hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "('d, 'e, 'g) FSM"   and xb :: "('d \<times> 'e) list set"   and xc :: "('d \<times> 'e) list"   and xd :: "('d \<times> 'e) list" assumes "Rep_Cov x xa xb xc xd" obtains M2 :: "('d, 'e, 'f) FSM" and M1 :: "('d, 'e, 'g) FSM" and V'' :: "('d \<times> 'e) list set" and vs :: "('d \<times> 'e) list" and xs :: "('d \<times> 'e) list" where "x = M2" and "xa = M1" and "xb = V''" and "xc = vs" and "xd = xs" and "\<exists>xs' vs'. xs' \<noteq> [] \<and> prefix xs' xs \<and> vs' \<in> V'' \<and> (\<exists>s2. io_targets M2 (initial M2) (vs @ xs') = {s2} \<and> io_targets M2 (initial M2) vs' = {s2}) \<and> (\<exists>s1. io_targets M1 (initial M1) (vs @ xs') = {s1} \<and> io_targets M1 (initial M1) vs' = {s1})"
cases: fixes a1 :: "instr \<times> int \<times> (char list \<Rightarrow> int) \<times> int list"   and a2 :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "a1 \<mapsto> a2" obtains i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (LOADI i__, pc__, s__, stk__)" and "a2 = (pc__ + 1, s__, i__ # stk__)"   | x :: "char list" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (LOAD x__, pca__, sa__, stka__)" and "a2 = (pca__ + 1, sa__, sa__ x__ # stka__)"   | pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (ADD, pcb__, sb__, stkb__)" and "a2 = (pcb__ + 1, sb__, (hd2 stkb__ + hd stkb__) # tl2 stkb__)"   | x :: "char list" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (STORE x, pcc__, sc__, stkc__)" and "a2 = (pcc__ + 1, sc__(x := hd stkc__), tl stkc__)"   | i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (JMP ia__, pcd__, sd__, stkd__)" and "a2 = (pcd__ + ia__ + 1, sd__, stkd__)"   | stk :: "int list" and i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" where "a1 = (JMPLESS ib__, pce__, se__, stke__)" and "a2 = (pce__ + ib__ + 1, se__, tl2 stke__)" and "hd2 stke__ < hd stke__"   | stk :: "int list" and i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" where "a1 = (JMPLESS ic__, pcf__, sf__, stkf__)" and "a2 = (pcf__ + 1, sf__, tl2 stkf__)" and "hd stkf__ \<le> hd2 stkf__"   | stk :: "int list" and i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" where "a1 = (JMPGE id__, pcg__, sg__, stkg__)" and "a2 = (pcg__ + id__ + 1, sg__, tl2 stkg__)" and "hd stkg__ \<le> hd2 stkg__"   | stk :: "int list" and i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" where "a1 = (JMPGE ie__, pch__, sh__, stkh__)" and "a2 = (pch__ + 1, sh__, tl2 stkh__)" and "hd2 stkh__ < hd stkh__"   | i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (JMPND if__, pci__, si__, stki__)" and "a2 = (pci__ + if__ + 1, si__, stki__)"   | i :: "int" and pc :: "int" and s :: "char list \<Rightarrow> int" and stk :: "int list" where "a1 = (JMPND i, pc, s, stk)" and "a2 = (pc + 1, s, stk)"
ideal_is_normal: fixes I :: "'a set" assumes "ideal I R" shows "I \<lhd> add_monoid R"
quasi_isometry_inverse: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_between X Y f" shows "\<exists>g. lambda (3 * C * lambda) -quasi_isometry_between Y X g \<and> (\<forall>x\<in>X. dist x (g (f x)) \<le> 3 * C * lambda) \<and> (\<forall>y\<in>Y. dist y (f (g y)) \<le> 3 * C * lambda)"
mssnth_cos_series_stream: fixes n :: "nat" shows "mssnth cos_series_stream n = (- 1) ^ n / fact (2 * n)"
clock_simObs_def: fixes envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "ClockView.clock_simObs envObs \<equiv> \<lambda>a. envObs a \<circ> ODList.hd \<circ> snd"
to_vec_eq_ith: fixes A :: "'a sq_mtx"   and i :: "'a" shows "to_vec A $ i = A $$ i"
singleton_inject: fixes a :: "'e"   and b :: "'e" assumes "{a} = {b}" shows "a = b"
FWdelay_bisimulation_lift_aux_flip: shows "FWdelay_bisimulation_lift_aux final2 r2 final1 r1 \<tau>move2 \<tau>move1"
count_roots_interval_rat_def: fixes p :: "int poly" shows "count_roots_interval_rat p = (let pp = real_of_int_poly p; (cr, nr) = count_roots_interval pp in Root_Info (\<lambda>a b. cr (real_of_rat a) (real_of_rat b)) (\<lambda>a. nr (real_of_rat a)))"
unprot: shows "\<not> (unprot \<sigma> \<or> protFw \<sigma>)"
rComm2: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and K :: "'a"   and P'' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<Psi> \<rhd> !P \<longmapsto> K\<lparr>N\<rparr> \<prec> P''"   and "\<And>C. Prop C \<Psi> (!P) (K\<lparr>N\<rparr> \<prec> P'')"   and "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> P'')"
tree_induct_consistent: fixes t :: "'a tree"   and P :: "'a tree \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "consistent t"   and "\<And>w\<^sub>b b a. P (Leaf w\<^sub>b b) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; a \<in> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; a \<notin> alphabet t\<^sub>1; a \<in> alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; a \<notin> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a" shows "P t a"
ht: fixes P :: "'a \<Rightarrow> real" assumes "unitary P" shows "unitary (t P)"
independence_invisible: fixes a :: "'action"   and b :: "'action" assumes "a \<in> visible"   and "b \<in> visible" shows "\<not> ind a b"
rel_FGf_pos_distr_cond_def: fixes Co1 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and Co1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Contra1 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Contra1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and uu :: "('y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'ak) itself" shows "rel_FGf_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1' L2 L2'. rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 OO rel_FGf L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_FGf (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2'))"
intfAPP_def: fixes f :: "interface \<Rightarrow> 'b"   and x :: "interface" shows "\<langle>x\<rangle>\<^sub>if \<equiv> f x"
map_sum_image_Plus: fixes f :: "'j \<Rightarrow> 'h"   and g :: "'k \<Rightarrow> 'i"   and A :: "'j set"   and B :: "'k set" shows "map_sum f g ` (A <+> B) = f ` A <+> g ` B"
aforms_err_to_aforms_ex: fixes x :: "real list"   and e :: "nat \<Rightarrow> real"   and X :: "((real \<times> real pdevs) \<times> real) list"   and n :: "nat" assumes "x \<in> aforms_err e X"   and "degree_aforms_err X \<le> n"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "\<exists>e'\<in>UNIV \<rightarrow> {- 1..1}. x = aform_vals e' (aforms_err_to_aforms n X) \<and> (\<forall>i<n. e' i = e i)"
ide_char: fixes f :: "'a \<times> 'a" shows "CC.ide f = (ide (fst f) \<and> ide (snd f))"
simple_rotations_def: shows "simple_rotations = {id, rotate_A, rotate_B, rotate_C, rotate_D}"
cases: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" assumes "a1 \<rhd> a2 \<approx> a3" obtains \<Psi> :: "'b" and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "\<Psi> \<rhd> P \<lessapprox><weakBisim> Q" and "\<Psi> \<rhd> P \<leadsto><weakBisim> Q" and "\<forall>\<Psi>''. \<Psi> \<otimes> \<Psi>'' \<rhd> P \<approx> Q" and "\<Psi> \<rhd> Q \<approx> P"
compact_imp_fip_image: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a set" assumes "class.topological_space open"   and "topological_space.compact open s"   and "\<And>i. i \<in> I \<Longrightarrow> topological_space.closed open (f i)"   and "\<And>I'. \<lbrakk>finite I'; I' \<subseteq> I\<rbrakk> \<Longrightarrow> s \<inter> \<Inter> (f ` I') \<noteq> {}" shows "s \<inter> \<Inter> (f ` I) \<noteq> {}"
one_neq_iffs_3: fixes x :: "one" shows "(x \<noteq> oneShot) = (x = notOneShot)"
A_def: shows "A = {k \<in> {0<..n}. [a ^ k = 1] (mod n)}"
m_assoc: fixes x :: "int \<times> int"   and y :: "int \<times> int"   and z :: "int \<times> int" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
added_ids_def: fixes es :: "('d \<times> ('d, 'b) operation) event list"   and p :: "'b" shows "added_ids es p \<equiv> List.map_filter (\<lambda>x. case x of Broadcast x \<Rightarrow> Map.empty x | Deliver (i, Add j e) \<Rightarrow> if e = p then Some j else None | Deliver (i, Rem set x) \<Rightarrow> Map.empty x) es"
Pi_def: fixes A :: "'ag set"   and B :: "'ag \<Rightarrow> 'ah set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
induct: fixes P :: "('a \<times> nat) list \<Rightarrow> ('a \<times> nat) list \<Rightarrow> bool"   and a0 :: "('a \<times> nat) list"   and a1 :: "('a \<times> nat) list" assumes "\<And>n. P [] n"   and "\<And>x p m n. \<lbrakk>\<And>x21 x22 xa y. \<lbrakk>n = x21 # x22; (xa, y) = x21; x = xa\<rbrakk> \<Longrightarrow> P m x22; \<And>x21 x22 xa y. \<lbrakk>n = x21 # x22; (xa, y) = x21; x \<noteq> xa; x < xa\<rbrakk> \<Longrightarrow> P m n; \<And>x21 x22 xa y. \<lbrakk>n = x21 # x22; (xa, y) = x21; x \<noteq> xa; \<not> x < xa\<rbrakk> \<Longrightarrow> P ((x, p) # m) x22\<rbrakk> \<Longrightarrow> P ((x, p) # m) n" shows "P a0 a1"
valid_intv'_distinct: fixes c :: "int"   and d :: "int"   and I :: "intv'"   and I' :: "intv'"   and x :: "'b"   and y :: "'b"   and u :: "'b \<Rightarrow> real" assumes "- c \<le> d"   and "valid_intv' c d I"   and "valid_intv' c d I'"   and "intv'_elem x y u I"   and "intv'_elem x y u I'" shows "I = I'"
nonzero_mult_divide_mult_cancel_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
closure_closed: fixes S :: "'a set" assumes "closed S" shows "closure S = S"
acquire_release_ownership_nth_update: fixes R :: "nat set"   and \<O> :: "nat set"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and xs' :: "nat \<Rightarrow> nat option"   and sb' :: "'b memref list"   and \<D>' :: "bool"   and A :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "R \<subseteq> \<O>"   and "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"   and "ownership_distinct ts" shows "(\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set (ts[i := (p', is', xs', sb', \<D>', \<O> \<union> A - R, \<R>')]). \<O>) = (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, uu_)\<in>set ts. \<O>) \<union> A - R"
is_coext_def: fixes x' :: "'a"   and f :: "'a"   and g :: "'a" shows "e.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow> x\<guillemotright> \<and> f = C e (local.prod g b)"
GB_alt_1: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. (red G)\<^sup>*\<^sup>* f 0)"
insert_before_locs_impl: shows "insert_before_locs = local.a_insert_before_locs"
Col_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Col A B C \<equiv> Bet A B C \<or> Bet B C A \<or> Bet C A B"
tconf_hext_mono: fixes h :: "'heap"   and t :: "'thread_id"   and h' :: "'heap" assumes "P,h \<turnstile> t \<surd>t"   and "h \<unlhd> h'" shows "P,h' \<turnstile> t \<surd>t"
Inf_fun_def: fixes A :: "('d \<Rightarrow> 'e) set" shows "Inf A = (\<lambda>x. INF f\<in>A. f x)"
exists_sorted_distinct_fset_of_list: fixes f :: "'b fset" shows "\<exists>l. sorted l \<and> distinct l \<and> f = fset_of_list l"
of_hypnat_less_0_iff: fixes m :: "nat star" shows "\<not> of_hypnat m < 0"
divide_eq_0_iff: 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 a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = zero) = (a = zero \<or> b = zero)"
urD_6: fixes y :: "real" shows "(y = real_of_1 (p, l, r)) = root_cond (p, l, r) y"
6: shows "(H * (d \<sqinter> - a \<squnion> (d \<sqinter> - a)\<^sup>T))\<^sup>\<star> * H * a * top \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
nvC_def: fixes c :: "('b, 'c) lit list" shows "nvC c \<equiv> \<Union> (set (map nvL c))"
ennreal_eq_0_iff: fixes x :: "real" shows "(ennreal x = 0) = (x \<le> 0)"
weak_unit_def: fixes a :: "'a" shows "weak_unit a \<equiv> a \<star> a \<cong> a \<and> fully_faithful_functor (local.Left a) (local.Left a) (H\<^sub>L a) \<and> fully_faithful_functor (local.Right a) (local.Right a) (H\<^sub>R a)"
reduced_run_decompose: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "w =\<^sub>F w\<^sub>1 @ w\<^sub>2"
cmod_power2: fixes z :: "complex" shows "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and comp_res :: "comp_res" shows "P (case comp_res of LESS \<Rightarrow> f1 | EQUAL \<Rightarrow> f2 | GREATER \<Rightarrow> f3) = (\<not> (comp_res = LESS \<and> \<not> P f1 \<or> comp_res = EQUAL \<and> \<not> P f2 \<or> comp_res = GREATER \<and> \<not> P f3))"
simps_11: fixes v :: "'b" shows "restricted_subformulas nprop\<^sub>n(v) = {}"
rel_compile_fundef: fixes fd :: "('a, 'b, 'c, 'op) Std.instr fundef" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd (compile_fundef fd)"
poly_poly_eq: fixes p :: "'a poly poly"   and x :: "'a"   and y :: "'a" shows "poly (poly p [:x:]) y = poly (eval_poly (\<lambda>p. [:poly p y:]) p [:0::'a, 1::'a:]) x"
in_mset_fset_fmember: fixes x :: "'c"   and N :: "'c fset" shows "(x \<in># mset_fset N) = (x |\<in>| N)"
differentiable_def: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f differentiable F) = (\<exists>D. (f has_derivative D) F)"
H_def: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_kat.H p x q = (rel_kat.t_op p ; x \<subseteq> x ; rel_kat.t_op q)"
left_neutral: fixes a :: "'b" shows "(0::'b) OR a = a"
WTrtAAss: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'addr expr"   and T :: "ty"   and i :: "'addr expr"   and e :: "'addr expr"   and T' :: "ty" assumes "P,E,h \<turnstile> a : T\<lfloor>\<rceil>"   and "P,E,h \<turnstile> i : Integer"   and "P,E,h \<turnstile> e : T'" shows "P,E,h \<turnstile> a\<lfloor>i\<rceil> := e : Void"
SQinf: shows "N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts) = Liminf_llist (lmap Q_of_state Sts)"
times_nat_less: fixes x :: "ordinal"   and n :: "nat" assumes "x < a" shows "x * ordinal_of_nat n < a"
hyps_2: shows "(\<Union>a\<in>toplevel_summands r2. toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>r2\<guillemotright>\<guillemotright>"
fib_strict: shows "fib\<cdot>\<bottom> = \<bottom>"
subst_typ'_stepwise: fixes instT :: "((variable \<times> String.literal set) \<times> typ) list"   and t :: "Core.term" assumes "distinct (map fst instT)"   and "\<And>x. x \<in> tvsT_Set (snd ` set instT) \<Longrightarrow> x \<notin> fst ` set instT" shows "subst_typ' instT t = fold (\<lambda>single. subst_typ' [single]) instT t"
delete_preserve_indices: fixes xs :: "('c \<times> 'd \<times> bool) list"   and i :: "'c"   and ys :: "('c \<times> 'd \<times> bool) list" assumes "delete xs i = Some ys" shows "fst ` set xs = fst ` set ys"
map_FGf_def: fixes l1 :: "'p \<Rightarrow> 'z"   and l2 :: "'q \<Rightarrow> 'aa"   and co1 :: "'s \<Rightarrow> 'ab"   and co2 :: "'t \<Rightarrow> 'ac"   and contra1 :: "'ad \<Rightarrow> 'v"   and contra2 :: "'ae \<Rightarrow> 'w" shows "map_FGf l1 l2 co1 co2 contra1 contra2 = map_F l1 l2 id co1 co2 id contra1 contra2 id"
Mp_Mp: fixes f :: "int poly" shows "q.eq_m (q.Mp f) f"
max_absorb1: fixes y :: "'d"   and x :: "'d" assumes "y \<le> x" shows "max x y = x"
strict_trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "c < b" shows "c < a"
clopenin_eq_frontier_of: fixes X :: "'b topology"   and S :: "'b set" shows "(closedin X S \<and> openin X S) = (S \<subseteq> topspace X \<and> X frontier_of S = {})"
ennreal_sum_less_top: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I < top) = (\<forall>i\<in>I. f i < top)"
prems: fixes i :: "nat"   and j :: "nat" shows "i < length (a # xs) \<and> j < length (a # xs) \<and> i \<noteq> j \<and> (a # xs) ! i = (a # xs) ! j \<longrightarrow> f ((a # xs) ! i) = (0::'a) \<and> f ((a # xs) ! j) = (0::'a)"
HNatInfinite_FreeUltrafilterNat_iff: fixes X :: "nat \<Rightarrow> nat" shows "(star_n X \<in> HNatInfinite) = (\<forall>u. \<forall>\<^sub>F n in \<U>. u < X n)"
get_nondet_def: fixes get :: "('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and f :: "'c \<Rightarrow> ('a, 'm) nondetT" shows "get_nondet get f = NondetT (get (\<lambda>s. run_nondet (f s)))"
L_def: shows "L \<equiv> map (\<lambda>k. \<phi> ` elts (E k)) [0..<length \<alpha>s]"
vec_to_iarray_nth': fixes A :: "('c, 'd) vec"   and i :: "'d" shows "vec_to_iarray A !! mod_type_class.to_nat i = A $ i"
LIM_offset_zero: fixes f :: "'e \<Rightarrow> 'f"   and L :: "'f"   and a :: "'e" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>h. f (a + h)) \<midarrow>0::'e\<rightarrow> L"
1: shows "t + pp_of_term u = s + pp_of_term v"
differentiable_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and t :: "'b set"   and s :: "'b set" assumes "f differentiable_on t"   and "s \<subseteq> t" shows "f differentiable_on s"
elims_2: fixes x :: "poly" assumes "isconstant x" obtains v :: "int \<times> int" where "x = C v__"   | v :: "nat" where "x = Bound va__"   | v :: "poly" and va :: "poly" where "x = Add vb__ vaa__"   | v :: "poly" and va :: "poly" where "x = Sub vc__ vab__"   | v :: "poly" and va :: "poly" where "x = Mul vd__ vac__"   | v :: "poly" where "x = Neg ve__"   | v :: "poly" and va :: "nat" where "x = Pw vf__ va"   | v :: "poly" and vc :: "nat" and vb :: "poly" where "x = CN v (Suc vc) vb"
leg_simps_3: shows "C.arr \<nu>\<pi>.leg1"
frontier_cbox: fixes a :: "'a"   and b :: "'a" shows "frontier (cbox a b) = cbox a b - box a b"
get_spec_def: fixes M :: "('a SecurityInvariant \<times> 'a SecurityInvariant_configured) list" shows "get_spec M \<equiv> map snd M"
strategy_attracts_VVp: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and A :: "'a set"   and W :: "'a set" assumes "strategy p \<sigma>"   and "strategy_attracts_via p \<sigma> v0 A W"   and "v0 \<in> A - W"   and "v0 \<in> VV p"   and "\<not> deadend v0" shows "\<sigma> v0 \<in> A \<union> W"
iso_fgraph_trans: fixes f :: "'a list list"   and g :: "'a list list"   and h :: "'a list list" assumes "f \<simeq> g"   and "g \<simeq> h" shows "f \<simeq> h"
apps_eq_tail_conv: fixes r :: "lam"   and ts :: "lam list"   and s :: "lam" shows "(r \<degree>\<degree> ts = s \<degree>\<degree> ts) = (r = s)"
ipurge_tr_aux_union: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and V :: "'c set"   and xs :: "'b list" shows "ipurge_tr_aux I D (U \<union> V) xs = ipurge_tr_aux I D V (ipurge_tr_aux I D U xs)"
fac_acc_body2_body1_eq: shows "fac_acc_body2 = fac_acc_body1"
bigD: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state" assumes "\<sigma>' \<in> big P \<sigma>" shows "\<exists>n. \<sigma>' \<in> small_nstep P \<sigma> n \<and> \<sigma>' \<in> endset"
assms_1: shows "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> vproduct n A"
appCheckcast: fixes C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Checkcast C, P, pc, mxs, T\<^sub>r, s) = (\<exists>T ST LT. s = (T # ST, LT) \<and> is_class P C \<and> is_refT T)"
lhd_def: fixes llist :: "'a llist" shows "lhd llist = (case llist of LCons x21 x22 \<Rightarrow> x21)"
post_meas1_bell00_snd: shows "post_meas1 2 |\<beta>\<^sub>0\<^sub>0\<rangle> 1 = |unit_vec 4 3\<rangle>"
S4_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S4 rmhist p = S rmhist True True True clkB rpcB histA histB p"
exp_1_approx: fixes n :: "nat" assumes "0 < n" shows "\<bar>exp 1 - euler_approx n\<bar> \<le> 1 / fact n"
util_imp_total: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "total_on carrier relation"
B'_def: shows "B' = \<langle>lb2, b, A'\<rangle>"
Frame_def: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> bool" shows "Frame P Q F \<equiv> \<forall>s. P s \<longrightarrow> (Q \<and>* F) s"
dns1'': fixes x :: "('b \<times> 'b) set" shows "x O rel_ar (rel_ar x) = x"
arr_Set_ArrVal_vrange: shows "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"
that: fixes q :: "('t \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" assumes "p = r + (\<Sum>g\<in>G. q g \<odot> g)"   and "\<And>g. q g \<in> punit.dgrad_p_set d m"   and "\<And>g. lt (q g \<odot> g) \<preceq>\<^sub>t lt p" shows "thesis"
induct: fixes P :: "com \<Rightarrow> bool"   and com :: "com" assumes "P Skip"   and "\<And>x1 x2. P (x1:=x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P x2; P x3\<rbrakk> \<Longrightarrow> P (if (x1) x2 else x3)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (while (x1) x2)" shows "P com"