hcju/isabelleps huggingface dataset:
tensor_mat_eval: fixes i :: "nat"   and dims :: "nat list"   and j :: "nat"   and vars :: "nat set"   and m1 :: "'a mat"   and m2 :: "'a mat" assumes "i < state_sig.d dims"   and "j < state_sig.d dims" shows "tensor_mat dims vars m1 m2 $$ (i, j) = m1 $$ (partial_state.encode1 dims vars i, partial_state.encode1 dims vars j) * m2 $$ (partial_state.encode2 dims vars i, partial_state.encode2 dims vars j)"
L_zero_2: fixes M :: "'a measure" shows "defect (\<LL> 0 M) = 1"
r': shows "r' \<le> r * cmod (deriv f a) / 12"
sources_char': fixes \<mu> :: "'a"   and a :: "'a" assumes "arr \<mu>" shows "(a \<in> sources \<mu>) = (some_src \<mu> \<cong> a)"
R'_simps_2: fixes y :: "nat"   and n :: "nat" assumes "y \<le> n" shows "((Suc n, y) \<in> R' n) = (\<exists>x'>n. (x', y) \<in> R)"
ConjunctiveD: fixes a :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "a \<in> Apply.Conjunctive" shows "a (Inf A) = Inf (a ` A)"
cring_simprules_11: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
assms_1: shows "[a] \<preceq>\<^sub>F\<^sub>I w"
shift_gt: fixes j :: "nat"   and i :: "nat"   and e :: "nat \<Rightarrow> 'a"   and T :: "'a" assumes "j < i" shows "(e\<langle>i:T\<rangle>) j = e j"
step_op_inl: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and op :: "'op"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and opinl :: "'opinl"   and x :: "'dyn"   and F' :: "'fenv_inca"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IOp op"   and "\<AA>\<rr>\<ii>\<tt>\<yy> op = ar"   and "ar \<le> length \<Sigma>"   and "\<II>\<nn>\<ll> op (take ar \<Sigma>) = Some opinl"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x"   and "Finca_add F f (rewrite_fundef_body fd pc (IOpInl opinl)) = F'" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F' H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
NF_def: fixes A :: "('c \<times> 'c) set" shows "NF A = {a. A `` {a} = {}}"
op_defs_6: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "until \<phi> \<psi> \<pi>l \<equiv> \<exists>i. \<psi> (map (sdrop i) \<pi>l) \<and> (\<forall>j\<in>{0..<i}. \<phi> (map (sdrop j) \<pi>l))"
compute_real_of_float: fixes m :: "int"   and e :: "int" shows "real_of_float (Float m e) = (if 0 \<le> e then real_of_int m * 2 ^ nat e else real_of_int m / 2 ^ nat (- e))"
orthogonal_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x y = (inner x y = 0)"
presAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = gAbs MOD xs x X (h X)"
equal_multiset_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "equal_class.equal A B = (A = B)"
IH: fixes s :: "'a wordinterval" assumes "valid_prefixes rs"   and "(a1, b1) \<in> set (routing_port_ranges rs s)"   and "(a2, b2) \<in> set (routing_port_ranges rs s)"   and "wordinterval_to_set b1 \<noteq> {}" shows "(b1 \<noteq> b2) = (wordinterval_to_set b1 \<inter> wordinterval_to_set b2 = {})"
simps_1: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and yU :: "char list \<times> Unary_PCF.type" shows "subst xT t \<langle>yU\<rangle> = (if xT = yU then t else \<langle>yU\<rangle>)"
diff_list_rec: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "diff_list (x # xs) ys = (if x \<in> set ys then diff_list xs ys else x # diff_list xs ys)"
igFreshClsSTR_imp_igFreshCls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igFreshClsSTR MOD" shows "FixSyn.igFreshCls Delta MOD"
mat_to_cols_list_is_mat: fixes A :: "complex Matrix.mat" assumes "0 < dim_col A" shows "Matrix_Legacy.mat (mult.row_length (mat_to_cols_list A)) (length (mat_to_cols_list A)) (mat_to_cols_list A)"
symE: fixes r :: "('g \<times> 'g) set"   and b :: "'g"   and a :: "'g" assumes "sym r"   and "(b, a) \<in> r" obtains "(a, b) \<in> r"
wt_s'_ge_s: shows "wt s' \<ge>\<^sub>p wt s"
map_nth: fixes xs :: "'a list" shows "map ((!) xs) [0..<length xs] = xs"
wf_ciD3: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "wf_ci ci"   and "ci ins P h stk loc C M pc frs"   and "ins_jump_ok ins pc'" shows "ci ins P h stk loc C M pc' frs"
ring_simprules_23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
outs_gpv_bind_gpv: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv" shows "outs_gpv \<I> (gpv \<bind> f) = outs_gpv \<I> gpv \<union> (\<Union>x\<in>results_gpv \<I> gpv. outs_gpv \<I> (f x))"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : \<I> \<otimes> \<I> \<rightarrow> \<I>\<guillemotright>"
finite_measure_Union: fixes A :: "'c set"   and M :: "'c pmf"   and B :: "'c set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "A \<inter> B = {}" shows "measure_pmf.prob M (A \<union> B) = measure_pmf.prob M A + measure_pmf.prob M B"
case1: assumes "0 \<le> x"   and "0 \<le> y" shows "x * y \<le> 4611686018427387904"
combine_inv1_2: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt" assumes "inv1 lt"   and "inv1 rt" shows "inv1l (combine lt rt)"
adds_term_splus: fixes u :: "'t"   and v :: "'t"   and s :: "'a" assumes "u adds\<^sub>t v" shows "u adds\<^sub>t s \<oplus> v"
i: shows "(LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) \<noteq> i"
init_fin_silent_move_RunningI: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and x' :: "'x"   and m' :: "'m" assumes "silent_move t (x, m) (x', m')" shows "\<tau>trsys.silent_move (init_fin t) init_fin_\<tau>move ((Running, x), m) ((Running, x'), m')"
one_step_flowpipe: fixes m :: "'c set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'c set \<Rightarrow> ('c set \<times> 'd) option nres"   and X0 :: "'c set"   and h :: "real" assumes "one_step_method m"   and "wd TYPE('c)" shows "one_step X0 h m \<le> SPEC (\<lambda>(h', uu_, RES_ivl, RES). 0 < h' \<and> h' \<le> h \<and> flowpipe0 X0 h' h' RES_ivl RES)"
pa_inv_final: fixes H1 :: "('a, 'b) hashedTa"   and H2 :: "('c, 'b) hashedTa" assumes "hashedTa H1"   and "hashedTa H2"   and "hta_has_idx_s H1"   and "hta_has_idx_sf H2" shows "\<forall>sc. sc \<in> wa_invar (det_wa_wa (pa_det_algo H1 H2)) \<and> sc \<notin> wa_cond (det_wa_wa (pa_det_algo H1 H2)) \<longrightarrow> (case pa_\<alpha> sc of (Q, W, \<delta>d) \<Rightarrow> \<lparr>ta_initial = ta_initial (hta_\<alpha> H1) \<times> ta_initial (hta_\<alpha> H2), ta_rules = \<delta>d\<rparr> = ta_fwd_reduce (ta_prod (hta_\<alpha> H1) (hta_\<alpha> H2)))"
prems: fixes x :: "'a" assumes "x \<in> poly_vars (m # p)" shows "\<alpha> x = \<beta> x"
Var: fixes xs :: "'varSort"   and x :: "'var" shows "phi (qVar xs x)"
pullback_arr_cod_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f" shows "category.inverse_arrows C (prj1 f (partial_magma.cod C f)) (elementary_category_with_pullbacks.tuple C prj0 prj1 (partial_magma.dom C f) f (partial_magma.cod C f) f)"
is_reduced_GB_homogeneous: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "punit.is_reduced_GB G"   and "ideal G = ideal F"   and "g \<in> G" shows "homogeneous g"
case_1: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "'e \<Rightarrow> 'b"   and x1 :: "'c" shows "(case In1 x1 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f1 x1"
RSadj_def: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec" shows "RSadj x y \<nu> = (\<chi>z. \<nu> $ swap x y z)"
pow_eq_id: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "(x [^] n = \<one>) = (local.ord x dvd n)"
wls_Abs_swap_ex: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<exists>y. y \<notin> {x, x'} \<and> fresh xs y X \<and> fresh xs y X' \<and> X #[y \<and> x]_xs = X' #[y \<and> x']_xs)"
norm_conv_dist: fixes x :: "'a" shows "norm x = dist x (0::'a)"
le_cases: fixes y :: "'b"   and x :: "'b" obtains "y \<le> x"   | "x \<le> y"
viterbi_sequence: fixes s :: "'s"   and t_end :: "'s"   and os :: "'t list" assumes "0 < snd (viterbi s t_end os)" shows "snd (viterbi s t_end os) = val_of s (fst (viterbi s t_end os)) os"
spectrum_non_empty: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "0 < n" shows "spectrum A \<noteq> {}"
inner_sum_left: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and y :: "'d" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. inner (f x) y)"
gnodes_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "gnodes A w V \<equiv> transition_system_initial.nodes (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V) (\<lambda>v. v \<in> ginitial A \<inter> V)"
int_of_uint64_mult: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x * y) = int_of_uint64 x * int_of_uint64 y mod 18446744073709551616"
decisive1_witness: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "profile A Is P"   and "j \<in> Is" obtains P' :: "'b \<Rightarrow> ('a \<times> 'a) set" where "profile A Is P'" and "x \<^bsub>(P' j)\<^esub>\<prec> y \<and> y \<^bsub>(P' j)\<^esub>\<prec> z" and "\<And>i. i \<noteq> j \<Longrightarrow> y \<^bsub>(P' i)\<^esub>\<prec> x \<and> y \<^bsub>(P' i)\<^esub>\<prec> z \<and> (x \<^bsub>(P' i)\<^esub>\<preceq> z) = (x \<^bsub>(P i)\<^esub>\<preceq> z) \<and> (z \<^bsub>(P' i)\<^esub>\<preceq> x) = (z \<^bsub>(P i)\<^esub>\<preceq> x)"
finite_if_eq_beyond_finite: fixes S :: "'a set"   and s' :: "'a set" assumes "finite S" shows "finite {s. s - S = s' - S}"
path2_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and invar :: "'b \<Rightarrow> bool"   and inEdges' :: "'b \<Rightarrow> 'c \<Rightarrow> ('c \<times> 'd) list"   and g :: "'b"   and n :: "'c"   and ns :: "'c list"   and m :: "'c" shows "graph_path_base.path2 \<alpha>n invar inEdges' g n ns m \<equiv> graph_path_base.path \<alpha>n invar inEdges' g ns \<and> n = hd ns \<and> m = last ns"
less_length_cut_same_Inl: fixes I :: "(nat + nat set) list"   and r :: "nat"   and p :: "nat"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "r < length I"   and "I ! r = Inl p" shows "p < length (cut_same (any, replicate (length I) False) (stream_enc (w, I)))"
cblinfun_eq_on_span: fixes x :: "'a"   and S :: "'a set"   and F :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and G :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "x \<in> cspan S"   and "\<And>s. s \<in> S \<Longrightarrow> F *\<^sub>V s = G *\<^sub>V s" shows "F *\<^sub>V x = G *\<^sub>V x"
initial_state_reachable: fixes e :: "('r, 'l, 'v) expr"   and r :: "'r" assumes "program_expr e" shows "reachable (\<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)))"
face_cycle_set_induct: fixes a :: "'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> face_cycle_set x"   and "P x"   and "\<And>y. \<lbrakk>y \<in> face_cycle_set x; P y\<rbrakk> \<Longrightarrow> P (face_cycle_succ y)" shows "P a"
pathF_def: shows "path PM pathF (initial PM) \<and> target pathF (initial PM) = FAIL"
closed_segment_imp_Re_Im_1: fixes x :: "complex"   and lb :: "complex"   and ub :: "complex" assumes "x \<in> closed_segment lb ub"   and "Re lb \<le> Re ub" shows "Re lb \<le> Re x \<and> Re x \<le> Re ub"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'d"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = ((bool = True \<longrightarrow> P f1) \<and> (bool = False \<longrightarrow> P f2))"
right_totalI: fixes A :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>y. \<exists>x. A x y" shows "right_total A"
induce_reachable_preserves_paths: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>(G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> {w. u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> w}\<^esub> v"
less_eq_ChildAct_def: fixes x :: "ChildAct"   and y :: "ChildAct" shows "x \<le> y \<equiv> x = y \<or> x < y"
ltl_to_rabin_base_axioms: shows "ltl_to_rabin_base \<up>af \<up>af\<^sub>G Abs Abs M_fin \<Sigma> w"
gen_dfs_outer_initial: fixes E :: "('a \<times> 'a) set"   and V0 :: "'a set"   and brk :: "bool" assumes "finite (E\<^sup>* `` V0)" shows "gen_dfs_outer E (E\<^sup>* `` V0) V0 V0 {} brk"
exec_move_Cond1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (if (e) e1 else e2) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
strongly_fair_def3: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "strongly_fair enabled taken = (\<lambda>s. (\<box>\<diamond>enabled) s \<longrightarrow> (\<box>\<diamond>taken) s)"
mask_eq_x_eq_0: fixes x :: "'b word"   and w :: "'b word" shows "(x AND w = x) = (x AND NOT w = 0)"
arr_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "arr f = (f = Zero \<or> f = One \<or> f = j0 \<or> f = j1)"
hyps: shows "\<forall>ma<n. gcd (fib m) (fib (ma mod m)) = gcd (fib m) (fib ma)"
finite_C: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" assumes "nf_invar c src dst PRED C N d" shows "finite C"
assoc_1: fixes shows_prec :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'a list \<Rightarrow> char list \<Rightarrow> char list"   and d :: "nat"   and x :: "'a"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_prec d x r @ s = shows_prec d x (r @ s)"
get_inter_def: fixes X :: "'d set" shows "get_inter X = SPEC (\<lambda>(Y, Z). X = Y \<inter> Z)"
continuous_on_def: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
abs_Gromov_completion_limit: fixes u :: "nat \<Rightarrow> 'a" assumes "Gromov_completion_rel u u" shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> abs_Gromov_completion u"
input_unconstrained_aval_register_swap: fixes a :: "vname aexp"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option"   and r' :: "nat \<Rightarrow>f value option" assumes "\<forall>i. \<not> aexp_constrains a (V (R i))" shows "aval a (join_ir i r) = aval a (join_ir i r')"
emptymi_def: shows "emptymi \<equiv> \<lparr>dpm = empty_pointermap, dcl = Map.empty\<rparr>"
path_connected_space_imp_path_component_of: fixes X :: "'a topology"   and a :: "'a"   and b :: "'a" assumes "path_connected_space X"   and "a \<in> topspace X"   and "b \<in> topspace X" shows "path_component_of X a b"
assm: shows "\<exists>va u. v=va--u \<and> (\<exists>v u. va=v\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)) \<and> (\<exists>v ua. u=v\<parallel>ua \<and> True \<and> (\<exists>v u. ua=v\<parallel>u \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1) \<and> True)) \<and> True))"
rep_eq: fixes x :: "('b, unit) RBT.rbt" shows "r_max x = rbt_max (rbt.impl_of x)"
wte: shows "P,E,s \<turnstile> e : T \<surd>"
lbound_of_def: fixes x :: "'a"   and y :: "'a"   and b :: "'a" shows "lbound_of x y b \<equiv> x \<succeq> b \<and> y \<succeq> b"
tstp_lt_def: fixes tstp :: "nat + nat"   and ts :: "nat"   and tp :: "nat" shows "tstp_lt tstp ts tp = (case tstp of Inl ts' \<Rightarrow> ts' \<le> ts | Inr tp' \<Rightarrow> tp' < tp)"
induct: fixes P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list" assumes "\<And>uu_. P 0 uu_"   and "\<And>v. P (Suc v) []"   and "\<And>v va vb. P (Suc v) (drop (Suc v) (va # vb)) \<Longrightarrow> P (Suc v) (va # vb)" shows "P a0 a1"
bind_def: fixes m :: "'c + 'e"   and f :: "'e \<Rightarrow> 'c + 'd" shows "m \<bind> f = (case m of Inl e \<Rightarrow> Inl e | Inr x \<Rightarrow> f x)"
prems_4: shows "a \<in> \<O> \<union> all_acquired (x # sb)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Left.in_hom f a b" shows "Left.in_hom (Left.L f) (Left.L a) (Left.L b)"
conj_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (Cooper.conj p q)"
content_smult: fixes c :: "'a"   and p :: "'a poly" shows "content (smult c p) = normalize c * content p"
concrete_category_axioms: shows "concrete_category (Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))) (\<lambda>F G. Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G)) (\<lambda>F. F) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) \<sigma> \<tau>)"
finprod_all1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (1::'b)" shows "prod f A = (1::'b)"
urel_integer_minus: fixes x :: "integer"   and y :: "int"   and x' :: "integer"   and y' :: "int" assumes "urel_integer x y"   and "urel_integer x' y'" shows "urel_integer (minus_p_integer pp x x') (minus_p p y y')"
help1: fixes a :: "nat"   and X :: "enat"   and Y :: "enat"   and b :: "nat"   and Z :: "enat" assumes "enat a + X \<le> Y"   and "enat b + Z \<le> X" shows "enat (a + b) + Z \<le> Y"
simps_5: fixes t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr (insert\<langle>t,s\<rangle> # A) D = tr A (List.insert (t, s) D)"
c_merge_all: fixes xss :: "'a list list" shows "int (local.c_merge_all xss) \<le> int (length (concat xss)) * \<lceil>log 2 (real (length xss))\<rceil>"
and_extract_pure_right_iff: fixes P :: "assn"   and b :: "bool" shows "P \<and>\<^sub>A \<up> b = (emp \<and>\<^sub>A P) * \<up> b"
numeral_power_less_of_nat_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and i :: "num"   and n :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (power.power one times (numeral.numeral one plus i) n) (semiring_1.of_nat one plus zero x) = (numeral i ^ n < x)"
rbt_lookup_rbt_union: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "('a, 'b) RBT_Impl.rbt"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less s"   and "ord.rbt_sorted less t" shows "ord.rbt_lookup less (ord.rbt_union less s t) = ord.rbt_lookup less s ++ ord.rbt_lookup less t"
veval'_welldefined': fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> v"   and "fdisjnt C (fmdom \<Gamma>)"   and "consts t |\<subseteq>| fmdom \<Gamma> |\<union>| C"   and "fmpred (\<lambda>_. vwelldefined') \<Gamma>"   and "pre_strong_term_class.wellformed t"   and "wellformed_venv \<Gamma>"   and "\<not> shadows_consts t"   and "not_shadows_vconsts_env \<Gamma>" shows "vwelldefined' v"
euler_fas'_def: fixes D :: "nat" shows "euler_fas' D = map fold_const_fa (euler_fas (map Var [0..<D]) (Var (2 * D)) (map Var [D..<2 * D]))"
lub_least: fixes A :: "'f llist set"   and z :: "'f llist" assumes "Complete_Partial_Order.chain lprefix A"   and "\<And>x. x \<in> A \<Longrightarrow> lprefix x z" shows "lprefix (lSup A) z"
not_outside_connected_component_le: fixes S :: "'a set" assumes "bounded S"   and "2 \<le> DIM('a)" shows "- outside S = {x. \<forall>B. \<exists>y. B \<le> norm y \<and> \<not> connected_component (- S) x y}"
assms: shows "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L"
PRelQ: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel"
closed_halfspace_ge: fixes b :: "real"   and a :: "'a" shows "closed {x. b \<le> a \<bullet> x}"