hcju/isabelleps huggingface dataset:
eq_exhaust: fixes y :: "cmsg" obtains x :: "agent" where "eq y (cAgent x__)"   | x :: "nat" where "eq y (cNumber xa__)"   | x :: "nonce_t" where "eq y (cNonce xb__)"   | x :: "ltkey" where "eq y (cLtK xc__)"   | x :: "ephkey" where "eq y (cEphK xd__)"   | x :: "cmsg" and x' :: "cmsg" where "eq y (cPair xe__ x'__)"   | x :: "cmsg" and x' :: "cmsg" where "eq y (cEnc xf__ x'a__)"   | x :: "cmsg" and x' :: "cmsg" where "eq y (cAenc xg__ x'b__)"   | x :: "cmsg" and x' :: "cmsg" where "eq y (cSign xh__ x'c__)"   | x :: "cmsg" where "eq y (cHash xi__)"   | x :: "tag" where "eq y (cTag xj__)"   | x :: "cmsg" and x' :: "cmsg" where "eq y (cExp x x')"
l3_inv4_trans: shows "{l3_inv4} TS.trans l3 {> l3_inv4}"
demod1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d (x \<cdot> y) \<le> d z" shows "x \<cdot> d y \<le> d z \<cdot> x"
alwaysWFI: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> WF(A)_v \<longrightarrow> \<box>WF(A)_v"
dghm_up_ArrMap_app_component: fixes a :: "V"   and \<CC> :: "V"   and i :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<lparr>i\<rparr> = \<phi> i\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"
under_Under_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "trans r"   and "a \<in> under r b"   and "b \<in> Under r C" shows "a \<in> Under r C"
surjective_conv_total: fixes x :: "'a" shows "surjective x = times_top_class.total (x\<^sup>T)"
a1: shows "\<Gamma> \<turnstile> Lam [x].t : T' \<rightarrow> T"
\<eta>_map_simp: fixes y :: "'d" assumes "D.ide y" shows "transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map \<eta>o y = \<phi> y (F y)"
apply_power_order: fixes f :: "'b perm"   and a :: "'b" shows "apply (f ^ order f a) a = a"
left_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). stk1.rel x y}" shows "a \<in> Sigma I \<FF>1"
simps_1: shows "diag_block_mat [] = 0\<^sub>m 0 0"
continuousI: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "\<And>f. OrdinalInduct.strict_mono f \<Longrightarrow> F (oLimit f) = oLimit (\<lambda>n. F (f n))"   and "\<And>x. F x \<le> F (oSuc x)" shows "continuous F"
\<psi>: fixes s :: "'s" assumes "P s" shows "s \<notin> \<psi>"
I_qe_FMo\<^sub>1: fixes as :: "atom list"   and xs :: "real list" shows "R.I (qe_FMo\<^sub>1 as) xs = R.I (qe_FM\<^sub>1 as) xs"
routing_action_next_hop_update_def: fixes h :: "'b word"   and pk :: "'b routing_rule" shows "routing_action_next_hop_update h pk = pk\<lparr>routing_action := routing_action pk\<lparr>next_hop := Some h\<rparr>\<rparr>"
simps_9: fixes x :: "name"   and P :: "pi" shows "summands (<\<nu>x>P) = (if \<exists>a P'. a \<noteq> x \<and> P = a{x}.P' then {<\<nu>x>P} else {})"
simps_3: fixes v :: "'a"   and va :: "'a"   and y :: "('a, 'b) Combinators list" shows "removeShadowRules2 (DenyAllFromTo v va # y) = DenyAllFromTo v va # removeShadowRules2 y"
IH: assumes "spec_ops (xs @ ys) deps"   and "\<forall>i\<in>set (map fst xs). i < oid"   and "\<forall>a\<in>set (map fst ys). oid < a"   and "\<forall>ref\<in>deps oper. ref < oid" shows "spec_ops (xs @ [(oid, oper)] @ ys) deps"
intros_1: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma>\<^sub>\<sigma> :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and state_update :: "'b \<Rightarrow> final_decision \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma>\<^sub>0 :: "'b"   and built_in_chain :: "char list"   and default_policy :: "action"   and ps :: "'c list" shows "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) ps [] \<sigma>\<^sub>0"
ts_inv_def: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and I :: "'g \<Rightarrow> 'f option"   and ts :: "'g \<Rightarrow> ('h \<times> 'j \<Rightarrow>f nat) option"   and m :: "'i" shows "ts_inv P I ts m \<equiv> \<forall>t. case ts t of None \<Rightarrow> True | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> \<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m"
mul_opt_def: fixes U\<^sub>1 :: "nat option"   and U\<^sub>2 :: "nat option" shows "U\<^sub>1 \<otimes> U\<^sub>2 \<equiv> case (U\<^sub>1, U\<^sub>2) of (None, b) \<Rightarrow> None | (Some u\<^sub>1, None) \<Rightarrow> None | (Some u\<^sub>1, Some u\<^sub>2) \<Rightarrow> Some (u\<^sub>1 * u\<^sub>2)"
fls_mult_commute_base_factor: fixes a :: "'a fls"   and b :: "'a fls" shows "fls_base_factor a * fls_base_factor b = fls_base_factor b * fls_base_factor a"
equivalence_in_bicategory_axioms: shows "internal_equivalence f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> local.inv \<phi>))"
radical_divide: fixes k :: "nat"   and r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a fps"   and b :: "'a fps" assumes "0 < k"   and "r k (fps_nth a 0) ^ k = fps_nth a 0"   and "r k (fps_nth b 0) ^ k = fps_nth b 0"   and "fps_nth a 0 \<noteq> (0::'a)"   and "fps_nth b 0 \<noteq> (0::'a)" shows "(r k (fps_nth a 0 / fps_nth b 0) = r k (fps_nth a 0) / r k (fps_nth b 0)) = (fps_radical r k (a / b) = fps_radical r k a / fps_radical r k b)"
neutr_eq_iff: fixes a :: "'c"   and b :: "'c" shows "(\<bottom> = a \<squnion> b) = (a = \<bottom> \<and> b = \<bottom>)"
is_Var_def: fixes x :: "hf" shows "is_Var x \<equiv> Ord x \<and> 0 \<^bold>\<in> x"
simps: fixes a1 :: "'c option list"   and a2 :: "'c option set"   and P :: "'c option process"   and Q :: "'c option process" shows "((a1, a2) \<in> seq_comp_failures P Q) = ((\<exists>xs X. a1 = xs \<and> a2 = X \<and> xs \<notin> sentences P \<and> (xs, X) \<in> failures P \<and> None \<notin> set xs) \<or> (\<exists>xs X Y. a1 = xs \<and> a2 = insert None X \<inter> Y \<and> xs \<in> sentences P \<and> (xs, X) \<in> failures P \<and> ([], Y) \<in> failures Q) \<or> (\<exists>xs ys Y. a1 = xs @ ys \<and> a2 = Y \<and> xs \<in> sentences P \<and> (ys, Y) \<in> failures Q \<and> ys \<noteq> []) \<or> (\<exists>xs X Y. a1 = xs \<and> a2 = X \<union> Y \<and> (xs, X) \<in> seq_comp_failures P Q \<and> (xs, Y) \<in> seq_comp_failures P Q))"
lt3_def: shows "lt3 \<equiv> less_than <*lex*> lt2"
serial_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>A. (x, y) \<in> r" shows "serial_on A r"
A: shows "D y \<notin> sources I D u (y # zs)"
less_le_not_le: fixes x :: "'t"   and y :: "'t" shows "(x \<prec>\<^sub>t y) = strict (\<preceq>\<^sub>t) x y"
bo_def: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" shows "bo a b = (m a b = Some b)"
atLeastLessThan_empty_iff2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "({} = subset_mset.atLeastLessThan a b) = (\<not> a \<subset># b)"
vimage_Collect_eq: fixes f :: "'q \<Rightarrow> 'r"   and P :: "'r \<Rightarrow> bool" shows "f -` Collect P = {y. P (f y)}"
R: assumes "lim F \<in> carrier"   and "F \<longlongrightarrow> lim F" shows "R"
simps: fixes a :: "presb" shows "lformula0 a = (\<exists>is i. a = presb.Eq is i 0)"
monom_closed: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom (UP R) a n \<in> carrier (UP R)"
limit_Append_path_nonelem_split': fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and p :: "(symbol \<times> character list) list"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "p \<in> limit (LocalLexing.Append \<NN> \<TT> \<RR> \<SS> (LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k (Suc u)) k) (LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u)"   and "p \<notin> LocalLexing.\<P> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u" shows "\<exists>q ts. p = q @ ts \<and> LocalLexing.is_continuation \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k u q ts"
ln_Gamma_def: fixes z :: "'a" shows "ln_Gamma z = lim (ln_Gamma_series z)"
fun_belowD: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
chine_is_identity: shows "C.ide \<mu>\<nu>_\<pi>.chine"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "(a \<cong>\<^sub>B a') = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f)"
cfun_eq_iff: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
bind_return_pmf': fixes N :: "'a pmf" shows "N \<bind> return_pmf = N"
im_diff: fixes g :: "'g"   and g' :: "'g" assumes "g \<in> UNIV"   and "g' \<in> UNIV" shows "T (g - g') = T g - T g'"
abs_eq: fixes x :: "('a, 'c, 'b) Tree" shows "S_transform_Tree\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = S_transform_Tree x"
intros_26: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Tee_local i] : [t] _> [t]"
coclop_dual_var: fixes f :: "'b \<Rightarrow> 'b" shows "coclop f = clop (\<partial>\<^sub>F f)"
norm_stacks: shows "\<Sigma>1 = norm_stack \<Sigma>2"
inst: shows "instance_of\<^sub>l\<^sub>s (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x))) C\<^sub>1"
domD: fixes a :: "'b"   and m :: "'b \<Rightarrow> 'c option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
Eats_Mem_sf: fixes t :: "tm"   and u :: "tm"   and i :: "name" assumes "\<And>j. Sigma_fm (Var j EQ t)"   and "\<And>k. Sigma_fm (Var k EQ u)" shows "Sigma_fm (Eats t u IN Var i)"
WT_sim: shows "\<I>_real, \<I>_middle \<turnstile>\<^sub>C sim_outer \<surd>"
h1b: shows "MPoly_Type.degree p var = 1 \<longrightarrow> isolate_variable_sparse p var 2 = 0"
hyps_1: shows "atom ` domA as \<sharp>* (\<Gamma>, L)"
quot_def: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "eqv.quot eqv = {{x. eqv a x} |a. True}"
simps: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and uu :: "nat"   and r :: "('b \<times> nat) tree" shows "split_min \<langle>l, (a, uu), r\<rangle> = (if l = \<langle>\<rangle> then (a, r) else let (x, l') = split_min l in (x, balanceL l' a r))"
hom: fixes T :: "'a \<Rightarrow> 'b"   and g :: "'a"   and g' :: "'a" assumes "UGroupHom T" shows "T (g + g') = T g + T g'"
noFaultE: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and f :: "'f"   and P :: "bool" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Fault f}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f" shows "P"
D_def: shows "D = diagvector (\<lambda>j. cis (arg (y $h j)))"
fm_image_measurable_finite: fixes M :: "'e measure"   and X :: "('i \<Rightarrow> 'e) set"   and n :: "nat" assumes "space M = UNIV"   and "X \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. M))" shows "fm n ` X \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. M))"
startsby_zero_power_nth_same: fixes a :: "'a fps"   and n :: "nat" assumes "fps_nth a 0 = (0::'a)" shows "fps_nth (a ^ n) n = fps_nth a 1 ^ n"
ACI_norm_idem: fixes r :: "'a rexp" shows "\<guillemotleft>\<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>r\<guillemotright>"
coclop_coextensive: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "f \<le> id"
lleq_less: fixes x :: "char list"   and y :: "char list" shows "less_charlist x y = (lleq_charlist x y \<and> \<not> lleq_charlist y x)"
pred_stream_stl: fixes \<phi> :: "'a \<Rightarrow> bool"   and xs :: "'a stream" shows "pred_stream \<phi> xs \<longrightarrow> pred_stream \<phi> (stl xs)"
bij_betw_same_card: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "card A = card B"
functions_of_msubset_of: fixes A :: "'c set"   and F :: "('c \<Rightarrow> 'd) set"   and B :: "'d set" assumes "finite A"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_permutation A B" shows "functions_of A (msubset_of A F) = F"
lfp_le_gfp: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "lfp f \<le> gfp f"
root_exp_cancel: fixes xa :: "real" assumes "0 \<le> xa" shows "root n xa ^ n = xa"
cSup_insert: fixes X :: "'a multiset set"   and a :: "'a multiset" assumes "X \<noteq> {}"   and "subset_mset.bdd_above X" shows "Sup (insert a X) = a \<union># Sup X"
Un_left_commute: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
simps: fixes f :: "'c \<Rightarrow> nat"   and u :: "nat"   and ns :: "nat list"   and xs :: "'c list" shows "count_inv f (u, ns, xs) = (\<forall>x. count (mset xs) x = f x)"
A: fixes x :: "'c list"   and y :: "'c list" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
q_def: fixes z :: "nat"   and u :: "nat" shows "q z u = p (c_fst z) (c_snd z) u"
assms_2: shows "Deliver (i1, Append i2 e) \<in> set xs"
vseq_implies_hpar_2: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.seq \<nu> \<mu>" shows "trg\<^sub>D \<nu> = trg\<^sub>D \<mu>"
enat_less_INF_I: fixes x :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> enat" assumes "x \<noteq> \<infinity>"   and "\<And>y. y \<in> S \<Longrightarrow> x < f y" shows "x < \<Sqinter> (f ` S)"
polyadd: fixes bs :: "'b list"   and p :: "poly"   and q :: "poly" shows "\<lparr>p +\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> + \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
elims_2: fixes x :: "'b queue" assumes "is_empty x" obtains q :: "'b queue" where "x = q" and "list q = []"
cases: fixes x :: "net_tree" obtains p1 :: "net_tree" and p2 :: "net_tree" where "x = p1 \<parallel> p2"   | i :: "nat" and R :: "nat set" where "x = \<langle>i; R\<rangle>"
b_pos: shows "\<forall>\<^sub>F x in at_top. 0 < b x"
is_weakly_dom_component_safe_def: shows "Shadow_DOM_Components.is_weakly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"
simply_connected_imp_Borsukian: fixes S :: "'a set" assumes "simply_connected S"   and "locally path_connected S" shows "Borsukian S"
rel_fset_fset: fixes \<chi> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A1 :: "'c fset"   and A2 :: "'d fset" shows "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
rank_iarrays_code: fixes A :: "'b iarray iarray" shows "rank_iarray A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (Gauss_Jordan_iarrays A)))"
norm_triangle_mono: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and r :: "real"   and b :: "'b"   and s :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm a \<le> r"   and "norm b \<le> s" shows "norm (plus a b) \<le> r + s"
mrw: fixes a :: "nat" assumes "enat a < r"   and "a \<in> read_actions E" shows "P,E \<turnstile> a \<leadsto>mrw ws a"
lasso_cycle_def: fixes L :: "('a, 'b) lasso_scheme" shows "lasso_cycle L = lasso_va L # lasso_cysfx L"
commit_correct: shows "Schnorr_\<Sigma>_commit.abstract_com.correct"
Cut: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and P :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> Call : P"   and "insert P G \<rhd> c : A" shows "G \<rhd> c : A"
trace_in_simps: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and ss :: "'d set"   and t :: "real"   and s :: "'d"   and x :: "real \<times> 'd"   and \<omega> :: "(real \<times> 'd) stream" assumes "transition_rates R" shows "transition_rates.trace_in ss t s (x ## \<omega>) = (if t < fst x then s \<in> ss else transition_rates.trace_in ss t (snd x) \<omega>)"
d_1: shows "root_main' x n = (let xpm = x ^ pm; xp = xpm * x in if xp \<le> n then (x, xp = n) else root_main' ((n div xpm + x * int pm) div int p) n)"
bex_Infinitesimal_iff2: fixes x :: "'c star"   and z :: "'c star" shows "(\<exists>y\<in>Infinitesimal. x = z + y) = (x \<approx> z)"
lookup_sum_list: fixes ps :: "('d \<Rightarrow>\<^sub>0 'c) list"   and a :: "'d" shows "lookup (sum_list ps) a = (\<Sum>p\<leftarrow>ps. lookup p a)"
univ_poly_is_domain: fixes K :: "'a list set" assumes "subring K (poly_ring R)" shows "domain (K [X]\<^bsub>poly_ring R\<^esub>)"
hn_rel_compI: fixes a :: "'a nres"   and b :: "'b nres"   and R2 :: "('b \<times> 'a) set"   and R1 :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and c :: "'c" assumes "nofail a"   and "(b, a) \<in> \<langle>R2\<rangle>nres_rel" shows "hn_rel R1 b c \<Longrightarrow>\<^sub>A hn_rel (hr_comp R1 R2) a c"
subst_compose_eq_or: fixes \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm"   and x1 :: "'var"   and x2 :: "'var" assumes "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<in> var"   and "x2 \<in> var"   and "x1 = x2 \<or> x2 \<notin> Fvars \<phi>" shows "subst (subst \<phi> t1 x1) t2 x2 = subst \<phi> (substT t1 t2 x2) x1"
comp_eq_dest: fixes a :: "'j \<Rightarrow> 'i"   and b :: "'h \<Rightarrow> 'j"   and c :: "'k \<Rightarrow> 'i"   and d :: "'h \<Rightarrow> 'k"   and v :: "'h" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
x: shows "primitive (Polynomial.monom 1 1) \<and> monic (Polynomial.monom 1 1) \<and> degree (Polynomial.monom 1 1) = 1"
inj_on_proj_arcs_H: shows "inj_on ((`) proj_arcs_H) H.face_cycle_sets"
sumPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto>[Rel] Q"   and "Rel \<subseteq> Rel'"   and "Id \<subseteq> Rel'" shows "P \<oplus> R \<leadsto>[Rel'] Q \<oplus> R"
diff_less_imp_swap: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "0 < i"   and "k - i < j" shows "k - j < i"
not_may_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "(\<not> may_lock l t) = (\<exists>t'. t' \<noteq> t \<and> has_lock l t')"
parent_child_rel_child_nodes2: fixes parent :: "(_) object_ptr"   and h :: "(_) heap"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr"   and child_obj :: "(_) object_ptr" assumes "known_ptr parent"   and "h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children"   and "child \<in> set children"   and "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 child = child_obj" shows "(parent, child_obj) \<in> parent_child_rel h"
eucl_Sup: fixes X :: "'a set" shows "Sup X = (\<Sum>i\<in>Basis. (SUP x\<in>X. x \<bullet> i) *\<^sub>R i)"
state_update_before_is_CSP: fixes sc :: "'d \<times> 'd \<Rightarrow> bool"   and Ac :: "('c, 'd) action" shows "is_CSP_process (R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A') ;; action.relation_of Ac)"
coll_add: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "coll 0 x y"   and "coll 0 x z" shows "coll 0 x (y + z)"
target_alt_def: fixes r :: "'action list"   and p :: "'state" shows "target r p = last (p # states r p)"
distinct_42: fixes x71 :: "key"   and x72 :: "msg"   and x61 :: "msg"   and x62 :: "msg" shows "Crypt x71 x72 \<noteq> \<lbrace>x61, x62\<rbrace>"
poly_eval_uminus: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a (- p) = - poly_eval a p"
rep_eq: fixes x :: "'b ltl set"   and xa :: "'b ltl_prop_equiv_quotient" shows "x \<up>\<Turnstile>\<^sub>P xa = x \<Turnstile>\<^sub>P Rep xa"
not_mod2_eq_Suc_0_eq_0: fixes n :: "nat" shows "(n mod 2 \<noteq> Suc 0) = is_even n"
equivp_reflp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "equivp R" shows "R x x"
sup_bot_right: fixes x :: "'b" shows "x + (0::'b) = x"
full_imp_bal: fixes n :: "nat"   and t :: "'b tree234" assumes "full n t" shows "bal t"
"1_1": shows "\<forall>m<c. 0 < n \<longrightarrow> (\<forall>x xa. x * xa = m ^ n \<longrightarrow> coprime x xa \<longrightarrow> (\<exists>k. x = k ^ n))"
unl_lift3: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "'e \<Rightarrow> 'b"   and y :: "'e \<Rightarrow> 'c"   and z :: "'e \<Rightarrow> 'd"   and w :: "'e" shows "(f<x, y, z>) w \<equiv> f (x w) (y w) (z w)"
not_inImage_iimg: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and B :: "'a set" assumes "a \<notin> f ` B" shows "iimg B f a = f a"
characteristic_iso_4: fixes f :: "'a" assumes "S.ide f" shows "S.iso ((f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a])"
HeronConf_interp_stepwise_instant_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
sets_rT: fixes A :: "'s stream set"   and x :: "'s" shows "(A \<in> sets (rT x)) = (A \<in> sets (stream_space (count_space UNIV)) \<and> A \<subseteq> {\<omega>. enabled x \<omega>})"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "TruePref.stable_on ds X" shows "inj_on Xd X"
subfield_iff_2: fixes K :: "'a set" assumes "subfield K (L\<lparr>carrier := A\<rparr>)" shows "field (L\<lparr>carrier := A, carrier := K\<rparr>)"
conv_radius_zero: shows "conv_radius (\<lambda>_. 0::'a) = \<infinity>"
smc_is_arrD_5: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
eq_wf_dfa: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "wf_dfa (eq_dfa n ks l) n"
interaction_any_bounded_by_map_gpv': fixes gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and h :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'f"   and g :: "'a \<Rightarrow> 'e" assumes "interaction_any_bounded_by gpv n"   and "surj h" shows "interaction_any_bounded_by (map_gpv' f g h gpv) n"
intro: fixes G :: "('a, 'b) gb_graph_rec_scheme" assumes "finite (gbg_F G)"   and "gbg_F G \<subseteq> Pow (g_V G)" shows "gb_graph_axioms G"
dom_eq_empty_conv: fixes f :: "'a \<Rightarrow> 'b option" shows "(dom f = {}) = (f = \<emptyset>)"
exec_step_ind_Putstatic_Done: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and ics :: "init_call_status"   and sh :: "sheap"   and sfs :: "sfields"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "\<forall>Cs. ics \<noteq> Called Cs"   and "sh D' = \<lfloor>(sfs, Done)\<rfloor>" shows "exec_step_ind (StepI (Putstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh(D' \<mapsto> (sfs(F \<mapsto> hd stk), Done)))"
dc_iso: fixes x :: "'a" shows "d (d x \<cdot> 1\<^sub>\<pi>) = d x"
continuous_on_path: fixes f :: "real \<Rightarrow> 'a"   and t :: "real set" assumes "path f"   and "t \<subseteq> {0..1}" shows "continuous_on t f"