xs ! i |i. i < length xs \<and> i \<in> I}"
subst_lit_def: fixes L :: "('a, 'b) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_lit L \<sigma> = L \<cdot>lit \<sigma>"
top_unique: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - x = - y \<squnion> - - y"
sincreasingI: fixes s :: "'b stream" assumes "\<And>x. \<exists>i. x < s !! i" shows "sincreasing s"
mono_transI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>sound P; sound Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "mono_trans t"
set_rev_mp: fixes c :: "'m"   and A :: "'m set"   and B :: "'m set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
intros_11: fixes E :: "env"   and C :: "qtname"   and dt :: "loc \<Rightarrow> ty option" assumes "is_class (prg E) C" shows "E,dt\<Turnstile>Init C\<Colon>\<surd>"
snd_echelon_form_of_column_k_det: fixes det_P :: "'c"   and A :: "'c iarray iarray"   and i :: "nat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and k :: "nat" shows "snd (echelon_form_of_column_k_det_iarrays (det_P, A, i, bezout) k) = echelon_form_of_column_k_iarrays (A, i, bezout) k"
is_valid_operator_sas_plus_then_6: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (effect_of op). \<forall>(v', a')\<in>set (effect_of op). v \<noteq> v' \<or> a = a'"
that: fixes A :: "int"   and B :: "int" assumes "0 < A"   and "local.beukers_integral2 = real_of_int A * Re (zeta 3) + real_of_int B / real (Lcm {1..n} ^ 3)" shows "thesis"
mat_diag_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "mat_diag n f \<in> carrier_mat n n"
uniformly_convergent_uniform_limit_iff: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "uniformly_convergent_on X f = uniform_limit X f (\<lambda>x. lim (\<lambda>n. f n x)) sequentially"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
vector_component_2: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(x + y) $ i = x $ i + y $ i"
add_left: fixes a :: "'b \<Rightarrow>\<^sub>L 'a"   and a' :: "'b \<Rightarrow>\<^sub>L 'a"   and b :: "'b" shows "blinfun_apply (a + a') b = blinfun_apply a b + blinfun_apply a' b"
R9_4: shows "pmf (sds R9) c = 1 - pmf (sds R35) a"
add_closed: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" assumes "a \<in> \<ww>"   and "b \<in> \<ww>" shows "add_local_ring_at a b \<in> \<ww>"
solutions_linorder: fixes D :: "nat"   and a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "pell D"   and "pell.solution D (a, b)"   and "pell.solution D (x, y)" shows "a \<le> x \<and> b \<le> y \<or> x \<le> a \<and> y \<le> b"
simps_3: fixes v :: "trm" shows "Geqo (Aterm v) undeft = undeff"
finite_simplex: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X"   and "x \<in> X" shows "finite x"
remove_all_cycles_neg_cycles: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and xs :: "nat list"   and ys :: "nat list" assumes "len m i j xs < len m i j (remove_all_cycles ys xs)" shows "\<exists>ys k. set ys \<subseteq> set xs \<and> k \<in> set xs \<and> len m k k ys < (0::'a)"
valid_length_mmuaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list" assumes "valid_mmuaux args cur aux auxlist" shows "length_mmuaux args aux = length auxlist"
lm081: fixes X :: "'c set"   and Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "X \<inter> Domain Q = {}" shows "(P +< Q) `` X = (P outside Domain Q) `` X"
lc: shows "\<bar>lead_coeff local.g\<bar> < p ^ local.ll"
nonempty_suffix_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<le>ns v \<equiv> u \<noteq> \<epsilon> \<and> u \<le>s v"
sub1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> k + a" shows "A - b \<equiv> k + (a - b)"
Nil_notin_tl_if_not_final: fixes r :: "(bool \<times> 'b) rexp" assumes "\<not> final r" shows "[] \<notin> tl ` Lm r"
hyps: shows "set (remdups (a' @ b)) = set (remdups a' @ remdups (list_diff b a'))"
ws: shows "ws = getFrN (y # map snd txs) (s # map fst txs) [\<phi>] (length txs)"
C6: shows "rel_respects_barb_set Rel (STCalWB SWB TWB) {success}"
test_compl_2: fixes x :: "'a" assumes "is_test x" shows "x \<cdot> tc x = 0"
subpath_image: fixes u :: "real"   and v :: "real"   and f :: "'c \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'c" shows "subpath u v (f \<circ> g) = f \<circ> subpath u v g"
exists_smooth_on: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "diff k charts1 charts2 f"   and "x \<in> manifold.carrier charts1" shows "\<exists>c1\<in>c_manifold.atlas charts1 k. \<exists>c2\<in>c_manifold.atlas charts2 k. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
cases: fixes a1 :: "('fenv_std, 'c, 'fun) prog"   and a2 :: "('fenv_std, 'c, ('fun, 'd) frame) state" assumes "Sstd.load a1 a2" obtains F :: "'fenv_std" and main :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op) Std.instr fundef" and H :: "'c" where "a1 = Prog F H main" and "a2 = State F H [Frame main 0 []]" and "Fstd_get F main = Some fd" and "arity fd = 0"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "r\<lparr>a\<rparr> = b"
r_one: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<otimes> \<one> = x"
polys_rel_def: shows "polys_rel = {(A, B). B = ran_m A}"
some_third_chamberD_facet: fixes X :: "'b set set"   and C :: "'b set"   and z :: "'b set"   and D :: "'b set" assumes "ThickChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "z \<lhd> ThickChamberComplex.some_third_chamber X C D z"
convex_connected: fixes S :: "'a set" assumes "convex S" shows "connected S"
Tietze_unbounded: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and U :: "'a set" assumes "continuous_on S f"   and "closedin (top_of_set U) S" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on U g" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
ta_seq_consist_nthI: fixes obs :: "('a, 'b) obs_event action llist"   and P :: "'c prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option" assumes "\<And>i ad al v. \<lbrakk>enat i < llength obs; lnth obs i = NormalAction (ReadMem ad al v); ta_seq_consist P vs (ltake (enat i) obs)\<rbrakk> \<Longrightarrow> \<exists>b. mrw_values P vs (list_of (ltake (enat i) obs)) (ad, al) = \<lfloor>(v, b)\<rfloor>" shows "ta_seq_consist P vs obs"
s_def: shows "s = sturm p ! Suc (Suc i)"
deeper_suc: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
cases: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" assumes "Lfilled a1 a2 a3 a4" obtains vs :: "e list" and lholed :: "Lholed" and es' :: "e list" and es :: "e list" where "a1 = 0" and "a2 = lholed__" and "a3 = esa__" and "a4 = vsa__ @ esa__ @ es'a__" and "const_list vsa__" and "lholed__ = LBase vsa__ es'a__"   | vs :: "e list" and lholed :: "Lholed" and n :: "nat" and es' :: "e list" and l :: "Lholed" and es'' :: "e list" and k :: "nat" and es :: "e list" and lfilledk :: "e list" where "a1 = k + 1" and "a2 = lholed" and "a3 = es" and "a4 = vs @ [Label n es' lfilledk] @ es''" and "const_list vs" and "lholed = LRec vs n es' l es''" and "Lfilled k l es lfilledk"
generalise_output_preserves_guard: fixes t :: "transition"   and r :: "nat"   and p :: "nat" shows "Guards (generalise_output t r p) = Guards t"
smember_def: fixes x :: "'d"   and s :: "'d stream" shows "smember x s = (x \<in> sset s)"
Seq: shows "aS\<^bsub>Ax\<^esub> = iS"
obj_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.obj (src\<^sub>B \<mu>)"
intro: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "reducedly_statically_complete_calculus_axioms Bot Inf entails Red_I Red_F" shows "reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F"
FP: fixes n :: "nat" assumes "n < length dl" shows "part {..<brn (dl ! n)} (F n ` P n) \<and> {} \<notin> F n ` P n"
retract_property: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "\<pi>\<^sub>* (\<pi>\<^sup>* x) \<sqsubseteq>\<^bsub>\<X>\<^esub> x"
simps: fixes p :: "float poly"   and e :: "float interval" shows "tm_norm_poly (TaylorModel p e) = TaylorModel (polynate p) e"
set_baliR: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.set_tree (baliR l a r) = Tree2.set_tree l \<union> {a} \<union> Tree2.set_tree r"
OrdmemD: fixes k :: "V"   and j :: "V" assumes "Ord k"   and "j \<in> elts k" shows "j < k"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list" assumes "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)"   and "P []" shows "P a0"
index_smult_vec_1: fixes i :: "nat"   and v :: "'d Matrix.vec"   and a :: "'d" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $v i = a * v $v i"
atms_of_union_mset: fixes A :: "'a literal multiset"   and B :: "'a literal multiset" shows "atms_of (A \<union># B) = atms_of A \<union> atms_of B"
outs: shows "\<I> \<turnstile>g gpv \<surd>"
x: shows "invariant_1_2 (p1, l1, r1)"
p: fixes C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list" assumes "C \<noteq> Object"   and "is_class P C"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> wf_cdecl wf_md P (C, D, fs, ms) \<and> P \<turnstile> C \<prec>\<^sup>1 D \<and> is_class P D \<and> Q D" shows "Q C"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and set_ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and alts' :: "'b set"   and agents' :: "'a set" assumes "strategyproof_scf agents alts scf set_ext"   and "liftable_set_extension alts' alts set_ext"   and "scf_lowering agents alts scf agents' alts'" shows "scf_lowering_strategyproof agents alts scf agents' alts' set_ext"
QChain: shows "Q \<Longrightarrow>\<^sub>\<tau> Q'''"
simps_2: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "invar (Node k v l r) = ((\<forall>p\<in>set_kdt l. p $ k \<le> v) \<and> (\<forall>p\<in>set_kdt r. v < p $ k) \<and> widest_spread_axis k UNIV (set_kdt l \<union> set_kdt r) \<and> invar l \<and> invar r)"
psimps_3: fixes c :: "'b"   and v :: "'b" assumes "unify_dom (Const c, Var v)" shows "unify (Const c) (Var v) = Some [(v, Const c)]"
get_M_pure: fixes ptr :: "(_) character_data_ptr"   and getter :: "'s option RCharacterData_scheme \<Rightarrow> 'v"   and h :: "(_) heap" shows "pure (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr getter) h"
INT_sA: shows "(\<Inter>n. snd (A n)) \<in> fmeasurable M"
character_data_ptr_kinds_eq_h: shows "character_data_ptr_kinds h' = character_data_ptr_kinds h"
Lp_monotone_limit_1: fixes p :: "ennreal"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and M :: "'b measure"   and C :: "real" assumes "0 < p"   and "AE x in M. incseq (\<lambda>n. f n x)"   and "\<And>n. Norm (\<LL> p M) (f n) \<le> C"   and "\<And>n. f n \<in> space\<^sub>N (\<LL> p M)" shows "AE x in M. convergent (\<lambda>n. f n x)"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> t T u U y. \<lbrakk>\<Gamma> \<turnstile> t : T; \<And>z. P z \<Gamma> t T; \<Gamma> \<turnstile> u : U; \<And>z. P z \<Gamma> u U\<rbrakk> \<Longrightarrow> P y \<Gamma> \<langle>t, u\<rangle> (T \<otimes> U)"   and "\<And>x T \<Gamma> t U y. \<lbrakk>x \<sharp> y; (x, T) # \<Gamma> \<turnstile> t : U; \<And>z. P z ((x, T) # \<Gamma>) t U\<rbrakk> \<Longrightarrow> P y \<Gamma> (\<lambda>x:T. t) (T \<rightarrow> U)"   and "\<And>\<Gamma> t T U u y. \<lbrakk>\<Gamma> \<turnstile> t : T \<rightarrow> U; \<And>z. P z \<Gamma> t (T \<rightarrow> U); \<Gamma> \<turnstile> u : T; \<And>z. P z \<Gamma> u T\<rbrakk> \<Longrightarrow> P y \<Gamma> (t \<cdot> u) U"   and "\<And>p t \<Gamma> T \<Delta> u U y. \<lbrakk>supp p \<sharp>* y; supp p \<sharp>* t; \<Gamma> \<turnstile> t : T; \<And>z. P z \<Gamma> t T; \<turnstile> p : T \<Rightarrow> \<Delta>; \<Delta> @ \<Gamma> \<turnstile> u : U; \<And>z. P z (\<Delta> @ \<Gamma>) u U\<rbrakk> \<Longrightarrow> P y \<Gamma> (LET p = t IN u) U" shows "P z x1 x2 x3"
defs: fixes \<sigma> :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "strategy_progression \<sigma> p = lappend (llist_of (prefixes p)) (ltl (iterates (augment_list \<sigma>) p))"
prime: fixes p :: "nat" assumes "prime p" shows "moebius_mu p = - (1::'b)"
subringE_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subring H R" shows "H \<subseteq> carrier R"
Pinv_3: fixes x1 :: "'a acom"   and x2 :: "'a acom" assumes "P x1"   and "P x2" shows "P (x1 \<nabla> x2)"
fold_matches_last_step_not_None: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c option"   and x :: "'a"   and y :: "'b"   and subs :: "'c"   and subs' :: "'c" assumes "length xs = length ys"   and "fold (\<lambda>(T, U) subs. Option.bind subs (f T U)) (zip (xs @ [x]) (ys @ [y])) (Some subs) = Some subs'" obtains point :: "'c" where "fold (\<lambda>(T, U) subs. Option.bind subs (f T U)) (zip xs ys) (Some subs) = Some point" and "f x y point = Some subs'"
doubles_Cons_same: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "double\<^sup>*\<^sup>* xs ys" shows "double\<^sup>*\<^sup>* (x # xs) (x # ys)"
f: shows "nodup_all lI \<and> (\<forall>j<length Sj. Sj ! j \<noteq> [] \<and> tl (Sj ! j) \<noteq> [] \<and> (Sj ! j ! 0, Sj ! j ! (length (Sj ! j) - 1)) = ({lI ! j}, s) \<and> (\<forall>i<length (Sj ! j) - 1. \<turnstile>\<N>\<^bsub>(Sj ! j ! i, Sj ! j ! (i + 1))\<^esub>))"