hcju/isabelleps huggingface dataset:
ji: shows "\<forall>x\<in>rvars (\<T> (l ! (j - 1))). \<langle>\<V> (l ! (j - 1))\<rangle> x = \<langle>\<V> (l ! i)\<rangle> x \<or> \<langle>\<V> (l ! (j - 1))\<rangle> x = the (\<B>\<^sub>l (l ! i) x) \<or> \<langle>\<V> (l ! (j - 1))\<rangle> x = the (\<B>\<^sub>u (l ! i) x)"
Src_Dom: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Src src (bicategorical_language.Dom V src trg t) = bicategorical_language.Src src t"
delta_pos_neg_least_1: fixes F :: "(('a \<times> 'b) admS \<times> ('c \<times> 'd) admS) dual \<times> ('a \<times> 'b) admS \<times> ('c \<times> 'd) admS \<Rightarrow> ('a \<times> 'b) admS \<times> ('c \<times> 'd) admS"   and ad :: "('a \<rightarrow> 'a) \<times> ('c \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'a) \<times> ('c \<rightarrow> 'c)"   and bd :: "('b \<rightarrow> 'b) \<times> ('d \<rightarrow> 'd) \<rightarrow> ('b \<rightarrow> 'b) \<times> ('d \<rightarrow> 'd)"   and rm :: "('a \<times> 'b) admS \<times> ('c \<times> 'd) admS"   and rp :: "('a \<times> 'b) admS \<times> ('c \<times> 'd) admS" assumes "DomSolP F ad bd"   and "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "DomSolP.delta_neg F \<le> dual rm"
swap_apply_1: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "Fun.swap a b f a = f b"
simps_5: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp" shows "vars (Div a\<^sub>1 a\<^sub>2) = vars a\<^sub>1 \<union> vars a\<^sub>2"
Stuck_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Stuck" shows "t = Stuck"
finite_supp_unique: fixes S :: "atom set"   and x :: "'a" assumes "S supports x"   and "finite S"   and "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x" shows "supp x = S"
assms_1: shows "compare_expansions C G = (EQ, c, c')"
filters_def: shows "filters = {X. X \<noteq> {} \<and> upset_setp X \<and> filtered X}"
9: shows "F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot"
qSwapInp_def: fixes xs :: "'i"   and x :: "'j"   and y :: "'j"   and qinp :: "'f \<Rightarrow> ('g, 'h, 'i, 'j, 'k) qTerm option" shows "qSwapInp xs x y qinp \<equiv> lift (qSwap xs x y) qinp"
ACI_norm_lderiv: fixes as :: "'a"   and r :: "'b rexp" shows "\<guillemotleft>lderiv as \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>lderiv as r\<guillemotright>"
satisfies_tableau_cong: fixes t :: "(nat \<times> linear_poly) list"   and v :: "nat \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'a" assumes "\<And>x. x \<in> tvars t \<Longrightarrow> v x = w x" shows "v \<Turnstile>\<^sub>t t = w \<Turnstile>\<^sub>t t"
i_drop_Suc_conv_tl: fixes f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "[f n] \<frown> f \<Up> Suc n = f \<Up> n"
init_imp_inv_imps_work_sum: fixes c :: "('loc, 't) Propagate.configuration" assumes "local.init_config c" shows "inv_imps_work_sum c"
joinable_simp: fixes x :: "'c"   and y :: "'c"   and R :: "('c \<times> 'd) set" shows "((x, y) \<in> joinable R) = (\<exists>z. (x, z) \<in> R \<and> (y, z) \<in> R)"
mem_ball_0: fixes x :: "'c"   and e :: "real" shows "(x \<in> ball (0::'c) e) = (norm x < e)"
invertible_inverse_invertible: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "additive.invertible (- u)"
A: shows "A \<equiv> P' \<inter> Ms \<union> Pw' \<inter> Ms \<union> R' \<union> Rw'"
least_le: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set"   and a :: "'a" assumes "least L x A"   and "a \<in> A" shows "x \<sqsubseteq>\<^bsub>L\<^esub> a"
run_bind_writer: fixes return :: "'c \<times> 'b list \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> ('c \<times> 'b list \<Rightarrow> 'a) \<Rightarrow> 'a"   and m :: "('b, 'c, 'a) writerT"   and f :: "'c \<Rightarrow> ('b, 'c, 'a) writerT" shows "run_writer (bind_writer return bind m f) = bind (run_writer m) (\<lambda>(a, ws). bind (run_writer (f a)) (\<lambda>(b, ws'). return (b, ws @ ws')))"
poly_to_row_index: fixes i :: "nat"   and ts :: "'c list"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "i < length ts" shows "poly_to_row ts p $ i = lookup p (ts ! i)"
midpoint_eq_endpoint_2: fixes a :: "'a"   and b :: "'a" shows "(midpoint a b = b) = (a = b)"
poincare_line_0_imag_is_y_axis: fixes y :: "complex_homo" assumes "y \<in> circline_set y_axis"   and "y \<noteq> 0\<^sub>h"   and "y \<noteq> \<infinity>\<^sub>h" shows "poincare_line 0\<^sub>h y = y_axis"
cSup_eq_maximum: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 z :: "'b"   and X :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "Sup X = z"
find_non0_all0: fixes l :: "nat"   and A :: "'a mat" assumes "local.find_non0 l A = None" shows "local.triangular_column l A"
intros_1: fixes \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "\<psi> \<omega>" shows "(\<phi> suntil \<psi>) \<omega>"
stutter_extend_simps_3: fixes G :: "('a, 'b) graph_rec_scheme" shows "g_V0 (stutter_extend G) = g_V0 G"
Predict_subset_Gen: fixes I :: "item set"   and P :: "(symbol \<times> character list) list set"   and k :: "nat" assumes "I \<subseteq> Gen P"   and "k \<le> length Doc" shows "Predict k I \<subseteq> Gen P"
Diff_disjoint: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> (B - A) = {}"
fun_mult_eval_assoc: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F"   and "a \<in> S" shows "(x \<otimes>\<^bsub>F\<^esub> y \<otimes>\<^bsub>F\<^esub> z) a = (x \<otimes>\<^bsub>F\<^esub> (y \<otimes>\<^bsub>F\<^esub> z)) a"
higher_deriv_comp_iter_lemma: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and i :: "nat"   and n :: "nat"   and m :: "nat" assumes "open s"   and "f holomorphic_on s"   and "f ` s \<subseteq> s"   and "z \<in> s"   and "f z = z"   and "i \<le> n"   and "deriv f z = 1"   and "\<And>i. \<lbrakk>1 < i; i \<le> n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0" shows "(deriv ^^ i) (f ^^ m) z = (deriv ^^ i) f z"
singletonD: fixes b :: "'b"   and a :: "'b" assumes "b \<in> {a}" shows "b = a"
axioms_1: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_pair_1_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1"
x: fixes x :: "'a" assumes "x \<in> A" shows "f x \<le> g x"
size_mset_singleton: fixes l :: "'c"   and r :: "'d" shows "size_mset {#(l, r)#} = size l + size r + 1"
Inf_from_def: fixes N :: "'f set" shows "no_labels.Inf_from N = {\<iota> \<in> Inf_F. set (prems_of \<iota>) \<subseteq> N}"
Process_eq_spec_optimized_seq: fixes P :: "'a process"   and Q :: "'a process"   and U :: "'a process"   and S :: "'a process" shows "((P `;` Q) = (U `;` S)) = (D (P `;` Q) = D (U `;` S) \<and> F_minus_D_seq P Q \<subseteq> F (U `;` S) \<and> F_minus_D_seq U S \<subseteq> F (P `;` Q))"
mbuf2t_take_eqD_2: fixes f :: "'b option list set \<Rightarrow> 'b option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "'b option list set list \<times> 'b option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "'b option list set list \<times> 'b option list set list"   and nts' :: "nat list"   and i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'b option list set \<Rightarrow> bool"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat" assumes "mbuf2t_take f z buf nts = (z', buf', nts')"   and "wf_mbuf2 i ja jb P Q buf"   and "list_all2 R [i..<j] nts"   and "ja \<le> j"   and "jb \<le> j" shows "list_all2 R [min ja jb..<j] nts'"
cut_same: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A"   and "insert A H \<turnstile> B" shows "H \<turnstile> B"
conj_chaos: fixes a :: "'b::refinement_lattice" shows "(a::'b::refinement_lattice) \<iinter> chaos = a"
pk: fixes z :: "real \<times> 'c" assumes "z \<in> {0..1} \<times> {undefined}" shows "p (k z) = (g \<circ> fst) z"
Ex1_def: fixes P :: "'a \<Rightarrow> bool" shows "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"
assms_4: shows "e2 \<in> out_arcs G v"
Call_invariant: fixes G :: "'a triple set"   and P :: "'a assn"   and pn :: "pname"   and Q :: "'a assn"   and s' :: "state"   and X :: "vname"   and I :: "'a \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> bool"   and k :: "'a \<Rightarrow> nat"   and a :: "state \<Rightarrow> nat" assumes "G|-{P}. BODY pn .{\<lambda>Z s. Q Z (setlocs s (loc s')[X::=s<Res>])}" shows "G|-{\<lambda>Z s. s' = s \<and> I Z (loc (s[X::=k Z])) \<and> P Z (setlocs s newlocs[Loc Arg::=a s])}. X:=CALL pn(a) .{\<lambda>Z s. I Z (loc (s[X::=k Z])) \<and> Q Z s}"
mapOf_lookup2: fixes i :: "int"   and t :: "(int \<times> 'b) Tree"   and j :: "int"   and a :: "'b" assumes "tlookup fst i t = Some (j, a)" shows "mapOf t i = Some a"
simps_12: shows "OUTfromV data12 = {stA2}"
size_last_index_conv: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "length xs = n" shows "(n = last_index xs x) = (x \<notin> set xs)"
HOLCF_trans_rules_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
line_suitable_junctors_3: fixes \<A> :: "'b set"   and F :: "'b formula"   and G :: "'b formula" shows "line_suitable \<A> (F \<^bold>\<or> G) = (line_suitable \<A> F \<and> line_suitable \<A> G)"
bi_unique_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique R" shows "bi_unique (list_all2 R)"
iso_iff_section_and_epi: fixes f :: "'s" shows "local.iso f = (section f \<and> epi f)"
inv_closed: fixes x :: "(((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))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
succ_inject_iff: fixes i :: "hf"   and j :: "hf" shows "(succ i = succ j) = (i = j)"
r_1: fixes l :: "nat"   and n :: "nat" assumes "l \<le> r n" shows "u l \<le> u (r n)"
ordIso_refl: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r =o r"
card_UNIV_uprod_def: shows "card_UNIV_class.card_UNIV = (let n = of_phantom card_UNIV_class.card_UNIV in Phantom('a uprod) (n * (n + 1) div 2))"
sorted_list_of_set_greaterThanLessThan: fixes i :: "nat"   and j :: "nat" assumes "Suc i < j" shows "sorted_list_of_set {i<..<j} = Suc i # sorted_list_of_set {Suc i<..<j}"
elims: fixes x :: "Core.term"   and xa :: "Core.term"   and y :: "Core.term" assumes "x \<bullet> xa = y" obtains uu_ :: "typ" and t :: "Core.term" and u :: "Core.term" where "x = Abs uu_ t" and "xa = u__" and "y = subst_bv u__ t"   | v :: "String.literal" and va :: "typ" and u :: "Core.term" where "x = Ct v__ va__" and "xa = ua__" and "y = Ct v__ va__ $ ua__"   | v :: "variable" and va :: "typ" and u :: "Core.term" where "x = Fv vb__ vaa__" and "xa = ub__" and "y = Fv vb__ vaa__ $ ub__"   | v :: "nat" and u :: "Core.term" where "x = Bv vc__" and "xa = uc__" and "y = Bv vc__ $ uc__"   | v :: "Core.term" and va :: "Core.term" and u :: "Core.term" where "x = v $ va" and "xa = u" and "y = v $ va $ u"
lin_redE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lin_red F p q" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" where "f \<in> F" and "red_single p q f (0::'a)"
le_mask_shiftl_le_mask: fixes s :: "nat"   and m :: "nat"   and n :: "nat"   and x :: "'b word" assumes "s = m + n"   and "x \<le> mask n" shows "x << m \<le> mask s"
iterate_is_empty_correct: fixes it :: "(bool \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> bool \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool"   and S0 :: "'c set" assumes "set_iterator it S0" shows "iterate_is_empty it = (S0 = {})"
expand: fixes gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" assumes "the_gpv gpv = the_gpv gpv'" shows "gpv = gpv'"
dd_un: shows "DD \<union> set_mset (old_side_prems_of \<gamma>) \<TTurnstile>e {old_concl_of \<gamma>}"
rep_eq: fixes x :: "nat"   and xa :: "uint" shows "Rep_uint (push_bit x xa) = push_bit x (Rep_uint xa)"
ccSup_subset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'a set"   and A :: "'a set" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable B"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
finite_vars_term: fixes t :: "('d, 'c) Term.term" shows "finite (vars_term t)"
execn_to_execn_strip': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "t \<notin> Fault ` F" shows "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
preserves_seq: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.seq (src\<^sub>C g) (src\<^sub>C f)"
same_length_different: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
BlockRedsFinal: fixes e\<^sub>0 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "final e\<^sub>2"   and "wf_prog wf_md P"   and "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None))" shows "P,E \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l\<^sub>0 V))\<rangle>"
Obj_Trg: fixes t :: "'c local.term" assumes "Arr t" shows "Obj (Trg t)"
assms: shows "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q"   and "\<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q" shows "Prop (P' \<parallel> Q)"
ord0_leq_trans: fixes x :: "ord0"   and y :: "ord0"   and z :: "ord0" assumes "(x, y) \<in> ord0_leq"   and "(y, z) \<in> ord0_leq" shows "(x, z) \<in> ord0_leq"
expectation_gpv_mono_strong: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and fail :: "ennreal"   and fail' :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "\<not> colossless_gpv \<I> gpv \<Longrightarrow> fail \<le> fail'"   and "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> f x \<le> g x" shows "expectation_gpv fail \<I> f gpv \<le> expectation_gpv fail' \<I> g gpv"
refinement1: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and I :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c" assumes "\<turnstile> P \<longrightarrow> Q"   and "|~ I \<and> \<circle>I \<and> _actrans A f \<longrightarrow> _actrans B g" shows "\<turnstile> P \<and> \<box>I \<and> \<box>[A]_f \<longrightarrow> Q \<and> \<box>[B]_g"
id_stake_snth_sdrop: fixes s :: "'a stream"   and i :: "nat" shows "s = stake i s @- s !! i ## sdrop (Suc i) s"
union_iff: fixes a :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" shows "(a \<in># A + B) = (a \<in># A \<or> a \<in># B)"
seq_reachable_in_cterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('f \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and \<xi> :: "'f"   and p :: "('f, 'g, 'e, 'h) seqp"   and I :: "'g seq_action \<Rightarrow> bool"   and p' :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = seqp_sos \<Gamma>"   and "(\<xi>, p) \<in> reachable A I"   and "p' \<in> sterms \<Gamma> p" shows "p' \<in> cterms \<Gamma>"
tiny_dghm_ArrMap_in_Vset: shows "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
sel_2: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "the_Fun (Fun x21 x22) = x21"
principal_subs_def: fixes As :: "'a set set"   and A :: "'a set" shows "principal_subs As A = Abs_filter (\<lambda>P. \<forall>x. x \<in> As \<and> x \<subseteq> A \<longrightarrow> P x)"
char_poly_matrix_def: fixes A :: "'a mat" shows "char_poly_matrix A = [:0::'a, 1::'a:] \<cdot>\<^sub>m 1\<^sub>m (dim_row A) + map_mat (\<lambda>a. [:- a:]) A"
assms: shows "((\<sigma>, NodeS i p R), a, \<sigma>', s') \<in> onode_sos S"
assms_1: shows "r \<in> local.mk_linear_orders C B"
A: shows "mC_C theta c1 c s1 s (lift P1 F1 ` P) (inv_into P (lift P1 F1))"
select_convs_2: fixes edge_rev :: "'c \<Rightarrow> 'c"   and edge_succ :: "'c \<Rightarrow> 'c"   and more :: "'d" shows "edge_succ \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_succ"
hyps: shows "ModPon y z x \<or> Exists y x \<or> Subst y x"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>A. g x = zero" shows "aggregation.sum_0 g A = zero"
b_least2_less_impl_eq: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "b_least2 f x y < y" shows "b_least2 f x y = (LEAST z. f x z \<noteq> 0)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "R.comp g f = (if right a f \<and> right a g \<and> cod f = local.dom g then g \<cdot> f else null)"
evaluate_match_total0: fixes pes :: "(pat \<times> exp0) list"   and s :: "'b state"   and env :: "v sem_env"   and v :: "v"   and v' :: "v" assumes "\<And>p e env s'. \<lbrakk>(p, e) \<in> set pes; clock s' \<le> clock s\<rbrakk> \<Longrightarrow> \<exists>s'' r. evaluate True env s' e (s'', r)" shows "\<exists>s' r. evaluate_match True env s v pes v' (s', r)"
angle_action_def: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<diamond>\<langle>P\<rangle>_v \<equiv> \<not> \<box>[\<not> P]_v"
mset_subset_eq_mono_add_left_cancel: fixes C :: "'b multiset"   and A :: "'b multiset"   and B :: "'b multiset" shows "(C + A \<subseteq># C + B) = (A \<subseteq># B)"
xs: shows "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) [Monitor.progress \<sigma> \<phi> j..<Monitor.progress \<sigma> \<phi> j'] xs"
remdups_adj_Nil_iff: fixes xs :: "'b list" shows "(remdups_adj xs = []) = (xs = [])"
measure_le_1: fixes X :: "'a set" shows "emeasure M X \<le> 1"
assms_2: shows "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
fA: shows "f (\<Sqinter>\<^bsub>L\<^esub>A) \<in> fps L f"
corec_sel_1: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'b"   and g21 :: "'a \<Rightarrow> 'c"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('c, 'b) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "p a" shows "terminal (corec_tllist p g1 g21 q22 g221 g222 a) = g1 a"
univ_poly_is_ring: fixes K :: "'a set" assumes "subring K R" shows "ring (K [X])"
inf_compl_bot: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (uminus x) = bot"