hcju/isabelleps huggingface dataset:
exists_sliced_intra_path_preds: fixes m :: "'node"   and as :: "'edge list"   and m' :: "'node"   and S :: "'node SDG_node set"   and cs :: "'edge list"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "m -as\<rightarrow>\<^sub>\<iota>* m'"   and "slice_edges S cs as = []"   and "m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "\<forall>c\<in>set cs. sourcenode c \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" obtains as' :: "'edge list" where "m -as'\<rightarrow>\<^sub>\<iota>* m'" and "preds (slice_kinds S as') (cf # cfs)" and "slice_edges S cs as' = []"
fun_FnCall: fixes F :: "nat"   and Xs :: "exp list" shows "fun (FnCall F Xs) = F"
resTy_mhead: fixes m :: "methd" shows "resTy (mhead m) = resTy m"
assms: shows "(p, p') \<in> unsorted_poly_rel_with0"
ccField_fup_CCexp: fixes e :: "exp"   and n :: "Arity\<^sub>\<bottom>" shows "ccField (ccExp_bot_syn n e) \<subseteq> fv e"
AB_eq: shows "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) = real_of_int A / real_of_int B"
Form_Body_nonempty: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body ka kb xs ys zs" shows "0 < length zs"
set_update_memI: fixes n :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "n < length xs" shows "x \<in> set (xs[n := x])"
InvariantConflictFlagCharacterization_def: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" shows "InvariantConflictFlagCharacterization conflictFlag F M \<equiv> conflictFlag = formulaFalse F (elements M)"
subst_fun_eq: fixes x :: "name"   and t\<^sub>1 :: "trm"   and y :: "name"   and t\<^sub>2 :: "trm"   and u :: "trm" assumes "[x].t\<^sub>1 = [y].t\<^sub>2" shows "t\<^sub>1[x::=u] = t\<^sub>2[y::=u]"
seqI: fixes f :: "'s"   and g :: "'s" assumes "S.arr f"   and "S.arr g"   and "S.dom g = S.cod f" shows "S.seq g f"
Cons_in_lists_iff: fixes x :: "'b"   and xs :: "'b list"   and A :: "'b set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
dis_log_break: fixes \<G> :: "'a cyclic_group"   and d' :: "nat"   and y :: "nat"   and m' :: "nat"   and d :: "nat"   and m :: "nat" assumes "pedersen \<G>"   and "\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> d' \<otimes>\<^bsub>\<G>\<^esub> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) [^]\<^bsub>\<G>\<^esub> m' = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> d \<otimes>\<^bsub>\<G>\<^esub> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) [^]\<^bsub>\<G>\<^esub> m"   and "y < order \<G>"   and "m' < m"   and "m < order \<G>" shows "y = nat ((int d' - int d) * Number_Theory_Aux.inverse (m - m') (order \<G>) mod int (order \<G>))"
eigenvalue_def: fixes A :: "'c mat"   and k :: "'c" shows "eigenvalue A k = (\<exists>v. eigenvector A v k)"
lm067: fixes f :: "('d \<times> 'c) set" shows "runiq (f\<inverse>) = (f \<in> converse ` runiqs)"
nth_item_covers: fixes x :: "('c, unit) Old_Datatype.node set" assumes "finite_item x" shows "\<exists>n. nth_item n = x"
det_swapcols: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and A :: "'b mat" assumes "k < n"   and "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (swapcols k l A) = - det A"
S_limit: shows "LIM k sequentially. S k \<bullet> (1::'a) :> at_top"
declC: shows "declC = invocation_declclass G mode s a' statT sig"
induct: fixes x1 :: "'c set"   and x2 :: "'c set"   and F :: "'c program"   and P :: "'c set \<Rightarrow> 'c set \<Rightarrow> bool" assumes "(x1, x2) \<in> leads F"   and "\<And>A B. F \<in> A ensures B \<Longrightarrow> P A B"   and "\<And>A B C. \<lbrakk>(A, B) \<in> leads F; P A B; (B, C) \<in> leads F; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>S B. \<forall>A\<in>S. (A, B) \<in> leads F \<and> P A B \<Longrightarrow> P (\<Union> S) B" shows "P x1 x2"
lcp_sym: fixes u :: "'b list"   and v :: "'b list" shows "u \<and>\<^sub>p v = v \<and>\<^sub>p u"
same_diag_def: fixes A :: "'a mat"   and B :: "'a mat" shows "local.same_diag A B \<equiv> \<forall>i<n. A $$ (i, i) = B $$ (i, i)"
abs_eq: shows "{#}\<^sub>z = abs_zmultiset ({#}, {#})"
2: fixes i :: "nat" shows "term_variants_pred P ((S @ t # T) ! i) ((S @ s # T) ! i)"
Appr: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and u :: "Lambda_Z.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "App u s \<rightarrow>\<^sub>\<beta> App u t"
simps_32: fixes vd :: "'b acom"   and ve :: "'b acom"   and v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b" shows "(vd;; ve \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
Poly_berlekamp_mat: fixes k :: "nat"   and u :: "'b mod_ring poly" assumes "k < degree u" shows "Poly (list_of_vec (row (berlekamp_mat u) k)) = [:0, 1:] ^ (CARD('b) * k) mod u"
content_image_stretch_interval: fixes m :: "'c \<Rightarrow> real"   and a :: "'c"   and b :: "'c" shows "content ((\<lambda>x. \<Sum>k\<in>Basis. (m k * (x \<bullet> k)) *\<^sub>R k) ` cbox a b) = \<bar>prod m Basis\<bar> * content (cbox a b)"
i_in_suc_n: shows "to_F \<iota> \<in> fst (lnth Ns n)"
syzygy_module_listI': fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and s :: "'t \<Rightarrow>\<^sub>0 'c" assumes "distinct bs"   and "sum_list (map2 (\<odot>) (cofactor_list_syz (length bs) s) bs) = 0"   and "component_of_term ` keys s \<subseteq> {0..<length bs}" shows "s \<in> syzygy_module_list bs"
P5: shows "\<nexists>x y. Contingent x \<and> Necessary y \<and> x explains y"
J2JVM_def: shows "J2JVM \<equiv> compP\<^sub>2 \<circ> compP\<^sub>1"
range: fixes k :: "natural"   and s :: "natural \<times> natural" assumes "0 < k" shows "fst (local.range k s) < k"
P_verifies_Signed_PReq: fixes MsgDualSign :: "msg"   and PIData :: "msg"   and HOIData :: "msg"   and PIHead :: "msg"   and PANData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and HOD :: "msg"   and PurchAmt :: "nat"   and M :: "agent"   and TransStain :: "msg"   and C :: "agent"   and evs :: "event list" assumes "MsgDualSign = \<lbrace>Hash PIData, HOIData\<rbrace>"   and "PIData = \<lbrace>PIHead, PANData\<rbrace>"   and "PIHead = \<lbrace>Number LID_M, Number XID, HOD, Number PurchAmt, Agent M, TransStain\<rbrace>"   and "Crypt (priSK C) (Hash MsgDualSign) \<in> parts (knows Spy evs)"   and "evs \<in> set_pur"   and "C \<notin> bad"   and "M \<notin> bad" shows "\<exists>OIData OrderDesc K j trans. HOD = Hash \<lbrace>Number OrderDesc, Number PurchAmt\<rbrace> \<and> HOIData = Hash OIData \<and> Notes M \<lbrace>Number LID_M, Agent (PG j), trans\<rbrace> \<in> set evs \<and> Says C M \<lbrace>\<lbrace>sign (priSK C) MsgDualSign, EXcrypt K (pubEK (PG j)) \<lbrace>PIHead, Hash OIData\<rbrace> PANData\<rbrace>, OIData, Hash PIData\<rbrace> \<in> set evs"
antisym_conv3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
l_sort_in_def: fixes xs :: "'b list" shows "l_sort_in xs \<equiv> (xs, [], [])"
simps_5: fixes Inv :: "'b"   and b :: "bexp"   and c :: "'b acom"   and P :: "'b"   and Inv' :: "'b"   and b' :: "bexp"   and c' :: "'b acom"   and P' :: "'b" shows "({Inv} WHILE b DO c {P} \<sqsubseteq> {Inv'} WHILE b' DO c' {P'}) = (b = b' \<and> c \<sqsubseteq> c' \<and> Inv \<sqsubseteq> Inv' \<and> P \<sqsubseteq> P')"
hn_refine_nofailI: fixes a :: "'a nres"   and \<Gamma> :: "assn"   and c :: "'b Heap"   and \<Gamma>' :: "assn"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "nofail a \<Longrightarrow> hn_refine \<Gamma> c \<Gamma>' R a" shows "hn_refine \<Gamma> c \<Gamma>' R a"
iteratei_abs_rule_P: assumes "set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; invar x; \<alpha> x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {\<alpha> x}. True; \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> I (S - {\<alpha> x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
borel_measurable_continuous_on_if: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "A \<in> sets borel"   and "continuous_on A f"   and "continuous_on (- A) g" shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel"
rationalizes_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set"   and r :: "('b \<times> 'b) set" shows "rationalizes_on A f r = (\<forall>B\<subseteq>A. f B = greatest r B)"
set_dropWhileD: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "x \<in> set (dropWhile P xs)" shows "x \<in> set xs"
sympartp_dual: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sympartp (\<lambda>x y. r y x) = sympartp r"
disjE2: fixes P :: "bool"   and Q :: "bool" assumes "P \<or> Q" obtains "P" and "\<not> Q"   | "Q"
bigtheta_inverse: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. inverse (f x)) \<in> \<Theta>[F](\<lambda>x. inverse (g x))) = (f \<in> \<Theta>[F](g))"
that: fixes a :: "'b" assumes "a \<in> A"   and "g a \<noteq> \<^bold>1" shows "thesis"
vle1: fixes x :: "V"   and y :: "V" assumes "x \<unlhd> y" shows "x \<le> y"
Id_onE: fixes c :: "'c \<times> 'c"   and A :: "'c set" assumes "c \<in> Id_on A" obtains x :: "'c" where "x \<in> A" and "c = (x, x)"
contour_integrable_reversepath: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g"   and "f contour_integrable_on g" shows "f contour_integrable_on reversepath g"
cadjoint_is_cadjoint: fixes G :: "'a \<Rightarrow> 'b" assumes "bounded_clinear G" shows "is_cadjoint (G\<^sup>\<dagger>) G"
c_def: shows "c = eps / max (real_of_rat (ir - il)) 1"
equivp_reflp_symp_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equivp R = (reflp R \<and> symp R \<and> transp R)"
noStuckn_exec_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "s \<noteq> Stuck"
ext_compat_list_strong_axioms: shows "ext_compat_list_strong ext"
iso_eval_Can: fixes t :: "'c local.term" assumes "Can t" shows "local.iso \<lbrace>t\<rbrace>"
rev_exhaust: fixes xs :: "'b list" obtains "xs = \<epsilon>"   | ys :: "'b list" and y :: "'b" where "xs = ys \<cdot> [y]"
ideI: fixes a :: "'a" assumes "left a a"   and "ide a" shows "Left_a.ide a"
min_height_size_log: fixes kdt :: "'b kdt" shows "real (min_height kdt) \<le> log 2 (real (KD_Tree.size_kdt kdt))"
simps_4: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Drop # rs) = Rule v Drop # process_ret rs"
Sup_insert: fixes a :: "'a"   and A :: "'a set" shows "Sup (insert a A) = a \<squnion> Sup A"
simple_packet_set_toSet_def: fixes ms :: "'b simple_match list" shows "simple_packet_set_toSet ms = {p. \<exists>m\<in>set ms. simple_matches m p}"
lim_proc_is_lub3a: fixes s :: "'c event list"   and P :: "'c process"   and c :: "'c event" assumes "front_tickFree s" shows "s \<notin> D P \<longrightarrow> (\<forall>t. t \<in> D P \<longrightarrow> \<not> t < s @ [c])"
Max_triangle_le: fixes z :: "nat" shows "Max_triangle z \<le> z"
emeasure_suntil_disj: fixes P :: "'s stream \<Rightarrow> bool"   and X :: "'s set"   and s :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P"   and "\<And>t. AE \<omega> in T t. \<not> (P \<sqinter> (HLD X \<sqinter> nxt (HLD X suntil P))) \<omega>" shows "emeasure (T s) {\<omega> \<in> space (T s). (HLD X suntil P) \<omega>} = lfp (\<lambda>F s. emeasure (T s) {\<omega> \<in> space (T s). P \<omega>} + set_nn_integral (measure_pmf (K s)) X F) s"
gsfw_valid_def: shows "gsfw_valid \<equiv> list_all (simple_match_valid \<circ> fst)"
algebraic_divide_cancel_left: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "x \<noteq> (0::'b)" shows "algebraic (x / y) = algebraic y"
valid: shows "valid_unMultigraph (del_unEdge v0 w v1 G)"
irsSp_Cons_param_out: fixes n'' :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var"   and p :: "'pname" assumes "n'' irs-ns\<rightarrow>\<^sub>d* n'"   and "n s-p:V\<rightarrow>\<^bsub>out\<^esub> n''" shows "n irs-n # ns\<rightarrow>\<^sub>d* n'"
interval_integral_const_2: fixes a :: "real"   and b :: "real"   and c :: "real" shows "LBINT x=ereal a..ereal b. c = c * (b - a)"
linorder_wlog: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less"   and "\<And>a b. less_eq a b \<Longrightarrow> P a b"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
Seq: fixes P\<^sub>1 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
inl_def: fixes p :: "(real \<times> real) pdevs" shows "inl p = independent_pdevs (map snd (list_of_pdevs (nlex_pdevs p)))"
rel_spmf_bind_reflI: fixes p :: "'a spmf"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'c spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (bind_spmf p f) (bind_spmf p g)"
uintNeq1: shows "uint (~~ w) \<noteq> 4294967295"
ring_iso_def: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "ring_iso R S = {h \<in> ring_hom R S. bij_betw h (carrier R) (carrier S)}"
sFTrue: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and n :: "nat" assumes "pbval b ps"   and "vars b \<subseteq> dom ps" shows "wp\<^sub>3\<^sub>' (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
infinite_Ico: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.atLeastLessThan less_eq less a b)"
wlsAll_implies_goodAll: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "(wls s X \<longrightarrow> good X) \<and> (wlsAbs (xs, s') A \<longrightarrow> goodAbs A) \<and> (wlsInp delta inp \<longrightarrow> goodInp inp) \<and> (wlsBinp delta binp \<longrightarrow> goodBinp binp)"
conf_NT: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and v :: "'c val" shows "heap_base.conf typeof_addr P h v NT = (v = Null)"
keys_phis'_codem': fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val"   and next' :: "'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_phis (uninst_code.ssa.phiNodes_of (p_g g s1_phis) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)" shows "Mapping.keys (uninst_code.phis'_codem (p_g g s1_phis) g next next' nodes_of_phis) = Mapping.keys (p_g g s1_phis g) - {next}"
"3_1": shows "\<forall>s\<in>set ss. is_root s"
mono_gamma_o: fixes sa :: "'av Abs_State.st option"   and sa' :: "'av Abs_State.st option" assumes "sa \<sqsubseteq> sa'" shows "\<gamma>\<^sub>o sa \<subseteq> \<gamma>\<^sub>o sa'"
differentiable_def: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'d filter" shows "(f differentiable F) = (\<exists>D. (f has_derivative D) F)"
mono: fixes x :: "'b" shows "option.mono_body (\<lambda>f. U (F (C f)) x)"
hinv_def: shows "Canon.hinv \<equiv> inv_into (UNIV // eq_app_right language) index_f"
F_unfold: fixes i :: "name" shows "F' i = (if i \<in> V' then Var (p' \<bullet> i) else Var i)"
induct: fixes P :: "'a MFOTL.formula \<Rightarrow> bool"   and a0 :: "'a MFOTL.formula" assumes "\<And>t1 t2. P (formula.Eq t1 t2)"   and "\<And>x y. P (formula.Neg (formula.Eq (trm.Const x) (trm.Const y)))"   and "\<And>x y. P (formula.Neg (formula.Eq (trm.Var x) (trm.Var y)))"   and "\<And>e ts. P (formula.Pred e ts)"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>; \<And>x3. \<psi> = formula.Neg x3 \<Longrightarrow> P x3\<rbrakk> \<Longrightarrow> P (formula.Neg (formula.Or (formula.Neg \<phi>) \<psi>))"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Or \<phi> \<psi>)"   and "\<And>\<phi>. P \<phi> \<Longrightarrow> P (formula.Exists \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Prev I \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Next I \<phi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>P \<phi>; \<And>x3. \<phi> = formula.Neg x3 \<Longrightarrow> P x3; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Since \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>P \<phi>; \<And>x3. \<phi> = formula.Neg x3 \<Longrightarrow> P x3; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Until \<phi> I \<psi>)"   and "\<And>va vb. P (formula.Neg (formula.Pred va vb))"   and "\<And>vc v. P (formula.Neg (formula.Eq (trm.Var vc) (trm.Const v)))"   and "\<And>v vc. P (formula.Neg (formula.Eq (trm.Const v) (trm.Var vc)))"   and "\<And>va. P (formula.Neg (formula.Neg va))"   and "\<And>v vc vb. P (formula.Neg (formula.Or (formula.Pred v vc) vb))"   and "\<And>v vc vb. P (formula.Neg (formula.Or (formula.Eq v vc) vb))"   and "\<And>v vc vb. P (formula.Neg (formula.Or (formula.Or v vc) vb))"   and "\<And>v vb. P (formula.Neg (formula.Or (formula.Exists v) vb))"   and "\<And>v vc vb. P (formula.Neg (formula.Or (formula.Prev v vc) vb))"   and "\<And>v vc vb. P (formula.Neg (formula.Or (formula.Next v vc) vb))"   and "\<And>v vc vd vb. P (formula.Neg (formula.Or (formula.Since v vc vd) vb))"   and "\<And>v vc vd vb. P (formula.Neg (formula.Or (formula.Until v vc vd) vb))"   and "\<And>va. P (formula.Neg (formula.Exists va))"   and "\<And>va vb. P (formula.Neg (formula.Prev va vb))"   and "\<And>va vb. P (formula.Neg (formula.Next va vb))"   and "\<And>va vb vc. P (formula.Neg (formula.Since va vb vc))"   and "\<And>va vb vc. P (formula.Neg (formula.Until va vb vc))" shows "P a0"
atin_def: fixes X :: "'b topology"   and a :: "'b" shows "atin X a \<equiv> inf (nhdsin X a) (principal (topspace X - {a}))"
elims: fixes x :: "'d state"   and xa :: "'b state \<times> 'c"   and y :: "'b state \<times> 'c" assumes "fix_clock x xa = y" obtains s :: "'d state" and s' :: "'b state" and res :: "'c" where "x = s" and "xa = (s', res)" and "y = (update_clock (\<lambda>_. if clock s' \<le> clock s then clock s' else clock s) s', res)"
intros_14: fixes c :: "i64"   and testop :: "testop" shows "\<lparr>[$C ConstInt64 c, $Testop T_i64 testop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_testop_i testop c))]\<rparr>"
cases: fixes a :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "a \<in> p_e R M N" obtains Ps :: "('a, 'b) sequent list" and c :: "('a, 'b) sequent" and \<Gamma> :: "('a, 'b) form multiset" and \<Delta> :: "('a, 'b) form multiset" where "a = extendRule ( M \<cdot> \<Gamma> \<Rightarrow>* N \<cdot> \<Delta>) (Ps, c)" and "(Ps, c) \<in> R" and "R \<subseteq> modRules2"
trans_def: fixes r :: "('d \<times> 'd) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
meet_self: fixes r :: "'c" shows "r \<sqinter> r = r"
Fvars_instInp_ptrm_1: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<tau> :: "'c"   and t :: "'b" assumes "Deduct_with_False_Disj_Rename var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "\<tau> \<in> Deduct_with_False_Disj_Rename.ptrm var fmla Var FvarsT Fvars eql cnj imp all exi prv (Suc 0)"   and "t \<in> trm" shows "Fvars (Generic_Syntax.instInp var subst \<tau> t) = insert (Generic_Syntax.Variable var 0) (FvarsT t)"
simps_13: fixes x3 :: "bool" shows "Bool x3 \<noteq> Null"
known_ptrs_known_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "NodeClass.known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "NodeClass.known_ptr ptr"
assms_2: shows "\<forall>nid\<in>actDn (t n\<^sub>s). length (bc (\<sigma>\<^bsub>nid\<^esub>t n\<^sub>s)) < length sbc"
rtrancl_into_rtrancl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
fresh_finite_atom_set: fixes S :: "atom set"   and a :: "atom" assumes "finite S" shows "a \<sharp> S = (a \<notin> S)"
self_append_conv2: fixes ys :: "'b list"   and xs :: "'b list" shows "(ys = append xs ys) = (xs = [])"
actionsD: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> actions E" shows "enat a < llength E"
drop_2_shift: fixes index :: "nat"   and L :: "'a list" assumes "0 < index"   and "1 < length L" shows "drop 2 L ! (index - Suc 0) = L ! Suc index"
S_imp_NS: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "fst (kbo.kbo w w0 scf least pr_strict pr_weak s t)" shows "snd (kbo.kbo w w0 scf least pr_strict pr_weak s t)"
assms: fixes b :: "'b"   and c :: "'b" shows "times_abc a (inf_b b c) = inf_c (times_abc a b) (times_abc a c)"
Right_Coset_memI: fixes h :: "'c"   and H :: "'c set"   and composition :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and x :: "'e" assumes "h \<in> H" shows "composition h x \<in> coset_notation.Right_Coset composition H x"
in_space_S: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "real \<times> 'd" assumes "transition_rates R" shows "x \<in> space (borel \<Otimes>\<^sub>M count_space UNIV)"