hcju/isabelleps huggingface dataset:
snd_Sup: fixes A :: "('d \<times> 'c) set" shows "snd (Sup A) = Sup (snd ` A)"
c: shows "Ide c \<and> Arr c \<and> Dom c = c \<and> Cod c = c \<and> ide \<lbrace>c\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>c\<^bold>\<rfloor>\<rbrace> \<and> \<guillemotleft>\<lbrace>c\<^bold>\<down>\<rbrace> : \<lbrace>c\<rbrace> \<rightarrow> \<lbrace>\<^bold>\<lfloor>c\<^bold>\<rfloor>\<rbrace>\<guillemotright>"
pell_power_Suc: fixes z :: "'a \<times> 'a"   and n :: "nat" shows "pell_power z (Suc n) = pell_mul z (pell_power z n)"
can_Ide_self: fixes t :: "'a local.term" assumes "Ide t" shows "can t t = \<lbrace>t\<rbrace>"
even_or_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (or a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<and> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
listsum_dims: fixes As :: "'b tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "dims (listsum ds As) = ds"
vars_uminus: fixes p :: "'b mpoly" shows "vars (- p) = vars p"
intro: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')" shows "l_set_child_nodes_get_disconnected_nodes_axioms set_child_nodes_locs get_disconnected_nodes_locs"
succs: fixes g' :: "'c"   and succs :: "'c \<Rightarrow> 'c list"   and g :: "'c"   and g'' :: "'c" assumes "g' \<in> set (succs g)"   and "RTranClp succs g' g''" shows "RTranClp succs g g''"
bst_mrbst_push_down: fixes t' :: "'a tree"   and t1 :: "'a tree"   and x :: "'a"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_push_down t1 x t2)"   and "bst t1"   and "bst t2"   and "\<And>y. y \<in> set_tree t1 \<Longrightarrow> y < x"   and "\<And>y. y \<in> set_tree t2 \<Longrightarrow> x < y" shows "bst t'"
id_is_gate: fixes n :: "nat" shows "gate n (Quantum.Id n)"
fin_s: shows "finite (next_marker_free_state t i ` {cid. channel cid \<noteq> None})"
"1_2": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "\<not> n < 0"   and "even n"   and "odd a" shows "Jacobi a (n div 2) = jacobi_code a (n div 2)"
bij_betw_empty1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set" assumes "bij_betw f {} A" shows "A = {}"
SUP_upper2: 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 i :: "'b"   and A :: "'b set"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "s2p A i"   and "less_eq u (f i)" shows "less_eq u (Sup (\<P> f A))"
mbta_MonoTran_disjunctive: shows "Rep_MonoTran ` mbt_algebra_class.disjunctive = Apply.disjunctive"
assms_2: shows "subgroup I (G\<lparr>carrier := J\<rparr>)"
inj_eq_image: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and Y :: "'b set" assumes "inj f"   and "f ` X = f ` Y" shows "X = Y"
fresh_variable_type_schemes: fixes sch :: "type_scheme" shows "\<exists>n. new_tv n sch"
lang_dual_lderiv_dual: fixes n :: "nat"   and r :: "'b rexp_dual"   and w :: "'a" assumes "wf_dual n r"   and "w \<in> \<Sigma> n" shows "lang_dual n (lderiv_dual w r) = lQuot w (lang_dual n r)"
neqE: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
abs_eq: fixes x :: "char list" shows "implode x = Abs_literal (map ascii_of x)"
simps_2: shows "op_sem Mult = lift_RealIntVal2 (*) (*)"
char_to_digit_simps_4: shows "char_to_digit CHR ''3'' = 3"
ASSFMT_2: shows "W' = W - {q} \<union> (br_dsq \<delta> q (Q, W) - Q)"
LeadsTo_Trans: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "F \<in> A \<longmapsto>w B"   and "F \<in> B \<longmapsto>w C" shows "F \<in> A \<longmapsto>w C"
F_def: shows "F \<equiv> \<lambda>n. \<Sum>k = 1..n. f k n"
ide_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "ide (local.dom f)"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "span_bicategory C prj0 prj1"
preserves_hom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.CC.in_hom f a b" shows "C.CC.in_hom (C.CC.map f) (C.CC.map a) (C.CC.map b)"
norm_triangle_sub: 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 x :: "'c"   and y :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm x \<le> norm y + norm (minus x y)"
dbl_inc_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one x = plus (plus x x) one"
sfv_def: fixes s :: "form list" shows "sfv s = flatten (map fv s)"
linord_of_list_Linear_order: fixes xs :: "'d list"   and ys :: "'d set" assumes "distinct xs"   and "ys = set xs" shows "linear_order_on ys (linord_of_list xs)"
indRelRST_to_SRel: fixes SP :: "'procS"   and SQ :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "SourceTerm SP \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> SourceTerm SQ" shows "(SP, SQ) \<in> SRel"
hom_add_eq_zero: fixes x :: "rat"   and y :: "rat" assumes "x + y = 0" shows "of_rat x + of_rat y = (0::'a)"
sup_ge1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) ((sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) x (y::'b))"
monom_Suc: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "Q (monom P a (Suc n))"
assms_2: shows "reconstruct_poly a as = reconstruct_poly a bs"
simps_2: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and ky :: "'a"   and vy :: "'e"   and ys :: "('a \<times> 'e) list" shows "prod_ord_pair f [] ((ky, vy) # ys) = (f ky (0::'d) vy \<and> prod_ord_pair f [] ys)"
FAss: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and h\<^sub>2' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "fs' = fs((F, D) \<mapsto> v)"   and "h\<^sub>2' = h\<^sub>2(a \<mapsto> (C, fs'))" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h\<^sub>2', l\<^sub>2)\<rangle>"
integral_iff: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier (UP R)"   and "b \<in> carrier (UP R)" shows "(a \<otimes>\<^bsub>UP R\<^esub> b = \<zero>\<^bsub>UP R\<^esub>) = (a = \<zero>\<^bsub>UP R\<^esub> \<or> b = \<zero>\<^bsub>UP R\<^esub>)"
abs_triangle_ineq3: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus (abs a) (abs b))) (abs (minus a b))"
Minimal_Truth_Soundness_Proof_Repr_def: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and isTrue :: "'f \<Rightarrow> bool"   and enc :: "'f \<Rightarrow> 'e"   and "proof" :: "'g set"   and prfOf :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and encPf :: "'g \<Rightarrow> 'e"   and Pf :: "'f" shows "Minimal_Truth_Soundness_Proof_Repr var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv isTrue enc proof prfOf encPf Pf \<equiv> CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf \<and> Minimal_Truth_Soundness var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num bprv isTrue"
final_finished: fixes s :: "'state1" assumes "final1 s" shows "finished step1 s"
result_def: fixes resumption :: "('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.result resumption = (case resumption of Resumption.resumption.Done x1 \<Rightarrow> x1)"
sets_measure_of: fixes A :: "'b set set"   and \<Omega> :: "'b set"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
mult_poly_0_left: fixes q :: "'b poly" shows "0 * q = 0"
labeled_lifting_intersection_axioms: shows "labeled_lifting_intersection Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL"
uminus: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "O[F](\<lambda>x. - g x) = O[F](g)"
pr1_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and \<delta> :: "'b \<Rightarrow> 'b"   and \<tau> :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "cartesian_monoidal_category C T \<alpha> \<iota> \<delta> \<tau>" shows "cartesian_monoidal_category.pr1 C T \<alpha> \<iota> \<tau> a b \<equiv> if partial_magma.ide C a \<and> partial_magma.ide C b then C (monoidal_category.lunit C T \<alpha> \<iota> b) (T (\<tau> a, b)) else partial_magma.null C"
simps: fixes p :: "nat"   and x :: "char list \<Rightarrow> char list"   and y :: "char list \<Rightarrow> char list" shows "pshowsp_prod p (x, y) = shows_string ''('' \<circ> x \<circ> shows_string '', '' \<circ> y \<circ> shows_string '')''"
simps_10: fixes v :: "'b"   and uu :: "'b ltln set" shows "prop\<^sub>n(v)[uu]\<^sub>\<mu> = prop\<^sub>n(v)"
cases: fixes a1 :: "'f"   and a2 :: "'f multiset"   and a3 :: "'g el_step"   and a4 :: "'f"   and a5 :: "'f multiset"   and S :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set" assumes "((a1, a2), a3, a4, a5) \<in> gtrp S" obtains s :: "'f" and c :: "'f multiset" and e :: "'g" and s' :: "'f" and c' :: "'f multiset" where "a1 = s__" and "a2 = cb__" and "a3 = LOC ea__" and "a4 = s'__" and "a5 = c'b__" and "((s__, cb__), ea__, s'__, c'b__) \<in> S"   | s :: "'f" and sl :: "'f" and c :: "'f multiset" and e :: "'g" and s' :: "'f" and c' :: "'f multiset" where "a1 = sl" and "a2 = add_mset s c" and "a3 = ENV e" and "a4 = sl" and "a5 = add_mset s' c'" and "((s, add_mset sl c), e, s', add_mset sl c') \<in> S"
suminf_zero: shows "(\<Sum>n. (0::'a)) = (0::'a)"
NSLIM_minus: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and L :: "'b" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "(\<lambda>x. - f x) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S - L"
top_left_mult_increasing: fixes x :: "'c" shows "x \<le> top \<sqinter> x"
A_def: shows "A = {n. nat_to_ce_set n \<in> C}"
discI: fixes interval_tree :: "interval_tree"   and x21 :: "interval_tree"   and x22 :: "nat idx_interval"   and x23 :: "nat"   and x24 :: "interval_tree" assumes "interval_tree = Node x21 x22 x23 x24" shows "interval_tree \<noteq> Tip"
wfG: shows "wf_graph \<lparr>nodes = V, edges = insert e E\<rparr>"
tarski_first5_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_first5 C B \<equiv> tarski_first3 C \<and> tarski_first5_axioms C B"
vec_setI: fixes v :: "'c vec"   and i :: "nat"   and a :: "'c" assumes "v $ i = a"   and "i < dim_vec v" shows "a \<in> set\<^sub>v v"
execn_sequence_flatten_to_execn: 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" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten c),s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
auxlist0_Nil: assumes "auxlist0 = []" shows "ne = 0 \<or> ne \<noteq> 0 \<and> (\<forall>t. t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> ne - t) \<le> right I \<longrightarrow> (\<nexists>i. \<tau> \<sigma> i = t))"
y: shows "\<bar>y\<bar> \<noteq> \<infinity>"
card_of_Times_cong: fixes A :: "'e set"   and B :: "'f set"   and C :: "'g set"   and D :: "'h set" assumes "|A| =o |B|"   and "|C| =o |D|" shows "|A \<times> C| =o |B \<times> D|"
segments_of_aform_def: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "segments_of_aform X = (let hs = half_segments_of_aform X in hs @ map (pairself (mirror_point (fst X))) hs)"
antecedent: shows "\<forall>w. P w z = (P w x \<and> \<not> O w y)"
inits_neq_Nil: fixes xs :: "['b]" shows "inits\<cdot>xs \<noteq> []"
perm_cont_simp: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<pi> :: "perm"   and x :: "'c"   and y :: "'c" assumes "class.cont_pt permute below" shows "(\<pi> \<bullet> x \<sqsubseteq> \<pi> \<bullet> y) = (x \<sqsubseteq> y)"
no_marker_after_RecvMarker: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "S t i \<turnstile> RecvMarker cid p q \<mapsto> S t (i + 1)"   and "channel cid = Some (q, p)" shows "Marker \<notin> set (msgs (S t (i + 1)) cid)"
invertibleD_1: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "invertible A" shows "matrix_inv A ** A = mat (1::'a)"
V0'_to_V0: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and v :: "'c"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd" assumes "g_rename_precond G f"   and "v \<in> g_V0 (fr_rename_ext ecnv f G)" shows "g_rename_precond.fi G f v \<in> g_V0 G"
Gromov_product_le_dist_2: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "Gromov_product_at e x y \<le> dist e y"
ab: shows "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
cons_conv_snoc: fixes x :: "'a"   and xs :: "'a list" shows "\<exists>ys y. x # xs = ys @ [y]"
simps: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list" shows "(Graph V \<Lambda> E)^\<Lambda> = \<Lambda>"
satA_snvA_transE: fixes at :: "('fsym, 'psym) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtA at"   and "I.wtE \<xi>"   and "\<And>\<sigma>. infTp \<sigma> \<or> (\<forall>x\<in>nvA at. tpOfV x \<noteq> \<sigma>)" shows "Ik.satA (transE \<xi>) at = I.satA \<xi> at"
cod_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" shows "CC.cod (CC.comp g f) = CC.cod g"
key_in_map1: fixes k :: "'a"   and M' :: "'a \<Rightarrow> 'b option"   and M :: "'a \<Rightarrow> 'b option" assumes "k \<notin> dom M'" shows "(M ++ M') k = M k"
Rep_zet_eq_explode: fixes A :: "ZF zet" shows "\<exists>z. Rep_zet A = explode z"
eq_act: shows "vpeq (Step.partition (current s1)) s1 s2"
tauFreshChainDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "xvec \<sharp>* P" shows "xvec \<sharp>* P'"
closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<UU>"   and "b \<in> \<UU>" shows "f a b \<in> \<UU>"
mini: assumes "distinct_indices cs" shows "distinct_indices_ns (set (to_ns cs))"
K_finite: fixes s :: "'s" assumes "s \<in> S" shows "finite (K s)"
UP_def: fixes x :: "'e" shows "SC.UP x \<equiv> concrete_category.MkArr {x} {x} (\<lambda>x\<in>{x}. x)"
convex_hull_singleton: fixes a :: "'a" shows "convex hull {a} = {a}"
cases: fixes a :: "cmsg" assumes "a \<in> cpayload" obtains A :: "agent" where "a = cAgent A"   | T :: "nat" where "a = cNumber T"   | N :: "nonce_t" where "a = cNonce N"   | K :: "ephkey" where "a = cEphK K"   | X :: "cmsg" where "a = cHash Xa__" and "Xa__ \<in> cpayload"   | X :: "cmsg" and Y :: "cmsg" where "a = cPair Xb__ Ya__" and "Xb__ \<in> cpayload" and "Ya__ \<in> cpayload"   | X :: "cmsg" and Y :: "cmsg" where "a = cEnc Xc__ Yb__" and "Xc__ \<in> cpayload" and "Yb__ \<in> cpayload"   | X :: "cmsg" and Y :: "cmsg" where "a = cAenc Xd__ Yc__" and "Xd__ \<in> cpayload" and "Yc__ \<in> cpayload"   | X :: "cmsg" and Y :: "cmsg" where "a = cSign Xe__ Yd__" and "Xe__ \<in> cpayload" and "Yd__ \<in> cpayload"   | X :: "cmsg" and Y :: "cmsg" where "a = cExp X Y" and "X \<in> cpayload" and "Y \<in> cpayload"
det3_switch: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "det3 a b c = - det3 a c b"
pp_prod_def: fixes X :: "'b ppath set"   and Y :: "'b ppath set" shows "X ; Y = {pp_fusion u v |u v. u \<in> X \<and> v \<in> Y \<and> pp_last u = pp_first v}"
cycle_iff: fixes r :: "'a"   and x :: "'a" assumes "point r" shows "(x ; r \<noteq> 0) = (r \<le> x\<^sup>T ; 1)"
card_inj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "finite B" shows "card A \<le> card B"
match_iface_case_wildcard_prefix: fixes i :: "char list"   and p_i :: "char list" assumes "iface_name_is_wildcard i" shows "match_iface (Iface i) p_i = (butlast i = take (length i - 1) p_i)"
shows_prec_unit_def: fixes d :: "nat"   and x :: "unit" shows "shows_prec d x = shows_string ''()''"
ND_lem: fixes F :: "'b formula" shows "{} \<turnstile> \<^bold>\<not> F \<^bold>\<or> F"
asprod_minus: fixes x :: "ant"   and z :: "int" assumes "x \<noteq> - \<infinity> \<and> x \<noteq> \<infinity>" shows "- z *\<^sub>a x = z *\<^sub>a (- x)"
dom_eqI: fixes a :: "'a2"   and f :: "'a2" assumes "C2.ide a"   and "C2.seq f a" shows "C2.dom f = a"
SN_rel_ext_trans: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set"   and M :: "'a \<Rightarrow> bool" assumes "SN_rel_ext P Pw R Rw M" shows "SN_rel_ext (((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>* O (P \<inter> {(s, t). M t}) O ((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>*) (((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>* O ((P \<union> Pw) \<inter> {(s, t). M t}) O ((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>*) (((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>* O ((P \<union> R) \<inter> {(s, t). M t}) O ((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>*) (((P \<union> Pw \<union> R \<union> Rw) \<inter> {(s, t). M t})\<^sup>*) M"
integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'b" assumes "integrable (count_space UNIV) f" shows "integral\<^sup>L (count_space UNIV) f = (\<Sum>x. f x)"
strong_exhaust: fixes y :: "dbfm" obtains x11 :: "dbtm" and x12 :: "dbtm" where "y = DBMem x11 x12"   | x21 :: "dbtm" and x22 :: "dbtm" where "y = DBEq x21 x22"   | x31 :: "dbfm" and x32 :: "dbfm" where "y = DBDisj x31 x32"   | x4 :: "dbfm" where "y = DBNeg x4"   | x5 :: "dbfm" where "y = DBEx x5"
image_mset_Diff: fixes B :: "'k multiset"   and A :: "'k multiset"   and f :: "'k \<Rightarrow> 'l" assumes "B \<subseteq># A" shows "image_mset f (A - B) = image_mset f A - image_mset f B"
that: fixes f1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "\<forall>x1 x2. P x1 x2 (f1 x1 x2) (f2 x1 x2)" shows "thesis"
valid_W_distinct_simps_1: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name"   and q :: "'g process_name" assumes "r \<in> ghost_honorary_grey (s p)"   and "valid_W_inv s" shows "(r \<in> ghost_honorary_grey (s q)) = (p = q)"
amalgamation_def: fixes i :: "nat"   and j :: "nat"   and x :: "nat" shows "amalgamation i j x \<equiv> if parallel i j x \<up> then None else Some (pdec2 (the (parallel i j x)))"
fun_cong: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and x :: "'b" assumes "f = g" shows "f x = g x"