hcju/isabelleps huggingface dataset:
wls_item_simps_3: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp delta inp" shows "goodInp inp"
sc0: shows "sgn (lead_coeff f) * content f \<noteq> 0"
DoI_eq_I: shows "\<F>C.D \<circ> \<F>C.inclusion_of_generators = \<F>C.inclusion_of_generators"
7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> y \<sqinter> z"
add_cancel_right_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
L_transform_formula_Not: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "L_transform_formula (FL_Formula.Not x) = Formula.Not\<^sub>\<alpha> (L_transform_formula x)"
insert_rga_preserves_order: fixes i :: "'a option"   and xs :: "'a list"   and e :: "'a" assumes "i = None \<or> (\<exists>i'. i = Some i' \<and> i' \<in> set xs)"   and "distinct xs" shows "\<exists>pre suf. xs = pre @ suf \<and> insert_rga xs (e, i) = pre @ e # suf"
cdim_raw_def: shows "cdim \<equiv> vector_space.dim (*\<^sub>C)"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>" shows "\<CC>\<lparr>Cod\<rparr> = []\<^sub>\<circ>"
progressing_Cons_Cons: fixes u :: "'b"   and v :: "'b"   and w :: "'b"   and es :: "('b \<times> 'b) list" shows "progressing ((u, v) # (v, w) # es) = (u \<noteq> w \<and> progressing ((v, w) # es))"
sq_norm_vec_greater_0: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'a) < \<parallel>v\<parallel>\<^sup>2) = (v \<noteq> 0\<^sub>v n)"
UN: shows "\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2 \<union> wrap {u}) = \<Union> (P\<^sub>2 \<union> wrap B\<^sub>2) \<union> {u}"
assms_3: fixes A :: "'a trm"   and B :: "'a trm" assumes "P S A"   and "P S B" shows "P S (App A B)"
real_of_int_def: shows "local.real_of_int = Real.real_of_int"
divide_nonpos_nonneg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq zero y" shows "less_eq (divide x y) zero"
hcomplex_mult_minus_one: fixes z :: "complex star" shows "- 1 * z = - z"
execute_while_unfolds_2: fixes h :: "heap"   and f :: "'b Heap" assumes "success p h"   and "\<not> cond p h" shows "execute (while p f) h = execute (return ()) h"
ex_imp_eqI: fixes P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and Q' :: "'b \<Rightarrow> bool" assumes "P = P'"   and "\<And>x. P x \<Longrightarrow> Q x = Q' x" shows "(\<exists>x. P x \<and> Q x) = (\<exists>x. P' x \<and> Q' x)"
isNSCont_LIM_iff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "isNSCont f a = f \<midarrow>a\<rightarrow> f a"
dgrad_p_set_le_minus: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "dgrad_p_set_le d {p - q} {p, q}"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and signalsInit2 :: "signal \<Rightarrow> nat"   and signalsResp2 :: "signal \<Rightarrow> nat"   and more :: "'a" shows "signalsResp2 \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, \<dots> = more\<rparr> = signalsResp2"
mop_prio_peek_min_def: fixes prio :: "'a \<Rightarrow> 'b"   and b :: "'a multiset" shows "mop_prio_peek_min prio b \<equiv> ASSERT (b \<noteq> {#}) \<bind> (\<lambda>_. SPEC (\<lambda>v. v \<in># b \<and> (\<forall>v'\<in>#b. prio v \<le> prio v')))"
PiE_mem2: fixes f :: "'h \<Rightarrow> 'i"   and S :: "'h set"   and T :: "'i set"   and x :: "'h" assumes "f \<in> S \<rightarrow>\<^sub>E T"   and "x \<in> S" shows "f x \<in> T"
doesnt_read_or_modify_vars_def: fixes c :: "'Com"   and X :: "'Var set" shows "doesnt_read_or_modify_vars c X = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> (\<forall>x\<in>X. \<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>))"
pref_mod_power: fixes u :: "'b list"   and w :: "'b list"   and l :: "nat" assumes "u <p w \<^sup>@ l" obtains k :: "nat" and z :: "'b list" where "k < l" and "z <p w" and "w \<^sup>@ k \<cdot> z = u"
length_enumerate: fixes n :: "nat"   and xs :: "'g list" shows "length (enumerate n xs) = length xs"
implmt1_methd: fixes C :: "qtname"   and I :: "qtname"   and G :: "prog"   and im :: "qtname \<times> mhead"   and sig :: "sig" assumes "G\<turnstile>C\<leadsto>1I"   and "wf_prog G"   and "im \<in> DeclConcepts.imethds G I sig" shows "? cm:methd G C sig: \<not> is_static cm \<and> \<not> is_static im \<and> G\<turnstile>resTy cm\<preceq>resTy im \<and> accmodi im = Public \<and> accmodi cm = Public"
acomplete_def: fixes t :: "'b tree" shows "acomplete t = (height t - min_height t \<le> 1)"
abc_acd_abd: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[a c d]]" shows "[[a b d]]"
simps_1: fixes c :: "'d" shows "get_const (Le c) = c"
subadditiveD: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> 'c"   and x :: "'b set"   and y :: "'b set" assumes "subadditive M f"   and "x \<inter> y = {}"   and "x \<in> M"   and "y \<in> M" shows "f (x \<union> y) \<le> f x + f y"
eff_def: fixes eff' :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset option"   and r :: "'a"   and s :: "'b"   and sl :: "'b fset" shows "i.eff eff' r s sl \<equiv> eff' r s = Some sl"
assms_2: shows "atom ` domA \<Delta> \<sharp>* \<Gamma>"
that: fixes b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s"   and "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
fold_fun_left_comm: fixes A :: "'a set"   and x :: "'a"   and z :: "'a" assumes "finite A" shows "sup x (Finite_Set.fold sup z A) = Finite_Set.fold sup (sup x z) A"
sup_mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "a \<squnion> b \<le> c \<squnion> d"
ord_isom_trans: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and F :: "('g, 'h) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g" assumes "Order D"   and "Order E"   and "Order F"   and "ord_isom D E f"   and "ord_isom E F g" shows "ord_isom D F (compose (carrier D) g f)"
decl: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt"   and mu :: "'a ModeUpd"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and E :: "('a, 'c, 'd) Stmt list" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c, sifum_lang.update_modes mu mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((sifum_lang.cxt_to_stmt E (c@[mu]), mds), mem), (sifum_lang.cxt_to_stmt E c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
m_inv_chain_group: fixes a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'b topology" assumes "Poly_Mapping.keys a \<subseteq> singular_simplex_set p X" shows "inv\<^bsub>chain_group p X\<^esub> a = - a"
cauchy_imp_bounded: fixes s :: "nat \<Rightarrow> 'c" assumes "Cauchy s" shows "bounded (range s)"
degree_u_mod_irreducible\<^sub>d_factor_0: fixes v :: "'b mod_ring poly"   and u :: "'b mod_ring poly"   and U :: "'b mod_ring poly set"   and fi :: "'b mod_ring poly" assumes "v \<in> {v. [v ^ CARD('b) = v] (mod u)}"   and "finite U"   and "u = \<Prod>U"   and "U \<subseteq> {q. irreducible q \<and> monic q}"   and "fi \<in> U" shows "degree (v mod fi) = 0"
assms: shows "finite {a. \<exists>b. f a b \<noteq> (0::'c)}"
less_eq_def: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "(x \<subseteq> y) = (x \<union> y = y)"
assms_2: shows "\<turnstile> imply (B # G) C"
list: fixes l :: "real list" assumes "length l = x1" shows "(drop (z + x1) l @ drop (z + x1 - length l) L)[var := B / C] = (drop z L)[var := B / C]"
backjumpLevelZero: fixes l :: "Literal"   and C :: "Literal list"   and M :: "(Literal \<times> bool) list" assumes "isLastAssertedLiteral (opposite l) (oppositeLiteralList C) (elements M)"   and "0 < elementLevel (opposite l) M"   and "set C = {l}" shows "isBackjumpLevel 0 l C M"
unit_imp_mod_eq_0: fixes b :: "'a"   and a :: "'a" assumes "is_unit b" shows "a mod b = (0::'a)"
pow_divides_pow_iff: fixes n :: "nat"   and a :: "'a"   and b :: "'a" assumes "0 < n" shows "(a ^ n dvd b ^ n) = (a dvd b)"
twoCisTypeI_10: shows "(\<lambda>y. twoC (0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
ub2: shows "\<forall>x. poly p x = 0 \<longrightarrow> x < ub2"
sum_of_unit_vec_length: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and c :: "complex" assumes "i < n"   and "j < n"   and "i \<noteq> j" shows "complex_of_real (\<parallel>unit_vec n i + c \<cdot>\<^sub>v unit_vec n j\<parallel>\<^sup>2) = 1 + cnj c * c"
initialE: fixes a :: "'a"   and b :: "'a" assumes "initial a"   and "ide b" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<And>f'. \<guillemotleft>f' : a \<rightarrow> b\<guillemotright> \<Longrightarrow> f' = f"
iMODb_iIN_subset_same: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<subseteq> [r\<dots>,m * c]"
simps_1: fixes x21 :: "'b"   and x22 :: "'b llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
UnionE: fixes A :: "'e"   and C :: "'e set set" assumes "A \<in> \<Union> C" obtains X :: "'e set" where "A \<in> X" and "X \<in> C"
cmult_in_iff: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. c * f x) \<in> L F g) = (f \<in> L F g)"
zero_in_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "0 \<in> punit.dgrad_p_set d m"
bb_aa_event: shows "\<forall>\<^sub>F k in sequentially. bb (k + 1) / aa (k + 1) < 1 / M * (bb k / aa k)"
Collect_mem_eq: fixes A :: "'b set" shows "\<lbrace>\<acute>(\<in>) A\<rbrace> = A"
simps_1: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d" shows "strips_operator.precondition_of \<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = precondition_of"
icard_finite: fixes A :: "'a set" assumes "finite A" shows "icard A = enat (card A)"
ctwo_ordLeq_Cinfinite: fixes r :: "('d \<times> 'd) set" assumes "Cinfinite r" shows "ctwo \<le>o r"
subst_domain_subst_compose: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain (\<sigma> \<circ>\<^sub>s \<tau>) = subst_domain \<sigma> - {x. \<exists>y. \<sigma> x = Var y \<and> \<tau> y = Var x} \<union> (subst_domain \<tau> - subst_domain \<sigma>)"
intro: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'c"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "RModule R smult M"   and "GroupHom M T"   and "RModuleHom_axioms R smult M smult' T" shows "RModuleHom R smult M smult' T"
add_diff_assoc: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "i + (j - k) = i + j - k"
ProofStepInduct_axioms: shows "ProofStepInduct G v0 v1 paths P_new sep_size"
finite_distinct_list: fixes A :: "'g set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> distinct xs"
assertionAux_axioms: shows "assertionAux (\<otimes>) SImp' (\<one>) SChanEq'"
secure_def: shows "secure \<equiv> \<forall>\<sigma> \<sigma>'. \<sigma> =\<^sub>L \<sigma>' \<longrightarrow> obs \<sigma> \<approx> obs \<sigma>' \<and> (terminates \<sigma> \<longrightarrow> obs \<sigma>' \<lesssim> obs \<sigma>)"
wf: shows "wf_formula n (FAnd \<phi> \<psi>)"
cofactorM_row_def: fixes A :: "(('c, 'd) vec, 'd) vec"   and i :: "'d" shows "cofactorM_row A i = (\<chi>j. cofactorM A $ i $ j)"
select_convs_3: fixes set_dom :: "'b set"   and set_func :: "'b \<Rightarrow> 'b"   and set_cod :: "'b set"   and more :: "'c" shows "set_cod \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = set_cod"
compute_SCC_def: shows "compute_SCC \<equiv> let so = ([], {}) in FOREACH\<^bsup>cscc_outer_invar\<^esup> V0 (\<lambda>v0 (l, D0). if v0 \<notin> D0 then let s = (l, initial v0 D0) in WHILE\<^sub>T\<^bsup>cscc_invar v0 D0\<^esup> (\<lambda>(l, p, D, pE). p \<noteq> []) (\<lambda>(l, p, D, pE). select_edge (p, D, pE) \<bind> (\<lambda>(vo, p, D, pE). ASSERT (p \<noteq> []) \<bind> (\<lambda>_. case vo of None \<Rightarrow> ASSERT (pE \<inter> last p \<times> UNIV = {}) \<bind> (\<lambda>_. let V = last p; (p, D, pE) = pop (p, D, pE); l = V # l in RETURN (l, p, D, pE)) | Some v \<Rightarrow> if v \<in> \<Union> (set p) then RETURN (l, collapse v (p, D, pE)) else if v \<notin> D then RETURN (l, push v (p, D, pE)) else RETURN (l, p, D, pE)))) s \<bind> (\<lambda>(l, p, D, pE). ASSERT (p = [] \<and> pE = {}) \<bind> (\<lambda>_. RETURN (l, D))) else RETURN (l, D0)) so \<bind> (\<lambda>(l, D). RETURN l)"
inject_1: fixes x1 :: "('b, 'c, 'd) Tree set['b]"   and y1 :: "('b, 'c, 'd) Tree set['b]" shows "(tConj x1 = tConj y1) = (x1 = y1)"
interior_of_subtopology_mono: fixes S :: "'c set"   and T :: "'c set"   and U :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T"   and "T \<subseteq> U" shows "subtopology X U interior_of S \<subseteq> subtopology X T interior_of S"
collision_trim_subset: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and S :: "real set"   and T :: "real set" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"   and "safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> S"   and "\<not> safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> T"   and "T \<subseteq> S" shows "safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> (S - T)"
select_def: fixes k :: "nat"   and xs :: "'b list" shows "select k xs = sort xs ! k"
gfp_ordinal_induct: fixes f :: "'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; gfp f \<le> S\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Inf M)" shows "P (gfp f)"
TVar_def: fixes x1 :: "nat" shows "TVar x1 \<equiv> Abs_ty (TVar_Rep x1)"
mult_zero_r: fixes a :: "nat \<Rightarrow> int" shows "\<zero> \<otimes> a = \<zero>"
at_set_avoiding_aux: fixes Xs :: "'a set"   and As :: "'a set"   and c :: "'b" assumes "at TYPE('a)"   and "Xs \<subseteq> As"   and "finite As"   and "finite (supp c)" shows "\<exists>pi. (pi \<bullet> Xs) \<sharp>* c \<and> pi \<bullet> Xs \<inter> As = {} \<and> set pi \<subseteq> Xs \<times> pi \<bullet> Xs"
app_def: fixes i :: "'c instr"   and P :: "'d prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "(ty list \<times> ty err list) option" shows "app i P mxs T\<^sub>r pc mpc xt t \<equiv> case t of None \<Rightarrow> True | \<lfloor>\<tau>\<rfloor> \<Rightarrow> app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt t). pc' < mpc)"
eqvt_def: fixes X :: "'d set" shows "eqvt X \<equiv> \<forall>x\<in>X. \<forall>p. p \<bullet> x \<in> X"
lspasl_iu_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.tern_rel plus sep_disj h1 h2 zero = (heap_sep_algebra.tern_rel plus sep_disj zero zero zero \<and> h1 = zero \<and> h2 = zero)"
af_ltl_continuation: fixes w :: "'a set list"   and w' :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "(w \<frown> w') \<Turnstile>\<^sub>n \<phi> = w' \<Turnstile>\<^sub>n af \<phi> w"
WT_gpv_inline1_2: fixes out :: "'call'"   and rpv :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and input :: "'ret'" assumes "Inr (out, rpv, rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "input \<in> responses_\<I> \<I>' out" shows "\<I>' \<turnstile>g rpv input \<surd>"
arr_cod: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.arr (CopxC.cod f)"
power_irr_def: fixes p :: "'a mod_ring poly"   and n :: "nat" shows "power_irr p n = p ^ n mod f"
Sup_pres_multr: fixes y :: "'b" shows "Sup_pres (\<lambda>z. z \<cdot> y)"
rep_eq: fixes x :: "'f \<Longrightarrow> 'e" shows "\<lbrakk>x\<rbrakk>\<^sub>e = get\<^bsub>x\<^esub>"
list_orderI: fixes ops :: "('b \<times> 'b option) list"   and xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and y :: "'b"   and zs :: "'b list" assumes "interp_ins ops = xs @ [x] @ ys @ [y] @ zs" shows "list_order ops x y"
CollectI: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "P a" shows "a \<in> {x. P x}"
moebius_pt_comp_inv_right: fixes M :: "moebius"   and z :: "complex_homo" shows "moebius_pt M (moebius_pt (moebius_inv M) z) = z"
inv_comp_left_2: fixes g :: "'a"   and f :: "'a" assumes "C.iso (g \<cdot>\<^sub>C f)"   and "C.iso g" shows "C.iso f"
f: shows "AE x in S \<Otimes>\<^sub>M T. 0 \<le> f x"
plus_vwb_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens x"   and "vwb_lens y"   and "x \<bowtie> y" shows "vwb_lens (x +\<^sub>L y)"
evalC_inj_on_hbase: fixes n :: "C"   and b :: "nat"   and m :: "C" assumes "n \<in> hbase b"   and "m \<in> hbase b"   and "evalC b n = evalC b m" shows "n = m"
neg_numeral_le_one: fixes 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus m)) one"
indspace_def: fixes smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set" shows "aezfun_scalar_mult.indspace smult G FH V = addfunset FH V \<inter> aezfun_scalar_mult.smultfunset smult G FH"
mwb_UNIV_src_is_vwb_lens: fixes X :: "'b \<Longrightarrow> 'c" assumes "mwb_lens X"   and "\<S>\<^bsub>X\<^esub> = UNIV" shows "vwb_lens X"
gcsort_count_intro: fixes f :: "'c \<Rightarrow> nat"   and t :: "nat \<times> nat list \<times> 'c list"   and x :: "'c" assumes "count_inv f t" shows "count (mset (gcsort_out t)) x = f x"
insert: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
assms: fixes x :: "'a"   and y :: "'b"   and \<pi> :: "perm" assumes "r x y" shows "r (\<pi> \<bullet> x) (\<pi> \<bullet> y)"
zero_stalk_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d"   and V :: "'d set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.zero_stalk \<FF> \<rho> zero_str I V \<equiv> direct_lim.class_of \<FF> \<rho> I V (zero_str V)"
is_sublist_add: fixes as :: "'a list"   and bs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "is_sublist as bs" shows "is_sublist as (xs @ bs @ ys)"
sol_nonstrict: shows "A\<^sub>1 *\<^sub>v x \<le> b\<^sub>1"
antisym: fixes b :: "'a"   and a :: "'a" assumes "less_eq0 b a"   and "less_eq0 a b" shows "a = b"
approx_1: shows "einterval a b = (\<Union>i. {l i..u i})"
mm_equiv_low_eq: fixes c\<^sub>1 :: "'Com\<^sub>A"   and mds :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com\<^sub>A"   and mem\<^sub>2 :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "abs.mm_equiv_abv \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A" shows "abs.low_mds_eq mds mem\<^sub>1 mem\<^sub>2"
ht_at_phl_cond: fixes u :: "'a"   and v :: "'a"   and x :: "'a"   and z :: "'a"   and w :: "'a"   and y :: "'a" assumes "\<lbrace>t u \<cdot> t v\<rbrace>x\<lbrace>z\<rbrace>"   and "\<lbrace>t u \<cdot> t w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>t u\<rbrace>t v \<cdot> x + t w \<cdot> y\<lbrace>z\<rbrace>"
is_serial_solution_for_problem_def: fixes \<Pi> :: "('a, 'b) strips_problem_scheme"   and \<pi> :: "'a strips_operator list" shows "is_serial_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_serial_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>)) \<pi>"
prems_2: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
simps_1: fixes sp :: "('b, 'a) sp\<^sub>\<nu>"   and s :: "'b stream" shows "shd (run\<^sub>\<nu> sp s) = fst (fst (run\<^sub>\<mu> (out sp) s))"
AXIOMSn_def: fixes B :: "ty" shows "AXIOMSn B \<equiv> {x:Ax y b |x y b. True}"
scaleHR_left_diff_distrib: fixes a :: "real star"   and b :: "real star"   and x :: "'a star" shows "scaleHR (a - b) x = scaleHR a x - scaleHR b x"
is_refT_def: shows "is_refT \<equiv> lfp (\<lambda>p x. x = NT \<or> (\<exists>C. x = Class C) \<or> (\<exists>A. x = A\<lfloor>\<rceil>))"
isomorphicE: fixes a :: "'d"   and a' :: "'d" assumes "D.isomorphic a a'" obtains f :: "'d" where "\<guillemotleft>f : a \<rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f"
run_converter_def: fixes converter :: "('a, 'b, 'c, 'd) converter" shows "run_converter converter = (case converter of Converter x \<Rightarrow> x)"
simps: fixes i :: "nat"   and n :: "nat" shows "local.b i n = (if n \<le> i then [:local.c n:] else local.b (Suc i) n * local.X i + [:local.c i:])"
interior_cbox: fixes a :: "'a"   and b :: "'a" shows "interior (cbox a b) = box a b"
atlas_is_atlas: fixes a1 :: "('a, 'b) chart"   and a2 :: "('a, 'b) chart" assumes "a1 \<in> atlas"   and "a2 \<in> atlas" shows "k-smooth_compat a1 a2"