hcju/isabelleps huggingface dataset:
lang_ENC_formula: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (ENC n (FOV \<phi>)) - {[]} = {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) \<phi>}"
Union_Pow_eq: fixes A :: "'d set" shows "\<Union> (Pow A) = A"
Proj_ortho_compl: fixes X :: "'a ccsubspace" shows "Proj (- X) = id_cblinfun - Proj X"
pos_less_divide_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "(0::'a) < c" shows "(a < b / c) = (a * c < b)"
Decision_approximating_bigstep_fun: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and X :: "final_decision" shows "approximating_bigstep_fun \<gamma> p rs (Decision X) = Decision X"
effect_for_insert_elt: fixes l :: "nat"   and h :: "heap"   and a :: "'b array"   and h' :: "heap"   and u :: "unit" assumes "l \<le> Array.length h a"   and "1 \<le> l"   and "effect (for [1..<l] (insert_elt a)) h h' u" shows "Array.length h a = Array.length h' a \<and> sorted (take l (Array.get h' a)) \<and> mset (Array.get h a) = mset (Array.get h' a)"
mem_assign_refinement_helper_const: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat"   and x :: "Eg1.var"   and c :: "nat" shows "mem\<^sub>A_of (mem\<^sub>C(Eg2_var\<^sub>C_of_Eg1 x := c)) = (mem\<^sub>A_of mem\<^sub>C)(x := c)"
add_top: fixes x :: "'c" assumes "(0::'c) \<le> x" shows "x + top = top"
IH: fixes z :: "(real \<times> real) \<times> real \<times> real"   and Pc :: "real \<times> real" assumes "ccw'.sortedP 0 Ps"   and "z \<in> set (polychain_of Pc Ps)" shows "list_all (\<lambda>a. case a of (xi, xj) \<Rightarrow> ccw xi xj (fst z) \<and> ccw xi xj (snd z)) (polychain_of Pc Ps)"
assms_1: fixes a :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "a \<in> A" shows "c \<cdot> a \<in> A"
kcdj: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
coeff_poly_to_mpoly: fixes v :: "nat"   and p :: "'b poly"   and k :: "nat" shows "MPoly_Type.coeff (poly_to_mpoly v p) (Poly_Mapping.single v k) = poly.coeff p k"
pivot_positions_2: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr nc"   and "pivot_fun A f nc" shows "distinct (map fst (pivot_positions A))"
1: shows "w1 = B \<circ> basic (make_vert_block (nat (codomain_wall B))) \<and> w2 = basic [] \<circ> B \<and> domain_wall B = 0 \<and> is_tangle_diagram B"
USUP_cong: fixes A :: "'d set"   and F :: "'d \<Rightarrow> ('e, 'f) uexpr"   and G :: "'d \<Rightarrow> ('e, 'f) uexpr" assumes "\<And>P. P \<in> A \<Longrightarrow> F P = G P" shows "(\<Squnion> P \<in> A \<bullet> F P) = (\<Squnion> P \<in> A \<bullet> G P)"
split_uv: shows "splits_at (u @ v) (i + length u) (u @ \<alpha>) N \<beta>"
bounded_linear_matrix_blinfun_apply: shows "bounded_linear (\<lambda>x. matrix (blinfun_apply x))"
prefix_take_eq_prefix_take_ex: fixes xs :: "'b list"   and ys :: "'b list" shows "(ys \<down> length xs = xs) = (\<exists>n. ys \<down> n = xs)"
independent_alt: fixes B :: "'b set" shows "independent B = (\<forall>X. finite {x. X x \<noteq> (0::'a)} \<longrightarrow> {x. X x \<noteq> (0::'a)} \<subseteq> B \<longrightarrow> (\<Sum>x | X x \<noteq> (0::'a). X x *s x) = (0::'b) \<longrightarrow> (\<forall>x. X x = (0::'a)))"
abs_eq: shows "(\<le>) = (\<lambda>X Y. Rep_upset Y \<subseteq> Rep_upset X)"
N: fixes F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a"   and fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"   and a :: "int"   and N :: "nat"   and C :: "'a"   and Y :: "real set" assumes "euler_maclaurin_int F f fs a N C Y" shows "0 < N"
assms: fixes v\<^sub>1 :: "'item list" assumes "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1" shows "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
injI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
gtr_step_cases: fixes s :: "'c"   and ce :: "'c multiset"   and e :: "'d"   and c' :: "'e multiset"   and gtrs :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" assumes "(add_mset s ce, e, c') \<in> gtr gtrs" obtains s' :: "'e" and ce' :: "'e multiset" where "c' = add_mset s' ce'__" and "((s, ce), e, s', ce'__) \<in> gtrs"   | cc :: "'c multiset" and ss :: "'c" and ss' :: "'e" and ce' :: "'e multiset" where "ce = add_mset ss cc" and "c' = add_mset ss' ce'" and "((ss, add_mset s cc), e, ss', ce') \<in> gtrs"
swapidseq_inverse_exists: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b" assumes "swapidseq n p" shows "\<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
adjoint_pair_antipar_2: fixes f :: "'b"   and g :: "'b" assumes "D.adjoint_pair f g" shows "D.ide g"
assms_5: fixes \<rho>' :: "complex mat" assumes "\<rho>' \<in> carrier_mat d d"   and "partial_density_operator \<rho>'" shows "positive (DS \<rho>') \<and> trace (DS \<rho>') \<le> trace \<rho>' \<and> DS \<rho>' \<in> carrier_mat d d"
wpc_least: fixes t :: "'d itself"   and Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and S :: "nat \<Rightarrow> ('e, 'g) labeled_graph" assumes "WPC t Rs S" shows "least t Rs (S 0) (chain_sup S)"
finals_app_map: fixes vs :: "val list"   and es :: "expr list" shows "finals (map Val vs @ es) = finals es"
hcomplex_eq_cancel_iff2: fixes x :: "real star"   and y :: "real star"   and xa :: "real star" shows "(HComplex x y = hcomplex_of_hypreal xa) = (x = xa \<and> y = 0)"
cfWrite_def: fixes cfile :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte" shows "cfWrite cfile byteIndex value \<equiv> if byteIndex div blockSize < numBlocks then if byteIndex < fileSize cfile then Some (cfWriteNoExtend cfile byteIndex value) else Some (cfWriteNoExtend (cfExtendFile cfile byteIndex) byteIndex value) else None"
export_Local_invariant: fixes G :: "nat triple set"   and Y :: "loc"   and a :: "state \<Rightarrow> nat"   and c :: "com" shows "G|-{\<lambda>Z s. Z = s<Y>}. LOCAL Y:=a IN c .{\<lambda>Z s. Z = s<Y>}"
res: shows "(xi, xj, p_ij) # (xsi, xsj, p_sisj) # rest = map (\<lambda>i. (x i, x (i + j), local.neville_aitken_main i (i + j))) [l..<Suc (l + k)]"
LIM_cong: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b"   and m :: "'b" assumes "a = b"   and "\<And>x. x \<noteq> b \<Longrightarrow> f x = g x"   and "l = m" shows "f \<midarrow>a\<rightarrow> l = g \<midarrow>b\<rightarrow> m"
hausdorff_distance_def: fixes A :: "'b set"   and B :: "'b set" shows "hausdorff_distance A B = (if A = {} \<or> B = {} \<or> \<not> bounded A \<or> \<not> bounded B then 0 else max (SUP x\<in>A. infdist x B) (SUP x\<in>B. infdist x A))"
slice_len: fixes "from" :: "nat"   and to :: "nat"   and xs :: "'a list" assumes "from \<le> to"   and "to \<le> length xs" shows "length (slice from to xs) = to - from"
assms_4: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> pmdl B"   and "f \<noteq> 0" shows "\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f"
prod_z_x2: shows "poly (prod_list qs) x2 = 0"
C: fixes x :: "hf"   and x' :: "hf"   and y :: "hf"   and y' :: "hf"   and z :: "hf"   and z' :: "hf" assumes "C x x' y y' z z'"   and "P y y'"   and "P z z'" shows "P x x'"
dmu': fixes j :: "nat"   and ii :: "nat" assumes "j < length fs'"   and "ii < length fs'" shows "rat_of_int (fs'.d\<mu> ii j) = rat_of_int (fs'.d fs' (Suc j)) * fs'.gs.\<mu> ii j"
B_def: shows "B \<equiv> {c. public c \<and> arity c = 0}"
pending_on_stack: fixes v :: "'v"   and w :: "'v" assumes "(v, w) \<in> pending s" shows "v \<in> set (stack s)"
subst_word_def: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list" shows "subst_word f xs = concat (map f xs)"
strip_guards_simps_4: fixes F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Seq c\<^sub>1 c\<^sub>2) = Seq (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
geq_arg_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "geq_arg f \<equiv> \<forall>x y. x \<le> f x y \<and> y \<le> f x y"
Im_Ln_pos_lt_imp: fixes z :: "complex" assumes "0 < Im z" shows "0 < Im (Ln z) \<and> Im (Ln z) < pi"
simps_6: fixes f1 :: "predicate \<Rightarrow> 'b list \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b" shows "(case Eq x21 x22 of predAtm x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
ub3_lb: shows "0 \<le> real_of_float (rapprox_rat (max prec 1) 1 3)"
stored_reused_is_generalised_output_of: fixes t' :: "transition"   and t :: "transition"   and p :: "nat"   and r :: "nat" assumes "stored_reused t' t = Some (p, r)" shows "is_generalised_output_of t' t p r"
sum_def: shows "sum \<equiv> comm_monoid_set.F (+) (0::'b)"
unary_wt_sym_0_gt\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b"   and g :: "'b" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "hmset_of_enat (arity_sym f) = 1"   and "wt_sym f = 0" shows "gt_sym f g \<or> g = f"
before_dist_r2: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "distinct vs"   and "before vs r1 r2" shows "r2 \<in> set (snd (splitAt r1 vs))"
G_soundness: shows "CM.Model GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf gPB intT GE_intF GE_intP"
read_only_unowned_nth_update: fixes i :: "nat"   and \<O> :: "nat set"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "\<O> \<inter> read_only \<S>\<^sub>s\<^sub>b = {}" shows "read_only_unowned \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
simps_2: fixes var :: "'f"   and t :: "('d, 'f) trm" shows "SIGP (var := t) = {Inl x |x. x \<in> SIGT t}"
vsubset_contraD: fixes A :: "V"   and B :: "V"   and c :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "c \<notin>\<^sub>\<circ> B" shows "c \<notin>\<^sub>\<circ> A"
arg_complex_of_real_times_i_positive: fixes k :: "real" assumes "0 < k" shows "arg (cor k * \<i>) = pi / 2"
advantage_nonneg: fixes \<eta> :: "nat"   and "oracle" :: "('b \<Rightarrow> 'grp \<Rightarrow> (bool list \<times> 'b) spmf) \<times> 'b"   and \<A> :: "('grp \<Rightarrow> ((bool list \<times> bool list) \<times> 'c, 'grp, bool list) gpv) \<times> ('grp \<times> bool list \<Rightarrow> 'c \<Rightarrow> (bool, 'grp, bool list) gpv)" shows "0 \<le> advantage \<eta> oracle \<A>"
not_Msg_eq: fixes m :: "'b message_af" shows "(\<forall>x. m \<noteq> Msg x) = (m = NoMsg)"
resumption_ord_resumeD: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "\<not> Resumption.resumption.is_Done r" shows "rel_fun (=) resumption_ord (Resumption.resumption.resume r) (Resumption.resumption.resume r')"
ba3_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
nAct_strict_mono_back: fixes c :: "'id"   and n :: "enat"   and t :: "cnf llist"   and n' :: "enat" assumes "\<langle>c #\<^bsub>n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>" shows "n < n'"
bisim1Sync11: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, Suc (Suc (length (compE2 e1))), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
f_has_derivative_at_neg: fixes x :: "real" assumes "x < 0" shows "(f has_derivative (\<lambda>x. 0)) (at x)"
as_needed_asses_submap_exec_xii: fixes v :: "'d"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "v \<in> fmdom' f" shows "fmlookup (f ++ g) v = fmlookup f v"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "abs_Tree\<^sub>\<alpha> (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
Expr2ZFfe: fixes f :: "ZF"   and e :: "(ZF, ZF) Expression" shows "Expr2ZF f E@ e = ZFTriple (SucNat (ZFTFst (Expr2ZF e))) (nat2Nat 1) (Opair f (Expr2ZF e))"
emptyci_def: shows "emptyci \<equiv> pointermap_empty \<bind> (\<lambda>ep. hm_new \<bind> (\<lambda>ehm. return \<lparr>dpmi = ep, dcli = ehm\<rparr>))"
simps_15: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
dom_char: fixes f :: "'a" shows "hom'.dom f = (if hom'.arr f then local.dom f else null)"
num_syms_num_unique_vars_clash: fixes f :: "nat \<Rightarrow> ('a, 'b) Term.term" assumes "\<forall>i. num_syms (f i) = num_syms (f (Suc i))"   and "\<forall>i. num_unique_vars (f i) < num_unique_vars (f (Suc i))" shows "False"
plus_1_eq: fixes x :: "'a" shows "g x (1::'a) = g1 x"
matches_rule_and_simp_help: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a' :: "action"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule (MatchAnd m m') a'], Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m' a'], Undecided\<rangle> \<Rightarrow> t"
assms_5: shows "r \<in> R1 \<or> r \<in> R2 \<or> r \<in> R3"
prv_fls: fixes \<phi> :: "fm" shows "{} \<turnstile> Fls IMP \<phi>"
f: shows "(\<Sum>i\<in>S. f i * i) = x"
endofunctor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a" shows "endofunctor A F \<equiv> functor A A F"
arr_inv: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.arr (VoV.inv f)"
acquired_reads_no_pending_write: fixes xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" shows "acquired_reads False xs A = acquired_reads False xs B"
bind_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "Option.bind x f = Option.bind y g"
monom_mult_eq_zero_iff: fixes c :: "'d"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "(monom_mult c t p = 0) = (c = (0::'d) \<or> p = 0)"
hsize_ge_num_args: fixes s :: "('a, 'b) tm" shows "hsize s \<le> hsize s"
blocking_on_CD_on': fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set"   and x :: "'a" assumes "Singleton_for_d.blocking_on ds X h X''"   and "x \<in> X''" shows "x \<in> Singleton_for_d.CD_on ds (X \<union> X'')"
indices_state_set_unsat: fixes I :: "'a list"   and s :: "('a, 'b) state" shows "indices_state (set_unsat I s) = indices_state s"
elims_3: fixes x :: "nat"   and xa :: "floatarith"   and xb :: "real list" assumes "\<not> isDERIV x xa xb" obtains xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xc__" and "xa = Add a__ b__" and "xb = vs__" and "\<not> (isDERIV xc__ a__ vs__ \<and> isDERIV xc__ b__ vs__)"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xca__" and "xa = Mult aa__ ba__" and "xb = vsa__" and "\<not> (isDERIV xca__ aa__ vsa__ \<and> isDERIV xca__ ba__ vsa__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcb__" and "xa = Minus ab__" and "xb = vsb__" and "\<not> isDERIV xcb__ ab__ vsb__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcc__" and "xa = Inverse ac__" and "xb = vsc__" and "\<not> (isDERIV xcc__ ac__ vsc__ \<and> interpret_floatarith ac__ vsc__ \<noteq> 0)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcd__" and "xa = Cos ad__" and "xb = vsd__" and "\<not> isDERIV xcd__ ad__ vsd__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xce__" and "xa = Arctan ae__" and "xb = vse__" and "\<not> isDERIV xce__ ae__ vse__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xcf__" and "xa = floatarith.Min af__ bb__" and "xb = vsf__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xcg__" and "xa = floatarith.Max ag__ bc__" and "xb = vsg__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xch__" and "xa = Abs ah__" and "xb = vsh__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xci__" and "xa = Sqrt ai__" and "xb = vsi__" and "\<not> (isDERIV xci__ ai__ vsi__ \<and> 0 < interpret_floatarith ai__ vsi__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcj__" and "xa = Exp aj__" and "xb = vsj__" and "\<not> isDERIV xcj__ aj__ vsj__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xck__" and "xa = Powr ak__ b" and "xb = vsk__" and "\<not> (isDERIV xck__ ak__ vsk__ \<and> isDERIV xck__ b vsk__ \<and> 0 < interpret_floatarith ak__ vsk__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcl__" and "xa = Ln al__" and "xb = vsl__" and "\<not> (isDERIV xcl__ al__ vsl__ \<and> 0 < interpret_floatarith al__ vsl__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcm__" and "xa = Floor am__" and "xb = vsm__" and "\<not> (isDERIV xcm__ am__ vsm__ \<and> interpret_floatarith am__ vsm__ \<notin> \<int>)"   | xc :: "nat" and a :: "floatarith" and n :: "nat" and vs :: "real list" where "x = xc" and "xa = Power a (Suc n)" and "xb = vs" and "\<not> isDERIV xc a vs"
sep_disj_prod_def: fixes m1 :: "'a \<times> 'b"   and m2 :: "'a \<times> 'b" shows "m1 ## m2 \<equiv> fst m1 ## fst m2 \<and> snd m1 ## snd m2"
simps_4: fixes e1 :: "expr"   and e2 :: "expr" shows "radicals (Addition e1 e2) = radicals e1 \<union> radicals e2"
simps_5: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "wellf (Pr n f g) = (arity g = Suc (Suc n) \<and> arity f = n \<and> wellf f \<and> wellf g)"
vmc_path_axioms: shows "vmc_path G P v p \<sigma>"
flowderiv_continuous_on: shows "continuous_on (Sigma X existence_ivl0) (\<lambda>(x, y). flowderiv x y)"
exec_gpv_eq_resource_on: fixes \<I> :: "('a, 'b) \<I>"   and res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource"   and gpv :: "('c, 'a, 'b) gpv" assumes "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<I> \<turnstile>res res \<surd>" shows "rel_spmf (rel_prod (=) (eq_resource_on (outs_\<I> \<I>))) (exec_gpv run_resource gpv res) (exec_gpv run_resource gpv res')"
Entry_unreachable: fixes g :: "'g" assumes "invar g" shows "inEdges g (Entry g) = []"
trivial_2: fixes b :: "'b" assumes "b \<in> C - B" shows "h b = \<^bold>1"
flist_inv: fixes xs :: "'a fset" shows "finset (flist xs) = xs"
submonoid_invertible: fixes u :: "'a \<Rightarrow> 'a" assumes "symmetric.sub.invertible u"   and "u \<in> G" shows "symmetric.invertible u"
assms_2: shows "\<forall>n. g (Suc n) \<le> g n"
simps_3: fixes S :: "'a ltl set"   and \<phi> :: "'a ltl" shows "eval\<^sub>G S (G \<phi>) = (if G \<phi> \<in> S then true else false)"
nontriv_solution_altdef: fixes z :: "'b \<times> 'b" shows "nontriv_solution z = (solution z \<and> z \<noteq> (1::'b, 0::'b))"
p_inf_pp: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - - y) = - (x \<squnion> y)"
simps_12: fixes p :: "fm"   and q :: "fm" shows "decr (Imp p q) = Imp (decr p) (decr q)"
assms: shows "Complete_Partial_Order.chain (\<sqsubseteq>) A"
monom_in_carrier: fixes a :: "'a list"   and n :: "nat" assumes "a \<in> carrier (K [X])" shows "set (UP.monom a n) \<subseteq> carrier (K [X])"
ex_Sup: fixes A :: "'b set" shows "\<exists>sup. is_Sup A sup"
SMPp_SMP_eq: fixes M :: "('fun, 'var) Term.term set" shows "SMPp (\<lambda>x. x \<in> M) = (\<lambda>x. x \<in> SMP M)"
simps_13: fixes x4 :: "nat"   and x2 :: "nat" shows "REL x4 \<noteq> WAIT x2"
assms_4: shows "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"
visits_via_visits: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set" assumes "visits_via P A W" shows "lset P \<inter> W \<noteq> {}"
simps: fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and buf :: "event_data option list set list list"   and ts :: "nat list" shows "mbufnt_take f z buf ts = (if [] \<in> set buf \<or> ts = [] then (z, buf, ts) else mbufnt_take f (f (map hd buf) (hd ts) z) (map tl buf) (tl ts))"
of_int_minus: fixes i :: "int" shows "\<guillemotleft>- i\<guillemotright> = \<ominus> \<guillemotleft>i\<guillemotright>"
assms_3: fixes while' :: "'a \<Rightarrow> 'a spmf" assumes "P while'" shows "P (\<lambda>s. if guard s then body s \<bind> while' else return_spmf s)"
2343: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<squnion> - - y = top"
rel_interior_real_semiline: fixes a :: "real" shows "rel_interior {a..} = {a<..}"
endpoints_shiftpath_1: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "pathfinish (shiftpath a g) = g a"
lepoll_imp_small: fixes X :: "'a set"   and A :: "'b set" assumes "X \<lesssim> A" shows "small X"
continuous_on_setdist: fixes T :: "'c set"   and S :: "'c set" shows "continuous_on T (\<lambda>y. setdist {y} S)"
map_scale_spmf: fixes f :: "'c \<Rightarrow> 'b"   and r :: "real"   and p :: "'c spmf" shows "map_spmf f (scale_spmf r p) = scale_spmf r (map_spmf f p)"
is_scc_closed: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and x :: "'a"   and y :: "'a" assumes "is_scc E U"   and "x \<in> U"   and "(x, y) \<in> E\<^sup>*"   and "(y, x) \<in> E\<^sup>*" shows "y \<in> U"
covector_inf_comp_3: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "times x top = x" shows "times (inf y (conv x)) z = times y (inf x z)"
chars_append: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "chars (a @ b) = chars a @ chars b"
errorT_map_def: shows "errorT_map \<equiv> \<Lambda> e a. \<mu> uu. errorT_abs oo fmap\<cdot>(error_map\<cdot>e\<cdot>a) oo errorT_rep"
D: shows "last (enum xs index key n mi ma) = offs_num n xs index key mi ma (n - Suc 0)"
L_eq: fixes x :: "'s"   and y :: "'t" assumes "R x y" shows "L y = map_pmf f (K x)"
complex_inner_i_right: fixes x :: "complex" shows "x \<bullet> \<i> = Im x"
simps_3: fixes \<pi> :: "'b prm"   and A :: "'b ptrm"   and B :: "'b ptrm" shows "\<pi> \<bullet> PApp A B = PApp (\<pi> \<bullet> A) (\<pi> \<bullet> B)"
intro: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "submonoid G M composition unit"   and "group G composition unit" shows "subgroup G M composition unit"
at_fin_set_fresh: fixes X :: "'a set"   and x :: "'a" assumes "at TYPE('a)"   and "finite X" shows "x \<sharp> X = (x \<notin> X)"
\<delta>x_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)"