hcju/isabelleps huggingface dataset:
ar_cos_same: fixes a :: "'a"   and I :: "'a set"   and x :: "'a" assumes "a \<in> carrier R"   and "ideal R I"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "x \<uplus>\<^bsub>R\<^esub> I = a \<uplus>\<^bsub>R\<^esub> I"
eq: shows "of_nat (x\<^sup>2 + 1) = (of_nat x + \<i>\<^sub>\<int>) * (of_nat x - \<i>\<^sub>\<int>)"
UnD1: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<union> B"   and "c \<notin> A" shows "c \<in> B"
IH: fixes t :: "('a, 'b) tm" assumes "position_of t (p @ [d])" shows "emb_step_at p d (subst \<rho> t) = subst \<rho> (emb_step_at p d t)"
cc_restr_cc_restr: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" shows "G G|` S' G|` S = G G|` (S \<inter> S')"
I1: fixes s :: "'d"   and x :: "'e" assumes "s \<in> S"   and "c d x \<le> SPEC (I {s})" shows "I {s} x"
distinct_insort_wrt: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "distinct (insort_wrt R xs) = distinct xs"
has_bottom_def: fixes P :: "'b set" shows "order.has_bottom P \<equiv> \<exists>z\<in>P. \<forall>x\<in>P. order.greater_eq x z"
rComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
Ax: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<in> \<Gamma>" shows "\<Gamma> \<turnstile> F"
simps_2: fixes x :: "'c"   and l :: "(('c \<times> 'd) \<times> nat) tree"   and a :: "'c"   and b :: "'d"   and lv :: "nat"   and r :: "(('c \<times> 'd) \<times> nat) tree" shows "AA_Map.delete x \<langle>l, ((a, b), lv), r\<rangle> = (case cmp x a of LT \<Rightarrow> adjust \<langle>AA_Map.delete x l, ((a, b), lv), r\<rangle> | EQ \<Rightarrow> if l = \<langle>\<rangle> then r else let (l', ab') = split_max l in adjust \<langle>l', (ab', lv), r\<rangle> | GT \<Rightarrow> adjust \<langle>l, ((a, b), lv), AA_Map.delete x r\<rangle>)"
gtr_preserve_s: fixes c :: "'c multiset"   and e :: "'d"   and c' :: "'c multiset"   and T :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set"   and P :: "'c multiset \<Rightarrow> bool" assumes "(c, e, c') \<in> gtr T"   and "P c"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P c'"
fold_mset_add_mset: fixes s :: "'m"   and x :: "'m"   and M :: "'m multiset" shows "fold_mset munionM s (add_mset x M) = munionM x (fold_mset munionM s M)"
g: fixes n :: "nat" shows "ff n \<in> Tr' \<and> infinite {t' \<in> Tr'. ff n < t'}"
init_sim: fixes a0 :: "'b" assumes "a0 \<in> b.V0" shows "\<exists>b0. b0 \<in> a.V0 \<and> (a0, b0) \<in> R\<inverse>"
mergeable: fixes \<phi> :: "'c MFOTL.formula"   and M :: "(char list \<times> 'c list) prefix \<Rightarrow> (nat \<times> 'c option list) set"   and strategy :: "'c option list \<Rightarrow> 'd set"   and k :: "'d" assumes "skip_inter \<phi> M strategy" shows "mergeable_envs (MFOTL.nfv \<phi>) {v. \<exists>v'. map the v' = v \<and> wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v' \<and> k \<in> strategy v'}"
strict_monoI: fixes f :: "'a list \<Rightarrow> 'b" assumes "\<And>x y. strict_border x y \<Longrightarrow> f x < f y" shows "border_order.strict_mono f"
SD_preorder: fixes q :: "'alt pmf"   and p :: "'alt pmf" shows "SD (Pareto R) q p = (p \<in> lotteries_on alts \<and> q \<in> lotteries_on alts \<and> (\<forall>x\<in>alts. measure_pmf.prob q (preferred_alts (Pareto R) x) \<le> measure_pmf.prob p (preferred_alts (Pareto R) x)))"
cut_le_Diff: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A - B) \<down>\<le> t = A \<down>\<le> t - B \<down>\<le> t"
hom_mult: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id (x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y) = hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (upper n)\<^esub> hom_induced (int n) (nsphere n) (equator n) (nsphere n) (upper n) id y"
lookup_minus: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and g :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "lookup (f - g) k = lookup f k - lookup g k"
Send: shows "ev = Send i' (occurs_on ev) r u u' m"
closure_minimal: fixes S :: "'d set"   and T :: "'d set" assumes "S \<subseteq> T"   and "closed T" shows "closure S \<subseteq> T"
f_bound_True_arbitrary: fixes ii :: "nat"   and fs :: "int vec list"   and outside :: "bool"   and j :: "nat" assumes "f_bound True ii fs" shows "f_bound outside j fs"
stake_0_1: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "stake\<cdot>0\<cdot>xs = [::]"
add_zero: fixes x :: "'b" shows "Sep_Eq.add2 x (0::'b) = Some x"
l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
no_valid_literal: fixes L :: "'a literal"   and C :: "'a literal set"   and t :: "'a trm"   and s :: "'a trm"   and \<sigma> :: "('a \<times> 'a trm) list"   and S :: "'a eclause set" assumes "L \<in> C"   and "orient_lit_inst L t s pos \<sigma>"   and "\<not> validate_ground_clause (int_clset S) (subst_cl C \<sigma>)" shows "trm_rep (t \<lhd> \<sigma>) S \<noteq> trm_rep (s \<lhd> \<sigma>) S"
distinct_replacefacesAt2_norm: fixes i :: "nat"   and Fss :: "face list list"   and "is" :: "nat list"   and newFs :: "face list"   and oldF :: "face" assumes "i < |Fss|"   and "i \<notin> set is"   and "distinct is"   and "distinct (normFaces (Fss ! i))"   and "distinct (normFaces newFs)"   and "set (normFaces (Fss ! i)) \<inter> set (normFaces newFs) \<subseteq> {}" shows "distinct (normFaces (replacefacesAt is oldF newFs Fss ! i))"
finite_induct: fixes P :: "'e\<cdot>('f, 'g) errorT \<Rightarrow> bool"   and n :: "nat"   and x :: "'e\<cdot>('f, 'g) errorT" assumes "P \<bottom>"   and "\<And>app. app \<noteq> \<bottom> \<Longrightarrow> P (ErrorT\<cdot>app)" shows "P (errorT_take n\<cdot>x)"
t1_exivl: shows "return_time {x \<in> S. s x = 0} x \<in> existence_ivl0 x"
atMost_Suc_eq_insert_0: fixes n :: "nat" shows "{..Suc n} = insert 0 (Suc ` {..n})"
PDG': shows "PDG src trg knd lve NewEntry lDef lUse state_val NewExit (Postdomination.standard_control_dependence src trg lve NewExit)"
after_const: fixes c :: "'c" shows "|~ #c$ = #c"
norm_cos_le: fixes z :: "complex" shows "cmod (cos z) \<le> exp (cmod z)"
B_Said_YM2: fixes B :: "agent"   and A :: "agent"   and NA :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>NB. Says B Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs"
in_eball_iff: fixes z :: "'a"   and z0 :: "'a"   and r :: "ereal" shows "(z \<in> eball z0 r) = (ereal (dist z0 z) < r)"
inorder_delete: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "bal t"   and "sorted1 (inorder t)" shows "inorder (Tree234_Map.delete x t) = AList_Upd_Del.del_list x (inorder t)"
bk: shows "eval (Cn 1 r_take [r_pdec2, r_pdec1]) [arg] \<down>= e_take k b"
preserves_cod: fixes f :: "'s" assumes "S.arr f" shows "S'.cod (\<Phi> f) = \<Phi> (S.cod f)"
fs_int': fixes k :: "nat"   and f :: "'a vec" assumes "k < n"   and "f \<in> set fs" shows "f $ k \<in> \<int>"
Aeta_expand_safe: fixes S :: "stack_elem list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" assumes "Astack S \<sqsubseteq> a" shows "(\<Gamma>, Aeta_expand a e, S) \<Rightarrow>\<^sup>* (\<Gamma>, e, S)"
sscan_const: fixes xs :: "'c stream"   and a :: "'c" shows "sscan (\<lambda>x _. x) xs a = xs"
Eq_RESUMPTION_refl: fixes r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "r \<in> resumption" shows "Eq_RESUMPTION r r"
at_within_t1_space_avoid: fixes x :: "'c"   and i :: "'c"   and X :: "'c set" assumes "x \<noteq> i" shows "at x within X - {i} = at x within X"
def_body: fixes x :: "'\<alpha>"   and y :: "'\<beta>" assumes "x \<noteq> \<bottom>"   and "y \<noteq> \<bottom>" shows "g x y \<noteq> \<bottom> \<and> g x y \<noteq> null"
disjoint_witness_correct_2: fixes s1 :: "'s1"   and s2 :: "'s2"   and a :: "'a" assumes "invar1 s1"   and "invar2 s2"   and "disjoint_witness s1 s2 = Some a" shows "a \<in> \<alpha>1 s1 \<inter> \<alpha>2 s2"
i_Exec_Stream_append_if: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream trans_fun xs c \<frown> i_Exec_Comp_Stream trans_fun input (if xs = [] then c else last (f_Exec_Comp_Stream trans_fun xs c))"
exec_gpv_invariant: fixes x :: "'a"   and s' :: "'b \<Rightarrow> '\<gamma> option"   and gpv :: "('a, 'b, '\<gamma>) gpv"   and s :: "'b \<Rightarrow> '\<gamma> option" assumes "(x, s') \<in> set_spmf (exec_gpv prf.random_oracle gpv s)"   and "finite (dom s)"   and "\<I>_full \<turnstile>g gpv \<surd>" shows "finite (dom s')"
regular_homogenous_system: fixes a11 :: "'b"   and a12 :: "'b"   and a21 :: "'b"   and a22 :: "'b"   and x1 :: "'b"   and x2 :: "'b" assumes "det2 a11 a12 a21 a22 \<noteq> (0::'b)"   and "a11 * x1 + a12 * x2 = (0::'b)"   and "a21 * x1 + a22 * x2 = (0::'b)" shows "x1 = (0::'b) \<and> x2 = (0::'b)"
funStart_prop1: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a set" shows "x \<in> funStar f x"
A: shows "acomp a\<^sub>1 @ acomp a\<^sub>2 @ (if f then [JMPLESS i] else [JMPGE i]) \<Turnstile> cfs\<box>"
geometric_sums_times_n: fixes c :: "'b" assumes "norm c < 1" shows "(\<lambda>n. c ^ n * of_nat n) sums (c / ((1::'b) - c)\<^sup>2)"
tp_cmul: fixes f :: "real \<Rightarrow> real"   and c :: "real" assumes "trigpoly f" shows "trigpoly (\<lambda>x. c * f x)"
par_beta_beta: fixes l :: "Label"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and u :: "sterm"   and u' :: "sterm" assumes "l \<in> dom f"   and "Obj f T => Obj f' T"   and "u => u'"   and "lc (Obj f T)"   and "lc u" shows "the (f l)\<^bsup>[Obj f T,u]\<^esup> => (the (f' l)\<^bsup>[Obj f' T,u']\<^esup>)"
inconsistent_eq_rank_less: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "inconsistent A b = (rank A < (if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b)) + 1 else 0))"
G: shows "(!) (map of_int_hom.vec_hom fs) ` {0..<i - 1} \<subseteq> Rn"
tmbound0_I: fixes a :: "tm"   and vs :: "'b list"   and b :: "'b"   and bs :: "'b list"   and b' :: "'b" assumes "tmbound0 a" shows "Itm vs (b # bs) a = Itm vs (b' # bs) a"
is_unit_smult_iff: fixes c :: "'a"   and p :: "'a poly" shows "(smult c p dvd 1) = (c dvd (1::'a) \<and> p dvd 1)"
card_Diff1_less_iff: fixes A :: "'c set"   and x :: "'c" shows "(card (A - {x}) < card A) = (finite A \<and> x \<in> A)"
QgrpUnit_1: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and H :: "'e set" assumes "Group G"   and "Ugp E"   and "G \<triangleright> H"   and "(G / H) \<cong> E" shows "carrier G = H"
l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> (known_ptr = DocumentClass.known_ptr \<and> type_wf = DocumentClass.type_wf) \<and> get_child_nodes = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes \<and> get_child_nodes_locs = l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs"
face_cycle_succ_def: shows "G'.face_cycle_succ \<equiv> edge_succ (rev_swap a b) \<circ> edge_rev (rev_swap a b)"
mu'_mu_large_row_im1: fixes ii :: "nat" assumes "i < ii"   and "ii < m" shows "gs2.\<mu> ii (i - 1) = fs.gs.\<mu> ii (i - 1) * gs2.\<mu> i (i - 1) + fs.gs.\<mu> ii i * \<parallel>fs.gs.gso i\<parallel>\<^sup>2 / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2"
netlimit_at_right: fixes x :: "real" shows "netlimit (at_right x) = x"
Image_singleton: fixes r :: "('f \<times> 'e) set"   and a :: "'f" shows "r `` {a} = {b. (a, b) \<in> r}"
good_item_simps_8: fixes xsa :: "'varSort"   and xa :: "'var" shows "good (Var xsa xa)"
simps_8: fixes P :: "('a, 'b, 'c) psi" shows "extractFrame (!P) = \<bottom>\<^sub>F"
of_complex_not_inf: fixes x :: "complex" shows "of_complex x \<noteq> \<infinity>\<^sub>h"
pdec2_le: fixes i :: "nat" shows "pdec2 i \<le> i"
chart_last_def: shows "chart_last \<equiv> (rep_proj ---> id) chart_last_nonzero"
add_mult_one_time_pad: fixes x :: "nat"   and q :: "nat"   and y :: "nat" assumes "coprime x q" shows "map_spmf (\<lambda>b. (y + x * b) mod q) (sample_uniform q) = sample_uniform q"
mult_zero_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.mult_zero times zero" shows "times a zero = zero"
one_plus_\<omega>_equals_\<omega>: shows "1 + \<omega> = \<omega>"
is_iso_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
inj_surj: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"   and "inj_on f {i. i \<le> n}" shows "f ` {i. i \<le> n} = {i. i \<le> n}"
impossible_Cons: fixes xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
wf_interp_for_formula_FExists: fixes I :: "(nat + nat set) list"   and \<phi> :: "'a formula"   and w :: "'a list" assumes "wf_formula (length I) (FExists \<phi>)"   and "w \<noteq> []" shows "wf_interp_for_formula (w, I) (FExists \<phi>) = (\<forall>p<length w. wf_interp_for_formula (w, Inl p # I) \<phi>)"
lens_indep_sym: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e" assumes "x \<bowtie> y" shows "y \<bowtie> x"
insert_spec_distinct: fixes xs :: "'b list"   and oid :: "'b"   and ref :: "'b option"   and r :: "'b" assumes "distinct xs"   and "\<And>x. x \<in> set xs \<Longrightarrow> x < oid"   and "ref = Some r \<longrightarrow> r < oid" shows "distinct (insert_spec xs (oid, ref))"
sf: shows "\<forall>i. simple_bochner_integrable M (sf i)"
gen_cfg_wf_extend_def: shows "gen_cfg_wf_extend \<equiv> map_fun Rep_gen_cfg_wf id gen_cfg.extend"
hcomp_simps_3: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.hseq \<nu> \<mu>" shows "Span.dom (\<nu> \<circ> \<mu>) = Span.dom \<nu> \<circ> Span.dom \<mu>"
impr_times: fixes a :: "'b N"   and b :: "'b N" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
is_path_undir_memb: fixes G :: "('a, 'b) graph"   and v :: "'a"   and p :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G"   and "is_path_undir G v p v'" shows "v \<in> nodes G \<and> v' \<in> nodes G"
le_sup_equiv_r: fixes a :: "'a"   and b :: "'a" shows "(a \<squnion> b = b) = (a \<le> b)"
\<Delta>\<Phi>_pass2: fixes hs :: "'b heap list" assumes "hs \<noteq> []"   and "no_Emptys hs" shows "\<Phi> (pass\<^sub>2 hs) - \<Phi>_hps hs \<le> log 2 (real (size_hps hs))"
equivP_pick_inj_on: fixes P :: "'d \<Rightarrow> bool"   and \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "equivP P \<phi>" shows "inj_on pick (Collect (P /// \<phi>))"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "P a b"
simps_3: fixes x1 :: "nat"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat" shows "Prime x1 \<noteq> Triple x21 x22 x23"
is_projection_on_eqI: fixes M :: "'b set"   and h :: "'b"   and x :: "'b"   and \<pi> :: "'b \<Rightarrow> 'b" assumes "closed_csubspace M"   and "h - x \<in> orthogonal_complement M"   and "x \<in> M"   and "is_projection_on \<pi> M" shows "\<pi> h = x"
sas_plus_equivalent_to_strips_i_a_XIII: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and G :: "'c \<Rightarrow> 'd option"   and I :: "'c \<Rightarrow> 'd option"   and \<pi> :: "('c \<times> 'd) strips_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "\<phi>\<^sub>S \<Psi> G \<subseteq>\<^sub>m execute_parallel_plan (execute_parallel_operator (\<phi>\<^sub>S \<Psi> I) ops') \<pi>" shows "\<phi>\<^sub>S \<Psi> G \<subseteq>\<^sub>m execute_parallel_plan (\<phi>\<^sub>S \<Psi> execute_parallel_operator_sas_plus I (map (strips_op_to_sasp \<Psi>) ops')) \<pi>"
dim_row_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a Matrix.vec" shows "dim_row (mat\<^sub>r nr nc g) = nr"
greaterThan_iff: fixes i :: "'a"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a" shows "(i \<in> ord.greaterThan less k) = less k i"
x_y_split: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<inter> NC \<cdot> y \<union> x \<cdot> {} = x \<cdot> y"
finite_group_def: fixes G :: "('e, 'f) monoid_scheme" shows "finite_group G \<equiv> Group.group G \<and> finite_group_axioms G"
up_eq: fixes x :: "'c"   and y :: "'c" shows "(up\<cdot>x = up\<cdot>y) = (x = y)"
commute: fixes a :: "'c"   and b :: "'c" shows "normalization_euclidean_semiring_class.gcd a b = normalization_euclidean_semiring_class.gcd b a"
seqE: fixes g :: "'a"   and f :: "'a" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
lemma_st_part_Ex: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<exists>t\<in>\<real>. \<forall>r\<in>\<real>. 0 < r \<longrightarrow> \<bar>x - t\<bar> < r"
elims: fixes x :: "'a Trace"   and y :: "nat" assumes "tLength x = y" obtains s :: "'a" where "x = tInit s__" and "y = 0"   | t :: "'a Trace" and s :: "'a" where "x = t \<leadsto> s" and "y = 1 + tLength t"
l_r: fixes n :: "nat" shows "l n \<le> r n"
trans_R: fixes x :: "'x"   and y :: "'x"   and z :: "'x" assumes "R x y"   and "R y z" shows "R x z"
proots_count_pCons_1_iff: fixes a :: "'a"   and s :: "'a set" shows "proots_count [:- a, 1::'a:] s = (if a \<in> s then 1 else 0)"
induced_modulus_def: fixes d :: "nat" shows "induced_modulus d = (d dvd n \<and> (\<forall>a. coprime a n \<and> [a = 1] (mod d) \<longrightarrow> \<chi> a = 1))"
Cardano_2: fixes y :: "real" assumes "discriminant_cubic_depressed e f \<noteq> 0"   and "y ^ 3 + e * y + f = 0" shows "y = root 3 u - e / (3 * root 3 u)"
llist_all2_cons1_def: fixes x :: "'a"   and sg' :: "'sg"   and sh :: "'sh" shows "local.llist_all2_cons1 x sg' sh = llist_all2 P (LCons x (lunstream g sg')) (lunstream h sh)"
hensel_fpxs1_def: fixes g :: "'b poly"   and h :: "'b poly"   and F :: "'b poly fps" assumes "fps_hensel_aux g h" shows "fps_hensel_aux.hensel_fpxs1 F g h = Abs_fps (fst \<circ> fps_hensel_aux.hensel_fpxs_aux F g h)"
ord_eq_less_subst: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a = f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
module_hom_ident: shows "Vector_Spaces.linear (*s) (*s) (\<lambda>x. x)"
undef_cong_mat: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and x :: "nat \<times> nat" assumes "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "undef_mat nr nc f x = undef_mat nr nc f' x"
assms: shows "z -\<^sub>h w = \<infinity>\<^sub>h"
col_minor: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "col (minor M i j) j = axis i (1::'c)"
mult_neg_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less a zero"   and "less b zero" shows "less zero (times a b)"
LNil: fixes xs :: "'a llist" assumes "lnull xs" shows "P xs"
epm_8: fixes w :: "'b"   and v :: "'b"   and e :: "'b" assumes "stone_algebra_class.comp_inf.acyclic w" shows "w \<sqinter> - stone_algebra_class.comp_inf.prim_EP w v e \<sqinter> id (stone_algebra_class.comp_inf.prim_P w v e) = bot"
module_def: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme" shows "Module.module R M \<equiv> cring R \<and> abelian_group M \<and> module_axioms R M"