hcju/isabelleps huggingface dataset:
iterateoi_rule_P: fixes m :: "'s"   and I :: "'u set \<Rightarrow> '\<sigma> \<Rightarrow> bool"   and \<sigma>0 :: "'\<sigma>"   and c :: "'\<sigma> \<Rightarrow> bool"   and f :: "'u \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>"   and P :: "'\<sigma> \<Rightarrow> bool" assumes "invar m"   and "I (\<alpha> m) \<sigma>0"   and "\<And>k it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; \<forall>j\<in>it. k \<le> j; \<forall>j\<in>\<alpha> m - it. j \<le> k; it \<subseteq> \<alpha> m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f k \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> \<alpha> m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. j \<le> k\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iterateoi m c f \<sigma>0)"
echelon_form_of_column_k_det_unit: fixes det_P :: "'b"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec"   and i :: "nat"   and k :: "nat" assumes "is_unit det_P" shows "is_unit (fst (echelon_form_of_column_k_det bezout (det_P, A, i) k))"
reparam_weak_def: fixes \<gamma>1 :: "real \<Rightarrow> 'b"   and \<gamma>2 :: "'c \<Rightarrow> 'b" shows "reparam_weak \<gamma>1 \<gamma>2 \<equiv> \<exists>\<phi>. (\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 \<circ> \<phi>) x) \<and> \<phi> piecewise_C1_differentiable_on {0..1} \<and> \<phi> 0 = (0::'c) \<and> \<phi> 1 = (1::'c) \<and> \<phi> ` {0..1} = {0::'c..1::'c}"
sum_to_tup_app: fixes f :: "('g + 'i) discr \<rightarrow> 'h" shows "sum_to_tup\<cdot>f = (\<Lambda> x. f\<cdot>(Discr (Inl (undiscr x))), \<Lambda> x. f\<cdot>(Discr (Inr (undiscr x))))"
sees_field_i_i_i_o_o_o_def: shows "sees_field_i_i_i_o_o_o \<equiv> \<lambda>x xa xb. pred.Pred (\<lambda>(xc, xd, xe). x \<turnstile> xa sees xb:xc (xd) in xe)"
to_struct_map_def: fixes T :: "('a, 'c) partial_object_scheme"   and f :: "'a \<Rightarrow> 'b" shows "to_struct_map T f = restrict f (carrier T)"
filter: shows "filter (\<lambda>document_ptr. |h \<turnstile> Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disconnected_nodes. return (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` set disconnected_nodes))|\<^sub>r) document_ptrs = [document_ptr]"
matched_intra_CFG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains as :: "'a list" where "CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n')"
assms_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) set" assumes "shift1_inv (qs, B)"   and "card B \<le> 1" shows "P (qs, B) (qs, B)"
hyps_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and u :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>2"   and "\<Gamma> \<turnstile> t is u : T\<^sub>2" shows "\<Gamma> \<turnstile> s is u : T\<^sub>2"
pr_tuple_1: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "span f g"   and "cod f = a"   and "cod g = b" shows "pr1 a b \<cdot> tuple f g = f"
out_in_indep: fixes x :: "'c \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'd" shows "$x\<acute> \<bowtie> $y"
subs: shows "\<omega>_limit_set x \<subseteq> K"
kbo_std_basis_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   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" shows "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf \<equiv> ground_heads gt_sym arity_sym arity_var ground_heads_var \<and> kbo_std_basis_axioms gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"
isSimplePath_cons: fixes s :: "nat"   and e :: "nat \<times> nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" shows "cf.isSimplePath s (e # p) t = (\<exists>u. e = (s, u) \<and> s \<noteq> u \<and> (s, u) \<in> cf.E \<and> cf.isSimplePath u p t \<and> s \<notin> set (cf.pathVertices_fwd u p))"
binomial_altdef_of_nat: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i))"
not_KeyCryptNonce_cardSK: fixes cardSK :: "nat"   and evs :: "event list"   and N :: "nat" assumes "cardSK \<notin> symKeys"   and "\<forall>C. cardSK \<noteq> priEK C"   and "evs \<in> set_cr"   and "Key cardSK \<notin> analz (knows Spy evs)" shows "\<not> KeyCryptNonce cardSK N evs"
harm_aux_ineq_2_lemma: fixes x :: "real" assumes "0 \<le> x" shows "1 < (x + 1) * ln (1 + 2 / (2 * x + 1))"
SN_restrict_SN_idemp: fixes A :: "('b \<times> 'b) set" shows "SN (restrict_SN A A)"
IP: fixes \<sigma> :: "'a"   and k :: "'u"   and v :: "'v"   and it :: "('u \<times> 'v) set" assumes "c \<sigma>"   and "(k, v) \<in> it"   and "it \<subseteq> map_to_set (\<alpha> m)"   and "I it \<sigma>" shows "I (it - {(k, v)}) (f (k, v) \<sigma>)"
iftrue: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and c1 :: "('b, 'c) MWLfCom"   and c2 :: "('b, 'c) MWLfCom" assumes "BMap (E b m) = True" shows "((if b then c1 else c2 fi, m), [c1], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
redT_acquire: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x1"   and ln :: "'l \<Rightarrow>f nat"   and n :: "'l"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln"   and "0 < ln $ n"   and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)" shows "s -1-t\<triangleright>(K$ [], [], [], [], [], convert_RA ln)\<rightarrow> s'"
PiE_arb: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f \<in> Pi\<^sub>E S T"   and "x \<notin> S" shows "f x = undefined"
P_def: fixes x :: "rat" shows "P x = (\<forall>y\<in>S. y \<le> of_rat x)"
must_syncE: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap" assumes "execd_mthr.must_sync P t x m" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and x' :: "'addr option \<times> 'addr frame list" and m' :: "'heap" and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "execd_mthr.r_syntax P t x m ta x' m'" and "exec_mthr.actions_ok s t ta" and "m = shr s"
map_add_comm: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" assumes "dom m1 \<inter> dom m2 = {}" shows "m2 \<Oplus> m1 = m1 \<Oplus> m2"
minus_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_minus [x, y] = x - y"
Rep_SkewBinomialHeap: fixes x :: "('c, 'd) SkewBinomialHeap" shows "Rep_SkewBinomialHeap x \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}"
Rep_uint64_inverse: fixes x :: "uint64" shows "Abs_uint64 (Rep_uint64 x) = x"
MergeGuardsD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P merge_guards c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
right_connected: fixes \<mu> :: "'a"   and \<mu>' :: "'a"   and \<nu> :: "'a" assumes "seq \<mu> \<mu>'" shows "(\<nu> \<star> \<mu> \<noteq> null) = (\<nu> \<star> \<mu>' \<noteq> null)"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b list \<Rightarrow> 'a"   and ta_rule :: "('b, 'c) ta_rule" shows "P (case ta_rule of x \<rightarrow> xa xb \<Rightarrow> f x xa xb) = (\<forall>x1 x2 x3. ta_rule = x1 \<rightarrow> x2 x3 \<longrightarrow> P (f x1 x2 x3))"
sepnMT: fixes p :: "('a, 'b) Combinators list" assumes "p \<noteq> []" shows "separate p \<noteq> []"
assms_2: assumes "vassn_tag (hn_ctxt A x y)" shows "P' x"
circ_sup_9: fixes x :: "'b"   and y :: "'b" shows "(x \<squnion> y)\<^sup>\<star> = (x\<^sup>\<star> * y)\<^sup>\<star> * x\<^sup>\<star>"
rel_refl_strong: fixes x :: "'a set['b]"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> Ra z z" shows "rel_bset Ra x x"
l3_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and KE :: "msg" shows "l3_step2 Rb A B KE \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xpkE = Some KE \<and> implAuth A B \<langle>Number 0, KE\<rangle> \<in> ik s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xpkE, xsk}), ik := {implAuth B A (Aenc (NonceF (Rb $ sk)) KE)} \<union> ik s, l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>) else l1_state.signals s, secret := {x. x = NonceF (Rb $ sk) \<and> Rb = test} \<union> secret s\<rparr>}"
kD_None: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<notin> kD rt" shows "rt dip = None"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'c set['e]"   and xb :: "'d set['e]" shows "rel_bset x xa xb = rel_set x (set_bset xa) (set_bset xb)"
has_subprob_densityD_1: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "f \<in> borel_measurable N"
less_eq_interval_def: fixes a :: "'a interval"   and b :: "'a interval" shows "(a \<le> b) = (lower b \<le> lower a \<and> upper a \<le> upper b)"
F_conv_f: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds F s = f s"
weak_bisim_def: fixes \<T>\<^sub>1 :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and \<T> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "weak_bisim \<T>\<^sub>1 \<T> \<equiv> \<forall>c\<^sub>1 c\<^sub>2 mds mem\<^sub>1 mem\<^sub>2 c\<^sub>1' mds' mem\<^sub>1'. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<T>\<^sub>1 \<and> \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> (\<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>, \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>) \<in> \<T>)"
simple_match_valid_alt_hlp1: fixes x :: "16 word \<times> 16 word" shows "({p. simple_match_port x p} \<noteq> UNIV) = (case x of (s, e) \<Rightarrow> s \<noteq> 0 \<or> e \<noteq> - 1)"
derivative_rhs: fixes x :: "'a" assumes "x \<in> X" shows "(f has_derivative blinfun_apply (f' x)) (at x)"
mplus_def: shows "mplus = COERCE(udom\<cdot>'c \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c, 'b\<cdot>'c \<rightarrow> 'b\<cdot>'c \<rightarrow> 'b\<cdot>'c)\<cdot>plusU"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q"   and "(\<Psi>, Q, R) \<in> Rel"   and "\<Psi> \<rhd> R \<sim> S" shows "(\<Psi>, P, S) \<in> Rel"
gd_term_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_term :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_term_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict \<equiv> ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
of_bool_eq_1: fixes one :: "'c"   and zero :: "'c" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero False = zero"
elements_mat_append_cols: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nc1 :: "nat" assumes "A \<in> carrier_mat n nc"   and "B \<in> carrier_mat n nc1" shows "elements_mat (A @\<^sub>c B) = elements_mat A \<union> elements_mat B"
comm_monoid_axioms: shows "comm_monoid (\<and>) True"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
irreducibility_hom_axioms: shows "irreducibility_hom poly_x_minus_y"
homogeneous_setD: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and a :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "homogeneous_set A"   and "a \<in> A" shows "hom_component a n \<in> A"
finish: shows "(length L - 1 = 0 \<or> ladder_prev_n L (length L - 1) < n') \<and> n' \<le> n \<and> LeftDerivation \<alpha> (take n' D) \<beta> \<and> LeftDerivation (\<alpha> @ \<delta>) (take n' D) (\<beta> @ \<delta>) \<and> derivation_ge (drop n' D) (length \<beta>) \<and> LeftDerivation \<delta> (derivation_shift (drop n' D) (length \<beta>) 0) \<delta>' \<and> ladder_\<gamma> (\<alpha> @ \<delta>) D L (length L - 1) = \<beta> @ \<delta>' \<and> ladder_j L (length L - 1) < length \<beta>"
convert_term_match: fixes t :: "term"   and u :: "'b"   and env :: "(name, 'b) fmap" assumes "match t u = Some env" shows "match t (convert_term u) = Some (fmmap convert_term env)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "order.greater_eq a b"   and "order.greater_eq b c" shows "order.greater_eq a c"
right: shows "\<forall>z. O z x \<longrightarrow> O z y"
H\<Phi>: fixes l :: "nat" shows "case H l of (n, N, j) \<Rightarrow> \<Phi> n N"
raw_induct: fixes P :: "('a, 'b) tllist \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('a, 'b) tllist"   and y :: "'b" assumes "\<And>terminal0 xs xsa. \<lbrakk>\<And>x y. \<lbrakk>terminal0 x = y; y \<noteq> undefined\<rbrakk> \<Longrightarrow> P x y; (if is_TNil xs then case xs of TNil x \<Rightarrow> id x | TCons x xa \<Rightarrow> undefined x xa else terminal0 (ttl xs)) = xsa; xsa \<noteq> undefined\<rbrakk> \<Longrightarrow> P xs xsa"   and "terminal0 xs = y"   and "y \<noteq> undefined" shows "P xs y"
prems_2: shows "\<forall>x\<in>{1..Suc m}. finite (A x)"
card_columns_le_ncols: fixes A :: "(('b, 'd) vec, 'c) vec" shows "card (columns A) \<le> ncols A"
simps_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Normal x1 = Normal y1) = (x1 = y1)"
distinct_match: fixes al :: "'b list"   and e :: "'b"   and bl :: "'b list"   and al' :: "'b list"   and bl' :: "'b list" assumes "distinct (al @ e # bl)" shows "(al @ e # bl = al' @ e # bl') = (al = al' \<and> bl = bl')"
finite_lists_length_le: fixes A :: "'b set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
simps_2: fixes uv :: "'b"   and uw :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "min_height (Node uv uw l r) = min (min_height l) (min_height r) + 1"
subsuming_states_def: fixes partial :: "nat \<Rightarrow> nat option" shows "subsuming_states partial \<equiv> {s \<in> valid_states. partial \<subseteq>\<^sub>m s}"
SCConc: fixes x :: "('b, 'c) Combinators list"   and y :: "('b, 'c) Combinators list" assumes "singleCombinators x"   and "singleCombinators y" shows "singleCombinators (x @ y)"
injectivity: fixes x :: "'d"   and y :: "'d" assumes "[:x:] = [:y:]" shows "x = y"
tinsert_d_amor: fixes n :: "nat"   and t :: "'a tree"   and a :: "'a"   and dl :: "nat"   and t' :: "'a tree"   and dl' :: "nat" assumes "n = size t"   and "insert_d a (t, dl) = (t', dl')"   and "bal_i (size t + dl) (height t)" shows "real (T_insert3_d a (t, n, dl)) + \<Phi> t' - \<Phi> t \<le> (6 * e + 2) * real (height t + 1) + 1"
tendsto_diff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
ux_1: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> ux x"
pmaxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "pmaxsimpchain (x # xs)" shows "pmaxsimpchain xs"
vec_component: fixes x :: "'c"   and i :: "'d" shows "vec x $ i = x"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(tInit x1 = tInit y1) = (x1 = y1)"
ordinal_plus_not_0: fixes x :: "ordinal"   and y :: "ordinal" shows "(0 < x + y) = (0 < x \<or> 0 < y)"
intro: assumes "\<And>x y. \<lbrakk>P x y; P y x\<rbrakk> \<Longrightarrow> x = y"   and "\<And>x y. PP x y = (P x y \<and> x \<noteq> y)" shows "M_axioms"
tensor_mat_positive: fixes m1 :: "complex mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "complex mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "positive m1"   and "positive m2" shows "positive (tensor_mat dims vars m1 m2)"
gen_length_def: fixes n :: "nat"   and xs :: "'a list" shows "List.gen_length n xs = n + \<parallel>xs\<parallel>"
Rep_Nat_inverse: fixes x :: "nat" shows "Abs_Nat (Rep_Nat x) = x"
small_by_big: fixes 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 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "final e'" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
goedel_numbering_P2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "goedel_numbering \<psi>" shows "\<psi> \<in> \<P>\<^sup>2"
replete_set_category_def: fixes S :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag" shows "replete_set_category S \<equiv> set_category S (cardSuc (cmax |UNIV| natLeq))"
alpha_in_var: fixes x :: "'d \<Longrightarrow> 'e" shows "x ;\<^sub>L fst\<^sub>L = $x"
nth_less_subdegree_zero: fixes n :: "nat"   and f :: "'a fps" assumes "n < subdegree f" shows "f $ n = (0::'a)"
Zfun_prod_Bfun: fixes f :: "'e \<Rightarrow> 'a"   and F :: "'e filter"   and g :: "'e \<Rightarrow> 'b" assumes "Zfun f F"   and "Bfun g F" shows "Zfun (\<lambda>x. f x ** g x) F"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> \<Rightarrow> Normal s"
c'': shows "c'' = mark_guards False (strip_guards (- F) c')"
assms: fixes a :: "'a" assumes "a \<in> A" shows "f a \<in> \<int>"
Suc_eq_plus1: fixes n :: "nat" shows "Suc n = n + 1"
SemSkip: fixes s :: "Var \<Rightarrow> Val" shows " s , Skip \<rightarrow>\<^sub>1  s "
errorT_induct: fixes P :: "'e\<cdot>('g, 'f) errorT \<Rightarrow> bool"   and y :: "'e\<cdot>('g, 'f) errorT" assumes "\<And>k. P (ErrorT\<cdot>k)" shows "P y"
indicator_eq_0_iff: fixes A :: "'l set"   and x :: "'l" shows "(indicator A x = (0::'k)) = (x \<notin> A)"
wtL_grdLit: fixes x :: "Preliminaries.var" assumes "prot (GE.tpOfV x) \<or> protFw (GE.tpOfV x)" shows "GE.wtL (grdLit x)"
mt: shows "ov # ovs = [] \<longrightarrow> n = 0 \<and> u = last (verticesFrom f r)"
inject_1: fixes x1 :: "prim_ty"   and y1 :: "prim_ty" shows "(PrimT x1 = PrimT y1) = (x1 = y1)"
between_trivial2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Bet A A B"
assms_5: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
inj_triple: fixes x :: "'a" assumes "is_inj x" shows "x = x ; x\<^sup>T ; x"
set_bd_3: fixes x :: "('l, 'm, 'n, 'k) F" shows "|F'set2 x| \<le>o F'bd"
sum_squares_eq_zero_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "(plus (times x x) (times y y) = zero) = (x = zero \<and> y = zero)"
order_subst2: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
\<eta>_eq: fixes x :: "'a" assumes "x \<in> G" shows "\<eta>' x = \<eta> x"
inverse_float_interval_eqI: fixes p :: "nat"   and X :: "float interval"   and IVL :: "float interval"   and x :: "real" assumes "inverse_float_interval p X = Some IVL"   and "x \<in>\<^sub>r X" shows "inverse x \<in>\<^sub>r IVL"
tellWT_def: shows "tellWT = (\<Lambda> x w. WriterT\<cdot>(return\<cdot>(Writer\<cdot>w\<cdot>x)))"