hcju/isabelleps huggingface dataset:
induct: fixes P :: "nat \<Rightarrow> ('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr \<Rightarrow> Unboxed.type option list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "('dyn, 'b, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and a2 :: "Unboxed.type option list" assumes "\<And>uu_ d \<Sigma>. P uu_ (Ubx.instr.IPush d) \<Sigma>"   and "\<And>uv_ n \<Sigma>. P uv_ (IPushUbx1 n) \<Sigma>"   and "\<And>uw_ b \<Sigma>. P uw_ (IPushUbx2 b) \<Sigma>"   and "\<And>ux_ uy_ \<Sigma>. P ux_ Ubx.instr.IPop (uy_ # \<Sigma>)"   and "\<And>n x \<Sigma>. P n (Ubx.instr.ILoad x) (None # \<Sigma>)"   and "\<And>uz_ \<tau> x \<Sigma>. P uz_ (ILoadUbx \<tau> x) (None # \<Sigma>)"   and "\<And>va_ x \<Sigma>. P va_ (Ubx.instr.IStore x) (None # None # \<Sigma>)"   and "\<And>vb_ x \<tau> \<Sigma>. P vb_ (Ubx.instr.IStore x) (None # Some \<tau> # \<Sigma>)"   and "\<And>vc_ \<tau>\<^sub>1 x \<tau>\<^sub>2 \<Sigma>. P vc_ (IStoreUbx \<tau>\<^sub>1 x) (None # Some \<tau>\<^sub>2 # \<Sigma>)"   and "\<And>vd_ op \<Sigma>. P vd_ (Ubx.instr.IOp op) \<Sigma>"   and "\<And>ve_ opinl \<Sigma>. P ve_ (Ubx.instr.IOpInl opinl) \<Sigma>"   and "\<And>vf_ opubx \<Sigma>. P vf_ (IOpUbx opubx) \<Sigma>"   and "\<And>vg_ f \<Sigma>. P vg_ (Ubx.instr.ICall f) \<Sigma>"   and "\<And>vh_. P vh_ Ubx.instr.IPop []"   and "\<And>vh_ v. P vh_ (Ubx.instr.ILoad v) []"   and "\<And>vh_ v vc vb. P vh_ (Ubx.instr.ILoad v) (Some vc # vb)"   and "\<And>vh_ v va. P vh_ (ILoadUbx v va) []"   and "\<And>vh_ v va vd vc. P vh_ (ILoadUbx v va) (Some vd # vc)"   and "\<And>vh_ v. P vh_ (Ubx.instr.IStore v) []"   and "\<And>vh_ v vc vb. P vh_ (Ubx.instr.IStore v) (Some vc # vb)"   and "\<And>vh_ v va. P vh_ (Ubx.instr.IStore v) [va]"   and "\<And>vh_ v vb ve vd. P vh_ (Ubx.instr.IStore v) (Some vb # Some ve # vd)"   and "\<And>vh_ v va. P vh_ (IStoreUbx v va) []"   and "\<And>vh_ v va vd vc. P vh_ (IStoreUbx v va) (Some vd # vc)"   and "\<And>vh_ v va vb. P vh_ (IStoreUbx v va) [vb]"   and "\<And>vh_ v va vb ve. P vh_ (IStoreUbx v va) (vb # None # ve)"   and "\<And>vh_ v vj_. P vh_ (Ubx.instr.ICJump v) vj_" shows "P a0 a1 a2"
sum_outgoing_alt_flow: fixes u :: "nat" assumes "u \<in> V" shows "sum f (outgoing u) = (\<Sum>v\<in>V. f (u, v))"
closed_segment_eq_real_ivl1: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "closed_segment a b = {a..b}"
strictly_preserves_unity: shows "F \<I>\<^sub>C = \<I>\<^sub>D"
st_mat_inject: fixes x :: "'a st_mat"   and y :: "'a st_mat" shows "(st_mat x = st_mat y) = (x = y)"
while: fixes cond :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and M :: "'b \<Rightarrow> (unit \<times> 'b) option"   and f :: "'b \<Rightarrow> nat" assumes "\<lbrace>\<lambda>\<sigma>. cond \<sigma> \<and> P \<sigma>\<rbrace> M \<lbrace>\<lambda>_. P\<rbrace>"   and "\<forall>\<sigma>. cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> \<noteq> None \<and> f (snd (the (M \<sigma>))) < f \<sigma>" shows "\<lbrace>P\<rbrace> _while_SE cond M \<lbrace>\<lambda>_ \<sigma>. \<not> cond \<sigma> \<and> P \<sigma>\<rbrace>"
bigo_mult7: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. f x \<noteq> (0::'b)" shows "O(f * g) \<subseteq> O(f) * O(g)"
invariant_def: fixes A :: "'d set" shows "invariant A \<equiv> {F. Init F \<subseteq> A} \<inter> stable A"
nsum_add: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A" shows "\<Sigma>\<^sub>e A (\<lambda>i. f i \<plusminus> g i) n = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A g n"
ik_rcv_map: fixes t :: "('a, 'b) Term.term"   and L :: "('a, 'b) Term.term list" assumes "t \<in> set L" shows "t \<in> ik\<^sub>s\<^sub>t (map Receive L)"
Ord_linear2: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k < l"   | "l \<le> k"
succ_of_entry0: shows "\<exists>s. (0, s) \<in> g_E G \<and> s \<noteq> 0"
advantage_def: fixes key_gen :: "'a spmf"   and hash :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and \<A> :: "(unit, 'b + 'b \<times> 'c, 'c + unit) gpv" shows "upf.advantage key_gen hash \<A> = spmf (upf.game key_gen hash \<A>) True"
owner_document: shows "h \<turnstile> get_owner_document (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) \<rightarrow>\<^sub>r owner_document"
black_def: fixes r :: "'h" shows "black r \<equiv> \<lambda>s. marked r s \<and> \<not> grey r s"
set_dom_cod_object_subset_2: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "Cod\<^bsub>set_cat U\<^esub> f \<in> Ob\<^bsub>set_cat U\<^esub>"
same: shows "set (find_consistent_signs_at_roots_R [:0, 1:] qs) = set (characterize_consistent_signs_at_roots [:0, 1:] qs)"
fold_keys_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and t :: "('d, 'e) RBT.rbt"   and A :: "'c" shows "fold_keys f t A = RBT.fold (\<lambda>k _. f k) t A"
prefix: shows "xs = pref @ [hd LTS, hd LTS]"
surjective_iff_injective_gen: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "finite T"   and "card S = card T"   and "f ` S \<subseteq> T" shows "(\<forall>y\<in>T. \<exists>x\<in>S. f x = y) = inj_on f S"
bintrunc_minus: fixes n :: "nat"   and w :: "int" assumes "0 < n" shows "take_bit (Suc (n - 1)) w = take_bit n w"
that: fixes q :: "int"   and p :: "int" assumes "0 < q"   and "x = real_of_int p / real_of_int q" shows "thesis"
strict_borderE: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_border xs ys" obtains "border xs ys" and "length xs < length ys"
neg_unique: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "are_complementary x y"   and "are_complementary x z" shows "y = z"
EqSim: fixes \<Psi>' :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "Eq \<Psi>' P Q" shows "\<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
simps_1: fixes uu :: "'b extB" shows "(extB.Bot \<le> uu) = True"
iso_incl_antisym_aux: fixes A :: "'c ring"   and B :: "'c ring" assumes "A \<lesssim> B"   and "B \<lesssim> A" shows "id \<in> ring_iso A B"
n_ivl_def: fixes ivl :: "ivl" shows "n_ivl ivl = 2 - m_ivl ivl"
superposition_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P1 :: "'b eclause"   and P2 :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.superposition trm_ord sel pos_ord filter_trms P1 P2 C \<sigma> k C' = (\<exists>L t s u v M p Cl_P1 Cl_P2 Cl_C polarity t' u' L' trms_C. L \<in> Cl_P1 \<and> M \<in> Cl_P2 \<and> basic_superposition.eligible_literal trm_ord sel L P1 \<sigma> \<and> basic_superposition.eligible_literal trm_ord sel M P2 \<sigma> \<and> variable_disjoint P1 P2 \<and> Cl_P1 = cl_ecl P1 \<and> Cl_P2 = cl_ecl P2 \<and> \<not> is_a_variable u' \<and> allowed_redex u' P1 \<sigma> \<and> trms_C = basic_superposition.get_trms filter_trms Cl_C (basic_superposition.dom_trms trm_ord Cl_C (subst_set (trms_ecl P1 \<union> trms_ecl P2 \<union> {r. \<exists>q. (q, p) \<in> pos_ord P1 t \<and> subterm t q r}) \<sigma>)) k \<and> C = Ecl Cl_C trms_C \<and> basic_superposition.orient_lit_inst trm_ord M u v pos \<sigma> \<and> basic_superposition.orient_lit_inst trm_ord L t s polarity \<sigma> \<and> u \<lhd> \<sigma> \<noteq> v \<lhd> \<sigma> \<and> subterm t p u' \<and> ck_unifier u' u \<sigma> k \<and> replace_subterm t p v t' \<and> (k = FirstOrder \<or> (subst_lit M \<sigma>, subst_lit L \<sigma>) \<in> basic_superposition.lit_ord trm_ord) \<and> (k = FirstOrder \<or> basic_superposition.strictly_maximal_literal trm_ord P2 M \<sigma>) \<and> L' = mk_lit polarity (Eq t' s) \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P1 - {L} \<union> (Cl_P2 - {M} \<union> {L'}))"
semiring_hom_axioms: shows "semiring_hom (inv hom)"
R: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
sublist_append_rightI: fixes xs :: "'a list"   and ss :: "'a list" shows "sublist xs (xs @ ss)"
csm: shows "cs\<^bsup>\<pi>'\<^esup> m = xs @ [\<pi> k]"
SET_CR5: fixes evs5 :: "event list"   and C :: "agent"   and k :: "nat"   and NC3 :: "nat"   and CardSecret :: "nat"   and KC2 :: "nat"   and KC3 :: "nat"   and SKi :: "nat"   and NC2 :: "nat"   and NCA :: "nat"   and i :: "nat"   and EKi :: "nat"   and KC1 :: "nat" assumes "evs5 \<in> set_cr"   and "C = Cardholder k"   and "Nonce NC3 \<notin> used evs5"   and "Nonce CardSecret \<notin> used evs5"   and "NC3 \<noteq> CardSecret"   and "Key KC2 \<notin> used evs5"   and "KC2 \<in> symKeys"   and "Key KC3 \<notin> used evs5"   and "KC3 \<in> symKeys"   and "KC2 \<noteq> KC3"   and "Gets C \<lbrace>sign (invKey SKi) \<lbrace>Agent C, Nonce NC2, Nonce NCA\<rbrace>, cert (CA i) EKi onlyEnc (priSK RCA), cert (CA i) SKi onlySig (priSK RCA)\<rbrace> \<in> set evs5"   and "Says C (CA i) (EXHcrypt KC1 EKi \<lbrace>Agent C, Nonce NC2\<rbrace> (Pan (pan C))) \<in> set evs5" shows "Says C (CA i) \<lbrace>Crypt KC3 \<lbrace>Agent C, Nonce NC3, Key KC2, Key (pubSK C), Crypt (priSK C) (Hash \<lbrace>Agent C, Nonce NC3, Key KC2, Key (pubSK C), Pan (pan C), Nonce CardSecret\<rbrace>)\<rbrace>, Crypt EKi \<lbrace>Key KC3, Pan (pan C), Nonce CardSecret\<rbrace>\<rbrace> # Notes C \<lbrace>Key KC2, Agent (CA i)\<rbrace> # Notes C \<lbrace>Key KC3, Agent (CA i)\<rbrace> # evs5 \<in> set_cr"
that: fixes cms\<^sub>2' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2' :: "'Var \<Rightarrow> 'Val"   and mems' :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" assumes "map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (cms\<^sub>2, mem\<^sub>2) \<leadsto>\<^bsub>k\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'" shows "thesis"
apex_def: shows "fg.apex \<equiv> C.dom fg.leg0"
Suc: fixes r :: "mregex"   and s :: "mregex" shows "LPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesR (\<Union>j\<le>i. LPDi j r) s \<union> (\<Union>j\<le>i. LPDi j s)"
oldest_token_distinct: fixes q :: "'b"   and q' :: "'b"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "q \<noteq> q'"   and "oldest_token q n = Some i"   and "oldest_token q' n = Some j" shows "i \<noteq> j"
simps_4: fixes x :: "real" shows "(ereal x < \<infinity>) = True"
of_real_Re: fixes k :: "nat" shows "complex_of_real (Re (\<chi> k)) = \<chi> k"
reduction_doesnt_break_things_invertibility_step1_R: fixes subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat_R signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs" shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) = length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
Image_Collect_case_prod: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e set" shows "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
norm_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "norm one = 1"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "filter P [] = []"
rep_eq: fixes x :: "uint64" shows "Rep_uint64 (- x) = - Rep_uint64 x"
perm_rem_simps_3: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "bij f"   and "y \<noteq> x"   and "f y \<noteq> x" shows "perm_rem x f y = f y"
iso_finfun_le: fixes A :: "'a \<Rightarrow>f 'b"   and B :: "'a \<Rightarrow>f 'b" shows "(($) A \<le> ($) B) = (A \<le> B)"
finite_dimensional_vector_space_on_def: fixes U\<^sub>V\<^sub>S :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_on U\<^sub>V\<^sub>S scale basis \<equiv> vector_space_on U\<^sub>V\<^sub>S scale \<and> finite_dimensional_vector_space_on_axioms U\<^sub>V\<^sub>S scale basis"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Normal x1 = Normal y1) = (x1 = y1)"
proper_weak_supplementation: fixes y :: "i"   and x :: "i" assumes "PP y x" shows "\<exists>z. PP z x \<and> \<not> O z y"
translate_origin: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "NO_MATCH (0::'b) p" shows "ccw p q r = ccw (0::'a) (q - p) (r - p)"
prob_space_axioms: fixes M :: "'b pmf" shows "prob_space (measure_pmf M)"
insort_is_Cons: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "class.linorder less_eq less"   and "\<forall>x\<in>set xs. less_eq (f a) (f x)" shows "linorder.insort_key less_eq f a xs = a # xs"
mod: shows "mod_hom K V2 (direct_sum V1 V2) (inj2 V1 V2)"
least_pr_weak': fixes f :: "'f"   and g :: "'f \<times> nat" assumes "least f" shows "kbo'.pr_weak' g (f, 0)"
pmf_of_set: fixes S :: "'c set"   and x :: "'c" assumes "S \<noteq> {}"   and "finite S" shows "pmf (pmf_of_set S) x = indicat_real S x / real (card S)"
strict_order_iff: fixes a :: "'b list"   and b :: "'b list" shows "strict_prefix a b = (a = gcp a b \<and> a \<noteq> b)"
rel_intros_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" assumes "R x2 y2" shows "rel_sort_key R (Final x2) (Final y2)"
of_nat_0_le_iff: fixes n :: "nat" shows "(0::'a) \<le> of_nat n"
pfun_graph_zero: shows "pfun_graph {}\<^sub>p = {}"
SN_on_trancl_SN_on_conv: fixes R :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_on (R\<^sup>+) T = SN_on R T"
Dom_preserves_Diag: fixes t :: "'c local.term" assumes "Diag t" shows "Diag (Dom t)"
hyps_1: fixes i :: "nat" assumes "x = i - 3"   and "3 \<le> i"   and "i \<le> k + 1"   and "j < n (i - 1)" shows "Polynomial.smult (\<Prod>l\<leftarrow>[3..<i]. \<alpha> l ^ (n (l - 1) - j)) (subresultant j (F 1) (F 2)) = Polynomial.smult (\<Prod>l\<leftarrow>[3..<i]. \<beta> l ^ (n (l - 1) - j) * f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)) * (- 1) ^ ((n (l - 2) - j) * (n (l - 1) - j))) (subresultant j (F (i - 2)) (F (i - 1)))"
pr0_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>hfsetcat.pr0 a b : hfsetcat.prod a b \<rightarrow> b\<guillemotright>"
beta_cases_2: fixes idn :: "variable"   and S :: "typ"   and t :: "Core.term"   and P :: "bool" assumes "Fv idn S \<rightarrow>\<^sub>\<beta> t" shows "P"
contains_def: fixes s :: "'a"   and r :: "'b \<times> 'c" shows "contains s r \<equiv> \<exists>rs. r \<in> set rs \<and> s = \<bottom> \<star> rs"
lr_of_tran_fbs_def: fixes rt :: "('b, 'c) routing_rule_scheme list"   and fw :: "'b simple_rule list"   and ifs :: "'d" shows "lr_of_tran_fbs rt fw ifs \<equiv> let gfw = map simple_rule_dtor fw; frt = lr_of_tran_s1 rt; prd = generalized_fw_join frt gfw in prd"
enough_flushs: shows "enough_flushs ts\<^sub>s\<^sub>b"
resr_superdist_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<rightarrow> z \<le> x \<rightarrow> z"
vimageD: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and A :: "'d set" assumes "a \<in> f -` A" shows "f a \<in> A"
monomial_decomp_shift: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs"   and "monomial_decomp qs" shows "monomial_decomp (local.shift k m qs)"
invariantQCharacterizationAfterApplyBackjump_1: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqC (getC state)"   and "getC state = [opposite (getCl state)]"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state'' = applyBackjump state in InvariantQCharacterization (getConflictFlag state'') (getQ state'') (getF state'') (getM state'')"
IH: assumes "distinct (map fst xs)" shows "distinct (map fst (remove1 x xs))"
smooth_on_def: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "k-smooth_on S f = (\<forall>n. enat n \<le> k \<longrightarrow> higher_differentiable_on S f n)"
assms: shows "local_iwf t (\<Gamma> \<turnstile> c)"
add1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ccw (0::'a) a b"   and "ccw (0::'a) c b" shows "ccw (0::'a) (a + c) b"
sfun_map_map: fixes f2 :: "'f \<rightarrow> 'e"   and g2 :: "'h \<rightarrow> 'g"   and f1 :: "'i \<rightarrow> 'f"   and g1 :: "'g \<rightarrow> 'j"   and p :: "'e \<rightarrow>! 'h" assumes "f2\<cdot>\<bottom> = \<bottom>"   and "g2\<cdot>\<bottom> = \<bottom>" shows "sfun_map\<cdot>f1\<cdot>g1\<cdot>(sfun_map\<cdot>f2\<cdot>g2\<cdot>p) = sfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
simps_1: fixes x :: "nat" shows "free_vars_expr_code (Var x) = {x}"
cong_mersenne_mod: fixes k :: "int"   and n :: "nat" shows "[mersenne_mod k n = k] (mod 2 ^ n - 1)"
col0: shows "col A' 0 = vec n (\<lambda>i. if i = 0 then e else (0::'a))"
Q: shows "Q \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
returns_heap_eq: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and h' :: "'l"   and h'' :: "'l" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
depthTransition_4: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "hnf P"   and "P \<longmapsto> a<\<nu>x> \<prec> P'" shows "depth P' < depth P"
ypos: fixes i :: "'n" shows "0 \<le> y $h i"
mem_lv_rel_set_rel_iff: fixes y :: "real list set"   and z :: "'b set" shows "((y, z) \<in> \<langle>lv_rel\<rangle>set_rel) = (z = eucl_of_list ` y \<and> (\<forall>x\<in>y. length x = DIM('b)))"
not_bn: fixes p :: "fm" assumes "bound0 p" shows "bound0 (not p)"
annir: fixes x :: "'a" shows "x \<cdot> (0::'a) = (0::'a)"
init: fixes vl :: "'value list"   and vl1 :: "'value list" assumes "B vl vl1" shows "\<Delta> istate vl istate vl1"
LSC_or_assoc: fixes F :: "'a formula"   and G :: "'a formula"   and H :: "'a formula" assumes "is_nnf F"   and "is_nnf G"   and "is_nnf H" shows "(F \<^bold>\<or> G) \<^bold>\<or> H, \<Gamma> \<Rightarrow>\<^sub>n = F \<^bold>\<or> (G \<^bold>\<or> H), \<Gamma> \<Rightarrow>\<^sub>n"
mono_neutral_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'d)" shows "sum g T = sum g S"
subseqs_length: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and n :: "nat"   and xs :: "'c list" assumes "\<And>x y z. f x (f y z) = f y (f x z)" shows "set (subseqs_length f a n xs) = {(foldr f ys a, ys) |ys. ys \<in> set (subseqs xs) \<and> length ys = n}"
m_inv_closed: fixes x :: "'a" assumes "x \<in> H" shows "inv x \<in> H"
Ag_ind_aGroup: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and D :: "'e set" assumes "aGroup A"   and "f \<in> carrier A \<rightarrow> D"   and "bij_to f (carrier A) D" shows "aGroup (Ag_ind A f)"
map_cong: fixes x :: "'c pmf"   and ya :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> set_pmf ya \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g ya"
exhaust: fixes y :: "'a + 'b" obtains a :: "'a" where "y = Inl a"   | b :: "'b" where "y = Inr b"
add_match_def: fixes m :: "'c match_expr"   and rs :: "'c rule list" shows "add_match m rs = map (\<lambda>r. case r of Rule m' a' \<Rightarrow> Rule (MatchAnd m m') a') rs"
parent_child_rel_child: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "(child \<in> set children) = ((ptr, 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) \<in> parent_child_rel h)"
unat_mono: fixes a :: "'a word"   and b :: "'a word" assumes "a < b" shows "unat a < unat b"
contour_integral_cong: fixes g :: "real \<Rightarrow> complex"   and g' :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex" assumes "g = g'"   and "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x" shows "contour_integral g f = contour_integral g' f'"