hcju/isabelleps huggingface dataset:
distinct_7: fixes x21 :: "'a"   and x22 :: "'a"   and x31 :: "'a"   and x32 :: "'a"   and x33 :: "'b" shows "DenyAllFromTo x21 x22 \<noteq> AllowPortFromTo x31 x32 x33"
characteristic_formula_eqvt: fixes p :: "perm"   and P :: "'state" shows "p \<bullet> characteristic_formula P = characteristic_formula (p \<bullet> P)"
simps_1: fixes as :: "'b list" shows "well_valued0 as Leaf = True"
injects_2: fixes discr :: "bool discr"   and discr' :: "bool discr" shows "(B\<cdot>discr = B\<cdot>discr') = (discr = discr')"
left_total_OO: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_total R"   and "left_total S" shows "left_total (R OO S)"
Sigma_empty_iff: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" shows "(Sigma I X = {}) = (\<forall>i\<in>I. X i = {})"
Affine_msum_aform: fixes X :: "'b \<times> 'b pdevs"   and d :: "nat"   and Y :: "'b \<times> 'b pdevs" assumes "degree_aform X \<le> d" shows "Affine (msum_aform d X Y) = {x + y |x y. x \<in> Affine X \<and> y \<in> Affine Y}"
comp_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and G :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> B C G)" shows "S (set_category_given_img.mkArr S img \<AA> B C G) (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkArr S img \<AA> A C (G \<circ> F)"
ceiling_correct: fixes x :: "'b" shows "of_int \<lceil>x\<rceil> - (1::'b) < x \<and> x \<le> of_int \<lceil>x\<rceil>"
agsum_add3mp: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier A"   and "n < 0"   and "0 < m" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
C_wf_trms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set C)"
anycast_simps_8: fixes P :: "'a \<Rightarrow> bool" shows "anycast P \<tau>\<^sub>s = True"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes "a1 \<longrightarrow>x a2" obtains E :: "ctx" and x :: "name" and t :: "lam" and t' :: "lam" where "a1 = E\<lbrakk>App (Lam [x].t) t'\<rbrakk>" and "a2 = E\<lbrakk>t[x::=t']\<rbrakk>"
Cart_lambda_cong: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. f x = g x" shows "vec_lambda f = vec_lambda g"
abs_eq: fixes x :: "int" shows "Uint64 (integer_of_int x) = Abs_uint64 (word_of_int x)"
pt: fixes l :: "nat" shows "positive (tensor_P (proj_psi_l l) (proj_k l))"
prems: shows "vars_change (a # as) vs s = s' # ss"
bounded_linear: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_clinear f" shows "bounded_linear f"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) (K\<langle>N\<rangle>) P [] (\<one>)"
simps_3: fixes v :: "int \<times> int"   and vb :: "int \<times> int"   and vc :: "(int \<times> int) list" shows "closest_pair_code (v # vb # vc) = (let (_, _, x, y) = closest_pair_rec_code (mergesort fst (v # vb # vc)) in (x, y))"
fds_nth_norm_mult_le: fixes f :: "'a fds"   and g :: "'a fds"   and n :: "nat" shows "norm (fds_nth (f * g) n) \<le> fds_nth (fds_norm f * fds_norm g) n"
po_eq_conv: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.po below" shows "(x = y) = (below x y \<and> below y x)"
isReasonAppend: fixes clause :: "Literal list"   and literal :: "Literal"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "isReason clause literal valuation" shows "isReason clause literal (valuation @ valuation')"
rule_sub_lem_1_e: fixes v :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v]" shows "[\<^bold>\<A>\<psi> \<^bold>\<equiv> \<^bold>\<A>\<chi> in v]"
all_forward: fixes P' :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<forall>x. P x"
splus_left_canc: fixes t :: "'a"   and v :: "'a"   and u :: "'a" shows "(local.punit.splus t v = local.punit.splus t u) = (v = u)"
support_not_const: fixes F :: "('b \<Rightarrow> 'a) set" assumes "F \<subseteq> FINFUNC"   and "const \<notin> F" shows "\<forall>f\<in>F. finite (SUPP f) \<and> SUPP f \<noteq> {} \<and> SUPP f \<subseteq> Field s"
rep_eq: fixes x :: "('c, 'd) lifted_boolean_algebra" shows "Rep_lifted_boolean_algebra (- x) = (\<lambda>f. - Rep_lifted_boolean_algebra x f)"
opt_bst_wpl_eq_pair': fixes i :: "int"   and j :: "int" shows "opt_bst_wpl i j = (local.opt_bst i j, local.min_wpl i j)"
is_OK_returns_result_E: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog" assumes "h \<turnstile> ok f" obtains x :: "'n" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
prefixI: fixes ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "ys = xs @ zs" shows "xs \<le> ys"
conj_disj_absorb: fixes P :: "bool"   and Q :: "bool" shows "(P \<and> Q \<or> Q) = Q"
rep_eq: shows "fromQ 1 = \<lparr>mag = 1::'a, dim = QD('c), unit_sys = unit\<rparr>"
trancl_mono_mp: fixes U :: "('a \<times> 'a) set"   and V :: "('a \<times> 'a) set"   and x :: "'a \<times> 'a" assumes "U \<subseteq> V"   and "x \<in> U\<^sup>+" shows "x \<in> V\<^sup>+"
equiv_allrel: fixes A :: "'a set" shows "equiv A (A \<times> A)"
induct: fixes P :: "rat \<Rightarrow> rat poly list \<Rightarrow> rat poly list \<Rightarrow> bool"   and a0 :: "rat"   and a1 :: "rat poly list"   and a2 :: "rat poly list" assumes "\<And>c irr. P c irr []"   and "\<And>c irr p ps. \<lbrakk>degree p = 0 \<Longrightarrow> P (c * coeff p 0) irr ps; \<lbrakk>degree p \<noteq> 0; rational_proper_factor p = None\<rbrakk> \<Longrightarrow> P c (p # irr) ps; \<And>x2. \<lbrakk>degree p \<noteq> 0; rational_proper_factor p = Some x2\<rbrakk> \<Longrightarrow> P c irr (x2 # p div x2 # ps)\<rbrakk> \<Longrightarrow> P c irr (p # ps)" shows "P a0 a1 a2"
pm_of_poly_in_Polys: fixes x :: "'a"   and p :: "'b poly" shows "pm_of_poly x p \<in> P[{x}]"
card_before: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs"   and "x \<in> set xs" shows "card (before x xs) = index xs x"
subst_upd_id_lam: fixes x :: "'b \<Longrightarrow> 'a"   and v :: "('b, 'a) uexpr" shows "(\<lambda>x. x)(x \<mapsto>\<^sub>s v) = [x \<mapsto>\<^sub>s v]"
h: fixes i :: "nat" assumes "k \<le> i"   and "i < n" shows "h i = f i"
stateIsDecreasedByValidTransitions: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "transition stateA stateB F0 decisionVars" shows "(stateB, stateA) \<in> terminationLess F0 decisionVars"
ranking_complement: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and f :: "nat \<times> 'c \<Rightarrow> nat" assumes "finite (nodes A)"   and "w \<in> streams (alphabet A)"   and "ranking A w f" shows "w \<in> language (complement A)"
f_ia: shows "f \<star> \<iota> \<noteq> null"
con_comp_sinks_aux: fixes U :: "'e option set"   and xs :: "'f list"   and p :: "'g \<Rightarrow> 'f"   and I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e"   and q :: "'h \<Rightarrow> 'f" assumes "U \<subseteq> range Some"   and "set xs \<subseteq> range p" shows "sinks_aux I D (the ` U) (map (inv p) xs) = the ` sinks_aux (con_comp_pol I) (con_comp_map D E p q) U xs"
blinfun_compose_assoc: fixes a :: "'e \<Rightarrow>\<^sub>L 'c"   and b :: "'d \<Rightarrow>\<^sub>L 'e"   and c :: "'b \<Rightarrow>\<^sub>L 'd" shows "a o\<^sub>L b o\<^sub>L c = a o\<^sub>L (b o\<^sub>L c)"
of_nat_eq_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = zero) = (m = 0)"
assms_3: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
f0: shows "(f \<longlongrightarrow> f0) (at_right 0)"
has_derivative_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> (0::'b)) (at x within s))"
c_top_welldefTr5: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "N \<bullet> (a \<cdot> b) \<subseteq> c_top G N (N \<bullet> a) (N \<bullet> b)"
sublens_pres_mwb: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e" assumes "mwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "mwb_lens X"
simps_2: fixes x :: "'a"   and e :: "'b \<Rightarrow> 'c"   and xs :: "('a \<times> ('b \<Rightarrow> 'c)) list"   and l :: "'a \<Rightarrow> 'c"   and s :: "'b" shows "postList ((x, e) # xs) l s = (l x = e s \<and> postList xs l s)"
that: fixes i :: "nat" assumes "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (Partial_Recursive.list_decode x)" shows "thesis"
sigma_sets_eqI: fixes A :: "'d set set"   and M :: "'d set"   and B :: "'d set set" assumes "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" shows "sigma_sets M A = sigma_sets M B"
maxsimpchain_append_reduce1: fixes X :: "'b set set"   and xs :: "'b set list"   and ys :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X (xs @ ys)" shows "SimplicialComplex.maxsimpchain X xs"
simpleDefs_phiDefs_disjoint: fixes n :: "'b"   and g :: "'a" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> phiDefs g n = {}"
set_relE2: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x' :: "'b" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x' \<in> s'" obtains x :: "'a" where "x \<in> s" and "(x, x') \<in> R"
simps_1: fixes f :: "'c" shows "Ide \<^bold>\<langle>f\<^bold>\<rangle> = C.ide f"
normalized_ports_singletonize_combine_rst: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b"   and rst :: "'b match_expr"   and m' :: "'b match_expr"   and pt :: "ipt_l4_ports" assumes "wf_disc_sel (disc, sel) C"   and "normalized_n_primitive (disc, sel) (case_ipt_l4_ports (\<lambda>x pts. length pts \<le> 1)) rst"   and "m' \<in> (\<lambda>spt. MatchAnd (Match (C spt)) rst) ` set (singletonize_L4Ports pt)" shows "normalized_n_primitive (disc, sel) (case_ipt_l4_ports (\<lambda>x pts. length pts \<le> 1)) m'"
prime_factorization_unit: fixes x :: "'a" assumes "is_unit x" shows "prime_factorization x = {#}"
inputChainFreshSet: fixes Xs :: "name set"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "Xs \<sharp>* inputChain xvec N P = (\<forall>x\<in>Xs. x mem xvec \<or> x \<sharp> N \<and> x \<sharp> P)"
simps_41: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "safe_formula (formula.Or \<phi> \<psi>) = (fv \<psi> = fv \<phi> \<and> safe_formula \<phi> \<and> safe_formula \<psi>)"
charact_cond_exp': fixes M :: "'b measure"   and Y :: "'b \<Rightarrow> 'c"   and X :: "'b \<Rightarrow> real"   and N :: "'c measure" assumes "finite_measure M"   and "disc_fct Y"   and "integrable M X"   and "Y \<in> M \<rightarrow>\<^sub>M N"   and "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}" shows "AEeq M (real_cond_exp M (fct_gen_subalgebra M N Y) X) (expl_cond_expect M Y X)"
vsubst_preserves_fresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "fresh zs z X"   and "zs \<noteq> ys \<or> z \<noteq> y1" shows "fresh zs z (X #[y1 // y]_ys)"
trancl_subset_Sigma: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "r \<subseteq> A \<times> A" shows "r\<^sup>+ \<subseteq> A \<times> A"
proj_psi_lR_le_post: shows "proj_psi_l R \<le>\<^sub>L post"
subdegree_power: fixes f :: "'a fps"   and n :: "nat" shows "subdegree (f ^ n) = n * subdegree f"
length_s_A': fixes n :: "nat" shows "length (s_A' n) = length init"
add_le_cancel_right: fixes a :: "'d multiset"   and c :: "'d multiset"   and b :: "'d multiset" shows "(a + c \<subseteq># b + c) = (a \<subseteq># b)"
preserves_modes_memD: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>A :: "'Com\<^sub>A"   and mds\<^sub>A :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>A :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "preserves_modes_mem \<R>"   and "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R>" shows "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C \<and> mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"
dual_inf: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (inf x y) = sup (dual x) (dual y)"
flow0_def: fixes x0 :: "'a"   and t :: "real" shows "rev.flow0 x0 t = ll_on_open.flow UNIV (\<lambda>_. - f) X 0 x0 t"
euclidean_size_gcd_le1: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "euclidean_size (gcd a b) \<le> euclidean_size a"
spmf_fast_uniform_ub: fixes n :: "nat"   and x :: "nat" shows "spmf (Fast_Dice_Roll.fast_uniform n) x \<le> (if x < n then 1 / real n else 0)"
surjective: fixes r :: "('a, 'b, 'c, 'd) HOMachine_scheme" shows "r = \<lparr>CinitState = CinitState r, sendMsg = sendMsg r, CnextState = CnextState r, HOcommPerRd = HOcommPerRd r, HOcommGlobal = HOcommGlobal r, \<dots> = HOMachine.more r\<rparr>"
induced_arrow_self: shows "induced_arrow (C.dom p0) (D.mkCone p0 p1) = C.dom p0"
r_e2i_def: shows "r_e2i \<equiv> Cn 1 r_pdec1 [r_e2frame]"
iWHILEIT_def: fixes bind :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and return :: "'d \<Rightarrow> 'b"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and s0 :: "'d" shows "iWHILEIT bind return I b f s0 \<equiv> REC\<^sub>T (WHILEI_body bind return I b f) s0"
pr_Hom_pres_nodes: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" assumes "is_pr_Hom \<phi> Fs\<^sub>1 Fs\<^sub>2" shows "\<phi> ` \<Union> (set ` Fs\<^sub>1) = \<Union> (set ` Fs\<^sub>2)"
GM: shows "get_min_rest ts = (Node r x ts\<^sub>1, ts\<^sub>2)"
assms_3: shows "y \<in> set (local.filter_min xs)"
fr_nGet1a: fixes nReturn :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and k :: "'b"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "'b \<Rightarrow> Config"   and nStore :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "'b \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat" assumes "FlexRayController (nReturn k) recv (nC k) (nStore k) (nSend k) (nGet k)"   and "t mod cycleLength (nC k) mem schedule (nC k)" shows "nGet k t = [t mod cycleLength (nC k)]"
spectral_radius_aux_5: fixes x :: "'b" shows "0 \<le> Inf {root n (norm (x ^ n)) |n. 0 < n}"
formula_of_cnf_def: fixes S :: "'a literal set set" shows "formula_of_cnf S \<equiv> form_of_cnf (sorted_list_of_set (sorted_list_of_set ` S))"
simps_1: fixes f :: "'r \<rightarrow> 's \<rightarrow> 'r"   and z :: "'r" shows "sfoldl\<cdot>f\<cdot>z\<cdot>[::] = z"
shows_sep_assoc: fixes xs :: "'c list"   and elt :: "'c \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list"   and r :: "char list"   and s :: "char list" assumes "\<And>r s. \<forall>x\<in>set xs. elt x r @ s = elt x (r @ s)"   and "\<And>r s. sep r @ s = sep (r @ s)" shows "shows_sep elt sep xs r @ s = shows_sep elt sep xs (r @ s)"
f_pos: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>join M \<noteq> \<infinity>"
topcontinuous_at_atin: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "topcontinuous_at X Y f x = (x \<in> topspace X \<and> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> limitin Y f (f x) (atin X x))"
swap_def: fixes a :: "'c \<Rightarrow> 'd"   and i :: "'c"   and j :: "'c" shows "swap a i j \<equiv> a(i := a j, j := a i)"
partition_onD4: fixes A :: "'c set"   and P :: "'c set set"   and p :: "'c set"   and q :: "'c set"   and x :: "'c" assumes "partition_on A P"   and "p \<in> P"   and "q \<in> P"   and "x \<in> p"   and "x \<in> q" shows "p = q"
in_carrier_4: shows "set (u # Us) \<subseteq> carrier R"
dist_fun_le_dist_first_terms: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd"   and N :: "nat" shows "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n)) |n. n \<le> N} + (1 / 2) ^ N"
proper_it_fold: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and it' :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b" assumes "proper_it it it'" shows "foldli (it (\<lambda>_. True) (\<lambda>x l. l @ [x]) []) = it'"
ring_in_Units_imp_not_zero: fixes a :: "nat \<Rightarrow> int" assumes "\<one> \<noteq> \<zero>"   and "a \<in> Units Zp" shows "a \<noteq> \<zero>"
Q1: shows "Q1 \<in> carrier_mat (dim_col A) (dim_col A)"
vsubstEnv_substEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "good Y" shows "rho &[Y / y]_ys &[y1 // y]_ys = rho &[Y #[y1 // y]_ys / y]_ys"
Graph1_aux: fixes M :: "node list"   and E :: "(nat \<times> nat) list"   and path :: "nat list" assumes "Roots \<subseteq> Blacks M"   and "\<And>i. i < length E \<Longrightarrow> \<not> BtoW (E ! i, M)"   and "1 < length path"   and "path ! (length path - 1) \<in> Roots"   and "\<And>i. i < length path - 1 \<Longrightarrow> \<exists>j<length E. E ! j = (path ! Suc i, path ! i)" shows "M ! (path ! 0) = Black"
LP: shows "(l, s) \<in> \<langle>Ra\<rangle>list_set_rel"
indep_in_indep: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E>.indep_in \<E> X" shows "indep_in \<E> X"
singular_face_restrict: fixes p :: "nat"   and i :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" assumes "0 < p"   and "i \<le> p" shows "singular_face p i (restrict f (standard_simplex p)) = singular_face p i f"
simps_2: fixes f :: "'d \<Rightarrow> 'd" shows "flatten (Basic f) = [Basic f]"
continuous_on_joinpaths_D1: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "continuous_on {0..1} (g1 +++ g2)" shows "continuous_on {0..1} g1"
homography_antihomography_exclusive: fixes f :: "complex_homo \<Rightarrow> complex_homo" assumes "is_antihomography f" shows "\<not> is_homography f"
uminus_closed: fixes a :: "'ag" assumes "a \<in> U" shows "-\<^sub>o\<^sub>w a \<in> U"
minus_to_eq: fixes x :: "'a"   and y :: "'a" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<ominus> y = \<zero>" shows "x = y"
Down_id_eq: fixes x :: "'a nres" shows "\<Down> Id x = x"
proj_set_subset: fixes n :: "'d"   and A :: "('d strand_label \<times> 'e) list" shows "set (proj n A) \<subseteq> set A"
invert_side_of_triangle_2: fixes h :: "'b"   and f :: "'b"   and g :: "'b" assumes "D.arr h"   and "f \<cdot>\<^sub>D g = h"   and "D.iso g" shows "D.seq h (D.inv g) \<and> f = h \<cdot>\<^sub>D D.inv g"
of_int_le_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) zero = (z \<le> 0)"
list_of_oalist_of_list_id: fixes xs :: "('a \<times> 'b) list \<times> 'o" assumes "oalist_inv xs" shows "list_of_oalistx (oalist_of_listx xs) = xs"
hom: shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) (path_image \<gamma>) (\<gamma> \<circ> (\<lambda>z. Arg2pi z / (2 * pi))) (\<gamma> \<circ> (\<lambda>z. Arg2pi z / (2 * pi)))"
ile_eSuc: fixes n :: "enat" shows "n \<le> eSuc n"
lub_upper: fixes M :: "'a set"   and x :: "'a" assumes "M \<in> Chains leq"   and "x \<in> M" shows "(x, lub M) \<in> leq"
deg_add: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "deg R (p \<oplus>\<^bsub>UP R\<^esub> q) \<le> max (deg R p) (deg R q)"
span_insertI: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "'c"   and B :: "'c set"   and r :: "'c" assumes "module scale"   and "p \<in> module.span scale B" shows "p \<in> module.span scale (insert r B)"
prj_simps_9: shows "C.cod trg_trg_f.Prj\<^sub>0 = f.apex"
i_contr: shows "i \<noteq> mod_type_class.to_nat (mod_type_class.from_nat i)"
true_conj_id: fixes P :: "'b \<Rightarrow> bool" shows "(P \<and> true) = P"
simps_2: fixes a :: "'a"   and ts :: "'a FOL_Fitting.term list"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "App a ts[s/k] = App a (ts[s/k])"
simps_9: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Max a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (max_aform_err p a1 a2)))"
assms: shows "GEN_OP ini is_Nil (\<langle>vrel\<rangle>lrel \<rightarrow> bool_rel)"
0: fixes x :: "nat"   and k :: "nat" shows "(\<Sum>n<Suc k. h * (z ^ x * (z ^ (k - n) * (h + z) ^ n))) = (\<Sum>j<Suc k. h * ((h + z) ^ j * z ^ (x + k - j)))"
val_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = val" shows "nsqn rt ip = sqn rt ip"
explicationGoodness2: shows "\<forall>c x xa xb. \<lfloor>Good xb x \<^bold>\<and> NeedsForPurpose xb xa x\<rfloor>\<^sub>c \<longrightarrow> \<lfloor>Good xb (xa xb)\<rfloor>\<^sub>c"
orthogonal_def: fixes vs :: "'b vec list" shows "orthogonal vs \<equiv> \<forall>i j. i < length vs \<longrightarrow> j < length vs \<longrightarrow> (vs ! i \<bullet> vs ! j = (0::'b)) = (i \<noteq> j)"
v': shows "(v0, v') = H.first_edge_of_walk (Q_hit_pre @ [y])"
narrow_st_def: fixes F1 :: "'a Abs_State.st"   and F2 :: "'a Abs_State.st" shows "F1 \<triangle> F2 = FunDom (\<lambda>x. fun F1 x \<triangle> fun F2 x) (inter_list (Abs_State.dom F1) (Abs_State.dom F2))"