hcju/isabelleps huggingface dataset:
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('ag \<Rightarrow> 'ag) \<Rightarrow> ('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> ('z, 'x, 'y, 'aa, 'ad option) RElement_scheme"   and v :: "'ag" assumes "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
scast_ucast_1: fixes a :: "'b word" assumes "is_down UCAST('b \<rightarrow> 'c)"   and "is_down UCAST('c \<rightarrow> 'd)" shows "SCAST('c \<rightarrow> 'd) (UCAST('b \<rightarrow> 'c) a) = UCAST('b \<rightarrow> 'd) a"
get_elements_by_tag_name_pure: fixes ptr :: "(_) object_ptr"   and tag :: "char list"   and h :: "(_) heap" shows "pure (get_elements_by_tag_name ptr tag) h"
not_zero: shows "u \<noteq> 0\<^sub>h"
hyps: fixes vs :: "'a list" shows "distinct (map f (remdups_wrt_rev f xs vs))"
finite_fold_rderiv_zeros: shows "finite {\<langle>fold rderiv (replicate k (zero idx)) \<langle>\<psi>\<rangle>\<rangle> |k. True}"
iotam_mHom: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "\<iota>m\<^bsub>M H,K\<^esub> \<in> mHom R (mdl M H) (mdl M (H \<minusplus> K))"
assms_2: fixes x :: "'a" assumes "is_unit x" shows "P x"
wf_inst_eq_aux': fixes x :: "PDDL_STRIPS_Semantics.term"   and T :: "PDDL_STRIPS_Semantics.type" assumes "ty_term Q objT x = Some T" shows "objT (subst_term f x) \<noteq> None"
simps_10: shows "push_forall (AllQ FalseF) = FalseF"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
meet_commute: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dlat.meet le x y = dlat.meet le y x"
init_zz': shows "local.adverse z i j \<triangleright> Suc n = local.adverse z' i j \<triangleright> Suc n"
f_eq_iff: fixes x :: "'a poly poly"   and y :: "'a poly poly" shows "(poly_y_x x = y) = (x = inv poly_y_x y)"
monad_prob_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_prob return bind sample" shows "monad_prob (return_writer return) (bind_writer return bind) (sample_writer sample)"
simps_3: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd"   and ps :: "char list list"   and p :: "char list" shows "containsCall procs (c\<^sub>1;; c\<^sub>2) ps p = (containsCall procs c\<^sub>1 ps p \<or> containsCall procs c\<^sub>2 ps p)"
run_catch: fixes m :: "'a + 'b"   and f :: "'a \<Rightarrow> 'c + 'b" assumes "isOK m" shows "projr (try m catch f) = projr m"
gr_zeroI: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
beta_preserves_term_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and r :: "Core.term"   and s :: "Core.term" assumes "term_ok \<Theta> r"   and "r \<rightarrow>\<^sub>\<beta> s" shows "term_ok \<Theta> s"
thetaParZOT_ZObisT: shows "thetaParZOT \<subseteq> ZObisT"
add_last: shows "arr_mset arr l (i + 1) = arr_mset arr l i + {#arr i#}"
Maclaurin: fixes h :: "real"   and n :: "nat"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "0 < h"   and "0 < n"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)" shows "\<exists>t>0. t < h \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
vars_iff_subtermeq_set: fixes x :: "'c"   and M :: "('d, 'c) Term.term set" shows "(x \<in> fv\<^sub>s\<^sub>e\<^sub>t M) = (Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M)"
qSwapAbs_preserves_qGoodAbs1: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGoodAbs A" shows "qGoodAbs (A $[[x \<and> y]]_zs)"
take_drop_update_first: fixes j :: "nat"   and ds :: "'c list"   and cs :: "'c list" assumes "j < length ds"   and "length cs = length ds" shows "(take j ds @ drop j cs)[j := ds ! j] = take (Suc j) ds @ drop (Suc j) cs"
wf_syscls_class_Thread: fixes P :: "'a prog" assumes "wf_syscls P" shows "\<exists>C fs ms. class P Thread = \<lfloor>(C, fs, ms)\<rfloor>"
s_1: fixes i :: "nat" shows "dist (f (s i)) p < 1 / real (Suc i)"
attractor_has_strategy_single: fixes W :: "'a set"   and v0 :: "'a"   and p :: "Player" assumes "W \<subseteq> V"   and "v0 \<in> attractor p W" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v0 (attractor p W) W"
complete_classes: fixes A :: "'a set" assumes "is_closed A" shows "A = closure_of A"
n_def: shows "n = vcard (xs #\<^sub>\<circ> x)"
eq_fract_2: fixes a :: "'c" shows "Fraction_Field.Fract a (0::'c) = Fraction_Field.Fract (0::'c) (1::'c)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" assumes "orthogonal_current \<Gamma> f S" obtains "\<And>x. \<lbrakk>x \<in> S; x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> weight \<Gamma> x \<le> d_IN f x" and "\<And>x. \<lbrakk>x \<in> S; x \<in> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> d_OUT f x = weight \<Gamma> x" and "\<And>u v. \<lbrakk>v \<in> RF\<^bsub>\<Gamma>\<^esub> S; u \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S\<rbrakk> \<Longrightarrow> f (u, v) = 0"
term_variants_Fun_4: shows "Fun f T = t \<cdot> \<delta>"
intros: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and tl :: "'b"   and s' :: "'a"   and \<tau>move :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "trsys s tl s'"   and "\<tau>move s tl s'" shows "\<tau>trsys.silent_move trsys \<tau>move s s'"
none_eqvt: fixes pi :: "('b \<times> 'b) list" shows "pi \<bullet> None = None"
adds_lcs: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus s (lcs s t)"
reset_trap_mod_def: fixes b :: "bool"   and s :: "'b sparc_state" shows "reset_trap_mod b s \<equiv> s\<lparr>state_var := write_reset_trap b (state_var s)\<rparr>"
set_mult_consistent: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and N :: "'c set"   and K :: "'c set" shows "N <#>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> K = N <#>\<^bsub>G\<^esub> K"
representation_ne_zero: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set"   and v :: "'e"   and b :: "'e" assumes "module scale"   and "module.representation scale basis v b \<noteq> (0::'d)" shows "b \<in> basis"
less_1: fixes p :: "'a list" assumes "length p < length pa__"   and "polynomial K p" shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> p = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)"
args_mk: fixes f :: "'b"   and ts :: "'a list" assumes "(f, length ts) \<in> F" shows "args (mk f ts) = ts"
assms: shows "z\<^sup>2 \<noteq> - 1"
causal_delivery: fixes m2 :: "'a"   and j :: "nat"   and m1 :: "'a" assumes "Deliver m2 \<in> set (history j)"   and "hb m1 m2" shows "Deliver m1 \<sqsubset>\<^sup>j Deliver m2"
field_differentiable_at_within: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and S :: "'a set" assumes "f field_differentiable at x" shows "f field_differentiable at x within S"
simps_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Le x1 = Le y1) = (x1 = y1)"
count_mset_set_3: fixes x :: "'d"   and A :: "'d set" assumes "x \<notin> A" shows "count (mset_set A) x = 0"
awalk_cyc_decompE: fixes p :: "'b list"   and q :: "'b list"   and r :: "'b list"   and s :: "'b list"   and u :: "'a"   and v :: "'a" assumes "awalk_cyc_decomp p = (q, r, s)"   and "awalk u p v"   and "\<not> distinct (awalk_verts u p)" obtains "p = q @ r @ s" and "distinct (awalk_verts u q)" and "\<exists>w. awalk u q w \<and> awalk w r w \<and> awalk w s v" and "closed_w r"
monitor_split: fixes v :: "'a option list"   and k :: "'b"   and i :: "nat"   and \<pi> :: "(char list \<times> 'a list) prefix" assumes "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"   and "k \<in> strategy v" shows "((i, v) \<in> M (splitter \<pi> k)) = ((i, v) \<in> M \<pi>)"
refl_on_domain: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "refl_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
fmpred_drop_fset: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and A :: "'a fset" assumes "fmpred P m" shows "fmpred P (fmdrop_fset A m)"
currentD_OUT': fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "countable_bipartite_web \<Gamma>"   and "current \<Gamma> f"   and "x \<notin> A \<Gamma>" shows "d_OUT f x = 0"
sin_lt_zero: fixes x :: "real" assumes "pi < x"   and "x < 2 * pi" shows "sin x < 0"
Cnotzero_imp_not_empty: fixes r :: "('c \<times> 'c) set" assumes "(r, czero) \<notin> ordIso \<and> Card_order r" shows "Field r \<noteq> {}"
dg_digraph_if_ge_Limit: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<beta> :: "V" assumes "digraph \<alpha> \<CC>"   and "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "digraph \<beta> \<CC>"
is_span: shows "C.span \<nu>.cod.leg0 \<nu>.cod.leg1"
inducts: fixes P :: "'d list \<Rightarrow> bool"   and list :: "'d list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
Goal: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>P' :: "'b"   and Qa :: "('a, 'b, 'c) psi"   and \<alpha>' :: "'a action"   and Q'a :: "('a, 'b, 'c) psi"   and Pa :: "('a, 'b, 'c) psi"   and A\<^sub>P' :: "name list" assumes "\<Psi>' \<otimes> \<Psi>\<^sub>P' \<rhd> Qa \<longmapsto> \<alpha>' \<prec> Q'a"   and "extractFrame Pa = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "bn \<alpha>' \<sharp>* Pa"   and "bn \<alpha>' \<sharp>* subject \<alpha>'"   and "A\<^sub>P' \<sharp>* \<Psi>'"   and "A\<^sub>P' \<sharp>* Qa"   and "A\<^sub>P' \<sharp>* \<alpha>'"   and "distinct A\<^sub>P'" shows "\<Psi>' \<rhd> Pa \<parallel> Qa \<longmapsto> \<alpha>' \<prec> Pa \<parallel> Q'a"
MtransC_induct2: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state"   and phi :: "('test, 'atom) com \<Rightarrow> 'state \<Rightarrow> ('test, 'atom) com \<Rightarrow> 'state \<Rightarrow> bool" assumes "(c, s) \<rightarrow>*c (c', s')"   and "\<And>c s. phi c s c s"   and "\<And>c s c' s' c'' s''. \<lbrakk>(c, s) \<rightarrow>*c (c', s'); phi c s c' s'; (c', s') \<rightarrow>c (c'', s'')\<rbrakk> \<Longrightarrow> phi c s c'' s''" shows "phi c s c' s'"
ac_Zp_def: fixes x :: "nat \<Rightarrow> int" shows "ac_Zp x = normalizer (nat (ord_Zp x)) x"
t_fourtimes_def: shows "t_fourtimes = adjust0 t_fourtimes_compile"
chart_inj_on: fixes x :: "'a"   and c :: "('a, 'b) chart"   and y :: "'a" assumes "x \<in> domain c"   and "y \<in> domain c" shows "(apply_chart c x = apply_chart c y) = (x = y)"
inject_3: fixes x31 :: "POPLmark.type"   and x32 :: "trm"   and y31 :: "POPLmark.type"   and y32 :: "trm" shows "((\<lambda><:x31. x32) = (\<lambda><:y31. y32)) = (x31 = y31 \<and> x32 = y32)"
elims_1: fixes x :: "'c set"   and xa :: "('d, 'c) strand_step list"   and y :: "bool" assumes "wf\<^sub>s\<^sub>t x xa = y" obtains V :: "'c set" where "x = V__" and "xa = []" and "y"   | V :: "'c set" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) strand_step list" where "x = Va__" and "xa = receive\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__" and "y = (fv t__ \<subseteq> Va__ \<and> wf\<^sub>s\<^sub>t Va__ S__)"   | V :: "'c set" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) strand_step list" where "x = Vb__" and "xa = send\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__" and "y = wf\<^sub>s\<^sub>t (Vb__ \<union> fv ta__) Sa__"   | V :: "'c set" and s :: "('d, 'c) Term.term" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) strand_step list" where "x = Vc__" and "xa = \<langle>assign: s__ \<doteq> tb__\<rangle>\<^sub>s\<^sub>t # Sb__" and "y = (fv tb__ \<subseteq> Vc__ \<and> wf\<^sub>s\<^sub>t (Vc__ \<union> fv s__) Sb__)"   | V :: "'c set" and s :: "('d, 'c) Term.term" and t :: "('d, 'c) Term.term" and S :: "('d, 'c) strand_step list" where "x = Vd__" and "xa = \<langle>check: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # Sc__" and "y = wf\<^sub>s\<^sub>t Vd__ Sc__"   | V :: "'c set" and uu_ :: "'c list" and uv_ :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" and S :: "('d, 'c) strand_step list" where "x = V" and "xa = \<forall>uu_\<langle>\<or>\<noteq>: uv_\<rangle>\<^sub>s\<^sub>t # S" and "y = wf\<^sub>s\<^sub>t V S"
coboundedI2: fixes c :: "'c"   and b :: "'c"   and a :: "'c" assumes "c \<le> b" shows "c \<le> sup a b"
inverse_unique: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v' :: "'c"   and v :: "'c" assumes "Group_Theory.monoid M composition unit"   and "composition u v' = unit"   and "composition v u = unit"   and "u \<in> M"   and "v \<in> M"   and "v' \<in> M" shows "v = v'"
hyps_3: assumes "labels prog l c" shows "\<exists>l'. prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto>* \<langle>c'',s'',l'\<rangle> \<and> labels prog l' c''"
subst_bv_no_loose_unchanged: fixes lev :: "nat"   and t :: "Core.term"   and v :: "Core.term" assumes "\<And>x. lev \<le> x \<Longrightarrow> \<not> loose_bvar1 t x"   and "is_variable v" shows "subst_bv1 t lev v = t"
assoc: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "dbm_add (dbm_add a b) c = dbm_add a (dbm_add b c)"
concrete_category_axioms: shows "concrete_category Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>x. x) (\<lambda>_ _ x _ _. x)"
simps_11: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" shows "af_letter (\<phi> M\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> and\<^sub>n af_letter \<phi> \<nu> or\<^sub>n (\<phi> M\<^sub>n \<psi>)"
fail_state_def: fixes fail :: "'d" shows "fail_state fail = StateT (\<lambda>s. fail)"
row_bra_vec: fixes v :: "complex vec" shows "row (bra_vec v) 0 = vec (dim_vec v) (\<lambda>i. cnj (v $ i))"
preserves_ide: fixes a :: "'a" assumes "S.ide a" shows "S.ide (R a)"
primitive_extractor_correct_5: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr"   and P :: "'d \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "\<forall>disc2 sel2. normalized_n_primitive (disc2, sel2) P m \<longrightarrow> normalized_n_primitive (disc2, sel2) P ms"
nonneg_lincomb_list_def: fixes n :: "nat"   and c :: "nat \<Rightarrow> 'b"   and Vs :: "'b vec list"   and b :: "'b vec" shows "gram_schmidt.nonneg_lincomb_list n c Vs b = (module.lincomb_list (module_vec TYPE('b) n) c Vs = b \<and> (\<forall>i<length Vs. (0::'b) \<le> c i))"
distinct_list_split: fixes xs :: "'a list"   and xa :: "'a list"   and x :: "'a"   and ya :: "'a list"   and xb :: "'a list"   and yb :: "'a list" assumes "distinct xs"   and "xs = xa @ x # ya"   and "xs = xb @ x # yb" shows "xa = xb \<and> ya = yb"
associated_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" shows "a \<sim>\<^bsub>G\<^esub> b = (a divides\<^bsub>G\<^esub> b \<and> b divides\<^bsub>G\<^esub> a)"
\<rho>_agreement: shows "MC.\<rho> = (\<lambda>f. if arr f then f \<cdot> \<r>[local.dom f] else null)"
case_eq_if: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and prod :: "'h \<times> 'i" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
injectivity: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" assumes "to_int_mod_ring x = to_int_mod_ring y" shows "x = y"
vertex_map: fixes x :: "'a" assumes "x \<in> \<Union> X" shows "g x \<in> \<Union> X"
cart2_append1_non_zero: fixes p :: "proj2" assumes "z_non_zero p" shows "cart2_append1 p \<noteq> 0"
try_spmf_bind_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c spmf"   and q :: "'c spmf" shows "TRY p \<bind> f ELSE q = p \<bind> (\<lambda>x. TRY f x ELSE q)"
gen_entries_simps_2: fixes kv :: "'c \<times> 'd"   and t :: "('c, 'd) rbt"   and kvts :: "(('c \<times> 'd) \<times> ('c, 'd) rbt) list" shows "RBT_Impl.gen_entries ((kv, t) # kvts) rbt.Empty = kv # RBT_Impl.gen_entries kvts t"
elims_2: fixes x :: "rat \<times> rat"   and xa :: "real" assumes "in_interval x xa" obtains l :: "rat" and r :: "rat" and x :: "real" where "xb__ = (l, r)" and "xa = x" and "real_of_rat l \<le> x \<and> x \<le> real_of_rat r"
tendsto_cong_limit: fixes f :: "'d \<Rightarrow> 'e"   and l :: "'e"   and F :: "'d filter"   and k :: "'e" assumes "(f \<longlongrightarrow> l) F"   and "k = l" shows "(f \<longlongrightarrow> k) F"
pxy: fixes j :: "nat" assumes "j < length x" shows "P (x ! j) (y ! j)"
set_us: shows "set us = snd ` set (zip (map fst txs) us)"
count_union: fixes M :: "'a multiset"   and N :: "'a multiset"   and a :: "'a" shows "count (M + N) a = count M a + count N a"
the_Class_eq: fixes C :: "qtname" shows "the_Class (Class C) = C"
simps_1: fixes x :: "'b"   and xs :: "'b list" shows "insert_nth 0 x xs = x # xs"
tObsC_abs_def: fixes t :: "('a, 'es, 'as) BEState Trace" shows "tObsC_abs t \<equiv> rel_ext (\<lambda>uu_. \<exists>t'. uu_ = (tFirst t', tLast t') \<and> t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t)"
that: shows "is_denormal_Float TYPE(('e, 'f) IEEE.float) x"
mult_oner: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c" assumes "class.ab_near_semiring_one plus times one" shows "times x one = x"
exists_exactly_one_following_state: fixes ev :: "('p, 's, 'm) event"   and c :: "('p, 's, 'm) configuration" assumes "can_occur ev c" shows "\<exists>!c'. c \<turnstile> ev \<mapsto> c'"
select_convs_3: fixes xc :: "int"   and yc :: "int"   and more :: "'a" shows "point2d.more \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = more"
Compl_anti_mono: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "- B \<subseteq> - A"
set_shadow_root_locs_def: shows "set_shadow_root_locs = (\<lambda>element_ptr. all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt_update))"
gauss_jordan_single_4: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C" shows "\<exists>P Q. C = P * A \<and> P \<in> carrier_mat nr nr \<and> Q \<in> carrier_mat nr nr \<and> P * Q = 1\<^sub>m nr \<and> Q * P = 1\<^sub>m nr"
atLeastLessThanSuc_atLeastAtMost: fixes l :: "nat"   and u :: "nat" shows "{l..<Suc u} = {l..u}"
normal_form_normal_form: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "normal_form F (normal_form F p) = normal_form F p"
psp2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> B co B'" shows "F \<in> B' \<inter> A \<longmapsto> B \<inter> A' \<union> (B' - B)"
post_fun_def: fixes p :: "'b"   and q :: "'b" shows "post_fun p q = (if p \<sqsubseteq> q then \<top> else \<bottom>)"
analz_Un_AencSet: fixes K :: "msg set"   and G :: "msg set"   and H :: "msg set" assumes "K \<subseteq> range LtK"   and "Enc_keys_clean (G \<union> H)" shows "analz (AencSet G K \<union> H) \<subseteq> AencSet G K \<union> analz (G \<union> H)"
axioms_1: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "symmetry_functor A1 A2" shows "category A1"
assms: shows "\<forall>(d, e, f)\<in>set les. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0"
bou: shows "bounded {x. \<not> (f x = x \<and> g x = x)}"
process_prop_6_def: fixes P :: "('b list \<times> 'b set) set \<times> 'b list set" shows "process_prop_6 P \<equiv> \<forall>xs X. xs \<in> snd P \<longrightarrow> (xs, X) \<in> fst P"
cSUP_const: fixes A :: "'d set"   and c :: "'e multiset" assumes "A \<noteq> {}" shows "(\<Squnion>x\<in>A. c) = c"
find_nu_measures_def: shows "find_nu_measures \<equiv> measures [(-) mu, \<lambda>\<nu>. LEAST i. seq (\<nu> + \<nu> + i) = seq \<nu>]"
subst_cls_add_mset: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and L :: "'a literal"   and C :: "'a literal multiset"   and \<sigma> :: "'b" shows "substitution_ops.subst_cls subst_atm (add_mset L C) \<sigma> = add_mset (substitution_ops.subst_lit subst_atm L \<sigma>) (substitution_ops.subst_cls subst_atm C \<sigma>)"
compact_imp_fip: fixes S :: "'a set"   and F :: "'a set set" assumes "compact S"   and "\<And>T. T \<in> F \<Longrightarrow> closed T"   and "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> F\<rbrakk> \<Longrightarrow> S \<inter> \<Inter> F' \<noteq> {}" shows "S \<inter> \<Inter> F \<noteq> {}"
vertical_composite_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "vertical_composite A B F G H \<sigma> \<tau> \<equiv> (category A \<and> category B \<and> functor A B F) \<and> (functor A B G \<and> functor A B H) \<and> natural_transformation A B F G \<sigma> \<and> natural_transformation A B G H \<tau>"
f3: fixes a :: "'a" shows "ad (ad (ad a)) = ad a"
divide_less_0_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
map_distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_map x" shows "x ; (y \<cdot> z) = x ; y \<cdot> x ; z"
snapshot_after_all_prerecording_events: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "trace init t final"   and "\<forall>i'\<ge>i. \<not> prerecording_event t i'"   and "\<forall>j'<i. \<not> postrecording_event t j'"   and "i \<le> length t" shows "state_equal_to_snapshot (S t i) final"